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| author | Thales Lima Oliveira <thaleslima.ufu@gmail.com> | 2020-07-29 18:14:14 -0300 |
|---|---|---|
| committer | Thales Lima Oliveira <thaleslima.ufu@gmail.com> | 2020-07-29 18:14:14 -0300 |
| commit | 51f475bd91d88e5060f500361bd9847bb719c422 (patch) | |
| tree | 184e16150eee9b019b843cf7f3a853fc1e1c7c88 /docs | |
| parent | 198ea803731dfb34226114c7230ba2dfae4f23c4 (diff) | |
| download | PSP.git-51f475bd91d88e5060f500361bd9847bb719c422.tar.gz PSP.git-51f475bd91d88e5060f500361bd9847bb719c422.tar.xz PSP.git-51f475bd91d88e5060f500361bd9847bb719c422.zip | |
Documentation update
Diffstat (limited to 'docs')
59 files changed, 1351 insertions, 201 deletions
diff --git a/docs/404.html b/docs/404.html index 091e4a7..6fc885d 100644 --- a/docs/404.html +++ b/docs/404.html @@ -6,14 +6,14 @@ <meta name="generator" content="Docusaurus v2.0.0-alpha.58"> <link rel="search" type="application/opensearchdescription+xml" title="PSP-UFU" href="/PSP/opensearch.xml"><title data-react-helmet="true">Page Not Found | PSP-UFU</title><meta data-react-helmet="true" property="og:title" content="Page Not Found | PSP-UFU"><meta data-react-helmet="true" name="twitter:card" content="summary_large_image"><link data-react-helmet="true" rel="shortcut icon" href="/PSP/img/favicon.ico"><link rel="stylesheet" href="/PSP/styles.68586bc0.css"> <link rel="preload" href="/PSP/styles.26e2fe91.js" as="script"> -<link rel="preload" href="/PSP/runtime~main.3ede30cf.js" as="script"> -<link rel="preload" href="/PSP/main.1c1af766.js" as="script"> +<link rel="preload" href="/PSP/runtime~main.57a83c0d.js" as="script"> +<link rel="preload" href="/PSP/main.1dbec03f.js" as="script"> </head> <body> <script>!function(){function t(t){document.documentElement.setAttribute("data-theme",t)}var e=window.matchMedia("(prefers-color-scheme: dark)"),n=function(){var t=null;try{t=localStorage.getItem("theme")}catch(t){}return t}();null!==n?t(n):e.matches&&t("dark")}()</script><div id="__docusaurus"> <nav class="navbar navbar--light navbar--fixed-top"><div class="navbar__inner"><div class="navbar__items"><div aria-label="Navigation bar toggle" class="navbar__toggle" role="button" tabindex="0"><svg xmlns="http://www.w3.org/2000/svg" width="30" height="30" viewBox="0 0 30 30" role="img" focusable="false"><title>Menu</title><path stroke="currentColor" stroke-linecap="round" stroke-miterlimit="10" stroke-width="2" d="M4 7h22M4 15h22M4 23h22"></path></svg></div><a class="navbar__brand" href="/PSP/"><img class="navbar__logo" src="/PSP/img/logo.svg" alt="PSP-UFU logo"><strong class="navbar__title">PSP-UFU</strong></a><a class="navbar__item navbar__link" href="/PSP/docs/">User 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searchInput_2eto"></div></div></div></div><div role="presentation" class="navbar-sidebar__backdrop"></div><div class="navbar-sidebar"><div class="navbar-sidebar__brand"><a class="navbar__brand" href="/PSP/"><img class="navbar__logo" src="/PSP/img/logo.svg" alt="PSP-UFU logo"><strong class="navbar__title">PSP-UFU</strong></a></div><div class="navbar-sidebar__items"><div class="menu"><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" href="/PSP/docs/">User Guide</a></li><li class="menu__list-item"><a class="menu__link" href="/PSP/blog">Changelog</a></li><li class="menu__list-item"><a href="https://github.com/Thales1330/PSP" target="_blank" rel="noopener noreferrer" class="menu__link">PSP-UFU Source Code</a></li></ul></div></div></div></nav><div class="main-wrapper"><div class="container margin-vert--xl"><div class="row"><div class="col col--6 col--offset-3"><h1 class="hero__title">Page Not Found</h1><p>We could not find what you were looking for.</p><p>Please contact the owner of the site that linked you to the original URL and let them know their link is broken.</p></div></div></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" 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Built with Docusaurus.</div></div></div></footer></div> <script src="/PSP/styles.26e2fe91.js"></script> -<script src="/PSP/runtime~main.3ede30cf.js"></script> -<script src="/PSP/main.1c1af766.js"></script> +<script src="/PSP/runtime~main.57a83c0d.js"></script> +<script src="/PSP/main.1dbec03f.js"></script> </body> </html>
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Essa barra \xe9 necessariamente geradora, uma vez que ela \xe9 respons\xe1vel pelo equil\xedbrio do balan\xe7o de pot\xeancia do sistema."))),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-heading"}),Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",Object(s.a)({parentName:"h5"},{className:"admonition-icon"}),Object(n.b)("svg",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"}),Object(n.b)("path",Object(s.a)({parentName:"svg"},{fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"})))),"Aten\xe7\xe3o!")),Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-content"}),Object(n.b)("p",{parentName:"div"},"O tipo de barra deve ser definido no ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"bus"}),"elemento barramento"),"."),Object(n.b)("p",{parentName:"div"},"Note que ",Object(n.b)("strong",{parentName:"p"},"o sistema deve possuir somente uma barra de refer\xeancia"),"."))),Object(n.b)("p",null,"Como mencionado anteriormente, as equa\xe7\xf5es s\xe3o n\xe3o-lineares e a solu\xe7\xe3o anal\xedtica n\xe3o \xe9 pr\xe1tica. As solu\xe7\xf5es dessas equa\xe7\xf5es seguem ",Object(n.b)("strong",{parentName:"p"},Object(n.b)("a",Object(s.a)({parentName:"strong"},{href:"https://en.wikipedia.org/wiki/Iterative_method"}),"processos iterativos")),", em que s\xe3o atribu\xeddos valores estimados (ou iniciais) para as barras com tens\xf5es desconhecidas e, baseado na pot\xeancia ativa e reativa e m\xf3dulo da tens\xe3o especificados, calcula-se por meio das equa\xe7\xf5es previamente apresentadas as novas tens\xf5es complexas em cada n\xf3 do sistema."),Object(n.b)("p",null,"Na sequ\xeancia, esse conjunto de valores para as tens\xf5es em cada barra \xe9 utilizado para novamente calcular outro grupo de tens\xf5es. Cada c\xe1lculo de um novo conjunto de tens\xf5es \xe9 chamado itera\xe7\xe3o. O processo iterativo \xe9 repetido at\xe9 que as mudan\xe7as em todas as barras sejam menores do que um valor pr\xe9-estipulado, obtendo assim a converg\xeancia."),Object(n.b)("h2",{id:"execu\xe7\xe3o-do-fluxo-de-carga-no-psp-ufu"},"Execu\xe7\xe3o do fluxo de carga no PSP-UFU"),Object(n.b)("p",null,"Ap\xf3s a constru\xe7\xe3o do diagrama unifilar no ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"powerEditor"}),"editor de pot\xeancia"),", a execu\xe7\xe3o do fluxo de carga \xe9 realizada no ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"mainScreen#menu-ribbon"}),"menu Simula\xe7\xe3o")," clicando no bot\xe3o ",Object(n.b)("strong",{parentName:"p"},"Fluxo de carga"),"."),Object(n.b)("div",null,Object(n.b)("center",null,Object(n.b)("img",{src:Object(b.a)("images/menuSimulationPF.svg"),alt:"Menu Simula\xe7\xe3o",title:"Menu Simula\xe7\xe3o"}))),Object(n.b)("div",{className:"admonition admonition-tip alert alert--success"},Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-heading"}),Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",Object(s.a)({parentName:"h5"},{className:"admonition-icon"}),Object(n.b)("svg",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"}),Object(n.b)("path",Object(s.a)({parentName:"svg"},{fillRule:"evenodd",d:"M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"})))),"Dica")),Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-content"}),Object(n.b)("p",{parentName:"div"},"Caso o fluxo de carga tenha sido executado com sucesso, as setas de pot\xeancia ser\xe3o exibidas, a barra de status indicar\xe1 sucesso na opera\xe7\xe3o e os ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"text"}),"elementos de texto")," ser\xe3o atualizados."))),Object(n.b)("p",null,Object(n.b)("strong",{parentName:"p"},"Outra possibilidade")," \xe9 a execu\xe7\xe3o por meio do c\xe1lculo cont\xednuo, tamb\xe9m presente no ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"mainScreen#menu-ribbon"}),"menu Simula\xe7\xe3o")," e seu acionamento \xe9 realizado co clicar no bot\xe3o ",Object(n.b)("strong",{parentName:"p"},"Habilitar solu\xe7\xe3o"),". Com essa op\xe7\xe3o, os c\xe1lculos est\xe1ticos selecionados nas ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"simulationConfig"}),"configura\xe7\xf5es de simula\xe7\xe3o")," s\xe3o automaticamente realizados ao modificar quaisquer par\xe2metros da rede, como dados el\xe9tricos e acionamento dos disjuntores dos elementos (remo\xe7\xe3o ou inser\xe7\xe3o)."),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-heading"}),Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",Object(s.a)({parentName:"h5"},{className:"admonition-icon"}),Object(n.b)("svg",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"}),Object(n.b)("path",Object(s.a)({parentName:"svg"},{fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"})))),"Aten\xe7\xe3o!")),Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-content"}),Object(n.b)("p",{parentName:"div"},"Evite construir o circuito com o c\xe1lculo cont\xednuo habilitado, uma vez que configura\xe7\xf5es tempor\xe1rias podem levar a erros de execu\xe7\xe3o da simula\xe7\xe3o."),Object(n.b)("p",{parentName:"div"},"Para desabilitar o c\xe1lculo cont\xednuo clique no bot\xe3o ",Object(n.b)("strong",{parentName:"p"},"Desabilitar solu\xe7\xe3o"),"."))),Object(n.b)("h3",{id:"erros-comuns-na-execu\xe7\xe3o-do-fluxo-de-carga"},"Erros comuns na execu\xe7\xe3o do fluxo de carga"),Object(n.b)("p",null,"A seguir s\xe3o apresentados os erros mais comuns relacionados ao fluxo de carga."),Object(n.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-o-n\xfamero-m\xe1ximo-de-itera\xe7\xf5es-foi-alcan\xe7ado"},'A seguinte mensagem de erro \xe9 exibida: "O n\xfamero m\xe1ximo de itera\xe7\xf5es foi alcan\xe7ado"'),Object(n.b)("p",null,"Essa mensagem de erro \xe9 exibida quando o m\xe9todo de solu\xe7\xe3o num\xe9rica selecionado nas ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"simulationConfig"}),"configura\xe7\xf5es de simula\xe7\xe3o")," atinge o n\xfamero m\xe1ximo de itera\xe7\xf5es inserido. As seguintes situa\xe7\xf5es podem ocasionar esse erro:"),Object(n.b)("ul",null,Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Os par\xe2metros do circuito est\xe3o incorretos"),". caso par\xe2metros muito fora dos valores padr\xe3o dos elementos el\xe9tricos sejam inseridos, o c\xe1lculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Algum elemento possui par\xe2metros discrepantes dos demais"),". Caso um dos elementos inseridos possua um valor de imped\xe2ncia muito distinto dos demais, como por exemplo uma linha com imped\xe2ncia muito elevada ou uma carga muito pequena, pode levar \xe0 diverg\xeancia do m\xe9todo num\xe9rico. Nesse caso, reconsidere a necessidade de representa\xe7\xe3o desses elementos no circuito e alterne entre os m\xe9todos num\xe9ricos de solu\xe7\xe3o dispon\xedveis."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"O n\xfamero m\xe1ximo de itera\xe7\xf5es est\xe1 muito baixo"),". Alguns circuitos exigem um n\xfamero maior de itera\xe7\xf5es, portanto altere o valor do m\xe1ximo de itera\xe7\xf5es nas ",Object(n.b)("a",Object(s.a)({parentName:"li"},{href:"simulationConfig"}),"configura\xe7\xf5es de simula\xe7\xe3o"),". Tamb\xe9m tente alternar entre os m\xe9todos num\xe9ricos de solu\xe7\xe3o dispon\xedveis."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Os par\xe2metros de simula\xe7\xe3o est\xe3o inadequados"),". Caso um par\xe2metro do m\xe9todo de solu\xe7\xe3o esteja inadequado, como fator de acelera\xe7\xe3o ou toler\xe2ncia, o c\xe1lculo pode n\xe3o alcan\xe7ar a converg\xeancia. Altere esses par\xe2metros nas ",Object(n.b)("a",Object(s.a)({parentName:"li"},{href:"simulationConfig"}),"configura\xe7\xf5es de simula\xe7\xe3o"),".")),Object(n.b)("h4",{id:"os-dados-de-sa\xedda-s\xe3o-exibidos-como-nan-ou-nan"},'Os dados de sa\xedda s\xe3o exibidos como "NaN" ou "nan"'),Object(n.b)("p",null,'Isso ocorre devido a erros de opera\xe7\xf5es matem\xe1ticas nos c\xe1lculos de fluxo de carga. "NaN" significa ',Object(n.b)("em",{parentName:"p"},"Not a Number"),"."),Object(n.b)("ul",null,Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Algum barramento est\xe1 isolado"),". Esse erro \xe9 bastante comum e pode ocorrer ao inserir um barramento sem conect\xe1-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solu\xe7\xe3o \xe9 eliminar essa barra do diagrama."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Os par\xe2metros do circuito est\xe3o incorretos"),". caso par\xe2metros muito fora dos valores padr\xe3o dos elementos el\xe9tricos sejam inseridos, o c\xe1lculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Algum elemento possui par\xe2metros discrepantes dos demais"),". Caso um dos elementos inseridos possua um valor de imped\xe2ncia muito distinto dos demais, como por exemplo uma linha com imped\xe2ncia muito elevada ou uma carga muito pequena, pode levar \xe0 diverg\xeancia do m\xe9todo num\xe9rico. Nesse caso, reconsidere a necessidade de representa\xe7\xe3o desses elementos no circuito e alterne entre os m\xe9todos num\xe9ricos de solu\xe7\xe3o dispon\xedveis.")),Object(n.b)("h2",{id:"m\xe9todos-de-solu\xe7\xe3o-num\xe9rica-do-fluxo-de-carga-no-psp-ufu"},"M\xe9todos de solu\xe7\xe3o num\xe9rica do fluxo de carga no PSP-UFU"),Object(n.b)("p",null,"Os m\xe9todos implementados no programa para solu\xe7\xe3o do problema de fluxo de carga no PSP-UFU s\xe3o ",Object(n.b)("strong",{parentName:"p"},"Gauss-Seidel (GS)")," e ",Object(n.b)("strong",{parentName:"p"},"Newton-Raphson (NR)"),". Al\xe9m desses m\xe9todos cl\xe1ssicos, um m\xe9todo h\xedbrido pode ser utilizado (definido nas ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"simulationConfig"}),"configura\xe7\xf5es de simula\xe7\xe3o"),"), em que \xe9 utilizado inicialmente o GS e na sequ\xeancia o NR, aumentando a chance de converg\xeancia do NR."),Object(n.b)("h3",{id:"gauss-seidel"},"Gauss-Seidel"),Object(n.b)("p",null,"O m\xe9todo de Gauss-Seidel tem sido bastante utilizado nas \xfaltimas d\xe9cadas para solu\xe7\xe3o do problema de fluxo de carga, uma vez que n\xe3o h\xe1 a necessidade de fatorar a matrizes, reduzindo o esfor\xe7o computacional. Atualmente, restri\xe7\xf5es computacionais s\xe3o menos problem\xe1ticas e outros m\xe9todos s\xe3o normalmente escolhidos, por\xe9m o Gauss-Seidel ainda possui valor did\xe1tico e, visto que o PSP-UFU tamb\xe9m possui fins educacionais, optou-se pela implementa\xe7\xe3o desse m\xe9todo."),Object(n.b)("p",null,"Para iniciar as itera\xe7\xf5es do m\xe9todo s\xe3o necess\xe1rios valores iniciais para as tens\xf5es (",Object(n.b)("span",Object(s.a)({parentName:"p"},{className:"math math-inline"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(n.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(n.b)("semantics",{parentName:"math"},Object(n.b)("mrow",{parentName:"semantics"},Object(n.b)("msubsup",{parentName:"mrow"},Object(n.b)("mover",Object(s.a)({parentName:"msubsup"},{accent:"true"}),Object(n.b)("mi",{parentName:"mover"},"V"),Object(n.b)("mo",{parentName:"mover"},"\u02d9")),Object(n.b)("mi",{parentName:"msubsup"},"i"),Object(n.b)("mn",{parentName:"msubsup"},"0"))),Object(n.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"\\dot{V}_i^0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"1.178854em",verticalAlign:"-0.258664em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord accent"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.9201900000000001em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.22222em"}}),"V"))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.25233em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"3em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"accent-body",style:{left:"-0.13889em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"\u02d9"))))))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"msupsub"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-t vlist-t2"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.8141079999999999em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-2.441336em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault mtight"}),"i"))),Object(n.b)("span",Object(s.a)({parentName:"span"},{style:{top:"-3.063em",marginRight:"0.05em"}}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"pstrut",style:{height:"2.7em"}})),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"sizing reset-size6 size3 mtight"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"mord mtight"}),"0")))),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-s"}),"\u200b")),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist-r"}),Object(n.b)("span",Object(s.a)({parentName:"span"},{className:"vlist",style:{height:"0.258664em"}}),Object(n.b)("span",{parentName:"span"})))))))))),") que devem ser calculadas. 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\xe9 o m\xf3dulo da tens\xe3o especificada para a barra PV. A barra de refer\xeancia tem o m\xf3dulo e \xe2ngulo de tens\xe3o fixos e n\xe3o participam no processo iterativo."),Object(n.b)("p",null,"O m\xe9todo de Gauss-Seidel mostra um ",Object(n.b)("strong",{parentName:"p"},"n\xfamero excessivo de itera\xe7\xf5es")," e, com o intuito reduzi-los, multiplicam-se as corre\xe7\xf5es de tens\xf5es por uma constante. Essa opera\xe7\xe3o amplia o valor da corre\xe7\xe3o, trazendo a tens\xe3o para mais perto do valor do valor final. Os multiplicadores que realizam essa converg\xeancia melhorada s\xe3o chamados de fatores de acelera\xe7\xe3o. Para qualquer sistema existem valores \xf3timos para os fatores de acelera\xe7\xe3o e uma escolha inadequada pode resultar em uma converg\xeancia mais lenta ou torn\xe1-la imposs\xedvel. Normalmente \xe9 utilizado um fator de acelera\xe7\xe3o igual a 1,6, valor definido como padr\xe3o no programa. 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tens\xe3o deixa de ser controlada partir de ent\xe3o."),Object(n.b)("p",null,"A verifica\xe7\xe3o de viola\xe7\xe3o e troca de tipo de barra pode ser realizada a cada itera\xe7\xe3o ou ao final da converg\xeancia do c\xe1lculo. No PSP-UFU foi implementada a \xfaltima estrat\xe9gia, uma vez que separa os conceitos de c\xe1lculo e de verifica\xe7\xe3o de limites, tornando mais f\xe1cil o desenvolvimento de novos m\xe9todos num\xe9ricos e limites. Nessa abordagem, ao ajustar o sistema para a nova situa\xe7\xe3o n\xe3o violadora o c\xe1lculo iterativo deve ser retomado at\xe9 que obtenha novamente a converg\xeancia."),Object(n.b)("h2",{id:"refer\xeancias"},"Refer\xeancias"),Object(n.b)("ul",null,Object(n.b)("li",{parentName:"ul"},"MONTICELLI, A. J. Fluxo de Carga em Redes de Energia El\xe9trica. S\xe3o Paulo: Edgar Bl\xfccher, 1983."),Object(n.b)("li",{parentName:"ul"},"STEVENSON JR.; WILLIAN, D. Elementos de An\xe1lise de Sistemas de Pot\xeancia. 2\xaa ed. S\xe3o Paulo: McGraw-Hill, 1986."),Object(n.b)("li",{parentName:"ul"},"MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: ",Object(n.b)("a",Object(s.a)({parentName:"li"},{href:"https://doi.org/10.1007/978-3-642-13669-6"}),"https://doi.org/10.1007/978-3-642-13669-6")),Object(n.b)("li",{parentName:"ul"},"ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley & Sons, New York, 2001. doi: ",Object(n.b)("a",Object(s.a)({parentName:"li"},{href:"https://doi.org/10.1002/9781118878286"}),"https://doi.org/10.1002/9781118878286")),Object(n.b)("li",{parentName:"ul"},"TINNEY, W. F.; HART, C. E. Power Flow Solution by Newton\u2019s Method. IEEE Transaction on Power Apparatus and Systems, v. 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Essa barra \xe9 necessariamente geradora, uma vez que ela \xe9 respons\xe1vel pelo equil\xedbrio do balan\xe7o de pot\xeancia do sistema."))),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-heading"}),Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",Object(s.a)({parentName:"h5"},{className:"admonition-icon"}),Object(n.b)("svg",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"}),Object(n.b)("path",Object(s.a)({parentName:"svg"},{fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"})))),"Aten\xe7\xe3o!")),Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-content"}),Object(n.b)("p",{parentName:"div"},"O tipo de barra deve ser definido no ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"bus"}),"elemento barramento"),"."),Object(n.b)("p",{parentName:"div"},"Note que ",Object(n.b)("strong",{parentName:"p"},"o sistema deve possuir somente uma barra de refer\xeancia"),"."))),Object(n.b)("p",null,"Como mencionado anteriormente, as equa\xe7\xf5es s\xe3o n\xe3o-lineares e a solu\xe7\xe3o anal\xedtica n\xe3o \xe9 pr\xe1tica. As solu\xe7\xf5es dessas equa\xe7\xf5es seguem ",Object(n.b)("strong",{parentName:"p"},Object(n.b)("a",Object(s.a)({parentName:"strong"},{href:"https://en.wikipedia.org/wiki/Iterative_method"}),"processos iterativos")),", em que s\xe3o atribu\xeddos valores estimados (ou iniciais) para as barras com tens\xf5es desconhecidas e, baseado na pot\xeancia ativa e reativa e m\xf3dulo da tens\xe3o especificados, calcula-se por meio das equa\xe7\xf5es previamente apresentadas as novas tens\xf5es complexas em cada n\xf3 do sistema."),Object(n.b)("p",null,"Na sequ\xeancia, esse conjunto de valores para as tens\xf5es em cada barra \xe9 utilizado para novamente calcular outro grupo de tens\xf5es. Cada c\xe1lculo de um novo conjunto de tens\xf5es \xe9 chamado itera\xe7\xe3o. O processo iterativo \xe9 repetido at\xe9 que as mudan\xe7as em todas as barras sejam menores do que um valor pr\xe9-estipulado, obtendo assim a converg\xeancia."),Object(n.b)("h2",{id:"execu\xe7\xe3o-do-fluxo-de-carga-no-psp-ufu"},"Execu\xe7\xe3o do fluxo de carga no PSP-UFU"),Object(n.b)("p",null,"Ap\xf3s a constru\xe7\xe3o do diagrama unifilar no ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"powerEditor"}),"editor de pot\xeancia"),", a execu\xe7\xe3o do fluxo de carga \xe9 realizada no ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"mainScreen#menu-ribbon"}),"menu Simula\xe7\xe3o")," clicando no bot\xe3o ",Object(n.b)("strong",{parentName:"p"},"Fluxo de carga"),"."),Object(n.b)("div",null,Object(n.b)("center",null,Object(n.b)("img",{src:Object(b.a)("images/menuSimulationPF.svg"),alt:"Menu Simula\xe7\xe3o",title:"Menu Simula\xe7\xe3o"}))),Object(n.b)("div",{className:"admonition admonition-tip alert alert--success"},Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-heading"}),Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",Object(s.a)({parentName:"h5"},{className:"admonition-icon"}),Object(n.b)("svg",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"}),Object(n.b)("path",Object(s.a)({parentName:"svg"},{fillRule:"evenodd",d:"M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"})))),"Dica")),Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-content"}),Object(n.b)("p",{parentName:"div"},"Caso o fluxo de carga tenha sido executado com sucesso, as setas de pot\xeancia ser\xe3o exibidas, a barra de status indicar\xe1 sucesso na opera\xe7\xe3o e os ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"text"}),"elementos de texto")," ser\xe3o atualizados."))),Object(n.b)("p",null,Object(n.b)("strong",{parentName:"p"},"Outra possibilidade")," \xe9 a execu\xe7\xe3o por meio do c\xe1lculo cont\xednuo, tamb\xe9m presente no ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"mainScreen#menu-ribbon"}),"menu Simula\xe7\xe3o")," e seu acionamento \xe9 realizado co clicar no bot\xe3o ",Object(n.b)("strong",{parentName:"p"},"Habilitar solu\xe7\xe3o"),". Com essa op\xe7\xe3o, os c\xe1lculos est\xe1ticos selecionados nas ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"simulationConfig"}),"configura\xe7\xf5es de simula\xe7\xe3o")," s\xe3o automaticamente realizados ao modificar quaisquer par\xe2metros da rede, como dados el\xe9tricos e acionamento dos disjuntores dos elementos (remo\xe7\xe3o ou inser\xe7\xe3o)."),Object(n.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-heading"}),Object(n.b)("h5",{parentName:"div"},Object(n.b)("span",Object(s.a)({parentName:"h5"},{className:"admonition-icon"}),Object(n.b)("svg",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"}),Object(n.b)("path",Object(s.a)({parentName:"svg"},{fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"})))),"Aten\xe7\xe3o!")),Object(n.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-content"}),Object(n.b)("p",{parentName:"div"},"Evite construir o circuito com o c\xe1lculo cont\xednuo habilitado, uma vez que configura\xe7\xf5es tempor\xe1rias podem levar a erros de execu\xe7\xe3o da simula\xe7\xe3o."),Object(n.b)("p",{parentName:"div"},"Para desabilitar o c\xe1lculo cont\xednuo clique no bot\xe3o ",Object(n.b)("strong",{parentName:"p"},"Desabilitar solu\xe7\xe3o"),"."))),Object(n.b)("p",null,"Os resultados do fluxo de carga s\xe3o exibidos nos ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"text"}),"elementos de texto vinculado"),", ao posicionar o mouse sobre os elementos e em ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"tabularReport"}),"relat\xf3rios tabulares"),"."),Object(n.b)("h3",{id:"erros-comuns-na-execu\xe7\xe3o-do-fluxo-de-carga"},"Erros comuns na execu\xe7\xe3o do fluxo de carga"),Object(n.b)("p",null,"A seguir s\xe3o apresentados os erros mais comuns relacionados ao fluxo de carga."),Object(n.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-o-n\xfamero-m\xe1ximo-de-itera\xe7\xf5es-foi-alcan\xe7ado"},'A seguinte mensagem de erro \xe9 exibida: "O n\xfamero m\xe1ximo de itera\xe7\xf5es foi alcan\xe7ado"'),Object(n.b)("p",null,"Essa mensagem de erro \xe9 exibida quando o m\xe9todo de solu\xe7\xe3o num\xe9rica selecionado nas ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"simulationConfig"}),"configura\xe7\xf5es de simula\xe7\xe3o")," atinge o n\xfamero m\xe1ximo de itera\xe7\xf5es inserido. As seguintes situa\xe7\xf5es podem ocasionar esse erro:"),Object(n.b)("ul",null,Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Os par\xe2metros do circuito est\xe3o incorretos"),". caso par\xe2metros muito fora dos valores padr\xe3o dos elementos el\xe9tricos sejam inseridos, o c\xe1lculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Algum elemento possui par\xe2metros discrepantes dos demais"),". Caso um dos elementos inseridos possua um valor de imped\xe2ncia muito distinto dos demais, como por exemplo uma linha com imped\xe2ncia muito elevada ou uma carga muito pequena, pode levar \xe0 diverg\xeancia do m\xe9todo num\xe9rico. Nesse caso, reconsidere a necessidade de representa\xe7\xe3o desses elementos no circuito e alterne entre os m\xe9todos num\xe9ricos de solu\xe7\xe3o dispon\xedveis."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"O n\xfamero m\xe1ximo de itera\xe7\xf5es est\xe1 muito baixo"),". Alguns circuitos exigem um n\xfamero maior de itera\xe7\xf5es, portanto altere o valor do m\xe1ximo de itera\xe7\xf5es nas ",Object(n.b)("a",Object(s.a)({parentName:"li"},{href:"simulationConfig"}),"configura\xe7\xf5es de simula\xe7\xe3o"),". Tamb\xe9m tente alternar entre os m\xe9todos num\xe9ricos de solu\xe7\xe3o dispon\xedveis."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Os par\xe2metros de simula\xe7\xe3o est\xe3o inadequados"),". Caso um par\xe2metro do m\xe9todo de solu\xe7\xe3o esteja inadequado, como fator de acelera\xe7\xe3o ou toler\xe2ncia, o c\xe1lculo pode n\xe3o alcan\xe7ar a converg\xeancia. Altere esses par\xe2metros nas ",Object(n.b)("a",Object(s.a)({parentName:"li"},{href:"simulationConfig"}),"configura\xe7\xf5es de simula\xe7\xe3o"),".")),Object(n.b)("h4",{id:"os-dados-de-sa\xedda-s\xe3o-exibidos-como-nan-ou-nan"},'Os dados de sa\xedda s\xe3o exibidos como "NaN" ou "nan"'),Object(n.b)("p",null,'Isso ocorre devido a erros de opera\xe7\xf5es matem\xe1ticas nos c\xe1lculos de fluxo de carga. "NaN" significa ',Object(n.b)("em",{parentName:"p"},"Not a Number"),"."),Object(n.b)("ul",null,Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Algum barramento est\xe1 isolado"),". Esse erro \xe9 bastante comum e pode ocorrer ao inserir um barramento sem conect\xe1-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solu\xe7\xe3o \xe9 eliminar essa barra do diagrama."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Os par\xe2metros do circuito est\xe3o incorretos"),". caso par\xe2metros muito fora dos valores padr\xe3o dos elementos el\xe9tricos sejam inseridos, o c\xe1lculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente."),Object(n.b)("li",{parentName:"ul"},Object(n.b)("strong",{parentName:"li"},"Algum elemento possui par\xe2metros discrepantes dos demais"),". Caso um dos elementos inseridos possua um valor de imped\xe2ncia muito distinto dos demais, como por exemplo uma linha com imped\xe2ncia muito elevada ou uma carga muito pequena, pode levar \xe0 diverg\xeancia do m\xe9todo num\xe9rico. Nesse caso, reconsidere a necessidade de representa\xe7\xe3o desses elementos no circuito e alterne entre os m\xe9todos num\xe9ricos de solu\xe7\xe3o dispon\xedveis.")),Object(n.b)("h2",{id:"m\xe9todos-de-solu\xe7\xe3o-num\xe9rica-do-fluxo-de-carga-no-psp-ufu"},"M\xe9todos de solu\xe7\xe3o num\xe9rica do fluxo de carga no PSP-UFU"),Object(n.b)("p",null,"Os m\xe9todos implementados no programa para solu\xe7\xe3o do problema de fluxo de carga no PSP-UFU s\xe3o ",Object(n.b)("strong",{parentName:"p"},"Gauss-Seidel (GS)")," e ",Object(n.b)("strong",{parentName:"p"},"Newton-Raphson (NR)"),". Al\xe9m desses m\xe9todos cl\xe1ssicos, um m\xe9todo h\xedbrido pode ser utilizado (definido nas ",Object(n.b)("a",Object(s.a)({parentName:"p"},{href:"simulationConfig"}),"configura\xe7\xf5es de simula\xe7\xe3o"),"), em que \xe9 utilizado inicialmente o GS e na sequ\xeancia o NR, aumentando a chance de converg\xeancia do NR."),Object(n.b)("h3",{id:"gauss-seidel"},"Gauss-Seidel"),Object(n.b)("p",null,"O m\xe9todo de Gauss-Seidel tem sido bastante utilizado nas \xfaltimas d\xe9cadas para solu\xe7\xe3o do problema de fluxo de carga, uma vez que n\xe3o h\xe1 a necessidade de fatorar a matrizes, reduzindo o esfor\xe7o computacional. 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\xe9 o m\xf3dulo da tens\xe3o especificada para a barra PV. A barra de refer\xeancia tem o m\xf3dulo e \xe2ngulo de tens\xe3o fixos e n\xe3o participam no processo iterativo."),Object(n.b)("p",null,"O m\xe9todo de Gauss-Seidel mostra um ",Object(n.b)("strong",{parentName:"p"},"n\xfamero excessivo de itera\xe7\xf5es")," e, com o intuito reduzi-los, multiplicam-se as corre\xe7\xf5es de tens\xf5es por uma constante. Essa opera\xe7\xe3o amplia o valor da corre\xe7\xe3o, trazendo a tens\xe3o para mais perto do valor do valor final. Os multiplicadores que realizam essa converg\xeancia melhorada s\xe3o chamados de fatores de acelera\xe7\xe3o. Para qualquer sistema existem valores \xf3timos para os fatores de acelera\xe7\xe3o e uma escolha inadequada pode resultar em uma converg\xeancia mais lenta ou torn\xe1-la imposs\xedvel. Normalmente \xe9 utilizado um fator de acelera\xe7\xe3o igual a 1,6, valor definido como padr\xe3o no programa. 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tens\xe3o deixa de ser controlada partir de ent\xe3o."),Object(n.b)("p",null,"A verifica\xe7\xe3o de viola\xe7\xe3o e troca de tipo de barra pode ser realizada a cada itera\xe7\xe3o ou ao final da converg\xeancia do c\xe1lculo. No PSP-UFU foi implementada a \xfaltima estrat\xe9gia, uma vez que separa os conceitos de c\xe1lculo e de verifica\xe7\xe3o de limites, tornando mais f\xe1cil o desenvolvimento de novos m\xe9todos num\xe9ricos e limites. Nessa abordagem, ao ajustar o sistema para a nova situa\xe7\xe3o n\xe3o violadora o c\xe1lculo iterativo deve ser retomado at\xe9 que obtenha novamente a converg\xeancia."),Object(n.b)("h2",{id:"refer\xeancias"},"Refer\xeancias"),Object(n.b)("ol",null,Object(n.b)("li",{parentName:"ol"},"MONTICELLI, A. J. Fluxo de Carga em Redes de Energia El\xe9trica. S\xe3o Paulo: Edgar Bl\xfccher, 1983."),Object(n.b)("li",{parentName:"ol"},"STEVENSON JR.; WILLIAN, D. Elementos de An\xe1lise de Sistemas de Pot\xeancia. 2\xaa ed. S\xe3o Paulo: McGraw-Hill, 1986."),Object(n.b)("li",{parentName:"ol"},"MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: ",Object(n.b)("a",Object(s.a)({parentName:"li"},{href:"https://doi.org/10.1007/978-3-642-13669-6"}),"https://doi.org/10.1007/978-3-642-13669-6")),Object(n.b)("li",{parentName:"ol"},"ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley & Sons, New York, 2001. doi: ",Object(n.b)("a",Object(s.a)({parentName:"li"},{href:"https://doi.org/10.1002/9781118878286"}),"https://doi.org/10.1002/9781118878286")),Object(n.b)("li",{parentName:"ol"},"TINNEY, W. F.; HART, C. E. Power Flow Solution by Newton\u2019s Method. IEEE Transaction on Power Apparatus and Systems, v. 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Nesse tipo de c\xe1lculo \xe9 poss\xedvel calcular faltas ",Object(m.b)("em",{parentName:"li"},"shunt")," nos ",Object(m.b)("a",Object(s.a)({parentName:"li"},{href:"bus"}),"barramentos")," balanceadas e desbalanceadas."),Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"N\xedvel de curto-circuito"),": Calcula o n\xedvel de curto-circuito (falta trif\xe1sica) em todos ",Object(m.b)("a",Object(s.a)({parentName:"li"},{href:"bus"}),"barramentos")," do sistema.")),Object(m.b)("p",null,"Ap\xf3s a constru\xe7\xe3o do diagrama unifilar no ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"powerEditor"}),"editor de pot\xeancia"),", a execu\xe7\xe3o do c\xe1lculo de curto-circuito \xe9 realizada no ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"mainScreen#menu-ribbon"}),"menu Simula\xe7\xe3o")," clicando no bot\xe3o ",Object(m.b)("strong",{parentName:"p"},"Falta"),". Para calcular o n\xedvel de curto-circuito (falta trif\xe1sica) em todos ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"bus"}),"barramentos")," do sistema, basta clicar no bot\xe3o ",Object(m.b)("strong",{parentName:"p"},"N\xedvel de curto-circuito"),"."),Object(m.b)("div",null,Object(m.b)("center",null,Object(m.b)("img",{src:Object(b.a)("images/menuSimulationFaulta.svg"),alt:"Execu\xe7\xe3o dos c\xe1lculos de curto-circuito",title:"Execu\xe7\xe3o dos c\xe1lculos de curto-circuito"}))),Object(m.b)("div",{className:"admonition admonition-caution alert alert--warning"},Object(m.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-heading"}),Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",Object(s.a)({parentName:"h5"},{className:"admonition-icon"}),Object(m.b)("svg",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"16",height:"16",viewBox:"0 0 16 16"}),Object(m.b)("path",Object(s.a)({parentName:"svg"},{fillRule:"evenodd",d:"M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"})))),"Aten\xe7\xe3o")),Object(m.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-content"}),Object(m.b)("p",{parentName:"div"},"\xc9 poss\xedvel calcular as faltas sem a execu\xe7\xe3o do ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"powerFlow"}),"fluxo de carga"),", por\xe9m ",Object(m.b)("strong",{parentName:"p"},"n\xe3o \xe9 recomend\xe1vel"),", visto que os valores das correntes de falta s\xe3o significativamente alteradas."))),Object(m.b)("p",null,Object(m.b)("strong",{parentName:"p"},"Outra possibilidade")," \xe9 a execu\xe7\xe3o por meio do c\xe1lculo cont\xednuo, tamb\xe9m presente no ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"mainScreen#menu-ribbon"}),"menu Simula\xe7\xe3o")," e seu acionamento \xe9 realizado co clicar no bot\xe3o ",Object(m.b)("strong",{parentName:"p"},"Habilitar solu\xe7\xe3o"),". Com essa op\xe7\xe3o, os c\xe1lculos est\xe1ticos selecionados nas ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"simulationConfig"}),"configura\xe7\xf5es de simula\xe7\xe3o")," s\xe3o automaticamente realizados ao modificar quaisquer par\xe2metros da rede, como dados el\xe9tricos e acionamento dos disjuntores dos elementos (remo\xe7\xe3o ou inser\xe7\xe3o)."),Object(m.b)("div",{className:"admonition admonition-warning alert alert--danger"},Object(m.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-heading"}),Object(m.b)("h5",{parentName:"div"},Object(m.b)("span",Object(s.a)({parentName:"h5"},{className:"admonition-icon"}),Object(m.b)("svg",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/2000/svg",width:"12",height:"16",viewBox:"0 0 12 16"}),Object(m.b)("path",Object(s.a)({parentName:"svg"},{fillRule:"evenodd",d:"M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"})))),"Cuidado!")),Object(m.b)("div",Object(s.a)({parentName:"div"},{className:"admonition-content"}),Object(m.b)("p",{parentName:"div"},"Os c\xe1lculos de curtos-circuitos n\xe3o s\xe3o habilitados por padr\xe3o no c\xe1lculo cont\xednuo e devem ser inseridos nas ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"simulationConfig"}),"configura\xe7\xf5es de simula\xe7\xe3o"),"."))),Object(m.b)("p",null,"Os resultados do c\xe1lculo de curto-circuito s\xe3o exibidos nos ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"text"}),"elementos de texto vinculado"),", ao posicionar o mouse sobre os barramentos e em ",Object(m.b)("a",Object(s.a)({parentName:"p"},{href:"tabularReport"}),"relat\xf3rios tabulares"),"."),Object(m.b)("h3",{id:"erros-comuns-na-execu\xe7\xe3o-do-c\xe1lculo-de-curto-circuito"},"Erros comuns na execu\xe7\xe3o do c\xe1lculo de curto-circuito"),Object(m.b)("p",null,"A seguir s\xe3o apresentados os erros mais comuns relacionados ao calculo de curto-circuito."),Object(m.b)("h4",{id:"a-seguinte-mensagem-de-erro-\xe9-exibida-falha-ao-inverter-a-matriz-admit\xe2ncia-de-sequ\xeancia-zero"},'A seguinte mensagem de erro \xe9 exibida: "Falha ao inverter a matriz admit\xe2ncia de sequ\xeancia zero"'),Object(m.b)("ul",null,Object(m.b)("li",{parentName:"ul"},Object(m.b)("strong",{parentName:"li"},"Impossibilidade de circula\xe7\xe3o da corrente de sequ\xeancia zero"),". Caso o gerador n\xe3o seja aterrado, n\xe3o circular\xe1 corrente de sequ\xeancia zero por ele. Nesse caso, dependendo da conex\xe3o do transformador pr\xf3ximo ao gerador sem aterramento, a matriz admit\xe2ncia de sequ\xeancia zero \xe9 singular. Para contornar esse problema escolha uma das duas solu\xe7\xf5es abaixo:",Object(m.b)("ul",{parentName:"li"},Object(m.b)("li",{parentName:"ul"},'Marque a op\xe7\xe3o "Neutro aterrado" e insira um alto valor de reat\xe2ncia de aterramento (',Object(m.b)("span",Object(s.a)({parentName:"li"},{className:"math math-inline"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(m.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mi",{parentName:"mrow"},"j"),Object(m.b)("mn",{parentName:"mrow"},"9999"),Object(m.b)("mtext",{parentName:"mrow"},"\xa0"),Object(m.b)("mi",{parentName:"mrow"},"p"),Object(m.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),"."),Object(m.b)("mi",{parentName:"mrow"},"u"),Object(m.b)("mi",Object(s.a)({parentName:"mrow"},{mathvariant:"normal"}),".")),Object(m.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"j9999~p.u.")))),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.85396em",verticalAlign:"-0.19444em"}})),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.05724em"}}),"j"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"9"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"9"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"9"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"9"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mspace nobreak"}),"\xa0"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"p"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"."),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"u"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"."))))),", por exemplo);"),Object(m.b)("li",{parentName:"ul"},"Ou, na barra do gerador, insira um reator de baixo valor de pot\xeancia reativa (",Object(m.b)("span",Object(s.a)({parentName:"li"},{className:"math math-inline"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"katex"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"katex-mathml"}),Object(m.b)("math",Object(s.a)({parentName:"span"},{xmlns:"http://www.w3.org/1998/Math/MathML"}),Object(m.b)("semantics",{parentName:"math"},Object(m.b)("mrow",{parentName:"semantics"},Object(m.b)("mn",{parentName:"mrow"},"1"),Object(m.b)("mo",Object(s.a)({parentName:"mrow"},{separator:"true",lspace:"0em",rspace:"0em"}),","),Object(m.b)("mn",{parentName:"mrow"},"0"),Object(m.b)("mtext",{parentName:"mrow"},"\xa0"),Object(m.b)("mi",{parentName:"mrow"},"v"),Object(m.b)("mi",{parentName:"mrow"},"a"),Object(m.b)("mi",{parentName:"mrow"},"r")),Object(m.b)("annotation",Object(s.a)({parentName:"semantics"},{encoding:"application/x-tex"}),"1{,}0~var")))),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"katex-html","aria-hidden":"true"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"base"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"strut",style:{height:"0.8388800000000001em",verticalAlign:"-0.19444em"}})),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"1"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mpunct"}),",")),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord"}),"0"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mspace nobreak"}),"\xa0"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.03588em"}}),"v"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault"}),"a"),Object(m.b)("span",Object(s.a)({parentName:"span"},{className:"mord mathdefault",style:{marginRight:"0.02778em"}}),"r"))))),", por exemplo).")))),Object(m.b)("h2",{id:"o-c\xe1lculo-de-curto-circuito"},"O c\xe1lculo de curto-circuito"),Object(m.b)("p",null,"Como j\xe1 foi apresentado anteriormente, as faltas que ocorrem com maior frequ\xeancia em sistemas de pot\xeancia s\xe3o assim\xe9tricas. Como qualquer falta assim\xe9trica provoca fluxo de corrente desequilibrada \xe9 necess\xe1rio empregar o m\xe9todo das componentes sim\xe9tricas. Esse m\xe9todo permite o estudo de sistemas balanceados com cargas desbalanceadas."),Object(m.b)("h3",{id:"m\xe9todo-das-componentes-sim\xe9tricas"},"M\xe9todo das componentes sim\xe9tricas"),Object(m.b)("p",null,"Esse m\xe9todo proposto por C. L. Fortescue, permite definir um sistema de n fasores desbalanceados em n \u2013 1 sistemas de n fases balanceados e um sistema de fase zero. O sistema de fase zero \xe9 definido por todas as fases de mesmo m\xf3dulo e \xe2ngulo.\nPara um sistema trif\xe1sico pode-se definir tr\xeas componentes de sequ\xeancia:"),Object(m.b)("ol",null,Object(m.b)("li",{parentName:"ol"},"Componentes de sequ\xeancia positiva, constituindo em tr\xeas fasores iguais em m\xf3dulo, 120\xba defasados entre si, e tendo a mesma sequ\xeancia de fase que os fasores originais;"),Object(m.b)("li",{parentName:"ol"},"Componentes de sequ\xeancia negativa, constituindo em tr\xeas fasores iguais em m\xf3dulo, 120\xba defasados entre si, e tendo a sequ\xeancia de fase oposta \xe0 dos fasores originais."),Object(m.b)("li",{parentName:"ol"},"Componentes de sequ\xeancia zero, constituindo em tr\xeas fasores iguais em m\xf3dulo e com defasagem nula entre si.")),Object(m.b)("p",null,"Com isso pode-se decompor as tens\xf5es de fase em componentes sim\xe9tricas pelas seguintes equa\xe7\xf5es:"),Object(m.b)("div",{className:"math 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\ No newline at end of file diff --git a/docs/blog/2019w42a-beta/index.html b/docs/blog/2019w42a-beta/index.html index f4ec250..c80d5aa 100644 --- a/docs/blog/2019w42a-beta/index.html +++ b/docs/blog/2019w42a-beta/index.html @@ -6,8 +6,8 @@ <meta name="generator" content="Docusaurus v2.0.0-alpha.58"> <link rel="search" type="application/opensearchdescription+xml" title="PSP-UFU" href="/PSP/opensearch.xml"><title data-react-helmet="true">2019w42a-beta | PSP-UFU</title><meta data-react-helmet="true" property="og:title" content="2019w42a-beta | PSP-UFU"><meta data-react-helmet="true" name="description" content="- Harmonic study implemented;"><meta data-react-helmet="true" property="og:description" content="- Harmonic study implemented;"><meta data-react-helmet="true" name="twitter:card" content="summary_large_image"><link data-react-helmet="true" rel="shortcut icon" href="/PSP/img/favicon.ico"><link rel="stylesheet" href="/PSP/styles.68586bc0.css"> <link rel="preload" 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href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>Harmonic study implemented;</li><li>Autolayout from importation implemented;</li><li>Importation of Matpower files;</li><li>Several minor bugfixes.</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><div><a href="https://github.com/Thales1330/PSP/tree/master/docs/blog/2019-10-13-2019w42a-beta.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit 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Built with Docusaurus.</div></div></div></footer></div> <script src="/PSP/styles.26e2fe91.js"></script> -<script src="/PSP/runtime~main.3ede30cf.js"></script> -<script src="/PSP/main.1c1af766.js"></script> +<script src="/PSP/runtime~main.57a83c0d.js"></script> +<script src="/PSP/main.1dbec03f.js"></script> <script src="/PSP/1.837a9375.js"></script> <script src="/PSP/2.19116c85.js"></script> <script src="/PSP/3.189ac30f.js"></script> diff --git a/docs/blog/2019w44a-beta/index.html b/docs/blog/2019w44a-beta/index.html index 6a16cb5..709c3b6 100644 --- a/docs/blog/2019w44a-beta/index.html +++ b/docs/blog/2019w44a-beta/index.html @@ -6,8 +6,8 @@ <meta name="generator" content="Docusaurus v2.0.0-alpha.58"> <link rel="search" type="application/opensearchdescription+xml" title="PSP-UFU" href="/PSP/opensearch.xml"><title data-react-helmet="true">22019w44a-beta | PSP-UFU</title><meta data-react-helmet="true" property="og:title" content="22019w44a-beta | PSP-UFU"><meta data-react-helmet="true" name="description" content="- Minor changes to 32bit MSW version for compatibility;"><meta data-react-helmet="true" property="og:description" content="- Minor changes to 32bit MSW version for compatibility;"><meta data-react-helmet="true" name="twitter:card" content="summary_large_image"><link data-react-helmet="true" rel="shortcut icon" href="/PSP/img/favicon.ico"><link rel="stylesheet" href="/PSP/styles.68586bc0.css"> <link rel="preload" href="/PSP/styles.26e2fe91.js" as="script"> -<link rel="preload" href="/PSP/runtime~main.3ede30cf.js" as="script"> -<link rel="preload" href="/PSP/main.1c1af766.js" as="script"> +<link rel="preload" href="/PSP/runtime~main.57a83c0d.js" as="script"> +<link rel="preload" href="/PSP/main.1dbec03f.js" as="script"> <link rel="preload" href="/PSP/1.837a9375.js" as="script"> <link rel="preload" href="/PSP/2.19116c85.js" as="script"> <link rel="preload" href="/PSP/3.189ac30f.js" as="script"> @@ -18,8 +18,8 @@ <script>!function(){function t(t){document.documentElement.setAttribute("data-theme",t)}var e=window.matchMedia("(prefers-color-scheme: dark)"),n=function(){var t=null;try{t=localStorage.getItem("theme")}catch(t){}return t}();null!==n?t(n):e.matches&&t("dark")}()</script><div id="__docusaurus"> <nav class="navbar navbar--light navbar--fixed-top"><div class="navbar__inner"><div class="navbar__items"><div aria-label="Navigation bar toggle" class="navbar__toggle" role="button" tabindex="0"><svg xmlns="http://www.w3.org/2000/svg" width="30" height="30" viewBox="0 0 30 30" role="img" focusable="false"><title>Menu</title><path stroke="currentColor" stroke-linecap="round" stroke-miterlimit="10" stroke-width="2" d="M4 7h22M4 15h22M4 23h22"></path></svg></div><a class="navbar__brand" href="/PSP/"><img class="navbar__logo" src="/PSP/img/logo.svg" alt="PSP-UFU logo"><strong class="navbar__title">PSP-UFU</strong></a><a class="navbar__item navbar__link" href="/PSP/docs/">User Guide</a><a aria-current="page" class="navbar__item navbar__link navbar__link--active" href="/PSP/blog">Changelog</a></div><div class="navbar__items navbar__items--right"><a href="https://github.com/Thales1330/PSP" target="_blank" rel="noopener noreferrer" class="navbar__item navbar__link">PSP-UFU Source Code</a><div class="react-toggle react-toggle--disabled displayOnlyInLargeViewport_1gtM"><div class="react-toggle-track"><div class="react-toggle-track-check"><span class="toggle_keGJ moon_1gwN"></span></div><div class="react-toggle-track-x"><span class="toggle_keGJ sun_3CPA"></span></div></div><div class="react-toggle-thumb"></div><input type="checkbox" disabled="" aria-label="Dark mode toggle" class="react-toggle-screenreader-only"></div><div class="navbar__search"><div class="searchWrapper_-F3-"><span aria-label="expand searchbar" role="button" class="searchIconButton_1Dnz" tabindex="0"></span><input type="search" id="search_input_react" placeholder="Search" aria-label="Search" class="navbar__search-input searchInput_2eto"></div></div></div></div><div role="presentation" class="navbar-sidebar__backdrop"></div><div class="navbar-sidebar"><div class="navbar-sidebar__brand"><a class="navbar__brand" href="/PSP/"><img class="navbar__logo" src="/PSP/img/logo.svg" alt="PSP-UFU logo"><strong class="navbar__title">PSP-UFU</strong></a></div><div class="navbar-sidebar__items"><div class="menu"><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" href="/PSP/docs/">User Guide</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link navbar__link--active" href="/PSP/blog">Changelog</a></li><li class="menu__list-item"><a href="https://github.com/Thales1330/PSP" target="_blank" rel="noopener noreferrer" class="menu__link">PSP-UFU Source Code</a></li></ul></div></div></div></nav><div class="main-wrapper"><div class="container margin-vert--lg"><div class="row"><div class="col col--8 col--offset-2"><article><header><h1 class="margin-bottom--sm blogPostTitle_2RZH">22019w44a-beta</h1><div class="margin-vert--md"><time datetime="2019-11-01T00:00:00.000Z" class="blogPostDate_3tRe">November 1, 2019 · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>Minor changes to 32bit MSW version for compatibility;</li><li>Travis CI (Continuous Integration) implemented (Linux).</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><div><a href="https://github.com/Thales1330/PSP/tree/master/docs/blog/2019-11-01-2019w44a-beta.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div><div class="margin-vert--xl"><nav class="pagination-nav" aria-label="Blog post page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/blog/2020w24a-beta"><div class="pagination-nav__sublabel">Previous Post</div><div class="pagination-nav__label">« 2020w24a-beta</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/blog/2019w42a-beta"><div class="pagination-nav__sublabel">Next Post</div><div 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Built with Docusaurus.</div></div></div></footer></div> <script src="/PSP/styles.26e2fe91.js"></script> -<script src="/PSP/runtime~main.3ede30cf.js"></script> -<script src="/PSP/main.1c1af766.js"></script> +<script src="/PSP/runtime~main.57a83c0d.js"></script> +<script src="/PSP/main.1dbec03f.js"></script> <script src="/PSP/1.837a9375.js"></script> <script src="/PSP/2.19116c85.js"></script> <script src="/PSP/3.189ac30f.js"></script> diff --git a/docs/blog/2020w24a-beta/index.html b/docs/blog/2020w24a-beta/index.html index 7c05465..d9f285e 100644 --- a/docs/blog/2020w24a-beta/index.html +++ b/docs/blog/2020w24a-beta/index.html @@ -6,8 +6,8 @@ <meta name="generator" content="Docusaurus v2.0.0-alpha.58"> <link rel="search" type="application/opensearchdescription+xml" title="PSP-UFU" href="/PSP/opensearch.xml"><title data-react-helmet="true">2020w24a-beta | PSP-UFU</title><meta data-react-helmet="true" property="og:title" content="2020w24a-beta | PSP-UFU"><meta data-react-helmet="true" name="description" content="- Device context render implemented (to avoid use OpenGL - some machines can't use it properly);"><meta data-react-helmet="true" property="og:description" content="- Device context render implemented (to avoid use OpenGL - some machines can't use it properly);"><meta data-react-helmet="true" name="twitter:card" content="summary_large_image"><link data-react-helmet="true" rel="shortcut icon" href="/PSP/img/favicon.ico"><link rel="stylesheet" href="/PSP/styles.68586bc0.css"> <link rel="preload" href="/PSP/styles.26e2fe91.js" as="script"> -<link rel="preload" href="/PSP/runtime~main.3ede30cf.js" as="script"> -<link rel="preload" href="/PSP/main.1c1af766.js" as="script"> +<link rel="preload" href="/PSP/runtime~main.57a83c0d.js" as="script"> +<link rel="preload" href="/PSP/main.1dbec03f.js" as="script"> <link rel="preload" href="/PSP/1.837a9375.js" as="script"> <link rel="preload" href="/PSP/2.19116c85.js" as="script"> <link rel="preload" href="/PSP/3.189ac30f.js" as="script"> @@ -18,8 +18,8 @@ <script>!function(){function t(t){document.documentElement.setAttribute("data-theme",t)}var e=window.matchMedia("(prefers-color-scheme: dark)"),n=function(){var t=null;try{t=localStorage.getItem("theme")}catch(t){}return t}();null!==n?t(n):e.matches&&t("dark")}()</script><div id="__docusaurus"> <nav class="navbar navbar--light navbar--fixed-top"><div class="navbar__inner"><div class="navbar__items"><div aria-label="Navigation bar toggle" class="navbar__toggle" role="button" tabindex="0"><svg xmlns="http://www.w3.org/2000/svg" width="30" height="30" viewBox="0 0 30 30" role="img" focusable="false"><title>Menu</title><path stroke="currentColor" stroke-linecap="round" stroke-miterlimit="10" stroke-width="2" d="M4 7h22M4 15h22M4 23h22"></path></svg></div><a class="navbar__brand" href="/PSP/"><img class="navbar__logo" src="/PSP/img/logo.svg" alt="PSP-UFU logo"><strong class="navbar__title">PSP-UFU</strong></a><a class="navbar__item navbar__link" href="/PSP/docs/">User Guide</a><a aria-current="page" class="navbar__item navbar__link navbar__link--active" href="/PSP/blog">Changelog</a></div><div class="navbar__items navbar__items--right"><a href="https://github.com/Thales1330/PSP" target="_blank" rel="noopener noreferrer" class="navbar__item navbar__link">PSP-UFU Source Code</a><div class="react-toggle react-toggle--disabled displayOnlyInLargeViewport_1gtM"><div class="react-toggle-track"><div class="react-toggle-track-check"><span class="toggle_keGJ moon_1gwN"></span></div><div class="react-toggle-track-x"><span class="toggle_keGJ sun_3CPA"></span></div></div><div class="react-toggle-thumb"></div><input type="checkbox" disabled="" aria-label="Dark mode toggle" class="react-toggle-screenreader-only"></div><div class="navbar__search"><div class="searchWrapper_-F3-"><span aria-label="expand searchbar" role="button" class="searchIconButton_1Dnz" tabindex="0"></span><input type="search" id="search_input_react" placeholder="Search" aria-label="Search" class="navbar__search-input searchInput_2eto"></div></div></div></div><div role="presentation" class="navbar-sidebar__backdrop"></div><div class="navbar-sidebar"><div class="navbar-sidebar__brand"><a class="navbar__brand" href="/PSP/"><img class="navbar__logo" src="/PSP/img/logo.svg" alt="PSP-UFU logo"><strong class="navbar__title">PSP-UFU</strong></a></div><div class="navbar-sidebar__items"><div class="menu"><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" href="/PSP/docs/">User Guide</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link navbar__link--active" href="/PSP/blog">Changelog</a></li><li class="menu__list-item"><a href="https://github.com/Thales1330/PSP" target="_blank" rel="noopener noreferrer" class="menu__link">PSP-UFU Source Code</a></li></ul></div></div></div></nav><div class="main-wrapper"><div class="container margin-vert--lg"><div class="row"><div class="col col--8 col--offset-2"><article><header><h1 class="margin-bottom--sm blogPostTitle_2RZH">2020w24a-beta</h1><div class="margin-vert--md"><time datetime="2020-06-13T00:00:00.000Z" class="blogPostDate_3tRe">June 13, 2020 · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>Device context render implemented (to avoid use OpenGL - some machines can't use it properly);</li><li>Added Visual Studio solution to the GitHub project;</li><li>Several major and minor bugfixes.</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><div><a href="https://github.com/Thales1330/PSP/tree/master/docs/blog/2020-06-13-2020w24a-beta.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div><div class="margin-vert--xl"><nav class="pagination-nav" aria-label="Blog post page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/blog/2020w28a-beta"><div class="pagination-nav__sublabel">Previous Post</div><div class="pagination-nav__label">« 2020w28a-beta</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/blog/2019w44a-beta"><div class="pagination-nav__sublabel">Next Post</div><div class="pagination-nav__label">22019w44a-beta »</div></a></div></nav></div></div></div></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. Built with Docusaurus.</div></div></div></footer></div> <script src="/PSP/styles.26e2fe91.js"></script> -<script src="/PSP/runtime~main.3ede30cf.js"></script> -<script src="/PSP/main.1c1af766.js"></script> +<script src="/PSP/runtime~main.57a83c0d.js"></script> +<script src="/PSP/main.1dbec03f.js"></script> <script src="/PSP/1.837a9375.js"></script> <script src="/PSP/2.19116c85.js"></script> <script src="/PSP/3.189ac30f.js"></script> diff --git a/docs/blog/2020w28a-beta/index.html b/docs/blog/2020w28a-beta/index.html index ab2a1c1..ffe5fb6 100644 --- a/docs/blog/2020w28a-beta/index.html +++ b/docs/blog/2020w28a-beta/index.html @@ -6,8 +6,8 @@ <meta name="generator" content="Docusaurus v2.0.0-alpha.58"> <link rel="search" type="application/opensearchdescription+xml" title="PSP-UFU" href="/PSP/opensearch.xml"><title data-react-helmet="true">2020w28a-beta | PSP-UFU</title><meta data-react-helmet="true" property="og:title" content="2020w28a-beta | PSP-UFU"><meta data-react-helmet="true" name="description" content="- Voltage heatmap implemented (OpenGL enabled);"><meta data-react-helmet="true" property="og:description" content="- Voltage heatmap implemented (OpenGL enabled);"><meta data-react-helmet="true" name="twitter:card" content="summary_large_image"><link data-react-helmet="true" rel="shortcut icon" href="/PSP/img/favicon.ico"><link rel="stylesheet" href="/PSP/styles.68586bc0.css"> <link rel="preload" href="/PSP/styles.26e2fe91.js" as="script"> -<link rel="preload" href="/PSP/runtime~main.3ede30cf.js" as="script"> -<link rel="preload" href="/PSP/main.1c1af766.js" as="script"> +<link rel="preload" href="/PSP/runtime~main.57a83c0d.js" as="script"> +<link rel="preload" href="/PSP/main.1dbec03f.js" as="script"> <link rel="preload" href="/PSP/1.837a9375.js" as="script"> <link rel="preload" href="/PSP/2.19116c85.js" as="script"> <link rel="preload" href="/PSP/3.189ac30f.js" as="script"> @@ -18,8 +18,8 @@ <script>!function(){function t(t){document.documentElement.setAttribute("data-theme",t)}var e=window.matchMedia("(prefers-color-scheme: dark)"),n=function(){var t=null;try{t=localStorage.getItem("theme")}catch(t){}return t}();null!==n?t(n):e.matches&&t("dark")}()</script><div id="__docusaurus"> <nav class="navbar navbar--light navbar--fixed-top"><div class="navbar__inner"><div class="navbar__items"><div aria-label="Navigation bar toggle" class="navbar__toggle" role="button" tabindex="0"><svg xmlns="http://www.w3.org/2000/svg" width="30" height="30" viewBox="0 0 30 30" role="img" focusable="false"><title>Menu</title><path stroke="currentColor" stroke-linecap="round" stroke-miterlimit="10" stroke-width="2" d="M4 7h22M4 15h22M4 23h22"></path></svg></div><a class="navbar__brand" href="/PSP/"><img class="navbar__logo" src="/PSP/img/logo.svg" alt="PSP-UFU logo"><strong class="navbar__title">PSP-UFU</strong></a><a class="navbar__item navbar__link" href="/PSP/docs/">User Guide</a><a aria-current="page" class="navbar__item 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blogPostTitle_2RZH">2020w28a-beta</h1><div class="margin-vert--md"><time datetime="2020-07-09T00:00:00.000Z" class="blogPostDate_3tRe">July 9, 2020 · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>Voltage heatmap implemented (OpenGL enabled);</li><li>Several major and minor bugfixes.</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><div><a href="https://github.com/Thales1330/PSP/tree/master/docs/blog/2020-07-09-2020w28a-beta.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div><div class="margin-vert--xl"><nav class="pagination-nav" aria-label="Blog post page navigation"><div class="pagination-nav__item"></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/blog/2020w24a-beta"><div class="pagination-nav__sublabel">Next Post</div><div class="pagination-nav__label">2020w24a-beta »</div></a></div></nav></div></div></div></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div 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Built with Docusaurus.</div></div></div></footer></div> <script src="/PSP/styles.26e2fe91.js"></script> -<script src="/PSP/runtime~main.3ede30cf.js"></script> -<script src="/PSP/main.1c1af766.js"></script> +<script src="/PSP/runtime~main.57a83c0d.js"></script> +<script src="/PSP/main.1dbec03f.js"></script> <script src="/PSP/1.837a9375.js"></script> <script src="/PSP/2.19116c85.js"></script> <script src="/PSP/3.189ac30f.js"></script> diff --git a/docs/blog/index.html b/docs/blog/index.html index 9cc0b4d..3377d4b 100644 --- a/docs/blog/index.html +++ b/docs/blog/index.html @@ -6,8 +6,8 @@ <meta name="generator" content="Docusaurus v2.0.0-alpha.58"> <link rel="search" type="application/opensearchdescription+xml" title="PSP-UFU" href="/PSP/opensearch.xml"><title data-react-helmet="true">Blog | PSP-UFU</title><meta data-react-helmet="true" property="og:title" content="Blog | PSP-UFU"><meta data-react-helmet="true" name="description" content="Blog"><meta data-react-helmet="true" property="og:description" content="Blog"><meta data-react-helmet="true" name="twitter:card" content="summary_large_image"><link data-react-helmet="true" rel="shortcut icon" href="/PSP/img/favicon.ico"><link rel="stylesheet" href="/PSP/styles.68586bc0.css"> <link rel="preload" href="/PSP/styles.26e2fe91.js" as="script"> -<link rel="preload" href="/PSP/runtime~main.3ede30cf.js" as="script"> -<link rel="preload" href="/PSP/main.1c1af766.js" as="script"> +<link rel="preload" href="/PSP/runtime~main.57a83c0d.js" as="script"> +<link rel="preload" href="/PSP/main.1dbec03f.js" as="script"> <link rel="preload" href="/PSP/1.837a9375.js" as="script"> <link rel="preload" href="/PSP/2.19116c85.js" as="script"> <link rel="preload" href="/PSP/3.189ac30f.js" as="script"> @@ -22,8 +22,8 @@ <script>!function(){function t(t){document.documentElement.setAttribute("data-theme",t)}var e=window.matchMedia("(prefers-color-scheme: dark)"),n=function(){var t=null;try{t=localStorage.getItem("theme")}catch(t){}return t}();null!==n?t(n):e.matches&&t("dark")}()</script><div id="__docusaurus"> <nav class="navbar navbar--light navbar--fixed-top"><div class="navbar__inner"><div class="navbar__items"><div aria-label="Navigation bar toggle" class="navbar__toggle" role="button" tabindex="0"><svg xmlns="http://www.w3.org/2000/svg" width="30" height="30" viewBox="0 0 30 30" role="img" focusable="false"><title>Menu</title><path stroke="currentColor" stroke-linecap="round" stroke-miterlimit="10" stroke-width="2" d="M4 7h22M4 15h22M4 23h22"></path></svg></div><a class="navbar__brand" href="/PSP/"><img class="navbar__logo" src="/PSP/img/logo.svg" alt="PSP-UFU logo"><strong class="navbar__title">PSP-UFU</strong></a><a class="navbar__item navbar__link" href="/PSP/docs/">User Guide</a><a aria-current="page" class="navbar__item navbar__link navbar__link--active" href="/PSP/blog">Changelog</a></div><div class="navbar__items navbar__items--right"><a 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class="navbar-sidebar__brand"><a class="navbar__brand" href="/PSP/"><img class="navbar__logo" src="/PSP/img/logo.svg" alt="PSP-UFU logo"><strong class="navbar__title">PSP-UFU</strong></a></div><div class="navbar-sidebar__items"><div class="menu"><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" href="/PSP/docs/">User Guide</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link navbar__link--active" href="/PSP/blog">Changelog</a></li><li class="menu__list-item"><a href="https://github.com/Thales1330/PSP" target="_blank" rel="noopener noreferrer" class="menu__link">PSP-UFU Source Code</a></li></ul></div></div></div></nav><div class="main-wrapper"><div class="container margin-vert--lg"><div class="row"><main class="col col--8 col--offset-2"><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_2RZH"><a href="/PSP/blog/2020w28a-beta">2020w28a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-07-09T00:00:00.000Z" class="blogPostDate_3tRe">July 9, 2020 · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>Voltage heatmap implemented (OpenGL enabled);</li><li>Several major and minor bugfixes.</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_2RZH"><a href="/PSP/blog/2020w24a-beta">2020w24a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-06-13T00:00:00.000Z" class="blogPostDate_3tRe">June 13, 2020 · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>Device context render implemented (to avoid use OpenGL - some machines can't use it properly);</li><li>Added Visual Studio solution to the GitHub project;</li><li>Several major and minor bugfixes.</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_2RZH"><a href="/PSP/blog/2019w44a-beta">22019w44a-beta</a></h2><div class="margin-vert--md"><time datetime="2019-11-01T00:00:00.000Z" class="blogPostDate_3tRe">November 1, 2019 · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>Minor changes to 32bit MSW version for compatibility;</li><li>Travis CI (Continuous Integration) implemented (Linux).</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_2RZH"><a href="/PSP/blog/2019w42a-beta">2019w42a-beta</a></h2><div class="margin-vert--md"><time datetime="2019-10-13T00:00:00.000Z" class="blogPostDate_3tRe">October 13, 2019 · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>Harmonic study implemented;</li><li>Autolayout from importation implemented;</li><li>Importation of Matpower files;</li><li>Several minor bugfixes.</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div></footer></article><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"></div><div class="pagination-nav__item pagination-nav__item--next"></div></nav></main></div></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. Built with Docusaurus.</div></div></div></footer></div> <script src="/PSP/styles.26e2fe91.js"></script> -<script src="/PSP/runtime~main.3ede30cf.js"></script> -<script src="/PSP/main.1c1af766.js"></script> +<script src="/PSP/runtime~main.57a83c0d.js"></script> +<script src="/PSP/main.1dbec03f.js"></script> <script src="/PSP/1.837a9375.js"></script> <script src="/PSP/2.19116c85.js"></script> <script src="/PSP/3.189ac30f.js"></script> diff --git a/docs/blog/tags/index.html b/docs/blog/tags/index.html index 1342599..e8e4e66 100644 --- a/docs/blog/tags/index.html +++ b/docs/blog/tags/index.html @@ -6,8 +6,8 @@ <meta name="generator" content="Docusaurus v2.0.0-alpha.58"> <link rel="search" type="application/opensearchdescription+xml" title="PSP-UFU" href="/PSP/opensearch.xml"><title data-react-helmet="true">Tags | PSP-UFU</title><meta data-react-helmet="true" property="og:title" content="Tags | PSP-UFU"><meta data-react-helmet="true" name="description" content="Blog Tags"><meta 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t(t){document.documentElement.setAttribute("data-theme",t)}var e=window.matchMedia("(prefers-color-scheme: dark)"),n=function(){var t=null;try{t=localStorage.getItem("theme")}catch(t){}return t}();null!==n?t(n):e.matches&&t("dark")}()</script><div id="__docusaurus"> <nav class="navbar navbar--light navbar--fixed-top"><div class="navbar__inner"><div class="navbar__items"><div aria-label="Navigation bar toggle" class="navbar__toggle" role="button" tabindex="0"><svg xmlns="http://www.w3.org/2000/svg" width="30" height="30" viewBox="0 0 30 30" role="img" focusable="false"><title>Menu</title><path stroke="currentColor" stroke-linecap="round" stroke-miterlimit="10" stroke-width="2" d="M4 7h22M4 15h22M4 23h22"></path></svg></div><a class="navbar__brand" href="/PSP/"><img class="navbar__logo" src="/PSP/img/logo.svg" alt="PSP-UFU logo"><strong class="navbar__title">PSP-UFU</strong></a><a class="navbar__item navbar__link" href="/PSP/docs/">User Guide</a><a aria-current="page" class="navbar__item navbar__link navbar__link--active" href="/PSP/blog">Changelog</a></div><div class="navbar__items navbar__items--right"><a href="https://github.com/Thales1330/PSP" target="_blank" rel="noopener noreferrer" class="navbar__item navbar__link">PSP-UFU Source Code</a><div class="react-toggle react-toggle--disabled displayOnlyInLargeViewport_1gtM"><div class="react-toggle-track"><div class="react-toggle-track-check"><span class="toggle_keGJ moon_1gwN"></span></div><div class="react-toggle-track-x"><span class="toggle_keGJ sun_3CPA"></span></div></div><div class="react-toggle-thumb"></div><input type="checkbox" disabled="" aria-label="Dark mode toggle" class="react-toggle-screenreader-only"></div><div class="navbar__search"><div class="searchWrapper_-F3-"><span aria-label="expand searchbar" role="button" class="searchIconButton_1Dnz" tabindex="0"></span><input type="search" id="search_input_react" placeholder="Search" aria-label="Search" class="navbar__search-input searchInput_2eto"></div></div></div></div><div role="presentation" class="navbar-sidebar__backdrop"></div><div class="navbar-sidebar"><div class="navbar-sidebar__brand"><a class="navbar__brand" href="/PSP/"><img class="navbar__logo" src="/PSP/img/logo.svg" alt="PSP-UFU logo"><strong class="navbar__title">PSP-UFU</strong></a></div><div class="navbar-sidebar__items"><div class="menu"><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" href="/PSP/docs/">User Guide</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link navbar__link--active" href="/PSP/blog">Changelog</a></li><li class="menu__list-item"><a href="https://github.com/Thales1330/PSP" target="_blank" rel="noopener noreferrer" class="menu__link">PSP-UFU Source Code</a></li></ul></div></div></div></nav><div class="main-wrapper"><div class="container margin-vert--lg"><div class="row"><main class="col col--8 col--offset-2"><h1>4 posts tagged with "psp-ufu"</h1><a href="/PSP/blog/tags">View All Tags</a><div class="margin-vert--xl"><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_2RZH"><a href="/PSP/blog/2020w28a-beta">2020w28a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-07-09T00:00:00.000Z" class="blogPostDate_3tRe">July 9, 2020 · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>Voltage heatmap implemented (OpenGL enabled);</li><li>Several major and minor bugfixes.</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div><div class="col text--right"><a aria-label="Read more about 2020w28a-beta" href="/PSP/blog/2020w28a-beta"><strong>Read More</strong></a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_2RZH"><a href="/PSP/blog/2020w24a-beta">2020w24a-beta</a></h2><div class="margin-vert--md"><time datetime="2020-06-13T00:00:00.000Z" class="blogPostDate_3tRe">June 13, 2020 · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>Device context render implemented (to avoid use OpenGL - some machines can't use it properly);</li><li>Added Visual Studio solution to the GitHub project;</li><li>Several major and minor bugfixes.</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div><div class="col text--right"><a aria-label="Read more about 2020w24a-beta" href="/PSP/blog/2020w24a-beta"><strong>Read More</strong></a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_2RZH"><a href="/PSP/blog/2019w44a-beta">22019w44a-beta</a></h2><div class="margin-vert--md"><time datetime="2019-11-01T00:00:00.000Z" class="blogPostDate_3tRe">November 1, 2019 · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>Minor changes to 32bit MSW version for compatibility;</li><li>Travis CI (Continuous Integration) implemented (Linux).</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div><div class="col text--right"><a aria-label="Read more about 22019w44a-beta" href="/PSP/blog/2019w44a-beta"><strong>Read More</strong></a></div></footer></article><article class="margin-bottom--xl"><header><h2 class="margin-bottom--sm blogPostTitle_2RZH"><a href="/PSP/blog/2019w42a-beta">2019w42a-beta</a></h2><div class="margin-vert--md"><time datetime="2019-10-13T00:00:00.000Z" class="blogPostDate_3tRe">October 13, 2019 · 1 min read</time></div><div class="avatar margin-vert--md"><a class="avatar__photo-link avatar__photo" href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener"><img src="https://avatars0.githubusercontent.com/u/16141330?s=460&v=4" alt="Thales"></a><div class="avatar__intro"><h4 class="avatar__name"><a href="https://github.com/thales1330" target="_blank" rel="noreferrer noopener">Thales</a></h4><small class="avatar__subtitle">PhD</small></div></div></header><section class="markdown"><ul><li>Harmonic study implemented;</li><li>Autolayout from importation implemented;</li><li>Importation of Matpower files;</li><li>Several minor bugfixes.</li></ul></section><footer class="row margin-vert--lg"><div class="col"><strong>Tags:</strong><a class="margin-horiz--sm" href="/PSP/blog/tags/psp-ufu">psp-ufu</a></div><div class="col text--right"><a aria-label="Read more about 2019w42a-beta" href="/PSP/blog/2019w42a-beta"><strong>Read More</strong></a></div></footer></article></div></main></div></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" 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\ No newline at end of file diff --git a/docs/docs/bus/index.html b/docs/docs/bus/index.html index 6126c67..9ea0407 100644 --- a/docs/docs/bus/index.html +++ b/docs/docs/bus/index.html @@ -6,8 +6,8 @@ <meta name="generator" content="Docusaurus v2.0.0-alpha.58"> <link rel="search" type="application/opensearchdescription+xml" title="PSP-UFU" href="/PSP/opensearch.xml"><title data-react-helmet="true">Barramento | PSP-UFU</title><meta data-react-helmet="true" name="twitter:card" content="summary_large_image"><meta data-react-helmet="true" property="og:title" content="Barramento | PSP-UFU"><meta data-react-helmet="true" name="description" content="Condutor de baixa impedância ao qual vários circuitos elétricos podem ser conectados em pontos separados."><meta data-react-helmet="true" property="og:description" content="Condutor de baixa impedância ao qual vários circuitos elétricos podem ser conectados em pontos separados."><meta data-react-helmet="true" property="og:url" 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menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulações</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_FFX1"><div class="container padding-vert--lg docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Barramento</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Condutor de baixa impedância ao qual vários circuitos elétricos podem ser conectados em pontos separados. Nota - Em muitos casos, o barramento consiste em uma barra. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=151-12-30" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="barramento-no-psp-ufu"></a>Barramento no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#barramento-no-psp-ufu" title="Direct link to heading">#</a></h2><p>O elemento <strong>barramento</strong>, ou simplesmente <strong>barra</strong>, é um conector ou nó do diagrama unifilar do PSP-UFU. Essa barra pode representar um PAC (Ponto de Acoplamento Comum), um poste de distribuição, uma subestação, um barramento da subestação, entre inúmeros outros tipos pontos de análise e conexão entre elementos.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="formulário-de-edição-dos-barramentos"></a>Formulário de edição dos barramentos<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-dos-barramentos" title="Direct link to heading">#</a></h2><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O barramento deve ser o <strong>primeiro elemento elétrico a ser inserido no diagrama de potência</strong>, uma vez que os demais componentes de potência são conectados nele.</p></div></div><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados das barras:</p><div><center><img src="/PSP/images/busForm.png" alt="Formulário dos barramentos no PSP-UFU" title="Formulário dos barramentos no PSP-UFU"></center></div><p>Esse formulário é subdividido em quatro contextos distintos:</p><ul><li><strong>Geral</strong>: no qual são inseridas informações gerais da barra e informações do fluxo de carga;</li><li><strong>Falta</strong>: local onde o curto-circuito <em>shunt</em> deve ser inserido;</li><li><strong>Estabilidade</strong>: contendo opções de visualização de dados da barra em gráficos no tempo e inserção de faltas trifásicas no cálculo de estabilidade transitória;</li><li><strong>Qualidade de energia</strong>: contém a opção de de visualização da impedância harmônica vista pela barra.</li></ul><hr><div><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_3zTA tabs__item--active">Geral</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_3zTA">Falta</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_3zTA">Estabilidade</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_3zTA">Qualidade de energia</li></ul><div role="tabpanel" class="margin-vert--md"><div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="nome"></a>Nome<a aria-hidden="true" tabindex="-1" class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="tensão-nominal"></a>Tensão nominal<a aria-hidden="true" tabindex="-1" class="hash-link" href="#tensão-nominal" title="Direct link to heading">#</a></h4><p>Utilizado para cálculo da impedância base de alguns elementos conectados, além do cálculo da relação de transformação dos transformadores conectados entre barras. Sua unidade pode ser selecionada, sendo expressa em V ou em kV.</p><p>A modificação desse parâmetro irá alterar toda a tensão do trecho conectado por linhas elétricas, sendo emitido um alerta ao usuário.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="tensão-controlada"></a>Tensão controlada<a aria-hidden="true" tabindex="-1" class="hash-link" href="#tensão-controlada" title="Direct link to heading">#</a></h4><p>Caracteriza o barramento como <strong>barra de tensão controlada (<a href="/PSP/docs/powerFlow">Barra PV</a>)</strong>,</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Essa opção é somente válida caso alguma máquina síncrona esteja conectado, caso contrário esse valor será ignorado. Caso o limite de potência reativa da máquina síncrona conectada seja ultrapassado esse valor também é ignorado.</p></div></div><p>O valor poderá ser inserido em p.u. ou em volts (ou kV caso a tensão nominal esteja nesta unidade).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="barra-de-referência"></a>Barra de referência<a aria-hidden="true" tabindex="-1" class="hash-link" href="#barra-de-referência" title="Direct link to heading">#</a></h4><p>Caracteriza o barramento como <strong>barra de referência (<a href="/PSP/docs/powerFlow">Barra de oscilação</a>)</strong>. Essa opção é somente válida caso esteja conectado um gerador síncrono, caso contrário uma mensagem de erro será exibida ao usuário ao realizar algum dos cálculos do programa.</p></div></div></div><hr><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Resumo de como definir o tipo de barra:</p><ul><li>Para definir o barramento como sendo uma <strong><a href="/PSP/docs/powerFlow">Barra PQ</a></strong>, as opções "Barra de referência" e "Tensão controlada" devem estar <strong>desmarcadas</strong>; </li><li>Para definir o barramento como sendo uma <strong><a href="/PSP/docs/powerFlow">Barra PV</a></strong>, deve-se <strong>marcar somente</strong> a opção "Tensão controlada", mantendo a opção "Barra de referência" desmarcada;</li><li>Para definir o barramento como sendo uma <strong><a href="/PSP/docs/powerFlow">Barra de Referência</a></strong>, deve-se marcar a opção "Barra de referência". Caso a opção "Tensão controlada" esteja desmarcada será admitido um valor de tensão controlada de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">1{,}0~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></li></ul></div></div><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p><strong>O sistema deve possuir somente uma barra de referência</strong>.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/bus.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/powerEditor"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Editor de Potência</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/line"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Linha »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_TbNY"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#barramento-no-psp-ufu" class="table-of-contents__link">Barramento no PSP-UFU</a></li><li><a href="#formulário-de-edição-dos-barramentos" class="table-of-contents__link">Formulário de edição dos barramentos</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_FFX1"><div class="container padding-vert--lg docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Ferramentas CAD</h1></header><div class="markdown"><blockquote><p>Desenho assistido por computador (DAC; em inglês: <em>computer aided design</em> - CAD) é o nome genérico de sistemas computacionais (<em>software</em>) utilizados pela engenharia, geologia, geografia, arquitetura e design para facilitar o projeto e desenho técnicos. No caso do <em>design</em>, este pode estar ligado especificamente a todas as suas vertentes (produtos como vestuário, eletroeletrônicos, automobilísticos, etc.), de modo que os jargões de cada especialidade são incorporados na interface de cada programa. <a href="https://pt.wikipedia.org/wiki/Desenho_assistido_por_computador" target="_blank" rel="noopener noreferrer">Wikipedia</a></p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="ferramentas-do-menu"></a>Ferramentas do menu<a aria-hidden="true" tabindex="-1" class="hash-link" href="#ferramentas-do-menu" title="Direct link to heading">#</a></h2><p>Todos os objetos criados contêm vários atributos gráficos, como: posição, orientação, cor, tamanho, etc. Para modificá-los foram inseridas algumas ferramentas intuitivas e de fácil utilização, as quais são: <a href="/PSP/docs/cadTools#selecionar"><strong>selecionar</strong></a>, <a href="/PSP/docs/cadTools#mover"><strong>mover</strong></a>, <a href="/PSP/docs/cadTools#girar"><strong>girar</strong></a>, <a href="/PSP/docs/cadTools#arrastar"><strong>arrastar</strong></a>, <a href="/PSP/docs/cadTools#zoom"><strong>zoom</strong></a>, <a href="/PSP/docs/cadTools#encaixar"><strong>encaixar</strong></a>, <a href="/PSP/docs/cadTools#copiar-e-colar"><strong>copiar, colar</strong></a> e <a href="/PSP/docs/cadTools#excluir"><strong>excluir</strong></a>.</p><p>Essas ferramentas podem ser acessadas por suas teclas de atalho assim como pelo <a href="/PSP/docs/mainScreen#menu-ribbon">submenu Ferramentas</a>.</p><p>Tais ferramentas permitem a <strong>personalização</strong>, <strong>navegação</strong> e auxiliam a <strong>criação</strong> de redes elétricas de potência, assim como diagramas de controle na plataforma.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="selecionar"></a>Selecionar<a aria-hidden="true" tabindex="-1" class="hash-link" href="#selecionar" title="Direct link to heading">#</a></h3><p>A seleção de um único objeto é realizada clicando com o botão esquerdo do mouse sobre o elemento, que será identificado pelo contorno em <span style="background-color:#80bfff;border-radius:2px;color:#000;padding:0.2rem">azul</span> do objeto.</p><p>Para a seleção de vários objetos simultaneamente é necessário clicar com o botão esquerdo do mouse em um local vazio da tela e arrastá-lo, criando assim um <strong>retângulo de seleção</strong> que irá acompanhar o ponteiro do mouse enquanto o botão esquerdo estiver pressionado. Ao soltá-lo todos os objetos que intercederem o retângulo serão selecionados.</p><p>Mantendo pressionado a tecla Control (Ctrl) do teclado pode-se manter seleções anteriores enquanto novos elementos são selecionados.</p><p>Para desmarcar todos os objetos basta clicar em uma área sem objetos.</p><video autoplay="" loop="" muted="" controls=""><source src="/PSP/videos/select.mp4" type="video/mp4"><source src="/PSP/videos/select.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="mover"></a>Mover<a aria-hidden="true" tabindex="-1" class="hash-link" href="#mover" title="Direct link to heading">#</a></h3><p>Um elemento poderá ser movido para qualquer local por meio da ação <em>drag-and-drop</em> (clicando e arrastando) ou pela ferramenta mover.</p><p>Ao habilitar o “modo mover”, movendo todos os componentes selecionados à medida que a posição do mouse é alterada. Para desabilitar esse modo basta pressionar a tecla Escape do teclado.</p><video autoplay="" loop="" muted="" controls=""><source src="/PSP/videos/move.mp4" type="video/mp4"><source src="/PSP/videos/move.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="girar"></a>Girar<a aria-hidden="true" tabindex="-1" class="hash-link" href="#girar" title="Direct link to heading">#</a></h3><p>Alguns objetos poderão ser rotacionados através da ferramenta girar, acessado pela tecla de atalho “R”, girando o objeto no sentido horário, ou “Shift + R”, para o sentido anti-horário (essa opção irá afetar todos os objetos selecionados). O elemento também pode ser rotacionado pelo <a href="/PSP/docs/mainScreen#menu-ribbon">submenu Ferramentas</a> ou ao acessar o menu de contexto clicando com o botão direito sobre o elemento.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O menu de contexto somente será exibido se o elemento estiver selecionado.</p></div></div><p>Ao utilizar essa ferramenta os objetos irão girar 45º a partir do seu ponto de origem. Para atingir a angulação pretendida basta repetir o processo de rotação.</p><video autoplay="" loop="" muted="" controls=""><source src="/PSP/videos/rotate.mp4" type="video/mp4"><source src="/PSP/videos/rotate.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="arrastar"></a>Arrastar<a aria-hidden="true" tabindex="-1" class="hash-link" href="#arrastar" title="Direct link to heading">#</a></h3><p>Três ferramentas muito úteis na navegação do circuito, principalmente em grandes redes, são: arrastar, zoom e encaixar.</p><p>A ferramenta arrastar possibilita mover todo o circuito <strong>preservando as posições relativas entre os elementos</strong>. Esta ferramenta pode ser acessada pelo submenu “Ferramentas” ou <strong>mantendo pressionada o <em>scroll</em> do mouse</strong>, ativando assim o “modo arrastar”. Ao ativá-lo todo o circuito será movido clicando e arrastando na direção desejada.</p><p>Para sair do “modo arrastar” basta pressionar a tecla Escape do teclado.</p><video autoplay="" loop="" muted="" controls=""><source src="/PSP/videos/drag.mp4" type="video/mp4"><source src="/PSP/videos/drag.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="zoom"></a>Zoom<a aria-hidden="true" tabindex="-1" class="hash-link" href="#zoom" title="Direct link to heading">#</a></h3><p>O zoom será aplicado ao utilizar o <strong>scroll do mouse</strong> para mais ou menos zoom. A ferramenta zoom irá aproximar ou distanciar o circuito a partir da posição do ponteiro do mouse na tela para encontrar objetos ou destaca-los.</p><video autoplay="" loop="" muted="" controls=""><source src="/PSP/videos/zoom.mp4" type="video/mp4"><source src="/PSP/videos/zoom.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="encaixar"></a>Encaixar<a aria-hidden="true" tabindex="-1" class="hash-link" href="#encaixar" title="Direct link to heading">#</a></h3><p>A ferramenta encaixar irá mover o circuito e aplicar o zoom necessário para que todos os componentes da rede sejam exibidos na tela. Esta ferramenta pode ser acessada através do submenu “Ferramentas” ou pela combinação de teclas “Shift + F”.</p><video autoplay="" loop="" muted="" controls=""><source src="/PSP/videos/fit.mp4" type="video/mp4"><source src="/PSP/videos/fit.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="copiar-e-colar"></a>Copiar e Colar<a aria-hidden="true" tabindex="-1" class="hash-link" href="#copiar-e-colar" title="Direct link to heading">#</a></h3><p>Duas ferramentas importantes durante o processo de criação da rede elétrica de potência no <em>software</em> são copiar e colar. Qualquer circuito criado no PSP-UFU poderá ser duplicado completamente ou parcialmente no mesmo projeto ou em múltiplos através dessas ferramentas.</p><p>Para copiar basta selecionar os elementos que se deseja duplicar e acessar a ferramenta no submenu “Ferramentas” ou pressionando a combinação das teclas “Ctrl + C”, então os dados do circuito serão copiados para a área de transferência. Ao copiar um circuito ele poderá ser colado no mesmo projeto ou em um projeto distinto, utilizando o mesmo submenu ou pressionando a combinação das teclas “Ctrl + V”.</p><video autoplay="" loop="" muted="" controls=""><source src="/PSP/videos/copyPaste.mp4" type="video/mp4"><source src="/PSP/videos/copyPaste.webm" type="video/webm"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="excluir"></a>Excluir<a aria-hidden="true" tabindex="-1" class="hash-link" href="#excluir" title="Direct link to heading">#</a></h3><p>Qualquer objeto poderá ser excluído do projeto por meio da ferramenta deletar, presente no menu Ribbon, pela tecla de atalho Delete ou pelo acesso ao menu de contexto clicando com o botão direito sobre o elemento selecionado.</p><video autoplay="" loop="" muted="" controls=""><source src="/PSP/videos/delete.mp4" type="video/mp4"><source src="/PSP/videos/delete.webm" type="video/webm"></video><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="personalização-gráfica-os-elementos"></a>Personalização gráfica os elementos<a aria-hidden="true" tabindex="-1" class="hash-link" href="#personalização-gráfica-os-elementos" title="Direct link to heading">#</a></h2><p>Alguns elementos permitem sua personalização gráfica, como as barras e linhas.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="barra"></a>Barra<a aria-hidden="true" tabindex="-1" class="hash-link" href="#barra" title="Direct link to heading">#</a></h3><p>No caso dos barramentos, pode-se alterar o <strong>comprimento da barra</strong> clicando e arrastando no <em>pickbox</em> exibido ao posicionar o ponteiro do mouse sobre esse elemento selecionado.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="linha"></a>Linha<a aria-hidden="true" tabindex="-1" class="hash-link" href="#linha" title="Direct link to heading">#</a></h3><p>As linhas podem ser inseridas com <strong>"nós" (pontos de ancoragem)</strong>, personalizando a disposição das linhas no diagrama unifilar. Esses nós são inseridos durante o processo de criação do elemento, anteriormente à seleção da segunda barra.</p><p>Após a sua inserção, novos nós podem ser anexados pelo menu de contexto acessado ao clicar com o botão direito sobre a linha selecionada. Sua remoção também é alcançada por uma opção do menu de contexto.</p><p>Assim como nos barramentos, a alteração da posição dos nós é obtida clicando e arrastando o <em>pickbox</em> exibido ao posicionar o ponteiro do mouse sobre esse elemento selecionado.</p><video autoplay="" loop="" muted="" controls=""><source src="/PSP/videos/graphicPropsChange.mp4" type="video/mp4"><source src="/PSP/videos/graphicPropsChange.webm" type="video/webm"></video><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="conexão-e-desconexão-de-elemetos-na-barra"></a>Conexão e desconexão de elemetos na barra<a aria-hidden="true" tabindex="-1" class="hash-link" href="#conexão-e-desconexão-de-elemetos-na-barra" title="Direct link to heading">#</a></h2><p>Os elementos podem ser desconectados e reconectados às barras após sua inserção somente pela ferramenta <em>drag-and-drop</em>. Para isso, o elemento deve ser selecionado e o seu nó de conexão deve ser arrastado para o local requerido.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>O elemento a ser desconectado/reconectado <strong>deve ser previamente selecionado</strong> para efetuar a operação. Caso contrário a barra será movida ou nenhuma operação será realizada.</p></div></div><p>No caso de desconexão, o elemento será automaticamente removido da simulação, sendo indicado pela sua cor (o elemento passa a ter cor <span style="background-color:#7f7f7f;border-radius:2px;color:#000;padding:0.2rem">cinza</span>).</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Ao reconectar o elemento ao circuito, deve-se <strong>atentar para estado do seu "disjuntor"</strong>, indicado por um quadrado próximo ao seu nó de conexão. Um elemento reconectado volta ao circuito com seu disjuntor aberto (<span style="background-color:#ff1919;border-radius:2px;color:#000;padding:0.2rem">vermelho</span>), sendo necessário clicar sobre o mesmo a fim de inserir o elemento no circuito.</p></div></div><video autoplay="" loop="" muted="" controls=""><source src="/PSP/videos/disconnect.mp4" type="video/mp4"><source src="/PSP/videos/disconnect.webm" type="video/webm"></video></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/cadTools.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/mainScreen"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Tela Principal</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/powerEditor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Editor de Potência »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_TbNY"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#ferramentas-do-menu" class="table-of-contents__link">Ferramentas do menu</a><ul><li><a href="#selecionar" class="table-of-contents__link">Selecionar</a></li><li><a href="#mover" class="table-of-contents__link">Mover</a></li><li><a href="#girar" class="table-of-contents__link">Girar</a></li><li><a href="#arrastar" class="table-of-contents__link">Arrastar</a></li><li><a href="#zoom" class="table-of-contents__link">Zoom</a></li><li><a href="#encaixar" class="table-of-contents__link">Encaixar</a></li><li><a href="#copiar-e-colar" class="table-of-contents__link">Copiar e Colar</a></li><li><a href="#excluir" class="table-of-contents__link">Excluir</a></li></ul></li><li><a href="#personalização-gráfica-os-elementos" class="table-of-contents__link">Personalização gráfica os elementos</a><ul><li><a href="#barra" class="table-of-contents__link">Barra</a></li><li><a href="#linha" class="table-of-contents__link">Linha</a></li></ul></li><li><a href="#conexão-e-desconexão-de-elemetos-na-barra" class="table-of-contents__link">Conexão e desconexão de elemetos na barra</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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class="docTitle_1vWb">Capacitor</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Um dispositivo constituído essencialmente por dois eletrodos separados por um dielétrico. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=436-01-03" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="capacitor-no-psp-ufu"></a>Capacitor no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#capacitor-no-psp-ufu" title="Direct link to heading">#</a></h2><p>O elemento capacitor representa, geralmente, um banco de capacitores <em>shunt</em> no circuito do PSP-UFU.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Um arranjo série de uma <a href="/PSP/docs/line">linha</a> sem as susceptâncias <em>shunt</em>, um <a href="/PSP/docs/bus">barramento</a> e um <a href="/PSP/docs/capacitor">capacitor</a> pode ser utilizado para fabricar um filtro passivo nos <a href="/PSP/docs/harmonics">estudos harmônicos</a>. Em versões futuras, um elemento de filtro passivo será implementado no PSP-UFU.</p></div></div><p>Os capacitores e reatores exigem somente sua potência reativa como parâmetro. Esses elementos passivos são somente inseridos na matriz admitância cuja impedância é calculada a partir de sua potência e tensão nominais.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="formulário-de-edição-dos-capacitores"></a>Formulário de edição dos capacitores<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-dos-capacitores" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos capacitores:</p><div><center><img src="/PSP/images/capacitorForm.png" alt="Formulário dos capacitores no PSP-UFU" title="Formulário dos capacitores no PSP-UFU"></center></div><p>Além do único contexto geral, pode ser observado o botão "Estabilidade" na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do capacitor durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/images/swCapacitor.png" alt="Formulário de chaveamento do capacitor" title="Formulário de chaveamento do capacitor"></center></div><div><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_3zTA tabs__item--active">Geral</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_3zTA">Botão Estabilidade</li></ul><div role="tabpanel" class="margin-vert--md"><div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="nome"></a>Nome<a aria-hidden="true" tabindex="-1" class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="potência-reativa"></a>Potência reativa<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potência-reativa" title="Direct link to heading">#</a></h4><p>Os capacitores e reatores exigem somente sua potência reativa como parâmetro, inserida em Mvar, kvar, var ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> (na base de potência do sistema). </p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/capacitor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/load"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Carga</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/inductor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Indutor »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_TbNY"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#capacitor-no-psp-ufu" class="table-of-contents__link">Capacitor no PSP-UFU</a></li><li><a href="#formulário-de-edição-dos-capacitores" class="table-of-contents__link">Formulário de edição dos capacitores</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Constante</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>As constantes são valores numéricos reais que permanecem inalteradas em todo o período de simulação. Esses elementos possuem somente um nó de saída, o qual poderá ser conectado a qualquer outro nó de entrada dos elementos de controle.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Esse elemento pode, por exemplo, ser utilizado para definir valores alvos ou deslocamentos de sinal.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/const.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/rateLimiter"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Limitador de taxa</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/gain"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Ganho »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_TbNY"></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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href="/PSP/docs/multiplier">Multiplicador</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/divider">Divisor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/exponential">Exponencial</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/limiter">Limitador absoluto</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/rateLimiter">Limitador de taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulações</a><ul class="menu__list"><li 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class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_FFX1"><div class="container padding-vert--lg docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Editor de Controle</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Os controles das <a href="/PSP/docs/syncGenerator">máquinas síncronas</a> são criados, modificados, importados e exportados em um editor distinto dos elementos de potência, apresentado na figura abaixo.</p><div><center><img src="/PSP/images/controlEditor.png" alt="Editor de elementos de controle do PSP-UFU" title="Editor de elementos de controle do PSP-UFU"></center></div><p>O acesso ao editor é realizado por meio de dois botões presentes no <a href="/PSP/docs/syncGenerator#formul%C3%A1rio-de-edi%C3%A7%C3%A3o-dos-geradores-s%C3%ADncronos">formulário de edição dos dados de estabilidade dos geradores síncronos</a>:</p><ul><li><a href="/PSP/docs/syncGenerator#utilizar-avr-e-regulador-de-velocidade">Editar AVR</a></li><li><a href="/PSP/docs/syncGenerator#utilizar-avr-e-regulador-de-velocidade">Editar regulador de velocidade</a></li></ul><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>No PSP-UFU a opção de editar o <strong>AVR</strong> engloba mais que somente o controle de tensão da máquina. Nele <strong>deve</strong> ser inserida a malha de controle da máquina assim como a <strong>excitatriz da máquina síncrona</strong>. Outras estratégias de controle (opcionais), como PSS (<em>Power System Stabilizer</em>) e/ou controles de sobre e sub excitação, são também implementadas em conjunto (como apresentado na figura anterior).</p></div></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Assim como no AVR, o <strong>Regulador de Velocidade</strong> engloba mais que a regulação primária da máquina. Nessa opção <strong>deve</strong> ser inserida ao menos a malha de controle da regulação primária de velocidade, assim como o <strong>modelo da turbina</strong>. Estratégias opcionais de controle da velocidade também são inseridas nessa opção.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="elementos-de-controle"></a>Elementos de Controle<a aria-hidden="true" tabindex="-1" class="hash-link" href="#elementos-de-controle" title="Direct link to heading">#</a></h2><p>O acesso aos elementos de controle é realizado em uma janela (removível e encaixável) por meio de ícones relacionados.</p><p>Uma vez que o usuário clicar no ícone desejado, o elemento de controle acompanhará o ponteiro do mouse até ser efetivamente inserido na posição desejada ao clicar novamente na área de trabalho. Os elementos são então conectados por “linhas de conexão” inseridas ao clicar nos nós dos componentes previamente adicionados, permitindo a construção da rede de controle genérica.</p><p>As ferramentas de manipulação e navegação, como arrastar, mover e excluir são <a href="/PSP/docs/cadTools">herdadas</a> do <a href="/PSP/docs/powerEditor">editor de elementos de potência</a>, possuindo comportamento idêntico. A <strong>edição dos dados</strong> dos componentes inseridos também é realizada com <strong>duplo clique</strong> sobre o elemento inserido, exibindo um formulário de edição de dados.</p><video autoplay="" loop="" muted="" controls=""><source src="/PSP/videos/timelapseControlEditor.mp4" type="video/mp4"><source src="/PSP/videos/timelapseControlEditor.webm" type="video/webm"></video><p>Os seguintes blocos de controle estão presentes no PSP-UFU:</p><ul><li><a href="/PSP/docs/io">Entrada e Saída</a></li><li><a href="/PSP/docs/transferFunction">Função Transferência</a></li><li>Matemática<ul><li><a href="/PSP/docs/sum">Somador</a></li><li><a href="/PSP/docs/multiplier">Multiplicador</a></li><li><a href="/PSP/docs/divider">Divisor</a></li><li><a href="/PSP/docs/exponential">Exponencial</a></li></ul></li><li>Limitadores<ul><li><a href="/PSP/docs/limiter">Limitador absoluto</a></li><li><a href="/PSP/docs/rateLimiter">Limitador de taxa</a></li></ul></li><li>Constantes<ul><li><a href="/PSP/docs/const">Constante</a></li><li><a href="/PSP/docs/gain">Ganho</a></li></ul></li><li><a href="/PSP/docs/mathExpression">Expressão Matemática</a></li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="inicialização-do-sistema-de-controle"></a>Inicialização do sistema de controle<a aria-hidden="true" tabindex="-1" class="hash-link" href="#inicialização-do-sistema-de-controle" title="Direct link to heading">#</a></h2><p>Caso as entradas do sistema de controle apresentem valores diferentes de zero ou a primeira operação entre elas não resulte em uma saída nula, o sistema necessita de inicialização, de forma a adequar os valores de entradas e saída dos blocos elementares e dos vetores de estado das <a href="/PSP/docs/transferFunction">funções transferências</a> presentes. Tal procedimento é realizado por meio da solução de toda rede de controle até que se obtenha sua convergência, ou seja, a diferença absoluta entre as mesmas saídas de uma solução anterior e uma atual deve ser nula ou muito próxima de zero.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>info</h5></div><div class="admonition-content"><p>O processo de <strong>inicialização é realizada automaticamente</strong> pelo PSP-UFU.</p><p>Uma vez que a inicialização é imposta pelas <a href="/PSP/docs/io">entradas</a> do controle, erros de convergência podem ocorrer devido à má parametrização dos elementos dos sistemas de <strong>potência e controle</strong>.</p></div></div><p>Para otimizar e melhorar a estabilidade do processo de inicialização utilizou-se um passo de integração variável dentro de limites, de forma que o passo aumenta em condições de diferenças menores entre as soluções do sistema de controle e diminui caso essa diferença se torne elevada. A implementação dessa abordagem reduziu significativamente o encerramento do processo com erro causado pela instabilidade numérica, além de acelerar a inicialização.</p><p>O fluxograma abaixo evidencia o processo de inicialização implementado no PSP-UFU:</p><div><center><img src="/PSP/images/controlInit.png" alt="Estrutura da inicialização do sistema de controle" title="Estrutura da inicialização do sistema de controle"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="teste-do-diagrama-de-controle"></a>Teste do diagrama de controle<a aria-hidden="true" tabindex="-1" class="hash-link" href="#teste-do-diagrama-de-controle" title="Direct link to heading">#</a></h2><p>Na parte inferior esquerda do Editor de Controle está presente o botão "<strong>Testar Sistema...</strong>". O botão acessa o formulário de teste de controles, como é indicado na figura abaixo:</p><div><center><img src="/PSP/images/testControl.png" alt="Formulário de teste de controles" title="Formulário de teste de controles"></center></div><p>Nesse formulário é possível inserir o comportamento de <strong>todas as entradas do diagrama</strong>:</p><ul><li><strong>Tipo de entrada</strong>: Define um dos tipos de entradas possíveis no PSP-UFU: "Passo (<em>Step</em>)", Rampa (<em>Ramp</em>) ou Quadrática (<em>Quadratic</em>);</li><li><strong>Tempo de início (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">t_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76508em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>)</strong>: Define o tempo de início na simulação da entrada definida. As entradas iniciais são sempre definidas como zero e assumem valores diferentes após o tempo de início;</li><li><strong>Inclinação (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span></span></span></span></span>)</strong>: Valor relacionado ao tipo de entrada:<ul><li>Para entrada do tipo "Passo" - O valor da inclinação define o valor final do passo:<div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo><</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo>≥</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} \text{Se~} t < t_i \rightarrow f(t) = 0,0\\ \text{Se~} t \ge t_i \rightarrow f(t) = \alpha \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord">0</span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></li><li>Para entrada do tipo "Rampa" - define o coeficiente de inclinação da reta:<div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo><</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo>≥</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mo>×</mo><mi>t</mi></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} \text{Se~} t < t_i \rightarrow f(t) = 0,0\\ \text{Se~} t \ge t_i \rightarrow f(t) = \alpha \times t \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord">0</span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord mathdefault">t</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></li><li>Para entrada do tipo "Quadrática" - define o coeficiente de crescimento da curva:<div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo><</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>Se </mtext><mi>t</mi><mo>≥</mo><msub><mi>t</mi><mi>i</mi></msub><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mo>×</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} \text{Se~} t < t_i \rightarrow f(t) = 0,0\\ \text{Se~} t \ge t_i \rightarrow f(t) = \alpha \times t^2 \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord">0</span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">Se</span><span class="mord nobreak"> </span></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></li></ul></li><li><strong>Passo de integração (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em"></span><span class="mord mathdefault">h</span></span></span></span></span>)</strong>: Passo do método de integração Trapezoidal Implícito;</li><li><strong>Tempo de simulação (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mi>f</mi></msub></mrow><annotation encoding="application/x-tex">t_f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9011879999999999em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>)</strong>: Tempo total de simulação.</li></ul><p>Após a simulação os resultados são exibidos no <a href="/PSP/docs/graphViewer">visualizador de gráficos</a>.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>A mesma parametrização de entrada é aplicada em todos os blocos de <a href="/PSP/docs/io">entrada</a> do sistema.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>É possível aplicar <strong>várias entradas distintas</strong> (inclusive com condicionais!) utilizando o bloco de <a href="/PSP/docs/mathExpression">expressão matemática</a>.</p></div></div><p>Utilizando o bloco de <a href="/PSP/docs/mathExpression">expressão matemática</a> é possível inserir entradas complexas, como por exemplo testar o diagrama de blocos (AVR DC1C + PSS1A - <a href="https://doi.org/10.1109/IEEESTD.2016.7553421">IEEE Std. 421.5-2016</a>) da figura abaixo:</p><div><center><img src="/PSP/images/exControl.svg" alt="Sistema de controle sob teste" title="Sistema de controle sob teste"></center></div><div><center><img src="/PSP/images/compControl.svg" alt="Comparação do sistema de controle sob teste" title="Comparação do sistema de controle sob teste"></center></div></div></article><div class="margin-vert--xl"><div class="row"><div 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href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_FFX1"><div class="container padding-vert--lg docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Divisor</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Esse elemento é bastante simples e cumpre com a função matemáticas de seu nome.</p><p>Possui duas entradas de valores reais e sua saída é computada na sequência do nó superior pelo inferior, ou seja, a saída será o valor presente no nó 1 dividido pelo nó 2.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Nós não conectados no divisor são considerados como valor nulo. Caso a divisão possua numerador zero, o valor da saída é definida como sendo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>15</mn></msup></mrow><annotation encoding="application/x-tex">10^{15}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></span></span>.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/divider.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/multiplier"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Multiplicador</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/exponential"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Exponencial »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_TbNY"></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_FFX1"><div class="container padding-vert--lg docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Exponencial</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>A função exponencial dos elementos de controle realiza essa operação com uma entrada real, obedecendo a expressão:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub><mo>=</mo><mi mathvariant="bold">A</mi><msup><mi>e</mi><mrow><mi mathvariant="bold">B</mi><msub><mi>u</mi><mi>n</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex">y_n = \bold{A} e^{\bold{B} u_n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8932769999999999em;vertical-align:0em"></span><span class="mord"><span class="mord mathbf">A</span></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8932769999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathbf mtight">B</span></span><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285719em"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">u_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">y_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a entrada e a saída do bloco exponencial, respectivamente;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">A</mi></mrow><annotation encoding="application/x-tex">\bold{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em"></span><span class="mord"><span class="mord mathbf">A</span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">B</mi></mrow><annotation encoding="application/x-tex">\bold{B}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em"></span><span class="mord"><span class="mord mathbf">B</span></span></span></span></span></span> são constantes definidas pelo usuário.</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="formulário-de-edição-de-dados-do-bloco-exponencial"></a>Formulário de edição de dados do bloco Exponencial<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-de-dados-do-bloco-exponencial" title="Direct link to heading">#</a></h2><p>A figura abaixo apresenta o formulário de edição de dados do bloco exponencial.</p><div><center><img src="/PSP/images/expForm.png" alt="Formulário de edição de dados do bloco exponencial no PSP-UFU" title="Formulário de edição de dados do bloco exponencial no PSP-UFU"></center></div><p>Esse bloco não linear é definido pelas constantes A e B, inseridas pelo usuário.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Tais blocos são úteis na representação de não linearidades, como, por exemplo, modelagem da saturação de máquinas de corrente contínuas presentes em alguns reguladores automáticos de tensão.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/exponential.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/divider"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Divisor</div></a></div><div class="pagination-nav__item 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De 70 a 80% das faltas em linhas de transmissão são entre uma fase e terra, as quais ocorrem devido ao centelhamento de apenas uma fase da linha para a torre e então para a terra.O menor número de faltas, cerca de 5%, envolve todas as três fases, chamadas de faltas trifásicas. Os outros tipos de faltas envolvem duas fases e duas fases e a terra."><meta data-react-helmet="true" property="og:description" content="O principal objetivo da análise de curto-circuito é o cálculo das correntes e tensões de falta para especificação de disjuntores, transformadores de corrente e a parametrização de relés de proteção. De 70 a 80% das faltas em linhas de transmissão são entre uma fase e terra, as quais ocorrem devido ao centelhamento de apenas uma fase da linha para a torre e então para a terra.O menor número de faltas, cerca de 5%, envolve todas as três fases, chamadas de faltas trifásicas. 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href="/PSP/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item"><a class="menu__link menu__link--sublist menu__link--active" href="#!">Simulações</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link menu__link--active active" tabindex="0" href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_FFX1"><div class="container padding-vert--lg docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Curto-Circuito</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>O principal objetivo da análise de curto-circuito é o cálculo das correntes e tensões de falta para especificação de disjuntores, transformadores de corrente e a parametrização de relés de proteção. De 70 a 80% das faltas em linhas de transmissão são entre uma fase e terra, as quais ocorrem devido ao centelhamento de apenas uma fase da linha para a torre e então para a terra.O menor número de faltas, cerca de 5%, envolve todas as três fases, chamadas de faltas trifásicas. Os outros tipos de faltas envolvem duas fases e duas fases e a terra.</p><p>Todas essas falhas, exceto a trifásica, são assimétricas e causam desequilíbrio entre as fases.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="cálculo-de-curto-circuito-no-psp-ufu"></a>Cálculo de Curto-Circuito no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#cálculo-de-curto-circuito-no-psp-ufu" title="Direct link to heading">#</a></h2><p>O primeiro estágio do cálculo de curto-circuito é a determinação das tensões pré-falta, das potências de geração e cargas do sistema. Esses dados são obtidos por meio do estudo de <a href="/PSP/docs/powerFlow">fluxo de carga</a>.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Atualmente, o PSP-UFU fornece resultado para os seguintes tipos de falta:</p><ul><li>Falta Trifásica (3F-T);</li><li>Falta Fase-Terra (F-T);</li><li>Falta Fase-Fase (F-F);</li><li>Falta Fase-Fase-Terra (F-F-T).</li></ul></div></div><p>Os modelos dos elementos elétricos que constituem um sistema de potência para o estudo de curto-circuito são semelhantes aos do fluxo de carga, apresentando algumas divergências para as faltas desbalanceadas (F-T, F-F e F-F-T).</p><p>As faltas que ocorrem com maior frequência em sistemas de potência são assimétricas. Como qualquer falta assimétrica provoca fluxo de corrente desequilibrada é necessário empregar o método das componentes simétricas. Esse método permite o estudo de sistemas balanceados em conjunto cargas desbalanceadas.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="execução-do-cálculo-de-curto-circuito-no-psp-ufu"></a>Execução do cálculo de curto-circuito no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#execução-do-cálculo-de-curto-circuito-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Existem duas formas de se calcular o curto-circuito no PSP-UFU:</p><ul><li><strong>Falta</strong>: Calcula a falta inserida nas <a href="/PSP/docs/bus">barras</a>. Nesse tipo de cálculo é possível calcular faltas <em>shunt</em> nos <a href="/PSP/docs/bus">barramentos</a> balanceadas e desbalanceadas.</li><li><strong>Nível de curto-circuito</strong>: Calcula o nível de curto-circuito (falta trifásica) em todos <a href="/PSP/docs/bus">barramentos</a> do sistema.</li></ul><p>Após a construção do diagrama unifilar no <a href="/PSP/docs/powerEditor">editor de potência</a>, a execução do cálculo de curto-circuito é realizada no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> clicando no botão <strong>Falta</strong>. Para calcular o nível de curto-circuito (falta trifásica) em todos <a href="/PSP/docs/bus">barramentos</a> do sistema, basta clicar no botão <strong>Nível de curto-circuito</strong>.</p><div><center><img src="/PSP/images/menuSimulationFaulta.svg" alt="Execução dos cálculos de curto-circuito" title="Execução dos cálculos de curto-circuito"></center></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>É possível calcular as faltas sem a execução do <a href="/PSP/docs/powerFlow">fluxo de carga</a>, porém <strong>não é recomendável</strong>, visto que os valores das correntes de falta são significativamente alteradas.</p></div></div><p><strong>Outra possibilidade</strong> é a execução por meio do cálculo contínuo, também presente no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> e seu acionamento é realizado co clicar no botão <strong>Habilitar solução</strong>. Com essa opção, os cálculos estáticos selecionados nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a> são automaticamente realizados ao modificar quaisquer parâmetros da rede, como dados elétricos e acionamento dos disjuntores dos elementos (remoção ou inserção).</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Os cálculos de curtos-circuitos não são habilitados por padrão no cálculo contínuo e devem ser inseridos nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>.</p></div></div><p>Os resultados do cálculo de curto-circuito são exibidos nos <a href="/PSP/docs/text">elementos de texto vinculado</a>, ao posicionar o mouse sobre os barramentos e em <a href="/PSP/docs/tabularReport">relatórios tabulares</a>.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="erros-comuns-na-execução-do-cálculo-de-curto-circuito"></a>Erros comuns na execução do cálculo de curto-circuito<a aria-hidden="true" tabindex="-1" class="hash-link" href="#erros-comuns-na-execução-do-cálculo-de-curto-circuito" title="Direct link to heading">#</a></h3><p>A seguir são apresentados os erros mais comuns relacionados ao calculo de curto-circuito.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="a-seguinte-mensagem-de-erro-é-exibida-falha-ao-inverter-a-matriz-admitância-de-sequência-zero"></a>A seguinte mensagem de erro é exibida: "Falha ao inverter a matriz admitância de sequência zero"<a aria-hidden="true" tabindex="-1" class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-falha-ao-inverter-a-matriz-admitância-de-sequência-zero" title="Direct link to heading">#</a></h4><ul><li><strong>Impossibilidade de circulação da corrente de sequência zero</strong>. Caso o gerador não seja aterrado, não circulará corrente de sequência zero por ele. Nesse caso, dependendo da conexão do transformador próximo ao gerador sem aterramento, a matriz admitância de sequência zero é singular. Para contornar esse problema escolha uma das duas soluções abaixo:<ul><li>Marque a opção "Neutro aterrado" e insira um alto valor de reatância de aterramento (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mn>9999</mn><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">j9999~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, por exemplo);</li><li>Ou, na barra do gerador, insira um reator de baixo valor de potência reativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mtext> </mtext><mi>v</mi><mi>a</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">1{,}0~var</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault" style="margin-right:0.03588em">v</span><span class="mord mathdefault">a</span><span class="mord mathdefault" style="margin-right:0.02778em">r</span></span></span></span></span>, por exemplo).</li></ul></li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="o-cálculo-de-curto-circuito"></a>O cálculo de curto-circuito<a aria-hidden="true" tabindex="-1" class="hash-link" href="#o-cálculo-de-curto-circuito" title="Direct link to heading">#</a></h2><p>Como já foi apresentado anteriormente, as faltas que ocorrem com maior frequência em sistemas de potência são assimétricas. Como qualquer falta assimétrica provoca fluxo de corrente desequilibrada é necessário empregar o método das componentes simétricas. Esse método permite o estudo de sistemas balanceados com cargas desbalanceadas.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="método-das-componentes-simétricas"></a>Método das componentes simétricas<a aria-hidden="true" tabindex="-1" class="hash-link" href="#método-das-componentes-simétricas" title="Direct link to heading">#</a></h3><p>Esse método proposto por C. L. Fortescue, permite definir um sistema de n fasores desbalanceados em n – 1 sistemas de n fases balanceados e um sistema de fase zero. O sistema de fase zero é definido por todas as fases de mesmo módulo e ângulo. +Para um sistema trifásico pode-se definir três componentes de sequência:</p><ol><li>Componentes de sequência positiva, constituindo em três fasores iguais em módulo, 120º defasados entre si, e tendo a mesma sequência de fase que os fasores originais;</li><li>Componentes de sequência negativa, constituindo em três fasores iguais em módulo, 120º defasados entre si, e tendo a sequência de fase oposta à dos fasores originais.</li><li>Componentes de sequência zero, constituindo em três fasores iguais em módulo e com defasagem nula entre si.</li></ol><p>Com isso pode-se decompor as tensões de fase em componentes simétricas pelas seguintes equações:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>a</mi></msub><mo>=</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>1</mn></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>2</mn></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>b</mi></msub><mo>=</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>b</mi><mn>1</mn></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>b</mi><mn>2</mn></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>b</mi><mn>0</mn></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>c</mi></msub><mo>=</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>c</mi><mn>2</mn></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>c</mi><mn>0</mn></mrow></msub></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} \dot{V}_a = \dot{V}_{a1} + \dot{V}_{a2} + \dot{V}_{a0}\\ \dot{V}_b = \dot{V}_{b1} + \dot{V}_{b2} + \dot{V}_{b0}\\ \dot{V}_c = \dot{V}_{c1} + \dot{V}_{c2} + \dot{V}_{c0} \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.9099999999999997em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t 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style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">c</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>A figura abaixo apresenta um exemplo de componentes simétricas e sua soma para obter os fasores desequilibrados.</p><div><center><img src="/PSP/images/fortescue.svg" alt="Exemplo de componentes simétricas e sua soma para obter os fasores desequilibrados" title="Exemplo de componentes simétricas e sua soma para obter os fasores desequilibrados"></center></div><p>Para simplificar os cálculos adota-se um operador “<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.63056em;vertical-align:0em"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">a</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span></span></span>”, com o intuito de indicar a rotação de um fasor. Tal operador é um número complexo de módulo unitário e ângulo de 120º:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo stretchy="true">‾</mo></mover><mo>=</mo><mn>1</mn><mi mathvariant="normal">∠</mi><mn>12</mn><msup><mn>0</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>=</mo><mn>1</mn><msup><mi>e</mi><mrow><mi>j</mi><mn>2</mn><mi>π</mi><mi mathvariant="normal">/</mi><mn>3</mn></mrow></msup><mo>=</mo><mo>−</mo><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>5</mn><mo>+</mo><mi>j</mi><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>866</mn></mrow><annotation encoding="application/x-tex">\overline{a} = 1 \angle 120^{\circ} = 1 e^{j2\pi/3} = -0{,}5 + j0{,}866</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.63056em;vertical-align:0em"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">a</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.724115em;vertical-align:0em"></span><span class="mord">1</span><span class="mord">∠</span><span class="mord">1</span><span class="mord">2</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.724115em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.938em;vertical-align:0em"></span><span class="mord">1</span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span><span class="mord mtight">2</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">π</span><span class="mord mtight">/</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">−</span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">8</span><span class="mord">6</span><span class="mord">6</span></span></span></span></span></div><p>Com isso pode-se utilizar as equações (de tensão apresentadas em conjunto com o operador “<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>a</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{a}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.63056em;vertical-align:0em"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">a</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span></span></span>” para construir a seguinte equação matricial:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>a</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>b</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>c</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mover><mover><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mover accent="true"><mi>a</mi><mo stretchy="true">‾</mo></mover><mn>2</mn></msup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mover accent="true"><mi>a</mi><mo stretchy="true">‾</mo></mover></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mover accent="true"><mi>a</mi><mo stretchy="true">‾</mo></mover></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mover accent="true"><mi>a</mi><mo stretchy="true">‾</mo></mover><mn>2</mn></msup></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo stretchy="true">⏞</mo></mover><mrow><mo fence="true">[</mo><mi mathvariant="bold">A</mi><mo fence="true">]</mo></mrow></mover><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>1</mn></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>2</mn></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} \dot{V}_a\\ \dot{V}_b\\ \dot{V}_c \end{bmatrix} = \overbrace{ \begin{bmatrix} 1 & 1 & 1\\ 1 & \overline{a}^2 & \overline{a}\\ 1 & \overline{a} & \overline{a}^2 \end{bmatrix} }^{\left[ \bold{A} \right]} \begin{bmatrix} \dot{V}_{a0}\\ \dot{V}_{a1}\\ \dot{V}_{a2} \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.8405700000000005em;vertical-align:-1.6702850000000007em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.170285em"><span style="top:-4.250095em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.9699049999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.6897149999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">c</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.6702850000000007em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:5.2692950000000005em;vertical-align:-1.6702850000000007em"></span><span class="mord mover"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.59901em"><span style="top:-4.6990099999999995em"><span class="pstrut" style="height:4.6990099999999995em"></span><span class="mord mover"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.69901em"><span style="top:-4.05101em"><span class="pstrut" style="height:4.05101em"></span><span class="mord"><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.0099999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-1.8099999999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.0099999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">a</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-1.8099999999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">a</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.0099999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">a</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span><span style="top:-1.8099999999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">a</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span></span></span></span><span class="svg-align" style="top:-6.202019999999999em"><span class="pstrut" style="height:4.05101em"></span><span class="stretchy" style="height:0.548em;min-width:1.6em"><span class="brace-left" style="height:0.548em"><svg width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMinYMin slice"><path d="M6 548l-6-6v-35l6-11c56-104 135.3-181.3 238-232 57.3-28.7 117 +-45 179-50h399577v120H403c-43.3 7-81 15-113 26-100.7 33-179.7 91-237 174-2.7 + 5-6 9-10 13-.7 1-7.3 1-20 1H6z"></path></svg></span><span class="brace-center" style="height:0.548em"><svg width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMidYMin slice"><path d="M200428 334 +c-100.7-8.3-195.3-44-280-108-55.3-42-101.7-93-139-153l-9-14c-2.7 4-5.7 8.7-9 14 +-53.3 86.7-123.7 153-211 199-66.7 36-137.3 56.3-212 62H0V214h199568c178.3-11.7 + 311.7-78.3 403-201 6-8 9.7-12 11-12 .7-.7 6.7-1 18-1s17.3.3 18 1c1.3 0 5 4 11 + 12 44.7 59.3 101.3 106.3 170 141s145.3 54.3 229 60h199572v120z"></path></svg></span><span class="brace-right" style="height:0.548em"><svg width="400em" height="0.548em" viewBox="0 0 400000 548" preserveAspectRatio="xMaxYMin slice"><path d="M400000 542l +-6 6h-17c-12.7 0-19.3-.3-20-1-4-4-7.3-8.3-10-13-35.3-51.3-80.8-93.8-136.5-127.5 +s-117.2-55.8-184.5-66.5c-.7 0-2-.3-4-1-18.7-2.7-76-4.3-172-5H0V214h399571l6 1 +c124.7 8 235 61.7 331 161 31.3 33.3 59.7 72.7 85 118l7 13v35z"></path></svg></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span><span style="top:-7.773019999999999em"><span class="pstrut" style="height:4.6990099999999995em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="minner mtight"><span class="mopen mtight delimcenter" style="top:0em"><span class="mtight">[</span></span><span class="mord mtight"><span class="mord mathbf mtight">A</span></span><span class="mclose mtight delimcenter" style="top:0em"><span class="mtight">]</span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.170285em"><span style="top:-4.250095em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.9699049999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.6897149999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.6702850000000007em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span></span></span></span></span></span></div><p>Considerando a matriz quadrada da equação anterior sendo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">[</mo><mi mathvariant="bold">A</mi><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">\left[ \bold{A} \right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">[</span><span class="mord"><span class="mord mathbf">A</span></span><span class="mclose delimcenter" style="top:0em">]</span></span></span></span></span></span>, pode-se encontrar as componentes simétricas pré-multiplicando ambos os lados dessa mesma equação por <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mo fence="true">[</mo><mi mathvariant="bold">A</mi><mo fence="true">]</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\left[ \bold{A} \right]^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.204008em;vertical-align:-0.25em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em">[</span><span class="mord"><span class="mord mathbf">A</span></span><span class="mclose delimcenter" style="top:0em">]</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954008em"><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></span>.</p><p>Da mesma forma que no estudo de fluxo de carga, a representação dos elementos do sistema para o estudo de curto-circuito é realizada por meio de circuitos equivalentes inseridos na matriz admitância de barras. Nas faltas assimétricas (F-T, F-F e F-F-T) é necessário formar três matrizes admitância de sequência: positiva, negativa e zero.</p><div class="admonition admonition-note alert alert--secondary"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M6.3 5.69a.942.942 0 0 1-.28-.7c0-.28.09-.52.28-.7.19-.18.42-.28.7-.28.28 0 .52.09.7.28.18.19.28.42.28.7 0 .28-.09.52-.28.7a1 1 0 0 1-.7.3c-.28 0-.52-.11-.7-.3zM8 7.99c-.02-.25-.11-.48-.31-.69-.2-.19-.42-.3-.69-.31H6c-.27.02-.48.13-.69.31-.2.2-.3.44-.31.69h1v3c.02.27.11.5.31.69.2.2.42.31.69.31h1c.27 0 .48-.11.69-.31.2-.19.3-.42.31-.69H8V7.98v.01zM7 2.3c-3.14 0-5.7 2.54-5.7 5.68 0 3.14 2.56 5.7 5.7 5.7s5.7-2.55 5.7-5.7c0-3.15-2.56-5.69-5.7-5.69v.01zM7 .98c3.86 0 7 3.14 7 7s-3.14 7-7 7-7-3.12-7-7 3.14-7 7-7z"></path></svg></span>Nota</h5></div><div class="admonition-content"><p>As informações a respeito das particularidades dos modelos para o estudo de curto-circuito são apresentados individualmente nos <a href="/PSP/docs/powerEditor#editando-dados-el%C3%A9tricos">elementos de potência</a>.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="equações-do-curto-circuito"></a>Equações do curto-circuito<a aria-hidden="true" tabindex="-1" class="hash-link" href="#equações-do-curto-circuito" title="Direct link to heading">#</a></h3><p>Primeiramente será tratado o equacionamento para faltas balanceadas e então os estudos serão estendidos para as faltas desbalanceadas por meio da utilização do método das componentes simétricas.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="faltas-balanceadas"></a>Faltas balanceadas<a aria-hidden="true" tabindex="-1" class="hash-link" href="#faltas-balanceadas" title="Direct link to heading">#</a></h4><p>Utiliza-se da matriz impedância de barras para o cálculo de curto-circuito, definida pela seguinte equação matricial:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><msub><mi>Z</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi></mrow></msub><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\dot{V}] = [Z_{bus}][\dot{I}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">]</span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>Z</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi></mrow></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[Z_{bus}]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></span> é a inversa da matriz admitância de barras, chamada de matriz impedância de barras.</li></ul><p>Por meio da expansão da equação anterior é possível calcular a corrente de falta trifásica na barra genérica <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi></msub><mo>=</mo><mfrac><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow></msub><mo>+</mo><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\dot{I}_f = \frac{\dot{E}_i}{\overline{z}_{ii}+\overline{z}_{f}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.206298em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.569298em;vertical-align:-0.972108em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5971899999999999em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.972108em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\dot{I}_f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.206298em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> é a corrente de falta trifásica na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\dot{E}_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.07019em;vertical-align:-0.15em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão pré-falta na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\overline{z}_{ii}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a impedância equivalente de Thevenin vista pela barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>, retirada da matriz impedância</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub></mrow><annotation encoding="application/x-tex">\overline{z}_{f}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.916668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> é a impedância de falta</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="faltas-desbalanceadas"></a>Faltas desbalanceadas<a aria-hidden="true" tabindex="-1" class="hash-link" href="#faltas-desbalanceadas" title="Direct link to heading">#</a></h4><p>O desenvolvimento das equações do cálculo de curto-circuito para faltas desbalanceadas é realizado seguindo o seguinte procedimento:</p><ol><li>Definir os diagramas no ponto da falta, mostrando as conexões de todas fases para a falta. Assume-se que apenas impedâncias balanceadas estão presentes em ambos os lados do ponto da falta e o equivalente Thevenin até esse ponto é conhecido;</li><li>Escrever as condições de contorno relacionando as tensões e corrente conhecidas para o tipo de falta estudada;</li><li>Transformar as correntes e tensões do item 2 de a-b-c para o sistema de coordenadas 0-1-2;</li><li>Encontrar a corrente do curto-circuito em estudo baseado no seguinte sistema de equações (para a fase A):<div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>1</mn></mrow></msub><mo>=</mo><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>a</mi></msub><mo>−</mo><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>1</mn></mrow></msub><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mn>1</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>2</mn></mrow></msub><mo>=</mo><mo>−</mo><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>2</mn></mrow></msub><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub><mo>=</mo><mo>−</mo><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mn>0</mn></msub></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} \dot{V}_{a1} = \dot{E}_a - \dot{I}_{a1} \overline{z}_1\\ \dot{V}_{a2} = - \dot{I}_{a2} \overline{z}_2\\ \dot{V}_{a0} = - \dot{I}_{a0} \overline{z}_0 \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.9099999999999997em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35002em"><span style="top:-2.19999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.19499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.20499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.15001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.2950099999999996em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.30501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.60002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.8500199999999998em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord">−</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord">−</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div></li></ol><p>A tabela abaixo apresenta as equações para as faltas desbalanceadas após a execução do procedimento apresenteado:</p><table><thead><tr><th><strong>Falta</strong></th><th align="center"><strong>Seq. Positiva (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{I}_{f}^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span>)</strong></th><th align="center"><strong>Seq. Negativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{I}_{f}^{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span>)</strong></th><th align="center"><strong>Seq. Zero (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>0</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{I}_{f}^{0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span>)</strong></th></tr></thead><tbody><tr><td><strong>F-T</strong></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mrow><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>0</mn></msubsup><mo>+</mo><mn>3</mn><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\dfrac{\dot{E}_i}{\overline{z}_{ii}^{1} + \overline{z}_{ii}^{2} + \overline{z}_{ii}^{0} + 3 \overline{z}_{f}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.6078659999999996em;vertical-align:-1.010676em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5971899999999999em"><span style="top:-2.275432em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.010676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{I}_{f}^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{I}_{f}^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span></td></tr><tr><td><strong>F-F</strong></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mrow><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><mo>+</mo><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\dfrac{\dot{E}_i}{\overline{z}_{ii}^{1} + \overline{z}_{ii}^{2} + \overline{z}_{f}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.6078659999999996em;vertical-align:-1.010676em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5971899999999999em"><span style="top:-2.275432em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.010676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">- \dot{I}_{f}^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3394059999999999em;vertical-align:-0.4192159999999999em"></span><span class="mord">−</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0{,}0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span></span></span></span></span></td></tr><tr><td><strong>F-F-T</strong></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mrow><mo fence="true">(</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>0</mn></msubsup><mo>+</mo><mn>3</mn><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub><mo fence="true">)</mo></mrow></mrow><mrow><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><mo>+</mo><mn>3</mn><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>0</mn></msubsup><mo>+</mo><mn>3</mn><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub><mo>+</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>0</mn></msubsup></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\dfrac{\dot{E}_i \left( \overline{z}_{ii}^{2} + \overline{z}_{ii}^{0} + 3 \overline{z}_{f} \right)}{\overline{z}_{ii}^{1} \overline{z}_{ii}^{2} + 3 \overline{z}_{ii}^{2} \overline{z}_{f} + \overline{z}_{ii}^{2} \overline{z}_{ii}^{0} + 3 \overline{z}_{ii}^{1} \overline{z}_{f} + \overline{z}_{ii}^{1} \overline{z}_{ii}^{0} }</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.670876em;vertical-align:-1.010676em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6602000000000001em"><span style="top:-2.275432em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.7400100000000003em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.010676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mo>−</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>2</mn></msubsup></mfrac></mstyle></mrow><annotation encoding="application/x-tex">- \dfrac{\dot{E}_i - \overline{z}_{ii}^{1} \dot{I}_{f}^{1}}{\overline{z}_{ii}^{2}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.700974em;vertical-align:-0.971568em"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.729406em"><span style="top:-2.275432em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.809216em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.971568em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td><td align="center"><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><msub><mover accent="true"><mi>E</mi><mo>˙</mo></mover><mi>i</mi></msub><mo>−</mo><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>1</mn></msubsup><msubsup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mi>f</mi><mn>1</mn></msubsup></mrow><mrow><msubsup><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow><mn>0</mn></msubsup><mo>+</mo><mn>3</mn><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>f</mi></msub></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">- \dfrac{\dot{E}_i - \overline{z}_{ii}^{1} \dot{I}_{f}^{1}}{\overline{z}_{ii}^{0} + 3 \overline{z}_{f}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.740082em;vertical-align:-1.010676em"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.729406em"><span style="top:-2.275432em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.809216em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.834568em"><span style="top:-2.4530000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.08346em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.010676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></td></tr></tbody></table><p>Para obter os valores em a-b-c é usada a equação matricial apresentada anteriormente, encerrando o cálculo de curto-circuito.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="referências"></a>Referências<a aria-hidden="true" tabindex="-1" class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley & Sons, New York, 2001. doi: <a href="https://doi.org/10.1002/9781118878286" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/9781118878286</a></li><li>STEVENSON JR.; WILLIAN, D. Elementos de Análise de Sistemas de Potência. 2ª ed. São Paulo: McGraw-Hill, 1986.</li><li>ANDERSON, P. M.; FOUAD, A. A. Power System Control and Stability. Wiley-IEEE Press, New York, 2002. doi: <a href="https://doi.org/10.1109/9780470545577" target="_blank" rel="noopener noreferrer">https://doi.org/10.1109/9780470545577</a></li><li>FORTESCUE, C. L. Method of Symmetrical Coordinates Applied to the Solution of Polyphase Networks. Trans. AIEE, v. 37, p.1027-1140, 1918. doi: <a href="https://doi.org/10.1109/T-AIEE.1918.4765570">https://doi.org/10.1109/T-AIEE.1918.4765570</a></li><li>ANDERSON, P. M. Analysis of faulted power systems. New York: IEEE Press, 1995.</li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/fault.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/powerFlow"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Fluxo de Potência</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/harmonics"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Harmônicos »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_TbNY"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#cálculo-de-curto-circuito-no-psp-ufu" class="table-of-contents__link">Cálculo de Curto-Circuito no PSP-UFU</a></li><li><a href="#execução-do-cálculo-de-curto-circuito-no-psp-ufu" class="table-of-contents__link">Execução do cálculo de curto-circuito no PSP-UFU</a><ul><li><a href="#erros-comuns-na-execução-do-cálculo-de-curto-circuito" class="table-of-contents__link">Erros comuns na execução do cálculo de curto-circuito</a></li></ul></li><li><a href="#o-cálculo-de-curto-circuito" class="table-of-contents__link">O cálculo de curto-circuito</a><ul><li><a href="#método-das-componentes-simétricas" class="table-of-contents__link">Método das componentes simétricas</a></li><li><a href="#equações-do-curto-circuito" class="table-of-contents__link">Equações do curto-circuito</a></li></ul></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulações</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_FFX1"><div class="container padding-vert--lg docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Fonte de Corrente Harmônica</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>(<strong>Fonte de perturbação harmônica</strong>) Parte do sistema elétrico de potência ou das instalações a ele conectadas, que causa distorção harmônica da forma de onda da corrente e / ou tensão. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=614-01-16" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="fonte-de-corrente-harmônica-no-psp-ufu"></a>Fonte de corrente harmônica no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#fonte-de-corrente-harmônica-no-psp-ufu" title="Direct link to heading">#</a></h2><p>As fontes de corrente harmônica são responsáveis pela injeção de correntes harmônicas e são utilizadas pela ferramenta de cálculo de <a href="/PSP/docs/harmonics">distorções harmônicas</a>. Uma lista de correntes harmônicas pode ser inserida em um mesmo elemento, conforme é exibido na o formulário de edição de dados.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>A presença da fonte de corrente harmônica <strong>não é considerada</strong> nos estudos de <a href="/PSP/docs/powerFlow">fluxo de carga</a>, <a href="/PSP/docs/fault">curto-circuito</a> e <a href="/PSP/docs/stability">estabilidade</a>.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="formulário-de-edição-das-fontes-de-corrente-harmônica"></a>Formulário de edição das fontes de corrente harmônica<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-das-fontes-de-corrente-harmônica" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados da fonte de corrente harmônica:</p><div><center><img src="/PSP/images/harmSourceForm.png" alt="Formulário dos indutores no PSP-UFU" title="Formulário dos indutores no PSP-UFU"></center></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="nome"></a>Nome<a aria-hidden="true" tabindex="-1" class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="ordem"></a>Ordem<a aria-hidden="true" tabindex="-1" class="hash-link" href="#ordem" title="Direct link to heading">#</a></h4><p>Especifica a ordem harmônica a ser inserida na lista de correntes harmônicas do elemento. Caso a ordem harmônica já exista na lista, ela será substituída pelos novos parâmetros.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="corrente"></a>Corrente<a aria-hidden="true" tabindex="-1" class="hash-link" href="#corrente" title="Direct link to heading">#</a></h4><p>Módulo da corrente harmônica.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="unidade"></a>Unidade<a aria-hidden="true" tabindex="-1" class="hash-link" href="#unidade" title="Direct link to heading">#</a></h4><p>Unidade do módulo de corrente harmônica, em A ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="ângulo"></a>Ângulo<a aria-hidden="true" tabindex="-1" class="hash-link" href="#ângulo" title="Direct link to heading">#</a></h4><p>Ângulo da corrente harmônica, em graus.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="adicionar--remover"></a>Adicionar / Remover<a aria-hidden="true" tabindex="-1" class="hash-link" href="#adicionar--remover" title="Direct link to heading">#</a></h4><p>Insere ou modifica a nova ordem harmônica ao clicar em “Adicionar”. Retira a ordem harmônica selecionada da lista ao clicar em “Remover”.</p></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/harmSource.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/inductor"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Indutor</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/syncGenerator"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Gerador Síncrono »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_TbNY"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#fonte-de-corrente-harmônica-no-psp-ufu" class="table-of-contents__link">Fonte de corrente harmônica no PSP-UFU</a></li><li><a href="#formulário-de-edição-das-fontes-de-corrente-harmônica" class="table-of-contents__link">Formulário de edição das fontes de corrente harmônica</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/powerEditor">Editor de Potência</a></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!" tabindex="-1">Elementos de Potência</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/bus">Barramento</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/line">Linha</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/transformer">Transformador</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/load">Carga</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/capacitor">Capacitor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/inductor">Indutor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" 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href="/PSP/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item"><a class="menu__link menu__link--sublist menu__link--active" href="#!">Simulações</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a aria-current="page" class="menu__link menu__link--active active" tabindex="0" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="0" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_FFX1"><div class="container padding-vert--lg docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Harmônicos</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Duas ferramentas relacionadas a harmônicos foram desenvolvidas no PSP-UFU:</p><ul><li>Distorção Harmônica Total de Tensão (THD, do inglês <em>Total Harmonic Distortion</em>);</li><li>Resposta na Frequência.</li></ul><p>A ferramenta de <strong><a href="https://en.wikipedia.org/wiki/Total_harmonic_distortion" target="_blank" rel="noopener noreferrer">Distorção Harmônica Total</a></strong> calcula as tensões harmônicas causadas por <a href="/PSP/docs/harmSource">fontes de corrente harmônicas</a>, assim como o THD de todos os barramentos do sistema.</p><p>A ferramenta de <strong>Resposta na Frequência</strong> (ou análise de varredura de frequência) envolve a variação da impedância da rede em um espectro de frequências observado a partir de um certo barramento.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>A análise de varredura de frequência é amplamente usada no projeto de filtros harmônicos.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Um arranjo série de uma <a href="/PSP/docs/line">linha</a> sem as susceptâncias <em>shunt</em>, um <a href="/PSP/docs/bus">barramento</a> e um <a href="/PSP/docs/capacitor">capacitor</a> pode ser utilizado para fabricar um filtro passivo nos <a href="/PSP/docs/harmonics">estudos harmônicos</a>. Em versões futuras, um elemento de filtro passivo será implementado no PSP-UFU.</p></div></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Na versão atual do programa não são consideradas as alterações das resistências do sistema causado pelo efeito pelicular. Versões futuras irão contemplar tal característica.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="execução-do-cálculo-de-harmônicos-no-psp-ufu"></a>Execução do cálculo de harmônicos no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#execução-do-cálculo-de-harmônicos-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Após a construção do diagrama unifilar no <a href="/PSP/docs/powerEditor">editor de potência</a>, a execução do cálculo das distorções harmônicas é realizada no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> clicando no botão <strong>Distorções Harmônicas</strong>. Para acessar a ferramenta de de varredura de frequência, basta clicar no botão <strong>Resposta na Frequência</strong>.</p><div><center><img src="/PSP/images/menuSimulationHamonics.svg" alt="Acesso às ferramentas de estudo harmônico" title="Acesso às ferramentas de estudo harmônico"></center></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="distorções-harmônicas"></a>Distorções Harmônicas<a aria-hidden="true" tabindex="-1" class="hash-link" href="#distorções-harmônicas" title="Direct link to heading">#</a></h3><p>Ao clicar sobre o botão "Distorções Harmônicas" as distorções causadas pelas <a href="/PSP/docs/harmSource">fontes de corrente harmônica</a> são calculadas em todos os barramentos do sistema.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Caso não forem inseridas <a href="/PSP/docs/harmSource">fontes de corrente harmônica</a> no sistema de potência, a distorção de tensão de todas as barras será <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mtext> </mtext><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">0{,}0~\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.94444em;vertical-align:-0.19444em"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord">%</span></span></span></span></span>.</p></div></div><p><strong>Outra possibilidade</strong> é a execução por meio do cálculo contínuo, também presente no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> e seu acionamento é realizado co clicar no botão <strong>Habilitar solução</strong>. Com essa opção, os cálculos estáticos selecionados nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a> são automaticamente realizados ao modificar quaisquer parâmetros da rede, como dados elétricos e acionamento dos disjuntores dos elementos (remoção ou inserção).</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Os cálculos dos níveis de THD <strong>não são habilitados por padrão</strong> no cálculo contínuo e devem ser inseridos nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>.</p></div></div><p>Os resultados das distorções harmônicas são exibidos nos <a href="/PSP/docs/text">elementos de texto vinculados</a> e ao posicionar o mouse sobre um barramento.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="resposta-na-frequência"></a>Resposta na Frequência<a aria-hidden="true" tabindex="-1" class="hash-link" href="#resposta-na-frequência" title="Direct link to heading">#</a></h3><p>Ao clicar sobre o botão "Resposta na Frequência" será exibido um formulário para inserção dos parâmetros da ferramenta:</p><div><center><img src="/PSP/images/injHarmCurrent.png" alt="Acesso à ferramentas resposta na frequência" title="Acesso à ferramentas resposta na frequência"></center></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="frequência-inicial"></a>Frequência inicial<a aria-hidden="true" tabindex="-1" class="hash-link" href="#frequência-inicial" title="Direct link to heading">#</a></h4><p>Define a frequência inicial da varredura.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="frequência-final"></a>Frequência final<a aria-hidden="true" tabindex="-1" class="hash-link" href="#frequência-final" title="Direct link to heading">#</a></h4><p>Define a frequência final da varredura.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="passo-de-frequência"></a>Passo de frequência<a aria-hidden="true" tabindex="-1" class="hash-link" href="#passo-de-frequência" title="Direct link to heading">#</a></h4><p>Define o passo de incremento da frequência. Passos menores irão gerar gráficos mais precisos, porém podem aumentar muito o tempo de execução.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="corrente-injetada"></a>Corrente injetada<a aria-hidden="true" tabindex="-1" class="hash-link" href="#corrente-injetada" title="Direct link to heading">#</a></h4><p>Define o barramento o qual será injetada a corrente harmônica para análise.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Esse campo pode ser entendido como: "<em>A resposta das impedâncias harmônicas vistas pelos barramentos do sistema caso tenha uma fonte de corrente harmônica na barra selecionada</em>".</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="botão-executar"></a>Botão Executar<a aria-hidden="true" tabindex="-1" class="hash-link" href="#botão-executar" title="Direct link to heading">#</a></h4><p>Ao clicar no botão executar a varredura será executada e os resultados exibidos no <a href="/PSP/docs/graphViewer">vizualizador de gráficos</a>, como mostra a imagem abaixo.</p><div><center><img src="/PSP/images/freqScanPlot.png" alt="Exemplo de resposta na frequência" title="Exemplo de resposta na frequência"></center></div><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Para visualizar a varredura da impedância harmônica de um barramento, essa opção deve ser habilitada em seu <a href="/PSP/docs/bus#imprimir-imped%C3%A2ncia-harm%C3%B4nica-da-barra">formulário de edição de dados</a>.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="cálculos-das-ferramentas-harmônicas"></a>Cálculos das ferramentas harmônicas<a aria-hidden="true" tabindex="-1" class="hash-link" href="#cálculos-das-ferramentas-harmônicas" title="Direct link to heading">#</a></h2><p>Ambos cálculos são realizados a partir da seguinte equação matricial:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mover accent="true"><mi>I</mi><mo>˙</mo></mover><msup><mo stretchy="false">]</mo><mi>h</mi></msup><mo>=</mo><mo stretchy="false">[</mo><msub><mi>Y</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi></mrow></msub><msup><mo stretchy="false">]</mo><mi>h</mi></msup><mo stretchy="false">[</mo><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msup><mo stretchy="false">]</mo><mi>h</mi></msup></mrow><annotation encoding="application/x-tex">[\dot{I}]^h= [Y_{bus}]^h [\dot{V}]^h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mover accent="true"><mi>I</mi><mo>˙</mo></mover><msup><mo stretchy="false">]</mo><mi>h</mi></msup></mrow><annotation encoding="application/x-tex">[\dot{I}]^h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span></span></span></span></span> Vetor das correntes harmônicas injetadas nas barras</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>Y</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi></mrow></msub><msup><mo stretchy="false">]</mo><mi>h</mi></msup></mrow><annotation encoding="application/x-tex">[Y_{bus}]^h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span></span></span></span></span> Matriz admitância harmônica de barras</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msup><mo stretchy="false">]</mo><mi>h</mi></msup></mrow><annotation encoding="application/x-tex">[\dot{V}]^h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.17019em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span></span></span></span></span></span></span></span></span> Vetor das tensões harmônicas nas barras</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em"></span><span class="mord mathdefault">h</span></span></span></span></span> Ordem harmônica</li></ul><p>O vetor das correntes harmônicas injetadas é definido no programa utilizando um elemento de potência, chamado “fonte de corrente harmônica”, em que o usuário pode criar uma lista de correntes injetadas (em A e/ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>) em um barramento.</p><p>O programa define automaticamente as ordens harmônicas na simulação analisando todas as listas de fontes de corrente harmônicas previamente aos cálculos. Com isso são calculadas as admitâncias harmônicas necessárias de cada elemento pela multiplicação das reatâncias indutivas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>l</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">x_{l}^1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.097216em;vertical-align:-0.2831079999999999em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span></span>) e susceptâncias capacitivas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>b</mi><mi>c</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">b_{c}^1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.061108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span></span>) fundamentais de cada elemento pelas ordens harmônicas:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>l</mi><mi>h</mi></msubsup><mo>=</mo><mi>h</mi><mo>×</mo><msubsup><mi>x</mi><mi>l</mi><mn>1</mn></msubsup><mspace linebreak="newline"></mspace><msubsup><mi>b</mi><mi>c</mi><mi>h</mi></msubsup><mo>=</mo><mi>h</mi><mo>×</mo><msubsup><mi>b</mi><mi>c</mi><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">x_{l}^h=h×x_l^1\\ b_{c}^h=h×b_c^1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.146108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999998em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em"></span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.1111079999999998em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.146108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999998em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em"></span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.1111079999999998em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">c</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Uma vez calculadas as admitâncias harmônicas, são utilizados os mesmos modelos e algoritmos convencionais para construção da <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Y</mi><mrow><mi>b</mi><mi>u</mi><mi>s</mi></mrow><mi>h</mi></msubsup></mrow><annotation encoding="application/x-tex">Y_{bus}^h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.132216em;vertical-align:-0.2831079999999999em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.4168920000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span></span>.</p><p>A equação matricial é resolvida usando o método de eliminação gaussiana para evitar a inversão de cada matriz de admitância harmônica. Este procedimento torna o cálculo das tensões harmônicas computacionalmente eficiente.</p><p>Uma vez calculadas as tensões harmônicas, a THD em uma barra genérica <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span> pode ser definida por:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>THD</mtext><mi>i</mi></msub><mo>=</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>h</mi><mo>=</mo><mn>2</mn></mrow><mi>n</mi></munderover><msubsup><mi>V</mi><mi>i</mi><mi>h</mi></msubsup></mrow><msubsup><mi>V</mi><mi>i</mi><mn>1</mn></msubsup></mfrac></mrow><annotation encoding="application/x-tex">\text{THD}_i=\dfrac{\sum_{h=2}^{n} V_{i}^h}{V_i^1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord text"><span class="mord">THD</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.5016819999999997em;vertical-align:-0.9628639999999999em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.538818em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959080000000001em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6897100000000003em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">h</span><span class="mrel mtight">=</span><span class="mord mtight">2</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">h</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9628639999999999em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Na ferramenta de Resposta na Frequência uma corrente senoidal é injetada na barra em uma faixa de frequências e o conjunto de equações matriciais é usado para calcular a resposta da tensão. Este cálculo é repetido em etapas discretas cobrindo o espectro de frequência especificado.</p><p>O programa usa a matriz admitância de sequência positiva e uma corrente injetada de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">1{,}0 p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> também de sequência positiva na equação matricial para calcular diretamente as impedâncias em p.u.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="referências"></a>Referências<a aria-hidden="true" tabindex="-1" class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>ARRILLAGA, J.; WATSON, N. R. Power System Harmonics. John Wiley & Sons; Chichester, 2003. doi: <a href="https://doi.org/10.1002/0470871229" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/0470871229</a></li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/harmonics.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/fault"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Curto-Circuito</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/stability"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Estabilidade »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_TbNY"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#execução-do-cálculo-de-harmônicos-no-psp-ufu" class="table-of-contents__link">Execução do cálculo de harmônicos no PSP-UFU</a><ul><li><a href="#distorções-harmônicas" class="table-of-contents__link">Distorções Harmônicas</a></li><li><a href="#resposta-na-frequência" class="table-of-contents__link">Resposta na Frequência</a></li></ul></li><li><a href="#cálculos-das-ferramentas-harmônicas" class="table-of-contents__link">Cálculos das ferramentas harmônicas</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_FFX1"><div class="container padding-vert--lg docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Motor de Indução</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Uma máquina assíncrona da qual apenas um enrolamento é energizado. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=411-31-10" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="motor-de-indução-trifásico-no-psp-ufu"></a>Motor de indução trifásico no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#motor-de-indução-trifásico-no-psp-ufu" title="Direct link to heading">#</a></h2><p>No PSP-UFU, os motores de indução são contemplados nos estudos de <a href="/PSP/docs/powerFLow">fluxo de carga</a> e no estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Os dados de estabilidade da máquina de indução são utilizados em conjunto com o fluxo de carga, <strong>calculando de forma correta a potência reativa das máquinas</strong> e consequentemente a tensão no barramento conectado.</p></div></div><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Ao marcar a opção "Calcular a potência reativa no fluxo de carga", devem ser inseridos os dados corretos na aba "Estabilidade", caso contrário, será atribuída uma potência reativa incorreta no motor.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="motor-de-indução-trifásico-no-cálculo-do-fluxo-de-carga"></a>Motor de indução trifásico no cálculo do fluxo de carga<a aria-hidden="true" tabindex="-1" class="hash-link" href="#motor-de-indução-trifásico-no-cálculo-do-fluxo-de-carga" title="Direct link to heading">#</a></h3><p>A figura abaixo apresenta o modelo do motor de indução trifásico (MIT) de gaiola simples.</p><div><center><img src="/PSP/images/indMotorModel.svg" alt="Circuito equivalente do motor de indução" title="Circuito equivalente do motor de indução"></center></div><p>As potências ativa e reativa podem ser calculadas em relação às variáveis e parâmetros do motor em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> como:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>=</mo><mfrac><mrow><msup><mi>V</mi><mn>2</mn></msup><mrow><mo fence="true">{</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><msub><mi>r</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub><mo fence="true">]</mo></mrow><mo>+</mo><msub><mi>K</mi><mn>1</mn></msub><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>m</mi></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub><mo fence="true">)</mo></mrow><mo>+</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo fence="true">]</mo></mrow><mo fence="true">}</mo></mrow></mrow><mrow><msup><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><msub><mi>r</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mi>m</mi></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup></mrow></mfrac><mspace linebreak="newline"></mspace><mi>Q</mi><mo>=</mo><mfrac><mrow><mo>−</mo><msup><mi>V</mi><mn>2</mn></msup><mrow><mo fence="true">{</mo><msub><mi>K</mi><mn>1</mn></msub><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><msub><mi>r</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub><mo fence="true">]</mo></mrow><mo>−</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>m</mi></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub><mo fence="true">)</mo></mrow><mo>+</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo fence="true">]</mo></mrow><mo fence="true">}</mo></mrow></mrow><mrow><msup><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><msub><mi>r</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo fence="true">[</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mi>m</mi></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo fence="true">]</mo></mrow><mn>2</mn></msup></mrow></mfrac><mspace linebreak="newline"></mspace><mtext> </mtext><mspace linebreak="newline"></mspace><msub><mi>K</mi><mn>1</mn></msub><mo>=</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">P = \frac{V^2 \left\{ \left( \frac{r_2}{s} \right) \left[\left(\frac{r_2}{s} \right) r_1-x_1 K_1-x_2 x_m \right]+K_1 \left[\left(\frac{r_2}{s} \right) \left(x_m+x_1 \right) + r_1 K_1 \right] \right\} }{\left[\left( \frac{r_2}{s} \right) r_1-x_1 K_1-x_2 x_m \right]^2+\left[\left( \frac{r_2}{s} \right)(x_m+x_1 )+r_1 K_1 \right]^2}\\ Q = \frac{-V^2 \left\{ K_1 \left[\left(\frac{r_2}{s} \right) r_1-x_1 K_1-x_2 x_m \right]- \left( \frac{r_2}{s} \right) \left[\left(\frac{r_2}{s} \right) \left(x_m+x_1 \right) + r_1 K_1 \right] \right\} }{\left[\left( \frac{r_2}{s} \right) r_1-x_1 K_1-x_2 x_m \right]^2+\left[\left( \frac{r_2}{s} \right)(x_m+x_1 )+r_1 K_1 \right]^2}\\ ~\\ K_1=x_2+x_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.884028em;vertical-align:-1.2940180000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.59001em"><span style="top:-2.11em"><span class="pstrut" style="height:3.054008em"></span><span class="mord"><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.054008em"><span style="top:-3.3029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.054008em"><span style="top:-3.3029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.284008em"><span class="pstrut" style="height:3.054008em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.7940180000000003em"><span class="pstrut" style="height:3.054008em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">{</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">}</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2940180000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">Q</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.884028em;vertical-align:-1.2940180000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.59001em"><span style="top:-2.11em"><span class="pstrut" style="height:3.054008em"></span><span class="mord"><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.054008em"><span style="top:-3.3029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.054008em"><span style="top:-3.3029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.284008em"><span class="pstrut" style="height:3.054008em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.7940180000000003em"><span class="pstrut" style="height:3.054008em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">{</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">}</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2940180000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0em;vertical-align:0em"></span><span class="mspace nobreak"> </span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Como pode ser observado nas equações acima, existem quatro variáveis e somente duas equações. Na prática, as variáveis podem ser reduzidas a três, uma vez que o módulo da tensão (V) é obtido nos resultados do <a href="/PSP/docs/powerFlow">fluxo de potência</a>. Para resolver as equações é necessário definir uma variável adicional. A variável escolhida como fixa no PSP-UFU é a potência ativa (P), por fornecer resultados numericamente corretos e adequados para motores em situações de estabilidade.</p><p>Portanto, nesse modelo estático, a potência ativa é mantida constante durante o cálculo do fluxo de carga e o escorregamento (s) é atualizado em cada iteração. A equação da potência ativa pode ser reescrita em relação ao escorregamento:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mn>2</mn></msup><mi>A</mi><mo>+</mo><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><mi>B</mi><mo>+</mo><mi>C</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\left(\frac{r_2}{s} \right)^2 A + \left(\frac{r_2}{s} \right) B + C = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.040008em;vertical-align:-0.686em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">s</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.3540079999999999em"><span style="top:-3.6029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.8359999999999999em;vertical-align:-0.686em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">s</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord mathdefault" style="margin-right:0.05017em">B</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">C</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">0</span></span></span></span></span></div><p>Em que:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><mi>P</mi><mrow><mo fence="true">(</mo><msubsup><mi>r</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>K</mi><mn>3</mn><mn>2</mn></msubsup><mo fence="true">)</mo></mrow><mo>−</mo><msup><mi>V</mi><mn>2</mn></msup><msub><mi>r</mi><mn>1</mn></msub><mspace linebreak="newline"></mspace><mi>B</mi><mo>=</mo><mn>2</mn><mi>P</mi><mo stretchy="false">(</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>2</mn></msub><mo>+</mo><msub><mi>K</mi><mn>3</mn></msub><msub><mi>K</mi><mn>4</mn></msub><mo stretchy="false">)</mo><mo>−</mo><msup><mi>V</mi><mn>2</mn></msup><mrow><mo fence="true">(</mo><msub><mi>K</mi><mn>2</mn></msub><mo>+</mo><msub><mi>K</mi><mn>1</mn></msub><msub><mi>K</mi><mn>3</mn></msub><mo fence="true">)</mo></mrow><mspace linebreak="newline"></mspace><mi>C</mi><mo>=</mo><mi>P</mi><mrow><mo fence="true">(</mo><msubsup><mi>K</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>K</mi><mn>4</mn><mn>2</mn></msubsup><mo fence="true">)</mo></mrow><mo>−</mo><msup><mi>V</mi><mn>2</mn></msup><msub><mi>K</mi><mn>1</mn></msub><msub><mi>K</mi><mn>4</mn></msub><mspace linebreak="newline"></mspace><msub><mi>K</mi><mn>2</mn></msub><mo>=</mo><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub><mspace linebreak="newline"></mspace><msub><mi>K</mi><mn>3</mn></msub><mo>=</mo><msub><mi>x</mi><mi>m</mi></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub><mspace linebreak="newline"></mspace><msub><mi>K</mi><mn>4</mn></msub><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>K</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A = P \left( r_1^2 + K_3^2 \right) - V^2 r_1\\ B = 2P(r_1 K_2 + K_3 K_4) - V^2 \left(K_2 + K_1 K_3 \right)\\ C = P \left( K_2^2 + K_4^2 \right) - V^2 K_1 K_4\\ K_2 = -x_1 K_1 -x_2 x_m\\ K_3 = x_m + x_1\\ K_4 = r_1 K_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.2141179999999998em;vertical-align:-0.35001em"></span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.0141079999999998em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.05017em">B</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.1141079999999999em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">C</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.2141179999999998em;vertical-align:-0.35001em"></span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.0141079999999998em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord">−</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Esse modelo pode ser inserido na solução do <a href="/PSP/docs/powerFlow">fluxo de carga</a> seguindo os seguintes passos:</p><ol><li>As constantes <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>K</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">K_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> a <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>K</mi><mn>4</mn></msub></mrow><annotation encoding="application/x-tex">K_4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são inicialmente calculadas . Esses valores são mantidos constantes durante toda a solução;</li><li>Em cada iteração são calculados os coeficientes <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault">A</span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.05017em">B</span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">C</span></span></span></span></span> utilizando o valor atualizado de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span></span></span>;</li><li>A equação quadrática é resolvida e dois valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\left(\frac{r_2}{s} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span></span></span></span></span> são obtidos, em que o maior deles é escolhido por estar na região estável da característica toque-escorregamento do motor;</li><li>Utilizando o novo valor de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mfrac><msub><mi>r</mi><mn>2</mn></msub><mi>s</mi></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\left(\frac{r_2}{s} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.20001em;vertical-align:-0.35001em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7114919999999999em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.4101em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em"><span style="top:-2.357em;margin-left:-0.02778em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span></span></span></span></span>, a potência reativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">Q</span></span></span></span></span>) é obtida. O vetor de potências é então atualizado e os procedimentos convencionais de solução do fluxo de potência são realizados.</li></ol><p>Os passos de 2 a 4 são repetidos até que se obtenha a convergência.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>No PSP-UFU, os motores de indução não são considerados no cálculo de curto-circuito.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="motor-de-indução-trifásico-no-estudo-de-estabilidade"></a>Motor de indução trifásico no estudo de estabilidade<a aria-hidden="true" tabindex="-1" class="hash-link" href="#motor-de-indução-trifásico-no-estudo-de-estabilidade" title="Direct link to heading">#</a></h3><p>Uma importante carga dinâmica são os motores de indução, uma vez que correspondem a uma parcela significativa das cargas presentes no sistema elétrico. O <a href="/PSP/docs/indMotor#motor-de-indu%C3%A7%C3%A3o-trif%C3%A1sico-no-c%C3%A1lculo-do-fluxo-de-carga">modelo da máquina de indução apresentado anteriormente</a>, a qual pode ser utilizada tanto como motor quanto como gerador, é bem estabelecida na literatura.</p><p>Como descrito na <a href="/PSP/docs/indMotor#motor-de-indu%C3%A7%C3%A3o-trif%C3%A1sico-no-c%C3%A1lculo-do-fluxo-de-carga">seção anterior</a>, a inicialização dessa máquina é realizada em conjunto com o fluxo de potência, visto que a potência reativa exigida pela máquina de indução é dependente dos parâmetros do motor, assim como a tensão do seu barramento. Essa abordagem é necessária, pois métodos convencionais conduzem a resultados errôneos em sistemas altamente carregados.</p><p>É necessário expressar a equação de movimento da máquina de indução em termos de torque e não potência, como é realizado com as máquinas síncronas. A simetria do rotor também faz com que a posição angular não seja importante e o escorregamento (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">s</span></span></span></span></span>) é utilizado no lugar da velocidade (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span></span></span></span></span>), em que:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mo>=</mo><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mrow><msub><mi mathvariant="normal">Ω</mi><mn>0</mn></msub><mo>−</mo><mi>ω</mi></mrow><msub><mi mathvariant="normal">Ω</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">s = frac{\Omega_0 - \omega}{\Omega_0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">s</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="mord mathdefault">a</span><span class="mord mathdefault">c</span><span class="mord"><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span></span><span class="mord"><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></span></div><p>Desprezando as perdas por atrito e ventilação e a potência no eixo suave, as equações mecânicas do motor são expressas da seguinte forma:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>m</mi></msub><mo>=</mo><mi>A</mi><mo>−</mo><mi>B</mi><mi>s</mi><mo>+</mo><mi>C</mi><msup><mi>s</mi><mn>2</mn></msup><mspace linebreak="newline"></mspace><msub><mi>T</mi><mi>e</mi></msub><mo>=</mo><mfrac><mrow><mi>R</mi><mi>e</mi><mrow><mo fence="true">{</mo><mover accent="true"><mi>E</mi><mo>˙</mo></mover><msup><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mo>∗</mo></msup><mo fence="true">}</mo></mrow></mrow><msub><mi mathvariant="normal">Ω</mi><mi>b</mi></msub></mfrac><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><mi>s</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo fence="true">(</mo><msub><mi>T</mi><mi>m</mi></msub><mo>−</mo><msub><mi>T</mi><mi>e</mi></msub><mo fence="true">)</mo></mrow><mrow><mn>2</mn><mi>H</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">T_m = A -Bs + Cs^2\\ T_e = \frac{Re\left\{ \dot{E}\dot{I}^* \right\}}{\Omega_b}\\ \frac{ds}{dt} = \frac{\left( T_m - T_e \right)}{2H}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault" style="margin-right:0.05017em">B</span><span class="mord mathdefault">s</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.8641079999999999em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">C</span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">e</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:3.02602em;vertical-align:-0.8360000000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.19002em"><span style="top:-2.464em"><span class="pstrut" style="height:3.15em"></span><span class="mord"><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.38em"><span class="pstrut" style="height:3.15em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-4.1900200000000005em"><span class="pstrut" style="height:3.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="mord mathdefault">e</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">{</span></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.055550000000000016em"><span class="mord">˙</span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.688696em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">}</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">s</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.08125em">H</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">e</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">T_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é o torque mecânico;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>e</mi></msub></mrow><annotation encoding="application/x-tex">T_e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">e</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é o torque elétrico;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.08125em">H</span></span></span></span></span> é a inércia do conjunto motor - carga mecânica</li></ul><p>Os termos <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault">A</span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.05017em">B</span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">C</span></span></span></span></span> são termos que definem o comportamento do torque mecânico da carga de acordo com o escorregamento. O torque mecânico normalmente varia com a <strong>velocidade</strong>, podendo ser expressa proporcionalmente com a seguinte equação quadrática:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>m</mi></msub><mo>∝</mo><mi>a</mi><mo>+</mo><mi>b</mi><mi>ω</mi><mo>+</mo><mi>c</mi><msup><mi>ω</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">T_m \propto a + b\omega + c\omega^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em"></span><span class="mord mathdefault">b</span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.8641079999999999em;vertical-align:0em"></span><span class="mord mathdefault">c</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>A</mi><mo>∝</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>B</mi><mo>∝</mo><mi>b</mi><mo>+</mo><mn>2</mn><mi>c</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>C</mi><mo>∝</mo><mi>c</mi></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} A \propto a + b + c \\ B \propto b + 2c \\ C \propto c \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.9099999999999997em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35002em"><span style="top:-2.19999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.19499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.20499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.15001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.2950099999999996em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.30501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.60002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.8500199999999998em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord mathdefault">c</span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05017em">B</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">2</span><span class="mord mathdefault">c</span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">C</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">∝</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault">c</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>As equações elétricas do motor de indução de gaiola simples são baseadas no circuito equivalente da <a href="/PSP/docs/indMotor#motor-de-indu%C3%A7%C3%A3o-trif%C3%A1sico-no-c%C3%A1lculo-do-fluxo-de-carga">figura anterior</a>. De forma semelhante ao modelo transitório da máquina síncrona, o motor de indução pode ser modelado pelo circuito equivalente de Thevenin de tensão transitória <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>E</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">E'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> atrás de uma resistência do estator <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">r_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e uma reatância transitória <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">x'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span>. A reatância transitória aparente de rotor bloqueado é dada por:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mfrac><mrow><msub><mi>x</mi><mn>2</mn></msub><msub><mi>x</mi><mi>m</mi></msub></mrow><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mi>m</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">x' = x_1 + \frac{x_2 x_m}{x_2 + x_m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.801892em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.94356em;vertical-align:-0.8360000000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>A constante de tempo de circuito aberto do rotor (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>T</mi><mn>0</mn><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><annotation encoding="application/x-tex">T_0'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.24810799999999997em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4518920000000004em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24810799999999997em"><span></span></span></span></span></span></span></span></span></span></span>) é:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>T</mi><mn>0</mn><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><msub><mi>x</mi><mi>m</mi></msub></mrow><mrow><msub><mi mathvariant="normal">Ω</mi><mi>b</mi></msub><msub><mi>r</mi><mn>2</mn></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">T_0' = \frac{x_2 + x_m}{\Omega_b r_2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.09633em;vertical-align:-0.8360000000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2603300000000002em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>E a reatância de circuito aberto é:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub><mo>=</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">x_0 = x_1 + x_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Uma vez que as reatâncias não são afetadas pela posição do rotor, as EADs do motor de indução podem ser expressar diretamente por componentes reais (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span></span></span></span></span>) e imaginárias (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">m</span></span></span></span></span>). Portanto, a descrição completa desse modelo é representada pelo seguinte sistema de equações algébrico-diferenciais:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>r</mi></msub><mo>−</mo><msubsup><mi>E</mi><mi>r</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>I</mi><mi>r</mi></msub><mo>−</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><msub><mi>I</mi><mi>m</mi></msub><mspace linebreak="newline"></mspace><msub><mi>V</mi><mi>m</mi></msub><mo>−</mo><msubsup><mi>E</mi><mi>m</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><msub><mi>r</mi><mn>1</mn></msub><msub><mi>I</mi><mi>m</mi></msub><mo>−</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><msub><mi>I</mi><mi>m</mi></msub><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>r</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mi mathvariant="normal">Ω</mi><mi>b</mi></msub><mi>s</mi><msubsup><mi>E</mi><mi>m</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><mfrac><mrow><msubsup><mi>E</mi><mi>r</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>+</mo><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>−</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>m</mi></msub></mrow><msubsup><mi>T</mi><mn>0</mn><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mfrac><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>m</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mi mathvariant="normal">Ω</mi><mi>b</mi></msub><mi>s</mi><msubsup><mi>E</mi><mi>r</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><mfrac><mrow><msubsup><mi>E</mi><mi>m</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mi>i</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>−</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>r</mi></msub></mrow><msubsup><mi>T</mi><mn>0</mn><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mfrac></mrow><annotation encoding="application/x-tex">V_r - E_r' = r_1 I_r - x' I_m\\ V_m - E_m' = r_1 I_m - x' I_m\\ \frac{dE_r'}{dt} = \Omega_b s E_m' - \frac{E_r' + \left( x_0 - x' \right) I_m}{T_0'}\\ \frac{dE_m'}{dt} = \Omega_b s E_r' - \frac{E_m' i \left( x_0 - x' \right) I_r}{T_0'}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.9518920000000001em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.9518920000000001em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.114892em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord mathdefault">s</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.3811999999999998em;vertical-align:-0.9523079999999999em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.433692em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9523079999999999em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.114892em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord mathdefault">s</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.3811999999999998em;vertical-align:-0.9523079999999999em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.433692em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9523079999999999em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="formulário-de-edição-dos-geradores-síncronos"></a>Formulário de edição dos geradores síncronos<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-dos-geradores-síncronos" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos motores de indução:</p><div><center><img src="/PSP/images/indMotorForm.png" alt="Formulário dos motores de indução no PSP-UFU" title="Formulário dos motores de indução no PSP-UFU"></center></div><p>No formulário pode ser observado o botão "Chaveamento" na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do motor durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/images/indMotorSw.png" alt="Formulário de chaveamento do motores de indução" title="Formulário de chaveamento do motores de indução"></center></div><div><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_3zTA tabs__item--active">Geral</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_3zTA">Estabilidade</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_3zTA">Botão Chaveamento</li></ul><div role="tabpanel" class="margin-vert--md"><div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="nome"></a>Nome<a aria-hidden="true" tabindex="-1" class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="potência-nominal"></a>Potência nominal<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potência-nominal" title="Direct link to heading">#</a></h4><p>Potência nominal do gerador, inserida em MVA, kVA ou VA.</p><p>Esse campo é especialmente importante caso a opção "Utilizar a potência nominal como base" esteja marcada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="potências-ativa-e-reativa"></a>Potências ativa e reativa<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potências-ativa-e-reativa" title="Direct link to heading">#</a></h4><p>Potências ativa (inserida em W, kW, MW ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>) e reativa (inserida em var, kvar, Mvar ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>) do motor.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Caso a opção “Calcular a potência reativa no fluxo de carga” esteja ativada, o campo de potência reativa é desativado para edição.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="calcular-a-potência-reativa-no-fluxo-de-carga"></a>Calcular a potência reativa no fluxo de carga<a aria-hidden="true" tabindex="-1" class="hash-link" href="#calcular-a-potência-reativa-no-fluxo-de-carga" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar os dados fornecidos no formulário de estabilidade para calcular a potência reativa do motor durante o processo iterativo do fluxo de carga.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Caso essa opção não seja utilizada o motor será considerado uma <a href="/PSP/docs/load">carga de potência constante</a> no estudo de <a href="/PSP/docs/powerFlow">fluxo de carga</a>.</p><p>A não utilização dessa opção poderá gerar erros de regime permamente no estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="utilizar-potência-nominal-como-base"></a>Utilizar potência nominal como base<a aria-hidden="true" tabindex="-1" class="hash-link" href="#utilizar-potência-nominal-como-base" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar a potência nominal do gerador como base para a conversão das unidades, inclusive aqueles no formulário de estabilidade, caso contrário será usada a potência base do sistema.</p></div></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="referências"></a>Referências<a aria-hidden="true" tabindex="-1" class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>SÁNCHEZ, J. C.; OLIVARES, T. I. A.; ORTIZ, G. R.; VEGA, D. R. Induction Motor Static Models for Power Flow and Voltage Stability Studies. In: IEEE Power and Energy Society General Meeting, 2012, San Diego. doi: <a href="https://doi.org/10.1109/PESGM.2012.6345618">https://doi.org/10.1109/PESGM.2012.6345618</a></li><li>IEEE Std 399-1997. IEEE Recommended Practice for Industrial and Commercial Power Systems Analysis (Brown Book). IEEE, New York, ago. 1998. doi: <a href="https://doi.org/10.1109/IEEESTD.1998.88568">https://doi.org/10.1109/IEEESTD.1998.88568</a></li><li>MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: <a href="https://doi.org/10.1007/978-3-642-13669-6" target="_blank" rel="noopener noreferrer">https://doi.org/10.1007/978-3-642-13669-6</a></li><li>ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley & Sons, New York, 2001. doi: <a href="https://doi.org/10.1002/9781118878286" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/9781118878286</a></li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/indMotor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/syncMotor"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Compensador Síncrono</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/controlEditor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Editor de Controle »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_TbNY"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#motor-de-indução-trifásico-no-psp-ufu" class="table-of-contents__link">Motor de indução trifásico no PSP-UFU</a><ul><li><a href="#motor-de-indução-trifásico-no-cálculo-do-fluxo-de-carga" class="table-of-contents__link">Motor de indução trifásico no cálculo do fluxo de carga</a></li><li><a href="#motor-de-indução-trifásico-no-estudo-de-estabilidade" class="table-of-contents__link">Motor de indução trifásico no estudo de estabilidade</a></li></ul></li><li><a href="#formulário-de-edição-dos-geradores-síncronos" class="table-of-contents__link">Formulário de edição dos geradores síncronos</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">PSP-UFU</h1></header><div class="markdown"><div class="admonition admonition-important alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>important</h5></div><div class="admonition-content"><p>This user guide was written in Brazilian Portuguese. If you want to help me translate this guide, contact-me on <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> or <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="sobre-o-psp-ufu"></a>Sobre o PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#sobre-o-psp-ufu" title="Direct link to heading">#</a></h2><p>O PSP-UFU (Plataforma de Sistemas de Potência da Universidade Federal de Uberlândia) é um software <strong>multi-plataforma</strong>, <strong>multilíngue</strong>, <strong>livre e de código aberto (FOSS)</strong> com <strong>recursos avançados de GUI (<em>Graphical User Interface</em>)</strong> e <strong>ferramentas CAD (<em>Computer Aided Design</em>)</strong> para estudos de sistemas elétricos de potência.</p><p>O software permite a construção de qualquer rede de transmissão elétrica e sistemas de controle através da inserção de elementos visuais. Para a visualização dos resultados, o programa oferece <strong>elementos de texto vinculados</strong> na tela principal e também <strong>editores de tabelas e gráficos</strong>.</p><p>O PSP-UFU tem como objetivo fornecer ferramentas eficientes de simulação para fins de <strong>pesquisa</strong> e <strong>educação</strong>, além de <strong>aplicações industriais</strong> em sistemas elétricos de potência.</p><p>Resumidamente, o <em>software</em> pode executar os seguintes estudos:</p><ul><li>Fluxo de potência</li><li>Cálculo de curto-circuito</li><li>Harmônicos</li><li>Estabilidade transitória e dinâmica</li></ul><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="publicações"></a>Publicações<a aria-hidden="true" tabindex="-1" class="hash-link" href="#publicações" title="Direct link to heading">#</a></h3><p>Os artigos abaixo possuem maiores detalhes a respeito do PSP-UFU:</p><blockquote><p>Oliveira, T. L., Guimarães, G. C., & Silva, L. R. C. (2019). PSP-UFU: An open-source, graphical, and multiplatform software for power system studies. <em>International Transactions on Electrical Energy Systems</em>, e12185. doi: <a href="https://doi.org/10.1002/2050-7038.12185">10.1002/2050-7038.12185</a></p></blockquote><blockquote><p>Oliveira, T. L., Guimarães, G. C., Silva, L. R., & Rezende, J. O. (2019). Power system education and research applications using free and open-source, graphical and multiplatform PSP-UFU software. <em>The International Journal of Electrical Engineering & Education</em>, 0020720919879058. doi: <a href="https://doi.org/10.1177/0020720919879058" target="_blank" rel="noopener noreferrer">10.1177/0020720919879058</a></p></blockquote><p>A Tese de Doutorado abaixo detalha em profundidade os cálculos envolvidos, arquitetura de software e ferramentas implementadas no PSP-UFU:</p><blockquote><p>Oliveira, T. L. (2019). Desenvolvimento de um programa computacional livre, gráfico, e multiplataforma para analisar sistemas elétricos de potência em regime permanente e dinâmico. doi: <a href="https://dx.doi.org/10.14393/ufu.te.2019.2444">10.14393/ufu.te.2019.2444</a></p></blockquote><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="contribuindo-com-o-projeto"></a>Contribuindo com o projeto<a aria-hidden="true" tabindex="-1" class="hash-link" href="#contribuindo-com-o-projeto" title="Direct link to heading">#</a></h3><p>Se você está interessado em contribuir com o PSP-UFU de alguma forma (desenvolvimento, críticas, sugestões, etc.), entre em contato pelo <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> ou <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="desenvolvedores"></a>Desenvolvedor<del>es</del><a aria-hidden="true" tabindex="-1" class="hash-link" href="#desenvolvedores" title="Direct link to heading">#</a></h3><p><a href="https://github.com/Thales1330" target="_blank" rel="noopener noreferrer">Thales Lima Oliveira</a></p></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/aboutPSP.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/installation"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Instalação »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_TbNY"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#sobre-o-psp-ufu" class="table-of-contents__link">Sobre o PSP-UFU</a><ul><li><a href="#publicações" class="table-of-contents__link">Publicações</a></li><li><a href="#contribuindo-com-o-projeto" class="table-of-contents__link">Contribuindo com o projeto</a></li><li><a href="#desenvolvedores" class="table-of-contents__link">Desenvolvedor<del>es</del></a></li></ul></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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class="docTitle_1vWb">Indutor</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Um reator destinado à conexão de derivação em uma rede para compensar a corrente capacitiva. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=421-01-03" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="indutor-no-psp-ufu"></a>Indutor no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#indutor-no-psp-ufu" title="Direct link to heading">#</a></h2><p>O elemento indutor (ou reator) representa, geralmente, reator <em>shunt</em> no circuito do PSP-UFU.</p><p>Os capacitores e reatores exigem somente sua potência reativa como parâmetro. Esses elementos passivos são somente inseridos na matriz admitância cuja impedância é calculada a partir de sua potência e tensão nominais.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="formulário-de-edição-dos-indutores"></a>Formulário de edição dos indutores<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-dos-indutores" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos indutores:</p><div><center><img src="/PSP/images/indForm.png" alt="Formulário dos indutores no PSP-UFU" title="Formulário dos indutores no PSP-UFU"></center></div><p>Além do único contexto geral, pode ser observado o botão "Estabilidade" na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do indutor durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/images/indSw.png" alt="Formulário de chaveamento do indutor" title="Formulário de chaveamento do indutor"></center></div><div><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_3zTA tabs__item--active">Geral</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_3zTA">Botão Estabilidade</li></ul><div role="tabpanel" class="margin-vert--md"><div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="nome"></a>Nome<a aria-hidden="true" tabindex="-1" class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="potência-reativa"></a>Potência reativa<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potência-reativa" title="Direct link to heading">#</a></h4><p>Os capacitores e reatores exigem somente sua potência reativa como parâmetro, inserida em Mvar, kvar, var ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> (na base de potência do sistema). </p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/inductor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/capacitor"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Capacitor</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/harmSource"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Fonte de Corrente Harmônica »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_TbNY"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#indutor-no-psp-ufu" class="table-of-contents__link">Indutor no PSP-UFU</a></li><li><a href="#formulário-de-edição-dos-indutores" class="table-of-contents__link">Formulário de edição dos indutores</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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href="/PSP/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulações</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_FFX1"><div class="container padding-vert--lg docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Instalação</h1></header><div class="markdown"><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="download"></a>Download<a aria-hidden="true" tabindex="-1" class="hash-link" href="#download" title="Direct link to heading">#</a></h2><p>O download do PSP-UFU pode ser realizado pelo <a href="https://github.com/Thales1330/PSP/releases/latest" target="_blank" rel="noopener noreferrer">"<em>last release</em>" do repositório do PSP-UFU no GitHub</a>.</p><p>Ao acessar ao link, role a página até encontrar os <em>Assets</em>:</p><div><center><img src="/PSP/images/download.png" alt="Assets download" title="Informações da Figura"></center></div><p>Como o PSP-UFU é um <em>software</em> multiplataforma, você deve baixar o arquivo correto:</p><ul><li>Para o sistema operacional <a href="/PSP/docs/installation#windows">Windows</a> baixe os arquivos <em>.exe</em> (recomendado) ou <em>.zip</em>.</li><li>Para os sistemas <a href="/PSP/docs/installation#linux">Linux</a> baseados no Debian (por exemplo, Ubuntu, Mint) <em>64 bits</em> baixe o pacote de instalação <em>.deb</em>.</li></ul><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Se você está interessado em contribuir com o PSP-UFU construindo-o para outros sistemas operacionais (principalmente macOS!) entre em contato pelo <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> ou <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>.</p></div></div><p>Na sequência são apresentados o passo-a-passo para instalação dos SOs <a href="/PSP/docs/installation#windows">Windows</a> e <a href="/PSP/docs/installation#linux">Linux</a>. Também são apresentadas as soluções para os erros mais recorrentes.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="windows"></a>Windows<a aria-hidden="true" tabindex="-1" class="hash-link" href="#windows" title="Direct link to heading">#</a></h2><p>Após baixar o arquivo <em>.exe</em> avance em cada passo do instalador. O executável irá solicitar automaticamente a instalação do <strong>Microsoft Visual C++ Redistribuível</strong> e, caso não esteja instalado no seu computador, realize os procedimentos do programa.</p><p>Caso o executável falhe em abrir, baixe o arquivo <em>.zip</em> e descompacte em alguma pasta do seu sistema. Nesse caso é necessário baixar e instalar manualmente o <a href="https://aka.ms/vs/16/release/vc_redist.x86.exe"><strong>Microsoft Visual C++ Redistribuível 32 bits</strong></a>.</p><p>Após a conclusão da instalação abra o PSP-UFU para verificar a correta instalação. O executável está presente na pasta "<em><local de instalação>/PSP-UFU/bin/PSP-UFU.exe</em>".</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="problemas-recorrentes-no-windows"></a>Problemas recorrentes no Windows<a aria-hidden="true" tabindex="-1" class="hash-link" href="#problemas-recorrentes-no-windows" title="Direct link to heading">#</a></h3><ul><li>Ao baixar o navegador pode <strong>falsamente</strong> sinalizar que é um arquivo perigoso, basta clicar em "Manter arquivo";</li><li>Ao abrir o instalador, novamente o Windows sinaliza que o arquivo é perigoso, basta clicar em "Executar assim mesmo";</li><li>Ao abrir o programa e alterar o seu idioma, uma mensagem de erro é exibida. Para resolver esse problema basta executar o PSP-UFU como administrador (clicar com botão direito no atalho do programa e posteriormente em executar como administrador). Altere novamente o idioma e reinicie o programa;</li><li>Ao criar um novo projeto a tela de trabalho não é exibida ou é exibida uma tela preta. Vá nas opções gerais e altere o renderizador para "Device Context". Caso apareça uma mensagem de erro, execute o programa como administrador e repita o processo. Reinicie o programa.</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="linux"></a>Linux<a aria-hidden="true" tabindex="-1" class="hash-link" href="#linux" title="Direct link to heading">#</a></h2><p>A instalação no sistema Linux é facilmente realizado pelo pacote de instalação automático <em>.deb</em>.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Note que essa versão para Linux está defasada e algumas funcionalidades do programa podem não estar presentes.</p><p>Caso queira contribuir com o projeto e construir a versão atual no sistema Linux baseado em Debian ou em outras distribuições entre em contato pelo <a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer">GitHub</a> ou <a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer">Twitter</a>.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="problemas-recorrentes-no-linux"></a>Problemas recorrentes no Linux<a aria-hidden="true" tabindex="-1" class="hash-link" href="#problemas-recorrentes-no-linux" title="Direct link to heading">#</a></h3><p>Pode ser necessário adicionar o <a href="https://wiki.wxwidgets.org/Compiling_and_getting_started#Running_wxWidgets_projects" target="_blank" rel="noopener noreferrer">diretório das bibliotecas do wxWidgets</a> nas variáveis de ambiente. Para isso, siga <strong>UM</strong> dos métodos apresentados abaixo:</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="método-i-recomendado"></a><em>Método I (recomendado)</em><a aria-hidden="true" tabindex="-1" class="hash-link" href="#método-i-recomendado" title="Direct link to heading">#</a></h4><p>Insira o seguinte comando no arquivo <strong>~/.bashrc</strong>:</p><div class="mdxCodeBlock_iHAB"><div class="codeBlockContent_32p_"><button type="button" aria-label="Copy code to clipboard" class="copyButton_1BYj">Copy</button><div tabindex="0" class="prism-code language-shell codeBlock_19pQ"><div class="codeBlockLines_2n9r" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token builtin class-name" style="color:rgb(255, 203, 107)">echo</span><span class="token plain"> </span><span class="token string" style="color:rgb(195, 232, 141)">"export LD_LIBRARY_PATH=</span><span class="token string variable" style="color:rgb(191, 199, 213)">$LD_LIBRARY_PATH</span><span class="token string" style="color:rgb(195, 232, 141)">:/usr/local/lib"</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">>></span><span class="token plain"> ~/.bashrc</span></div></div></div></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="método-ii-não-recomendado"></a><em>Método II (não recomendado)</em><a aria-hidden="true" tabindex="-1" class="hash-link" href="#método-ii-não-recomendado" title="Direct link to heading">#</a></h4><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Utilizando esse método os passos abaixo devem ser executados <strong>todas as vezes</strong> que você quiser abrir o PSP-UFU.</p></div></div><ol><li>Abra o terminal e acesse o diretório onde está instalado o PSP-UFU:</li></ol><div class="mdxCodeBlock_iHAB"><div class="codeBlockContent_32p_"><button type="button" aria-label="Copy code to clipboard" class="copyButton_1BYj">Copy</button><div tabindex="0" class="prism-code language-shell codeBlock_19pQ"><div class="codeBlockLines_2n9r" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token builtin class-name" style="color:rgb(255, 203, 107)">cd</span><span class="token plain"> /usr/local/bin</span></div></div></div></div></div><ol start="2"><li>Insira as bibliotecas compartilhadas do wxWidgets nas variáveis de ambiente:</li></ol><div class="mdxCodeBlock_iHAB"><div class="codeBlockContent_32p_"><button type="button" aria-label="Copy code to clipboard" class="copyButton_1BYj">Copy</button><div tabindex="0" class="prism-code language-shell codeBlock_19pQ"><div class="codeBlockLines_2n9r" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token builtin class-name" style="color:rgb(255, 203, 107)">export</span><span class="token plain"> </span><span class="token assign-left variable" style="color:rgb(191, 199, 213)">LD_LIBRARY_PATH</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token variable" style="color:rgb(191, 199, 213)">$LD_LIBRARY_PATH</span><span class="token plain">:/usr/local/lib</span></div></div></div></div></div><ol start="3"><li>Então, execute o PSP-UFU na <strong>mesma seção</strong> do terminal:</li></ol><div class="mdxCodeBlock_iHAB"><div class="codeBlockContent_32p_"><button type="button" aria-label="Copy code to clipboard" class="copyButton_1BYj">Copy</button><div tabindex="0" class="prism-code language-shell codeBlock_19pQ"><div class="codeBlockLines_2n9r" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">./PSP-UFU</span></div></div></div></div></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Verifique se o computador possui as unidades de vídeo instaladas corretamente, caso contrário, o programa não exibirá a área de trabalho para criar e editar diagramas de linha única ou diagramas de blocos de controle.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/installation.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav 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Built with Docusaurus.</div></div></div></footer></div> <script src="/PSP/styles.26e2fe91.js"></script> -<script src="/PSP/runtime~main.3ede30cf.js"></script> -<script src="/PSP/main.1c1af766.js"></script> +<script src="/PSP/runtime~main.57a83c0d.js"></script> +<script src="/PSP/main.1dbec03f.js"></script> <script src="/PSP/1.837a9375.js"></script> <script src="/PSP/2.19116c85.js"></script> <script src="/PSP/64.2a8261e0.js"></script> diff --git a/docs/docs/io/index.html b/docs/docs/io/index.html index 7ab2e13..4f39f5b 100644 --- a/docs/docs/io/index.html +++ b/docs/docs/io/index.html @@ -6,8 +6,8 @@ <meta name="generator" content="Docusaurus v2.0.0-alpha.58"> <link rel="search" type="application/opensearchdescription+xml" title="PSP-UFU" href="/PSP/opensearch.xml"><title data-react-helmet="true">Entrada / Saída | PSP-UFU</title><meta data-react-helmet="true" name="twitter:card" content="summary_large_image"><meta data-react-helmet="true" property="og:title" content="Entrada / Saída | PSP-UFU"><meta data-react-helmet="true" name="description" content="As entradas e saída são variáveis relacionadas às máquinas síncronas. 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tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_FFX1"><div class="container padding-vert--lg docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Entrada / Saída</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>As entradas e saída são variáveis relacionadas às máquinas síncronas. Cada tipo de entrada e saída está relacionada a um regulador da máquina: <strong>Regulador de Tensão (AVR, do inglês <em>Automatic Voltage Regulator</em>)</strong> e <strong>Regulador de Velocidade (RV)</strong>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>O sistema de controle deve ter <strong>ao menos uma entrada e uma saída</strong>.</p></div></div><p>As entradas e saídas do sistema de controle são definidas por esses blocos, os quais são distintos para cada tipo de escopo. A figura abaixo apresenta um exemplo de um formulário de edição das entradas e saídas de um AVR:</p><div><center><img src="/PSP/images/ioForm.png" alt="Formulário de edição de entradas e saídas de sistemas de controle de um AVR" title="Formulário de edição de entradas e saídas de sistemas de controle de um AVR"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="variáveis-de-entrada-e-saída"></a>Variáveis de entrada e saída<a aria-hidden="true" tabindex="-1" class="hash-link" href="#variáveis-de-entrada-e-saída" title="Direct link to heading">#</a></h2><p>O usuário deve sinalizar o tipo de bloco entre entrada e saída e, então, selecionar em uma lista a variável desejada. As seguintes variáveis de controle estão atualmente disponíveis no programa:</p><ul><li><p><strong>Tensão terminal</strong> (<em>entrada: AVR</em>): Módulo da tensão no barramento da máquina síncrona, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, variável no tempo. Essa variável é normalmente utilizada no cálculo do erro da tensão de referência do AVR;</p></li><li><p><strong>Velocidade</strong> (<em>entrada: AVR e RV</em>): Velocidade da máquina síncrona, em rad/s, variável no tempo. Normalmente utilizada no cálculo do erro de velocidade nos reguladores de velocidade, além de entrada do PSS em AVRs;</p></li><li><p><strong>Potência ativa e reativa</strong> (<em>entrada: AVR</em>): Potência ativa fornecida pela máquina síncrona, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, variável no tempo. Normalmente utilizada como entrada do PSS (potência ativa) e controle de sub e sobrecorrente de excitação nos AVRs;</p></li><li><p><strong>Tensão terminal inicial</strong> (<em>entrada: AVR</em>): Módulo da tensão no barramento da máquina síncrona prévia ao estudo dinâmico originado do fluxo de carga, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, fixo no tempo. Essa variável está normalmente associada à referência de tensão do AVR;</p></li><li><p><strong>Velocidade inicial</strong> (<em>entrada: AVR e RV</em>): Velocidade do sistema (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>π</mi><msub><mi>f</mi><mrow><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mrow><annotation encoding="application/x-tex">2 \pi f_{ref}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em"></span><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.03588em">π</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>), definida nas <a href="/PSP/docs/simulationConfig">opções de simulação</a>, em rad/s, fixa no tempo. Normalmente utiliza-se essa variável como referência de velocidade em RVs e normalização da velocidade;</p></li><li><p><strong>Potência mecânica inicial</strong> (<em>entrada: RV</em>): Potência mecânica inicial, calculada após a inicialização das máquinas síncronas com os dados originados do fluxo de carga, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, fixa no tempo. Normalmente é utilizada como referência de potência mecânica nos reguladores de velocidade;</p></li><li><p><strong>Variação de velocidade e potência ativa</strong> (<em>entrada: AVR</em>): Cálculo da variação dessas entradas entre os passos de integração normalizada pelo passo de integração, conforme a equação:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>−</mo><msub><mi>x</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><mi>h</mi></mfrac></mrow><annotation encoding="application/x-tex">\Delta x = \frac{x_n-x_{n-1}}{h}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.9463300000000001em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.26033em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">h</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault">x</span></span></span></span></span> é a variação da entrada (velocidade ou potência ativa);</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">x_{n-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span></span> é a variável no passo atual e anterior, respectivamente;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em"></span><span class="mord mathdefault">h</span></span></span></span></span> é o passo de integração.</li></ul><div class="admonition admonition-note alert alert--secondary"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M6.3 5.69a.942.942 0 0 1-.28-.7c0-.28.09-.52.28-.7.19-.18.42-.28.7-.28.28 0 .52.09.7.28.18.19.28.42.28.7 0 .28-.09.52-.28.7a1 1 0 0 1-.7.3c-.28 0-.52-.11-.7-.3zM8 7.99c-.02-.25-.11-.48-.31-.69-.2-.19-.42-.3-.69-.31H6c-.27.02-.48.13-.69.31-.2.2-.3.44-.31.69h1v3c.02.27.11.5.31.69.2.2.42.31.69.31h1c.27 0 .48-.11.69-.31.2-.19.3-.42.31-.69H8V7.98v.01zM7 2.3c-3.14 0-5.7 2.54-5.7 5.68 0 3.14 2.56 5.7 5.7 5.7s5.7-2.55 5.7-5.7c0-3.15-2.56-5.69-5.7-5.69v.01zM7 .98c3.86 0 7 3.14 7 7s-3.14 7-7 7-7-3.12-7-7 3.14-7 7-7z"></path></svg></span>Nota</h5></div><div class="admonition-content"><p>A normalização é necessária para a correta utilização da razão de passo de controle, definida nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>.</p></div></div></li><li><p><strong>Tensão de campo</strong> (<em>saída: AVR</em>): Define a tensão aplicada ao campo na máquina síncrona, em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> Utilizada como saída dos AVRs, controlando principalmente tensão no barramento conectado e/ou fator de potência da máquina;</p></li><li><p><strong>Potência mecânica</strong> (<em>saída: RV</em>): Define qual a potência mecânica aplicada no eixo da máquina síncrona. Utilizada como saída dos reguladores de velocidade, controlando principalmente a potência ativa injetada pela máquina e sua frequência.</p></li></ul></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/io.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/controlEditor"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Editor de Controle</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/transferFunction"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Função Transferência »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_TbNY"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#variáveis-de-entrada-e-saída" class="table-of-contents__link">Variáveis de entrada e saída</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Limitador absoluto</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Os blocos de limite absoluto cumprem a função de restringir os valores de entrada de acordo com os limites máximos e mínimos inseridos pelo usuário. Caso a entrada ultrapasse algum dos limites impostos, a saída será igual a esse valor limite até que a entrada retorne para a faixa permitida.</p><p>As restrições máximas e mínimas de valores dos sistemas de controle são modeladas utilizando um Limitador, cuja implementação é bastante simples e segue a seguinte expressão:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>L</mi><mrow><mi>s</mi><mi>u</mi><mi>p</mi></mrow></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><msub><mi>u</mi><mi>n</mi></msub><mo>></mo><msub><mi>L</mi><mrow><mi>s</mi><mi>u</mi><mi>p</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>u</mi><mi>n</mi></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><msub><mi>L</mi><mrow><mi>i</mi><mi>n</mi><mi>f</mi></mrow></msub><mo>≤</mo><msub><mi>u</mi><mi>n</mi></msub><mo>≤</mo><msub><mi>L</mi><mrow><mi>s</mi><mi>u</mi><mi>p</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>L</mi><mrow><mi>i</mi><mi>n</mi><mi>f</mi></mrow></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><msub><mi>u</mi><mi>n</mi></msub><mo><</mo><msub><mi>L</mi><mrow><mi>i</mi><mi>n</mi><mi>f</mi></mrow></msub></mrow></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">y_n = \begin{cases} L_{sup}{,} & \text{se } u_n > L_{sup} \\ u_n{,} & \text{se } L_{inf} \le u_n \le L_{sup}\\ L_{inf}{,} & \text{se } u_n < L_{inf} \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.9099999999999997em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35002em"><span style="top:-2.19999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.19499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.20499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.15001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.2950099999999996em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.30501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.60002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.8500199999999998em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mord"><span class="mpunct">,</span></span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mpunct">,</span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mord"><span class="mpunct">,</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">y_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> Éé o valor de saída atual do bloco</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>s</mi><mi>u</mi><mi>p</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{sup}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mrow><mi>i</mi><mi>n</mi><mi>f</mi></mrow></msub></mrow><annotation encoding="application/x-tex">L_{inf}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> são os limites superior e inferior, respectivamente</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">u_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é o valor de entrada atual do bloco</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="formulário-de-edição-do-bloco-limitador"></a>Formulário de edição do bloco Limitador<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-do-bloco-limitador" title="Direct link to heading">#</a></h2><p>A figura abaixo apresenta o formulário de edição de dados do bloco limitador.</p><div><center><img src="/PSP/images/limiterForm.png" alt="Formulário de edição de dados do bloco limitador no PSP-UFU" title="Formulário de edição de dados do bloco limitador no PSP-UFU"></center></div><p>Esse bloco não linear é definido pelo <strong>limite superior e inferior</strong>, inseridos pelo usuário.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Tais blocos são bastante encontrados nos sistemas de controle para limitar a atuação do controlador, além de representar limites físicos de modelos implementados, como, por exemplo, valores máximos de corrente de excitação ou potência mecânica fornecida pelas turbinas.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/limiter.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/exponential"><div 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taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulações</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_FFX1"><div class="container padding-vert--lg docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Linha</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Um meio de transmissão fabricado usado para transmitir energia eletromagnética entre dois pontos com um mínimo de radiação. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=704-02-02" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="linha-no-psp-ufu"></a>Linha no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#linha-no-psp-ufu" title="Direct link to heading">#</a></h2><p>As linhas no PSP-UFU são modelos <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span> equilibradas. Podem ser utilizadas como linhas de transmissão e distribuição de sistemas elétricos de potência.</p><p>A figura abaixo mostra o modelo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span> implementado no PSP-UFU:</p><div><center><img src="/PSP/images/lineModel.svg" alt="Modelo PI da linha" title="Modelo PI da linha"></center></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">r</mi><mi mathvariant="bold">L</mi></msub></mrow><annotation encoding="application/x-tex">\bold{r_L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.59444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord"><span class="mord mathbf">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33027699999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">L</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></span> é a resistência da linha;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">x</mi><mi mathvariant="bold">L</mi></msub></mrow><annotation encoding="application/x-tex">\bold{x_L}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.59444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33027699999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">L</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></span> é a reatância indutiva da linha;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi mathvariant="bold">b</mi><mi mathvariant="bold">L</mi><mi mathvariant="bold">d</mi></msubsup></mrow><annotation encoding="application/x-tex">\bold{b_{L}^{d}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.126385em;vertical-align:-0.277277em"></span><span class="mord"><span class="mord"><span class="mord mathbf">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.4227230000000004em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">L</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">d</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.277277em"><span></span></span></span></span></span></span></span></span></span></span></span> é a susceptância capacitiva <em>shunt</em> da linha.</li></ul><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>O modelo utilizado no PSP-UFU é <strong>equilibrado</strong> e <strong>não possui impedâncias mútuas</strong> entre as fases.</p></div></div><p>A linha pode ser inseridas com pontos de ancoragem, ou "nós", para maior personalização gráfica do elemento, como apresentado nas <a href="/PSP/docs/cadTools#linha">Ferramentas CAD</a>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>A linha deve ser inserida entre duas barras de <strong>mesma tensão nominal</strong>. Caso você tente inseri-la entre barras de tensão diferentes, uma mensagem de erro será exibida.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Um arranjo série de uma <a href="/PSP/docs/line">linha</a> sem as susceptâncias <em>shunt</em>, um <a href="/PSP/docs/bus">barramento</a> e um <a href="/PSP/docs/capacitor">capacitor</a> pode ser utilizado para fabricar um filtro passivo nos <a href="/PSP/docs/harmonics">estudos harmônicos</a>. Em versões futuras, um elemento de filtro passivo será implementado no PSP-UFU.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="formulário-de-edição-das-linhas"></a>Formulário de edição das linhas<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-das-linhas" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados das linhas de transmissão:</p><div><center><img src="/PSP/images/lineForm.png" alt="Formulário das linhas no PSP-UFU" title="Formulário das linhas no PSP-UFU"></center></div><p>Esse formulário é subdividido em dois contextos distintos:</p><ul><li><strong>Geral</strong>: no qual são inseridas informações gerais da linha e informações do fluxo de carga;</li><li><strong>Falta</strong>: local onde as impedâncias de sequência zero são inseridas.</li></ul><div class="admonition admonition-note alert alert--secondary"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M6.3 5.69a.942.942 0 0 1-.28-.7c0-.28.09-.52.28-.7.19-.18.42-.28.7-.28.28 0 .52.09.7.28.18.19.28.42.28.7 0 .28-.09.52-.28.7a1 1 0 0 1-.7.3c-.28 0-.52-.11-.7-.3zM8 7.99c-.02-.25-.11-.48-.31-.69-.2-.19-.42-.3-.69-.31H6c-.27.02-.48.13-.69.31-.2.2-.3.44-.31.69h1v3c.02.27.11.5.31.69.2.2.42.31.69.31h1c.27 0 .48-.11.69-.31.2-.19.3-.42.31-.69H8V7.98v.01zM7 2.3c-3.14 0-5.7 2.54-5.7 5.68 0 3.14 2.56 5.7 5.7 5.7s5.7-2.55 5.7-5.7c0-3.15-2.56-5.69-5.7-5.69v.01zM7 .98c3.86 0 7 3.14 7 7s-3.14 7-7 7-7-3.12-7-7 3.14-7 7-7z"></path></svg></span>Nota</h5></div><div class="admonition-content"><p>Os parâmetros necessários para construção da linha segundo seu modelo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span> são inseridos na aba Geral, utilizados para construção da matriz admitância de sequência positiva e negativa.</p><p>Dados adicionais de impedâncias de sequência zero necessário para o cálculo de curtos-circuitos desbalanceados são editados na aba Falta, utilizados na construção da matriz admitância de sequência zero.</p></div></div><p>Além desses dois contextos, pode ser observado o botão "Estabilidade" na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção da linha durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/images/lineSw.png" alt="Formulário de chaveamento da linha" title="Formulário de chaveamento da linha"></center></div><div><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_3zTA tabs__item--active">Geral</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_3zTA">Falta</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_3zTA">Botão Estabilidade</li></ul><div role="tabpanel" class="margin-vert--md"><div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="nome"></a>Nome<a aria-hidden="true" tabindex="-1" class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="tensão-nominal"></a>Tensão nominal<a aria-hidden="true" tabindex="-1" class="hash-link" href="#tensão-nominal" title="Direct link to heading">#</a></h4><p>Campo de informação <em>não editável</em> que apresenta a tensão nominal da linha. Para alterar esse campo é necessário editar o campo correspondente do <a href="/PSP/docs/bus#geral">barramento</a> conectado.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="potência-nominal"></a>Potência nominal<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potência-nominal" title="Direct link to heading">#</a></h4><p>Potência nominal da linha, inserida em MVA, kVA ou VA.</p><p>Esse campo é especialmente importante caso a opção "Utilizar a potência nominal como base" esteja marcada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="parâmetros-do-modelo-pi"></a>Parâmetros do modelo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span><a aria-hidden="true" tabindex="-1" class="hash-link" href="#parâmetros-do-modelo-pi" title="Direct link to heading">#</a></h4><p>Resistência (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mi>L</mi></msub></mrow><annotation encoding="application/x-tex">r_L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>), Reatância indutiva (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>L</mi></msub></mrow><annotation encoding="application/x-tex">x_L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) série e Susceptância capacitiva shunt <strong>total</strong> (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><msubsup><mi>b</mi><mi>L</mi><mi>d</mi></msubsup></mrow><annotation encoding="application/x-tex">2b_{L}^{d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.124439em;vertical-align:-0.275331em"></span><span class="mord">2</span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.424669em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em"><span></span></span></span></span></span></span></span></span></span></span>) presentes no modelo <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.03588em">π</span></span></span></span></span> da linha.</p><p>Esses parâmetros são inseridos em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Ω</span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi><mi mathvariant="normal">/</mi><mi>k</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">\Omega/km</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">Ω</span><span class="mord">/</span><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="mord mathdefault">m</span></span></span></span></span> (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.05764em">S</span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi mathvariant="normal">/</mi><mi>k</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">S/km</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.05764em">S</span><span class="mord">/</span><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="mord mathdefault">m</span></span></span></span></span> para <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>b</mi><mi>L</mi><mi>d</mi></msubsup></mrow><annotation encoding="application/x-tex">b_{L}^{d}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.124439em;vertical-align:-0.275331em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.424669em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.275331em"><span></span></span></span></span></span></span></span></span></span></span>).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="comprimento-da-linha"></a>Comprimento da linha<a aria-hidden="true" tabindex="-1" class="hash-link" href="#comprimento-da-linha" title="Direct link to heading">#</a></h4><p>Utilizado para calcular o valor dos parâmetros da linha inseridos em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi><mi mathvariant="normal">/</mi><mi>k</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">\Omega/km</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord">Ω</span><span class="mord">/</span><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="mord mathdefault">m</span></span></span></span></span> (ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi mathvariant="normal">/</mi><mi>k</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">S/km</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.05764em">S</span><span class="mord">/</span><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="mord mathdefault">m</span></span></span></span></span>).</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>O comprimento da linha é ignorado caso não sejam utilizadas as unidades por quilometro.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="utilizar-potência-nominal-como-base"></a>Utilizar potência nominal como base<a aria-hidden="true" tabindex="-1" class="hash-link" href="#utilizar-potência-nominal-como-base" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar a potência nominal da linha como base para a conversão das unidades, caso contrário será usada a <a href="/PSP/docs/simulationConfig">potência base do sistema</a>.</p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/line.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this 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href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_FFX1"><div class="container padding-vert--lg docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Carga</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><ol><li>a potência ativa, reativa ou aparente gerada, transmitida ou distribuída dentro de um sistema;</li><li>a potência demandada por um grupo de consumidores classificados de acordo com suas particularidades e características, por exemplo, carga de aquecimento, carga reativa diurna, etc. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=601-01-15" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</li></ol></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="carga-no-psp-ufu"></a>Carga no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#carga-no-psp-ufu" title="Direct link to heading">#</a></h2><p>As cargas são os elementos de potência consumidores de potência genéricos. Nos estudos de fluxo de carga é possível modelar a carga como <strong>potência ou impedância constante</strong> e em estabilidade pode-se compor a carga na proporção desejada em três tipos (<strong>carga ZIP</strong>): potência constante, impedância constante e corrente constante.</p><p>As cargas de potência constante nos estudos de fluxo de carga são inseridos no vetor de potências e permanecem fixos durante toda a simulação.</p><p>Os elementos shunt, como reatores, banco de capacitores e cargas de impedância constante, são representados simplesmente por suas admitâncias em derivação. Como os parâmetros de entrada geralmente se constituem de suas potências ativa (P) e reativa (Q) nominais (no caso de reatores e capacitores, P=0), deve-se, portanto, encontrar sua admitância com a equação abaixo para inseri-la na matriz admitância:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mo>=</mo><mfrac><mrow><mi>P</mi><mo>−</mo><mi>j</mi><mi>Q</mi></mrow><msup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mn>2</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">\overline{y} = \frac{P-jQ}{\dot{V}^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.825em;vertical-align:-0.19444em"></span><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.17052em;vertical-align:-0.8101900000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603299999999998em"><span style="top:-2.18981em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord mathdefault">Q</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8101900000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Para os estudos de <strong>curto-circuito</strong> e <strong>harmônicos</strong> as cargas são modeladas como impedância constante, independentemente da sinalização indicada em seu formulário de edição de dados.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="carga-no-estudo-de-estabilidade"></a>Carga no estudo de estabilidade<a aria-hidden="true" tabindex="-1" class="hash-link" href="#carga-no-estudo-de-estabilidade" title="Direct link to heading">#</a></h3><p>A modelagem das cargas do sistema de forma exata se torna impraticável em um problemas de estabilidade, visto à grande quantidade e variedade dos componentes envolvidos, além de apresentar modificações em suas composições causado por vários fatores, como tempo, condições climáticas e economia. Portanto várias aproximações devem ser utilizadas na formulação de um modelo de carga nos estudos aqui postos.</p><p>Uma forma interessante de se representar cargas dependentes da tensão são as cargas ZIP, as quais são funções quadráticas, possuindo três parcelas: impedância constante (Z), corrente constante (I) e potência constante (P). As potências ativa e reativa são obtidas pelas seguintes equações:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>L</mi></msub><mo>=</mo><msub><mi>P</mi><mrow><mi>Z</mi><mn>0</mn></mrow></msub><msup><mrow><mo fence="true">(</mo><mfrac><msub><mi>V</mi><mi>i</mi></msub><msub><mi>V</mi><mrow><mi>i</mi><mn>0</mn></mrow></msub></mfrac><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msub><mi>P</mi><mrow><mi>I</mi><mn>0</mn></mrow></msub><mrow><mo fence="true">(</mo><mfrac><msub><mi>V</mi><mi>i</mi></msub><mrow><msub><mi>V</mi><mi>i</mi></msub><mn>0</mn></mrow></mfrac><mo fence="true">)</mo></mrow><mo>+</mo><msub><mi>P</mi><mrow><mi>P</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">P_L=P_{Z0} \left(\frac{V_i}{V_{i0}} \right)^2+P_{I0} \left(\frac{V_i}{V_i0} \right)+P_{P0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.604038em;vertical-align:-0.95003em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.6540080000000001em"><span style="top:-3.9029000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-0.95003em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord">0</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>L</mi></msub><mo>=</mo><msub><mi>Q</mi><mrow><mi>Z</mi><mn>0</mn></mrow></msub><msup><mrow><mo fence="true">(</mo><mfrac><msub><mi>V</mi><mi>i</mi></msub><msub><mi>V</mi><mrow><mi>i</mi><mn>0</mn></mrow></msub></mfrac><mo fence="true">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msub><mi>Q</mi><mrow><mi>I</mi><mn>0</mn></mrow></msub><mrow><mo fence="true">(</mo><mfrac><msub><mi>V</mi><mi>i</mi></msub><mrow><msub><mi>V</mi><mi>i</mi></msub><mn>0</mn></mrow></mfrac><mo fence="true">)</mo></mrow><mo>+</mo><msub><mi>Q</mi><mrow><mi>P</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Q_L=Q_{Z0} \left(\frac{V_i}{V_{i0}} \right)^2+Q_{I0} \left(\frac{V_i}{V_i0} \right)+Q_{P0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.604038em;vertical-align:-0.95003em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.6540080000000001em"><span style="top:-3.9029000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-0.95003em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord">0</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>L</mi></msub></mrow><annotation encoding="application/x-tex">P_L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>L</mi></msub></mrow><annotation encoding="application/x-tex">Q_L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são as potências ativa e reativa da carga ZIP, respectivamente, conectadas em uma barra genérica i;</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão atual de uma barra genérica i;</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mrow><mi>i</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">V_{i0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão inicial de uma barra genérica i, obtida do estudo de fluxo de carga;</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>Z</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">P_{Z0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mrow><mi>Z</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Q_{Z0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são as parcelas de impedância constante para potências ativa e reativa, respectivamente;</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>I</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">P_{I0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mrow><mi>I</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Q_{I0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são as parcelas de corrente constante para potências ativa e reativa, respectivamente;</p><p><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>P</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">P_{P0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mrow><mi>P</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Q_{P0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são as parcelas de potência constante para potências ativa e reativa, respectivamente.</p><p>Os valores das parcelas de potência da carga ZIP são obtidos utilizando as potências da carga após a convergência do cálculo do fluxo de carga (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mrow><mi>L</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">P_{L0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>, para a potência ativa e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mrow><mi>L</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Q_{L0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>, para a potência reativa), utilizando as seguintes expressões:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>P</mi><mrow><mi>Z</mi><mn>0</mn></mrow></msub><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>k</mi><mrow><mi>P</mi><mi>Z</mi></mrow></msub><mn>100</mn></mfrac><mfrac><msub><mi>P</mi><mrow><mi>L</mi><mn>0</mn></mrow></msub><msubsup><mi>V</mi><mrow><mi>i</mi><mn>0</mn></mrow><mn>2</mn></msubsup></mfrac></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>P</mi><mrow><mi>I</mi><mn>0</mn></mrow></msub><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>k</mi><mrow><mi>P</mi><mi>I</mi></mrow></msub><mn>100</mn></mfrac><mfrac><msub><mi>P</mi><mrow><mi>L</mi><mn>0</mn></mrow></msub><msub><mi>V</mi><mrow><mi>i</mi><mn>0</mn></mrow></msub></mfrac></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>P</mi><mrow><mi>P</mi><mn>0</mn></mrow></msub><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>k</mi><mrow><mi>P</mi><mi>P</mi></mrow></msub><mn>100</mn></mfrac><msub><mi>P</mi><mrow><mi>L</mi><mn>0</mn></mrow></msub></mstyle></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} P_{Z0}=\displaystyle \frac{k_{PZ}}{100} \frac{P_{L0}}{V_{i0}^2}\\ P_{I0}=\displaystyle \frac{k_{PI}}{100} \frac{P_{L0}}{V_{i0}}\\ P_{P0}=\displaystyle \frac{k_{PP}}{100} P_{L0} \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6.60004em;vertical-align:-3.05002em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5500200000000004em"><span style="top:-0.9999899999999999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-0.99499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-1.2899900000000002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-1.5849900000000003em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-1.8799900000000005em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.1749900000000006em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.2049900000000004em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.1500100000000004em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.29501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.59001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.885010000000001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-5.180010000000001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-5.475010000000001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-5.50501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-5.80002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.05002em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5495919999999996em"><span style="top:-5.549592em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959080000000001em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9628639999999999em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.215288em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-1.007848em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">L</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.0495919999999996em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Os valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>P</mi><mi>Z</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{PZ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>P</mi><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{PI}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>P</mi><mi>P</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{PP}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> representam a composição em impedância, corrente e potência constantes, respectivamente, da parcela ativa da carga, os quais são inseridos pelo usuário na forma de porcentagem, sendo a soma desses três valores necessariamente igual a 100%. Os parâmetros <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>Q</mi><mi>Z</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{QZ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">Q</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">Z</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>Q</mi><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{QI}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">Q</span><span class="mord mathdefault mtight" style="margin-right:0.07847em">I</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>Q</mi><mi>P</mi></mrow></msub></mrow><annotation encoding="application/x-tex">k_{QP}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">Q</span><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span> podem ser interpretados de forma análoga, porém compõe a parcela reativa da carga.</p><p>O comportamento das potências, retiradas de simulações no PSP-UFU, de cada parcela que compõe a carga ZIP são evidenciadas na figura abaixo.</p><div><center><img src="/PSP/images/loadModel.svg" alt="Comportamento da carga ZIP implementada no software: (a) curva de Tensão x Potência; (b) Curva de Tensão x Corrente" title="Comportamento da carga ZIP implementada no software: (a) curva de Tensão x Potência; (b) Curva de Tensão x Corrente"></center></div><p>Cargas de corrente e potência constantes possuem problemas em tensões muito baixas. À medida que a tensão diminui as correntes dessas cargas não reduz, como pode ser observado na figura acima, resultando em perda de precisão e problemas na convergência de processos iterativos. Para contornar esse problema utiliza-se uma tensão pré-definida pelo usuário (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mrow><mi>l</mi><mi>o</mi><mi>w</mi></mrow></msub></mrow><annotation encoding="application/x-tex">V_{low}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight" style="margin-right:0.02691em">w</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>), a qual as cargas (ou parcelas) de corrente e potência constantes são modeladas como impedância constante, resultando no comportamento de potência e corrente apresentados na figura acima.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="formulário-de-edição-das-cargas"></a>Formulário de edição das cargas<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-das-cargas" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados das cargas:</p><div><center><img src="/PSP/images/loadForm.png" alt="Formulário das cargas no PSP-UFU" title="Formulário das cargas no PSP-UFU"></center></div><p>Esse formulário é subdividido em dois contextos distintos:</p><ul><li><strong>Geral</strong>: no qual são inseridas informações gerais das cargas, informações e o tipo de carga no fluxo de carga;</li><li><strong>Estabilidade</strong>: contendo opções de visualização de dados da carga em gráficos no tempo e opções de parametrização da carga ZIP.</li></ul><p>Além desses dois contextos, pode ser observado o botão "Chaveamento" na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção da carga durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/images/loadSw.png" alt="Formulário de chaveamento da carga" title="Formulário de chaveamento da carga"></center></div><div><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_3zTA tabs__item--active">Geral</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_3zTA">Estabilidade</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_3zTA">Botão Chaveamento</li></ul><div role="tabpanel" class="margin-vert--md"><div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="nome"></a>Nome<a aria-hidden="true" tabindex="-1" class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="potência-ativa"></a>Potência ativa<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potência-ativa" title="Direct link to heading">#</a></h4><p>Parcela de potência ativa da carga. Pode ser inserido em MW, kW W ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> (na base do sistema).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="potência-reativa"></a>Potência reativa<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potência-reativa" title="Direct link to heading">#</a></h4><p>Parcela de potência reativa da carga. Pode ser inserido em Mvar, kvar var ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> (na base do sistema).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="tipo-de-carga-fluxo-de-carga"></a>Tipo de carga (fluxo de carga)<a aria-hidden="true" tabindex="-1" class="hash-link" href="#tipo-de-carga-fluxo-de-carga" title="Direct link to heading">#</a></h4><p>Tipo da carga para o estudo de fluxo de carga, podendo ser selecionado dois tipos: Potência constante e Impedância constante. As cargas de potência constantes são inseridas nos vetores de potência e permanecem invariantes no cálculo, enquanto que para as cargas de impedância constante determina-se o valor da impedância utilizando a potência e tensão nominal, a qual é inserida na matriz admitância.</p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/load.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/transformer"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Transformador</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/capacitor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Capacitor »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_TbNY"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#carga-no-psp-ufu" class="table-of-contents__link">Carga no PSP-UFU</a><ul><li><a href="#carga-no-estudo-de-estabilidade" class="table-of-contents__link">Carga no estudo de estabilidade</a></li></ul></li><li><a href="#formulário-de-edição-das-cargas" class="table-of-contents__link">Formulário de edição das cargas</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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Controle</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/io">Entrada / Saída</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/transferFunction">Função Transferência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/sum">Somador</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/multiplier">Multiplicador</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/divider">Divisor</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/exponential">Exponencial</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/limiter">Limitador absoluto</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/rateLimiter">Limitador de taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulações</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_FFX1"><div class="container padding-vert--lg docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Tela Principal</h1></header><div class="markdown"><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="organização-da-tela-principal"></a>Organização da tela principal<a aria-hidden="true" tabindex="-1" class="hash-link" href="#organização-da-tela-principal" title="Direct link to heading">#</a></h2><p>A tela principal do PSP-UFU é dividida em três seções: <em><a href="/PSP/docs/mainScreen#menu-ribbon">Menu "Ribbon"</a></em>, <em><a href="/PSP/docs/mainScreen#%C3%A1rea-de-trabalho">Área de Trabalho</a></em> e <em><a href="/PSP/docs/mainScreen#barra-de-status">Barra de Status</a></em>.</p><div><center><img src="/PSP/images/mainScreen.svg" alt="Tela principal" title="Tela principal do PSP-UFU no Windows"></center></div><p>Resumidamente, no <strong>Menu Ribbon</strong> são acessadas as ferramentas do PSP-UFU; na <strong>Área de Trabalho</strong> são criados os diagramas unifilares de potência; e na <strong>Barra de Status</strong> são exibidas importantes informações acerca do estado da Área de Trabalho.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="menu-ribbon"></a>Menu Ribbon<a aria-hidden="true" tabindex="-1" class="hash-link" href="#menu-ribbon" title="Direct link to heading">#</a></h3><blockquote><p>Em informática o ribbon (faixa, em inglês) é um formato de apresentação de interface baseada na GUI onde a barra de ferramentas é mostrada através de uma barra mais larga com icones maiores possibilitando o uso dos aplicativos por dispositivos <em>touch screen</em>. <a href="https://pt.wikipedia.org/wiki/Ribbon_(inform%C3%A1tica)" target="_blank" rel="noopener noreferrer">Wikipedia</a></p></blockquote><p>O menu Ribbon é composto de três submenus: “Arquivo”, “Ferramentas” e “Simulação”.</p><div><center><img src="/PSP/images/menuFile.png" alt="Menu Arquivo" title="Menu Arquivo"></center></div><p>Em <strong>Arquivo</strong> estão as ferramentas gerais do programa, as quais permitem ao usuário criar e salvar novos projetos, além daquelas que possibilitam carregar os projetos já existentes e importar arquivos de outros programas. Esse submenu também possui opções gerais do programa e acesso a informações sobre o programa.</p><div><center><img src="/PSP/images/menuTools.png" alt="Menu Ferramentas" title="Menu Ferramentas"></center></div><p>Em <strong>Ferramentas</strong> estão presentes os itens específicos, os quais estão relacionados à criação e manipulação da rede elétrica de potência, relatórios tabulares de saída de dados e acesso ao formulário de criação de gráficos no tempo. Algumas configurações específicas do projeto ativo são acessadas por esse menu.</p><div><center><img src="/PSP/images/menuSimulation.png" alt="Menu Simulação" title="Menu Simulação"></center></div><p>Finalmente, o submenu <strong>Simulação</strong> proporciona o acesso do usuário aos cálculos realizados pelo <em>software</em> e suas configurações, além da ferramenta de “solução contínua”, a qual resolve os cálculos estáticos (fluxo de potência, curto-circuito e harmônicos) após quaisquer mudanças na rede de potência, como remoção de componente ou alteração de seus dados. Todas as configurações de simulação do projeto ativo são acessadas por esse menu.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="área-de-trabalho"></a>Área de Trabalho<a aria-hidden="true" tabindex="-1" class="hash-link" href="#área-de-trabalho" title="Direct link to heading">#</a></h3><p>Na área de trabalho estão presentes os projetos em execução, os quais são divididos por abas identificadas pelo nome do projeto (caso já esteja gravado no disco). É nessa área que são inseridos e excluídos os elementos elétricos, cujas ferramentas são acessadas por meio do <a href="/PSP/docs/mainScreen#menu-ribbon">Menu Ribbon</a>.</p><p>As edições gráficas dos elementos, como posição, conexão e tamanho dos barramentos, são realizadas por meio da ação <em>drag-and-drop</em> (arrastar e soltar) com a utilização do mouse. A edição dos dados elétricos é feita por meio de formulários de dados acessados clicando duas vezes sobre os elementos.</p><div><center><img src="/PSP/images/workspace.png" alt="Área de Trabalho" title="Área de Trabalho"></center></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="barra-de-status"></a>Barra de Status<a aria-hidden="true" tabindex="-1" class="hash-link" href="#barra-de-status" title="Direct link to heading">#</a></h3><p>A barra de status é responsável por informações interessantes sobre as circunstâncias atuais do programa: modo de operação do mouse (editar, mover, arrastar), zoom aplicado e posição do mouse na área de trabalho, além de dados sobre as ações dos usuários, como: inserção de elementos, informação sobre copiar e colar, etc.</p><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p><strong>Sempre fique atendo à barra de status</strong>, pois são fornecidas informações importantes acerca da operação do programa. Caso tenha alguma dúvida siga as instruções apresentadas nesse componente.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/mainScreen.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/installation"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Instalação</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/cadTools"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Ferramentas CAD »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_TbNY"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#organização-da-tela-principal" class="table-of-contents__link">Organização da tela principal</a><ul><li><a href="#menu-ribbon" class="table-of-contents__link">Menu Ribbon</a></li><li><a href="#área-de-trabalho" class="table-of-contents__link">Área de Trabalho</a></li><li><a href="#barra-de-status" class="table-of-contents__link">Barra de Status</a></li></ul></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. Built with Docusaurus.</div></div></div></footer></div> <script src="/PSP/styles.26e2fe91.js"></script> -<script src="/PSP/runtime~main.3ede30cf.js"></script> -<script src="/PSP/main.1c1af766.js"></script> +<script src="/PSP/runtime~main.57a83c0d.js"></script> +<script src="/PSP/main.1dbec03f.js"></script> <script src="/PSP/1.837a9375.js"></script> <script src="/PSP/2.19116c85.js"></script> <script src="/PSP/64.2a8261e0.js"></script> diff --git a/docs/docs/mathExpression/index.html b/docs/docs/mathExpression/index.html index 732621d..e4cfa4f 100644 --- a/docs/docs/mathExpression/index.html +++ b/docs/docs/mathExpression/index.html @@ -6,8 +6,8 @@ <meta name="generator" content="Docusaurus v2.0.0-alpha.58"> <link rel="search" type="application/opensearchdescription+xml" title="PSP-UFU" href="/PSP/opensearch.xml"><title data-react-helmet="true">Expressão Matemática | PSP-UFU</title><meta data-react-helmet="true" name="twitter:card" content="summary_large_image"><meta 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data-react-helmet="true" rel="canonical" href="https://thales1330.github.io/PSP/docs/mathExpression"><link rel="stylesheet" href="/PSP/styles.68586bc0.css"> <link rel="preload" href="/PSP/styles.26e2fe91.js" as="script"> -<link rel="preload" href="/PSP/runtime~main.3ede30cf.js" as="script"> -<link rel="preload" href="/PSP/main.1c1af766.js" as="script"> +<link rel="preload" href="/PSP/runtime~main.57a83c0d.js" as="script"> +<link rel="preload" href="/PSP/main.1dbec03f.js" as="script"> <link rel="preload" href="/PSP/1.837a9375.js" as="script"> <link rel="preload" href="/PSP/2.19116c85.js" as="script"> <link rel="preload" href="/PSP/64.2a8261e0.js" as="script"> @@ -38,8 +38,8 @@ c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5 c53.7,-170.3,84.5,-266.8,92.5,-289.5z M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.502996em"><span></span></span></span></span></span></span></span><span style="top:-3.5670040000000003em"><span class="pstrut" style="height:3.3370040000000003em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-4.014004em"><span class="pstrut" style="height:3.3370040000000003em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9559em"><span style="top:-3.2047920000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.73em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>P</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><msub><mi>x</mi><mi>q</mi></msub><mo>×</mo><mfrac><mi>P</mi><mi>V</mi></mfrac></mrow><annotation encoding="application/x-tex">P' = x_q \times \frac{P}{V}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8694379999999999em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.217331em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.872331em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em">V</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.394em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">P</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>Q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><msub><mi>x</mi><mi>q</mi></msub><mo>×</mo><mfrac><mi>Q</mi><mi>V</mi></mfrac></mrow><annotation encoding="application/x-tex">Q' = x_q \times \frac{Q}{V}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.946332em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8694379999999999em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.269439em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.924439em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em">V</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.446108em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">Q</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></li></ul><p>Primeiramente, deve-se inserir no campo "variáveis de entrada" as grandezas fornecidas pelo PSP-UFU. Nesse caso, como pode ser observado no bloco de <a href="/PSP/docs/io">entrada / saída</a>, todas as variáveis necessárias são fornecidas: <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.13889em">P</span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">Q</span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span></span></span>. Essas variáveis devem ser inseridas separadas por espaço: <code>p q v</code>.</p><p>Utilizando três blocos de <a href="/PSP/docs/io">entrada / saída</a> pode-se fornecer tais dados ao conectá-los ao bloco de expressão matemática.</p><p>As reatâncias transitórias de eixo direto (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>d</mi></msub></mrow><annotation encoding="application/x-tex">x_d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) e em quadratura (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>q</mi></msub></mrow><annotation encoding="application/x-tex">x_q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>) podem ser definidas diretamente na expressão matemática:</p><div class="mdxCodeBlock_iHAB"><div class="codeBlockContent_32p_"><button type="button" aria-label="Copy code to clipboard" class="copyButton_1BYj">Copy</button><div tabindex="0" class="prism-code language-cpp codeBlock_19pQ"><div class="codeBlockLines_2n9r" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">xd </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">0.146</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">xq </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">0.0969</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span></div></div></div></div></div><p>Os valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>P</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">P'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>Q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">Q'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.946332em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span> podem ser também calculados:</p><div class="mdxCodeBlock_iHAB"><div class="codeBlockContent_32p_"><button type="button" aria-label="Copy code to clipboard" class="copyButton_1BYj">Copy</button><div tabindex="0" class="prism-code language-cpp codeBlock_19pQ"><div class="codeBlockLines_2n9r" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">yp </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> p </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> v</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">yq </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> q </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> v</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span></div></div></div></div></div><p>Com todos os dados necessários pode-se calcular a corrente de campo:</p><div class="mdxCodeBlock_iHAB"><div class="codeBlockContent_32p_"><button type="button" aria-label="Copy code to clipboard" class="copyButton_1BYj">Copy</button><div tabindex="0" class="prism-code language-cpp codeBlock_19pQ"><div class="codeBlockLines_2n9r" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">i_f </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> p</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">xd </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">1.0</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain"> </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">yq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yp</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> p</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span></div></div></div></div></div><p>Finalmente, pode-se definir a saída do bloco (não inserindo <code>;</code>):</p><div class="mdxCodeBlock_iHAB"><div class="codeBlockContent_32p_"><button type="button" aria-label="Copy code to clipboard" class="copyButton_1BYj">Copy</button><div tabindex="0" class="prism-code language-cpp codeBlock_19pQ"><div class="codeBlockLines_2n9r" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">i_f</span></div></div></div></div></div><p>Portanto, para as entradas definidas como <code>p q v</code>, a expressão matemática total para cálculo da corrente de campo será:</p><div class="mdxCodeBlock_iHAB"><div class="codeBlockContent_32p_"><button type="button" aria-label="Copy code to clipboard" class="copyButton_1BYj">Copy</button><div tabindex="0" class="prism-code language-cpp codeBlock_19pQ"><div class="codeBlockLines_2n9r" style="color:#bfc7d5;background-color:#292d3e"><div class="token-line" style="color:#bfc7d5"><span class="token plain">xd </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">0.146</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">xq </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">0.0969</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">yp </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> p </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> v</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">yq </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> q </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> v</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">i_f </span><span class="token operator" style="color:rgb(137, 221, 255)">:</span><span class="token operator" style="color:rgb(137, 221, 255)">=</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> p</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">xd </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> xq </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> </span><span class="token number" style="color:rgb(247, 140, 108)">1.0</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain"> </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">yq </span><span class="token operator" style="color:rgb(137, 221, 255)">*</span><span class="token plain"> </span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yp</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">/</span><span class="token plain"> </span><span class="token function" style="color:rgb(130, 170, 255)">sqrt</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token punctuation" style="color:rgb(199, 146, 234)">(</span><span class="token plain">v </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> yq</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token plain"> </span><span class="token operator" style="color:rgb(137, 221, 255)">+</span><span class="token plain"> p</span><span class="token operator" style="color:rgb(137, 221, 255)">^</span><span class="token number" style="color:rgb(247, 140, 108)">2</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">)</span><span class="token punctuation" style="color:rgb(199, 146, 234)">;</span><span class="token plain"></span></div><div class="token-line" style="color:#bfc7d5"><span class="token plain">i_f</span></div></div></div></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="referências"></a>Referências<a aria-hidden="true" tabindex="-1" class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ul><li>NIEMINEN, J.; YLILUOMA, J. Function Parser for C++ v4.5.2. Disponível em: <a href="http://warp.povusers.org/FunctionParser/" target="_blank" rel="noopener noreferrer">http://warp.povusers.org/FunctionParser/</a>. </li></ul></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/mathExpression.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/gain"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Ganho</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/powerFlow"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Fluxo de Potência »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_TbNY"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#sintaxe-das-expressões-matemáticas" class="table-of-contents__link">Sintaxe das expressões matemáticas</a></li><li><a href="#formulário-de-edição-de-dados-do-bloco-de-expressão-matemática" class="table-of-contents__link">Formulário de edição de dados do bloco de Expressão Matemática</a></li><li><a href="#exemplo-de-expressão-matemática" class="table-of-contents__link">Exemplo de Expressão Matemática</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Nós não conectados no multiplicador são assumidos com o valor unitário e para o divisor é considerado valor nulo.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/multiplier.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this 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class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_FFX1"><div class="container padding-vert--lg docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Editor de Potência</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>O Editor de Elementos de Potência no PSP-UFU permite a inserção, manipulação e edição de elementos elétricos de potência por meio de um diagrama unifilar.</p><blockquote><p>(<strong>Diagrama unifilar</strong>) Um diagrama do sistema no qual as linhas polifásicas são representados por sua única linha equivalente. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=601-02-04" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Todos os elementos inseridos pode ser manipulados e editados utilizando as <a href="/PSP/docs/cadTools">Ferramentas CAD</a>.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="novo-projeto"></a>Novo projeto<a aria-hidden="true" tabindex="-1" class="hash-link" href="#novo-projeto" title="Direct link to heading">#</a></h2><p>A criação de um novo projeto é realizada clicando no item <strong>Novo projeto</strong> no <a href="/PSP/docs/mainScreen#menu-ribbon">submenu <strong>Arquivo</strong></a>. Essa ação cria um sistema em branco na área de trabalho, local onde é possível inserir os elementos elétricos por meio do <a href="/PSP/docs/mainScreen#menu-ribbon">submenu <strong>Ferramentas</strong></a> ou pelas <a href="/PSP/docs/powerEditor#teclas-de-atalho">teclas de atalho</a>.</p><p>Abaixo são apresentados os elementos elétricos de potência que podem ser inseridos no PSP-UFU:</p><ul><li><a href="/PSP/docs/bus">Barramento</a></li><li>Elementos de ramo<ul><li><a href="/PSP/docs/line">Linha</a></li><li><a href="/PSP/docs/transformer">Transformador</a></li></ul></li><li>Elementos Shunt<ul><li><a href="/PSP/docs/load">Carga</a></li><li><a href="/PSP/docs/capacitor">Capacitor</a></li><li><a href="/PSP/docs/inductor">Indutor</a></li><li><a href="/PSP/docs/harmSource">Fonte de Corrente Harmônica</a></li></ul></li><li>Máquinas<ul><li><a href="/PSP/docs/syncGenerator">Gerador Síncrono</a></li><li><a href="/PSP/docs/syncMotor">Compensador Síncrono</a></li><li><a href="/PSP/docs/indMotor">Motor de Indução</a></li></ul></li></ul><div><center><img src="/PSP/images/powerElements.svg" alt="Elementos de potência" title="Elementos de potência"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="inserindo-elementos"></a>Inserindo elementos<a aria-hidden="true" tabindex="-1" class="hash-link" href="#inserindo-elementos" title="Direct link to heading">#</a></h2><p>Por meio do submenu <strong>Ferramentas</strong> pode-se inserir os elementos de potência, como é apresentado na imagem abaixo.</p><div><center><img src="/PSP/images/addPowerElements.png" alt="Adicionar elementos de potência" title="Adicionar elementos de potência"></center></div><p>No caso de inserção de um barramento, o usuário deve apenas clicar na posição desejada, já para os outros componentes será solicitado que o usuário selecione em uma ou duas barras, o qual o novo componente será conectado.</p><p>Sempre que são inseridos novos elementos, um formulário de dados é exibido e pode-se editar os dados elétricos e acessar formulários adicionais, como dados de estabilidade.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>O primeiro elemento que a ser inserido no projeto <strong>deve</strong> ser um barramento. Os demais elementos elétricos devem ser conectados em uma ou mais barras.</p></div></div><p>Além dos elementos de potência pode-se inserir os elementos de <em>texto vinculado</em>, os quais estão associados a uma grandeza que se deseja vizualizar diretamente na <a href="/PSP/docs/mainScreen#%C3%A1rea-de-trabalho">Área de Trabalho</a>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>Ao inserir um barramento, a mesma irá acompanhar o ponteiro do mouse até que você clique na posição desejada. Para os demais elementos <strong>deve-se clicar em uma mais barras</strong> para exibir o elemento inserido.</p><p><strong>Sempre fique atento às instruções da <a href="/PSP/docs/mainScreen#barra-de-status">Barra de Status</a>.</strong></p></div></div><video autoplay="" loop="" muted="" controls=""><source src="/PSP/videos/timelapseBuild.mp4" type="video/mp4"></video><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="teclas-de-atalho"></a>Teclas de atalho<a aria-hidden="true" tabindex="-1" class="hash-link" href="#teclas-de-atalho" title="Direct link to heading">#</a></h3><p>É possível inserir todos os elementos de potência por meio de teclas de atalho, facilitando e agilizando a criação dos diagramas unifilares no PSP-UFU. A tabela abaixo apresenta tais atalhos:</p><table><thead><tr><th>Elemento</th><th align="center">Atalho</th><th>Elemento</th><th align="center">Atalho</th></tr></thead><tbody><tr><td>Barramento</td><td align="center">B</td><td>Carga</td><td align="center">Shift + L</td></tr><tr><td>Linha</td><td align="center">L</td><td>Capacitor</td><td align="center">Shift + C</td></tr><tr><td>Transformador</td><td align="center">T</td><td>Indutor</td><td align="center">Shift + I</td></tr><tr><td>Gerador</td><td align="center">G</td><td>Corrente harmônica</td><td align="center">Shift + H</td></tr><tr><td>Motor de indução</td><td align="center">I</td><td>Texto</td><td align="center">A</td></tr><tr><td>Compensador síncrono</td><td align="center">K</td><td></td><td align="center"></td></tr></tbody></table><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="editando-dados-elétricos"></a>Editando dados elétricos<a aria-hidden="true" tabindex="-1" class="hash-link" href="#editando-dados-elétricos" title="Direct link to heading">#</a></h2><p>Ao inserir um elemento, seu respectivo formulário de edição é exibido para inserção dos dados elétricos d elemento. 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Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_FFX1"><div class="container padding-vert--lg docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Fluxo de Potência</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Um estudo fundamental no planejamento da expansão e operação de um sistema elétrico é o <strong>fluxo de potência (ou fluxo de carga)</strong> uma vez que a operação satisfatória desse sistema depende do conhecimento dos efeitos da interligação, de novas cargas, de novas centrais geradoras e de novas linhas antes que elas sejam instaladas. Esse estudo tem como objetivo encontrar o fluxo de potência nos ramos e as tensões nodais do Sistema Elétrico de Potência (SEP) impostas pela geração e a carga.</p><p>As equações formuladas a partir dos modelos dos elementos presentes no estudo de fluxo de carga são não lineares e não possuem solução analítica explícita. Portanto é necessária a utilização de métodos numéricos iterativos para solução do problema de fluxo de carga.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="formulação-do-problema-de-fluxo-de-carga"></a>Formulação do problema de fluxo de carga<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulação-do-problema-de-fluxo-de-carga" title="Direct link to heading">#</a></h2><p>O problema de fluxo de carga pode ser representado por um sistema de equações e inequações algébricas não-lineares que correspondem às leis de Kirchhoff e a um conjunto de restrições de operação impostos pelos componentes de uma rede elétrica. -Na formulação do problema a cada barra da rede são associadas quatro variáveis, sendo que duas delas surgem como dados e duas como incógnitas (em uma barra de índice <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>):</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a magnitude da tensão na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">θ_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é o ângulo da tensão na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a injeção líquida de potência ativa na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a injeção líquida de potência reativa na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>.</li></ul><p>Relativo às variáveis que são incógnitas e os dados do sistema, podem ser definidos <strong>três tipos de barras</strong>:</p><ul><li><strong>Barra PQ</strong>: <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são dados, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">θ_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são calculados;</li><li><strong>Barra PV</strong>: <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são dados, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">θ_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são calculados;</li><li><strong>Barra de Referência</strong> <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">θ_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são dados, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são calculados.</li></ul><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>As <strong>barras PQ</strong> geralmente são representadas pelos barramentos de carga, os quais não há geração e controle de tensão.</p><p>As <strong>barras PV</strong> se caracterizam pelo controle de tensão mediante a injeção ou absorção de potência reativa por meio do controle da excitação de uma máquina síncrona.</p><p>A <strong>barra de Referência</strong> (ou de folga, de oscilação) tem como função, assim como o próprio nome diz, servir de referência de tensão e ângulo do sistema. Essa barra é necessariamente geradora, uma vez que ela é responsável pelo equilíbrio do balanço de potência do sistema.</p></div></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O tipo de barra deve ser definido no <a href="/PSP/docs/bus">elemento barramento</a>.</p><p>Note que <strong>o sistema deve possuir somente uma barra de referência</strong>.</p></div></div><p>Como mencionado anteriormente, as equações são não-lineares e a solução analítica não é prática. As soluções dessas equações seguem <strong><a href="https://en.wikipedia.org/wiki/Iterative_method" target="_blank" rel="noopener noreferrer">processos iterativos</a></strong>, em que são atribuídos valores estimados (ou iniciais) para as barras com tensões desconhecidas e, baseado na potência ativa e reativa e módulo da tensão especificados, calcula-se por meio das equações previamente apresentadas as novas tensões complexas em cada nó do sistema.</p><p>Na sequência, esse conjunto de valores para as tensões em cada barra é utilizado para novamente calcular outro grupo de tensões. Cada cálculo de um novo conjunto de tensões é chamado iteração. O processo iterativo é repetido até que as mudanças em todas as barras sejam menores do que um valor pré-estipulado, obtendo assim a convergência.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="execução-do-fluxo-de-carga-no-psp-ufu"></a>Execução do fluxo de carga no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#execução-do-fluxo-de-carga-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Após a construção do diagrama unifilar no <a href="/PSP/docs/powerEditor">editor de potência</a>, a execução do fluxo de carga é realizada no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> clicando no botão <strong>Fluxo de carga</strong>.</p><div><center><img src="/PSP/images/menuSimulationPF.svg" alt="Menu Simulação" title="Menu Simulação"></center></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Caso o fluxo de carga tenha sido executado com sucesso, as setas de potência serão exibidas, a barra de status indicará sucesso na operação e os <a href="/PSP/docs/text">elementos de texto</a> serão atualizados.</p></div></div><p><strong>Outra possibilidade</strong> é a execução por meio do cálculo contínuo, também presente no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> e seu acionamento é realizado co clicar no botão <strong>Habilitar solução</strong>. Com essa opção, os cálculos estáticos selecionados nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a> são automaticamente realizados ao modificar quaisquer parâmetros da rede, como dados elétricos e acionamento dos disjuntores dos elementos (remoção ou inserção).</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Evite construir o circuito com o cálculo contínuo habilitado, uma vez que configurações temporárias podem levar a erros de execução da simulação.</p><p>Para desabilitar o cálculo contínuo clique no botão <strong>Desabilitar solução</strong>.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="erros-comuns-na-execução-do-fluxo-de-carga"></a>Erros comuns na execução do fluxo de carga<a aria-hidden="true" tabindex="-1" class="hash-link" href="#erros-comuns-na-execução-do-fluxo-de-carga" title="Direct link to heading">#</a></h3><p>A seguir são apresentados os erros mais comuns relacionados ao fluxo de carga.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="a-seguinte-mensagem-de-erro-é-exibida-o-número-máximo-de-iterações-foi-alcançado"></a>A seguinte mensagem de erro é exibida: "O número máximo de iterações foi alcançado"<a aria-hidden="true" tabindex="-1" class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-o-número-máximo-de-iterações-foi-alcançado" title="Direct link to heading">#</a></h4><p>Essa mensagem de erro é exibida quando o método de solução numérica selecionado nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a> atinge o número máximo de iterações inserido. As seguintes situações podem ocasionar esse erro:</p><ul><li><strong>Os parâmetros do circuito estão incorretos</strong>. caso parâmetros muito fora dos valores padrão dos elementos elétricos sejam inseridos, o cálculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente.</li><li><strong>Algum elemento possui parâmetros discrepantes dos demais</strong>. Caso um dos elementos inseridos possua um valor de impedância muito distinto dos demais, como por exemplo uma linha com impedância muito elevada ou uma carga muito pequena, pode levar à divergência do método numérico. Nesse caso, reconsidere a necessidade de representação desses elementos no circuito e alterne entre os métodos numéricos de solução disponíveis.</li><li><strong>O número máximo de iterações está muito baixo</strong>. Alguns circuitos exigem um número maior de iterações, portanto altere o valor do máximo de iterações nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>. Também tente alternar entre os métodos numéricos de solução disponíveis.</li><li><strong>Os parâmetros de simulação estão inadequados</strong>. Caso um parâmetro do método de solução esteja inadequado, como fator de aceleração ou tolerância, o cálculo pode não alcançar a convergência. Altere esses parâmetros nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="os-dados-de-saída-são-exibidos-como-nan-ou-nan"></a>Os dados de saída são exibidos como "NaN" ou "nan"<a aria-hidden="true" tabindex="-1" class="hash-link" href="#os-dados-de-saída-são-exibidos-como-nan-ou-nan" title="Direct link to heading">#</a></h4><p>Isso ocorre devido a erros de operações matemáticas nos cálculos de fluxo de carga. "NaN" significa <em>Not a Number</em>.</p><ul><li><strong>Algum barramento está isolado</strong>. Esse erro é bastante comum e pode ocorrer ao inserir um barramento sem conectá-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solução é eliminar essa barra do diagrama.</li><li><strong>Os parâmetros do circuito estão incorretos</strong>. caso parâmetros muito fora dos valores padrão dos elementos elétricos sejam inseridos, o cálculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente.</li><li><strong>Algum elemento possui parâmetros discrepantes dos demais</strong>. Caso um dos elementos inseridos possua um valor de impedância muito distinto dos demais, como por exemplo uma linha com impedância muito elevada ou uma carga muito pequena, pode levar à divergência do método numérico. Nesse caso, reconsidere a necessidade de representação desses elementos no circuito e alterne entre os métodos numéricos de solução disponíveis.</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="métodos-de-solução-numérica-do-fluxo-de-carga-no-psp-ufu"></a>Métodos de solução numérica do fluxo de carga no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#métodos-de-solução-numérica-do-fluxo-de-carga-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Os métodos implementados no programa para solução do problema de fluxo de carga no PSP-UFU são <strong>Gauss-Seidel (GS)</strong> e <strong>Newton-Raphson (NR)</strong>. Além desses métodos clássicos, um método híbrido pode ser utilizado (definido nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>), em que é utilizado inicialmente o GS e na sequência o NR, aumentando a chance de convergência do NR.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="gauss-seidel"></a>Gauss-Seidel<a aria-hidden="true" tabindex="-1" class="hash-link" href="#gauss-seidel" title="Direct link to heading">#</a></h3><p>O método de Gauss-Seidel tem sido bastante utilizado nas últimas décadas para solução do problema de fluxo de carga, uma vez que não há a necessidade de fatorar a matrizes, reduzindo o esforço computacional. Atualmente, restrições computacionais são menos problemáticas e outros métodos são normalmente escolhidos, porém o Gauss-Seidel ainda possui valor didático e, visto que o PSP-UFU também possui fins educacionais, optou-se pela implementação desse método.</p><p>Para iniciar as iterações do método são necessários valores iniciais para as tensões (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mn>0</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_i^0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.178854em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.441336em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) que devem ser calculadas. Normalmente para barras do tipo PQ tem-se <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mn>0</mn></msubsup><mo>=</mo><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mo>+</mo><mi>j</mi><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\dot{V}_i^0=1{,}0+j0{,}0~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.178854em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.441336em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> e para barras do tipo PV <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mn>0</mn></msubsup><mo>=</mo><msubsup><mi>V</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup><mo>+</mo><mi>j</mi><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\dot{V}_i^0=V_i^{esp}+j0{,}0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.178854em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.441336em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span></span></span></span></span> [p.u.], em que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>V</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">V_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span> é o módulo da tensão especificada para a barra PV. A barra de referência tem o módulo e ângulo de tensão fixos e não participam no processo iterativo.</p><p>O método de Gauss-Seidel mostra um <strong>número excessivo de iterações</strong> e, com o intuito reduzi-los, multiplicam-se as correções de tensões por uma constante. Essa operação amplia o valor da correção, trazendo a tensão para mais perto do valor do valor final. Os multiplicadores que realizam essa convergência melhorada são chamados de fatores de aceleração. Para qualquer sistema existem valores ótimos para os fatores de aceleração e uma escolha inadequada pode resultar em uma convergência mais lenta ou torná-la impossível. Normalmente é utilizado um fator de aceleração igual a 1,6, valor definido como padrão no programa. A utilização do fator de aceleração é realizada por meio da equação:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mi>α</mi><mrow><mo fence="true">(</mo><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>−</mo><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mo fence="true">)</mo></mrow><mo>+</mo><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i_{AC}}^{(v + 1)} = \alpha \left( \dot{V}_{i}^{(v + 1)}- \dot{V}_{i_{AC}}^{(v)} \right) + \dot{V}_{i_{AC}}^{(v)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i_{AC}}^{(v + 1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão complexa da iteração atual com fator de aceleração aplicado</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i}^{(v + 1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.27686399999999994em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão complexa da iteração atual</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i_{AC}}^{(v)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão complexa da iteração anterior com fator de aceleração aplicado</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span></span></span></span></span> é o fator de aceleração</li></ul><p>O fluxograma abaixo demonstra como o método de Gauss-Seidel foi implementado no PSP-UFU:</p><div><center><img src="/PSP/images/gaussSeidel.svg" alt="Método numérico de Gauss-Seidel para fluxo de carga" title="Método numérico de Gauss-Seidel para fluxo de carga"></center></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="newton-raphson"></a>Newton-Raphson<a aria-hidden="true" tabindex="-1" class="hash-link" href="#newton-raphson" title="Direct link to heading">#</a></h3><p>O método de Newton-Raphson (também conhecido como método de Newton ou Newton-Fourier) para solução do fluxo de carga é descrito em vários livros e artigos. Atualmente é o algoritmo mais utilizado para solução do fluxo de carga. Para casos bem condicionados, esse método geralmente converge em 4 a 5 iterações, porém existe a possibilidade da técnica contornar o ponto da solução sem nunca atingi-la, o que também justifica implementação do método de Gauss-Seidel no <em>software</em>.</p><p>A expansão da <a href="https://en.wikipedia.org/wiki/Taylor_series" target="_blank" rel="noopener noreferrer">série de Taylor</a> para uma função de duas ou mais variáveis é a base do método de Newton-Raphson para resolver o problema do fluxo de carga. Fazendo a expansão em série de Taylor, para duas equações e duas incógnitas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mn>1</mn></msub><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">f_1\left( x_{1}, x_{2} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mn>2</mn></msub><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">f_2\left( x_{1}, x_{2} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span></span>), sem listar as derivadas parciais maiores que 1 na forma matricial, obtêm-se:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>f</mi><mn>1</mn></msub><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo separator="true">,</mo><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>K</mi><mn>2</mn></msub><mo>−</mo><msub><mi>f</mi><mn>2</mn></msub><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo separator="true">,</mo><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>f</mi><mn>1</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>f</mi><mn>1</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>f</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>f</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} K_1 - f_1\left( x_{1}^{(0)}, x_{2}^{(0)} \right)\\ K_2 - f_2\left( x_{1}^{(0)}, x_{2}^{(0)} \right) \end{bmatrix} = \begin{bmatrix} \displaystyle\frac{\partial f_1}{\partial x_1} & \displaystyle\frac{\partial f_1}{\partial x_2}\\ \displaystyle\frac{\partial f_2}{\partial x_1} & \displaystyle\frac{\partial f_2}{\partial x_2} \end{bmatrix} \begin{bmatrix} \Delta x_{1}^{(0)}\\ \Delta x_{2}^{(0)} \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.6010299999999997em;vertical-align:-1.55002em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05002em"><span style="top:-4.05002em"><span class="pstrut" style="height:3.15em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.15em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:4.41488em;vertical-align:-1.95744em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.45744em"><span style="top:-4.45744em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.95744em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.45744em"><span style="top:-4.45744em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.95744em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6548em"><span style="top:-3.6548em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-2.2499999999999996em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.1548000000000005em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size4">]</span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">K</span></span></span></span></span> é o resultado da equação <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">f\left( x_{1}, x_{2} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span></span> é a estimativa iniciais de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">x</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault">x</span></span></span></span></span> é o valor acrescido de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span></span> que resulta em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">x</span></span></span></span></span>, ou seja: <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow><mo>=</mo><msub><mi>f</mi><mn>1</mn></msub><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">f\left( x_{1}, x_{2} \right) = f_1\left( x_{1}^{(0)} + \Delta x_1, x_{2}^{(0)} + \Delta x_2 \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span></span></span></span></span></li></ul><p>Essa expressão pode ser resumida em:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>K</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>K</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><msup><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>J</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} \Delta K_1^{(0)}\\ \Delta K_2^{(0)} \end{bmatrix} = \begin{bmatrix} J \end{bmatrix}^{(0)} \begin{bmatrix} \Delta x_{1}^{(0)}\\ \Delta x_{2}^{(0)} \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6548em"><span style="top:-3.6548em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-2.2499999999999996em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.1548000000000005em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size4">]</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.09618em">J</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.1279000000000001em"><span style="top:-3.3029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6548em"><span style="top:-3.6548em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-2.2499999999999996em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.1548000000000005em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size4">]</span></span></span></span></span></span></span></div><p>Em que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>J</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[J]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord mathdefault" style="margin-right:0.09618em">J</span><span class="mclose">]</span></span></span></span></span> é a <strong>Matriz Jacobiana</strong></p><p>Com a equação acima é possível calcular os valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta x_{1}^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.26630799999999993em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta x_{2}^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.26630799999999993em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span></span>. Entretanto, esses valores somados às estimativas iniciais não determinam a solução correta, sendo necessário repetir o processo de determinação das constantes, formação da matriz jacobiana e solução da equação acima, o qual será refeito determinando novas estimativas <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta x_{1}^{(1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.26630799999999993em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta x_{2}^{(1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.26630799999999993em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span></span>.</p><p>Esse processo é repetido até que as correções se tornem tão pequenas que satisfaçam uma precisão escolhida.</p><p>Para aplicar o método de Newton-Raphson à solução das equações do fluxo de carga, utiliza-se as equações que representam as potências ativa e reativa injetadas em uma barra.</p><p>Assim como no método de Gauss-Seidel, a barra de referência é omitida da solução iterativa para determinar as tensões, pois a tensão complexa dessa barra é especificada. Como é conhecido o valor da potência ativa injetada (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>P</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">P_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span>) nas barras do tipo PQ e PV, além da potência reativa injetada (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span>) nas barras PQ pode-se definir <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>P</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">P_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span> como os valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">K</span></span></span></span></span>. Os valores estimados do módulo e ângulo da tensão correspondem aos valores estimados para <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>.</p><p>O jacobiano consiste nas derivadas parciais de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> em relação a cada uma das variáveis das equações injeção de potência líquida na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>. A matriz coluna formada por <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\Delta x_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Delta x_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> corresponde às correções de ângulo (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\Delta \theta_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) e módulo (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\Delta V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) das tensões de barra.</p><p>Com isso pode-se escrever a equação matricial de um sistema de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">n</span></span></span></span></span> barras, em que a barra número 1 corresponde à barra de referência e as barras de número 2 a n são barras do tipo PQ:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtable rowspacing="0.15999999999999992em" columnalign="right" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>NPQ</mtext><mo>+</mo><mtext>NPV</mtext><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>NPQ</mtext><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr></mtable><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>P</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>P</mi><mi>n</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{array}{r} \text{NPQ} + \text{NPV} \begin{cases} ~\\ ~ \end{cases}\\ ~\\ \text{NPQ} \begin{cases} ~\\ ~ \end{cases} \end{array} \begin{bmatrix} \Delta P_2\\ \vdots\\ \Delta P_n\\ \Delta Q_2\\ \vdots\\ \Delta Q_n\\ \end{bmatrix} = \begin{bmatrix} \displaystyle\frac{\partial P_2}{\partial \theta_2} & \dots & \displaystyle\frac{\partial P_2}{\partial \theta_n} & \displaystyle\frac{\partial P_2}{\partial V_2} & \dots & \displaystyle\frac{\partial P_2}{\partial V_n}\\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\ \displaystyle\frac{\partial P_n}{\partial \theta_2} & \dots & \displaystyle\frac{\partial P_n}{\partial \theta_n} & \displaystyle\frac{\partial P_n}{\partial V_2} & \dots & \displaystyle\frac{\partial P_n}{\partial V_n}\\ \displaystyle\frac{\partial Q_2}{\partial \theta_2} & \dots & \displaystyle\frac{\partial Q_2}{\partial \theta_n} & \displaystyle\frac{\partial Q_2}{\partial V_2} & \dots & \displaystyle\frac{\partial Q_2}{\partial V_n}\\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\ \displaystyle\frac{\partial Q_n}{\partial \theta_2} & \dots & \displaystyle\frac{\partial Q_n}{\partial \theta_n} & \displaystyle\frac{\partial Q_n}{\partial V_2} & \dots & \displaystyle\frac{\partial Q_n}{\partial V_n} \end{bmatrix} \begin{bmatrix} \Delta \theta_2\\ \vdots\\ \Delta \theta_2\\ \Delta V_2\\ \vdots\\ \Delta V_n\\ \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:8.520000000000003em;vertical-align:-4.010000000000001em"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.8500299999999994em"><span style="top:-5.850029999999999em"><span class="pstrut" style="height:3.75em"></span><span class="mord"><span class="mord text"><span class="mord">NPQ</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord text"><span class="mord">NPV</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.7599999999999993em"><span class="pstrut" style="height:3.75em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span><span style="top:-1.6499999999999997em"><span class="pstrut" style="height:3.75em"></span><span class="mord"><span class="mord text"><span class="mord">NPQ</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.3500300000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.458970000000001em"><span style="top:0.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-0.9999600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.5959600000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.191960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.787960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.3839600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.9799600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.57596em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.17196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.217950000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.458970000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.9500599999999997em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.510000000000002em"><span style="top:-7.357500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-5.497500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.297500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.0975000000000006em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.2375000000000005em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-0.03749999999999937em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:4.010000000000001em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.458970000000001em"><span style="top:0.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-0.9999600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.5959600000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.191960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.787960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.3839600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.9799600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.57596em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.17196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.217950000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.458970000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.9500599999999997em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:12.61603em;vertical-align:-6.050095em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.5659350000000005em"><span style="top:2.2500749999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:1.1000749999999995em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:0.5040749999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-0.0919250000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-0.6879250000000008em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.2839250000000009em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.879925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.475925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.071925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6679250000000008em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.263925em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.8599250000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.455925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.051925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.647925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-7.243925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-7.324915000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-8.565935000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:6.050095em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.84094em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-6.5049399999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.773499999999999em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.5660599999999993em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.23005999999999965em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:1.5013799999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.65344em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-6.3174399999999995em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:-4.585999999999999em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-2.3785599999999993em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-0.042559999999999654em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:1.6888799999999995em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.84094em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-6.5049399999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.773499999999999em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.5660599999999993em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.23005999999999965em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:1.5013799999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.84094em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-6.5049399999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.773499999999999em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.5660599999999993em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.23005999999999965em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:1.5013799999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.65344em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-6.3174399999999995em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:-4.585999999999999em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-2.3785599999999993em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-0.042559999999999654em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:1.6888799999999995em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.84094em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-6.5049399999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.773499999999999em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.5660599999999993em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.23005999999999965em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:1.5013799999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.5659350000000005em"><span style="top:2.2500749999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:1.1000749999999995em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:0.5040749999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-0.0919250000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-0.6879250000000008em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.2839250000000009em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.879925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.475925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.071925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6679250000000008em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.263925em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.8599250000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.455925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.051925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.647925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-7.243925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-7.324915000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-8.565935000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:6.050095em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.458970000000001em"><span style="top:0.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-0.9999600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.5959600000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.191960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.787960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.3839600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.9799600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.57596em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.17196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.217950000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.458970000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.9500599999999997em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.510000000000002em"><span style="top:-7.357500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-5.497500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.297500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.0975000000000006em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.2375000000000005em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-0.03749999999999937em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:4.010000000000001em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.458970000000001em"><span style="top:0.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-0.9999600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.5959600000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.191960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.787960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.3839600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.9799600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.57596em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.17196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.217950000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.458970000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.9500599999999997em"><span></span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>NPQ</mtext></mrow><annotation encoding="application/x-tex">\text{NPQ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord text"><span class="mord">NPQ</span></span></span></span></span></span> é o número de barras PQ</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>NPV</mtext></mrow><annotation encoding="application/x-tex">\text{NPV}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord text"><span class="mord">NPV</span></span></span></span></span></span> é o número de barras PV</li></ul><p>O processo iterativo se inicia calculando as potências ativas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>P</mi><mi>i</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">P_i^{calc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.107772em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) para as barras PQ e PV e as potências reativas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{calc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.107772em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) para as barras PQ, ambas as equações utilizando as estimativas iniciais das tensões complexas. Calcula-se, então, as correções de potência (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">\Delta P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault" style="margin-right:0.13889em">P</span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>Q</mi></mrow><annotation encoding="application/x-tex">\Delta Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord">Δ</span><span class="mord mathdefault">Q</span></span></span></span></span>):</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>P</mi><mo>=</mo><msubsup><mi>P</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup><mo>−</mo><msubsup><mi>P</mi><mi>i</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow></msubsup><mspace linebreak="newline"></mspace><mi mathvariant="normal">Δ</mi><mi>Q</mi><mo>=</mo><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup><mo>−</mo><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta P = P_i^{esp} - P_i^{calc}\\ \Delta Q = Q_i^{esp} - Q_i^{calc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7822999999999999em"><span style="top:-2.4231360000000004em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.180908em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.146108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999998em"><span style="top:-2.4530000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord">Δ</span><span class="mord mathdefault">Q</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7822999999999999em"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.180908em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.146108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999998em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span></span></div><p>O passo seguinte é a formação da matriz jacobiana. Com isso é possível calcular as correções de módulo e ângulo das tensões de todas as barras (com exceção da barra de referência). Para tanto, no PSP-UFU utiliza-se o método de <a href="https://en.wikipedia.org/wiki/Gaussian_elimination" target="_blank" rel="noopener noreferrer">Eliminação Gaussiana</a> e em sequência a substituição regressiva. Esse procedimento diminui o esforço computacional, uma vez que a inversão da matriz jacobiana em todas as iterações é evitada.</p><p>Com as correções de módulo e ângulo das tensões das barras calculados, aplicam-se as seguintes equações:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>θ</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><msubsup><mi>θ</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msubsup><mi>θ</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mspace linebreak="newline"></mspace><msubsup><mi>V</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><msubsup><mi>V</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msubsup><mi>V</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\theta_i^{(v+1)}=\theta_i^{(v)}+\Delta \theta_i^{(v)}\\ V_i^{(v+1)}=V_i^{(v)}+\Delta V_i^{(v)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span></div><p>O processo é então reiniciado e será repetido até que se obtenha a convergência, quando as correções se tornam tão pequenas que satisfaçam uma tolerância pré-estipulada.</p><p>O fluxograma abaixo mostra o método de Newton-Raphson para solução do fluxo de carga implementado.</p><div><center><img src="/PSP/images/newtonRaphson.svg" alt="Método numérico de Newton-Raphson para fluxo de carga" title="Método numérico de Newton-Raphson para fluxo de carga"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="controles-e-limites-em-um-problema-de-fluxo-de-carga"></a>Controles e limites em um problema de fluxo de carga<a aria-hidden="true" tabindex="-1" class="hash-link" href="#controles-e-limites-em-um-problema-de-fluxo-de-carga" title="Direct link to heading">#</a></h2><p>Nas barras de <a href="/PSP/docs/syncGenerator">geração</a> e também naquelas em que se encontra um <a href="/PSP/docs/syncMotor">compensador síncrono</a> conectado, o controle da magnitude da tensão no barramento é realizado por meio do ajuste da corrente de campo das máquinas síncronas, as quais podem operar sobrexcitadas (injetando reativos) ou subexcitadas (absorvendo reativos). Os valores limites de potência reativa que podem ser injetadas ou absorvidas dependem da máquina síncrona em estudo. Esses limites são incluídos no fluxo de carga com a criação de dois novos parâmetros, a potência reativa máxima (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>m</mi><mover accent="true"><mi>a</mi><mo>ˊ</mo></mover><mi>x</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{máx}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.107772em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.69444em"><span style="top:-2.7em"><span class="pstrut" style="height:2.7em"></span><span class="mord mathdefault mtight">a</span></span><span style="top:-2.7em"><span class="pstrut" style="height:2.7em"></span><span class="accent-body" style="left:-0.25em"><span class="mord mtight">ˊ</span></span></span></span></span></span></span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) e potência reativa mínima (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{min}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0833279999999998em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.824664em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>), sendo esses valores a soma dos limites individuais das máquinas em uma mesma barra genérica <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>.</p><p>A manutenção da potência reativa dentro dos limites é realizada pela troca do tipo de barra, ou seja, as barras violadoras que controlam a tensão (PV), passam a ser barras de carga (PQ), cuja potência é fixada como o limite que seria ultrapassado (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>m</mi><mover accent="true"><mi>a</mi><mo>ˊ</mo></mover><mi>x</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{máx}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.107772em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.69444em"><span style="top:-2.7em"><span class="pstrut" style="height:2.7em"></span><span class="mord mathdefault mtight">a</span></span><span style="top:-2.7em"><span class="pstrut" style="height:2.7em"></span><span class="accent-body" style="left:-0.25em"><span class="mord mtight">ˊ</span></span></span></span></span></span></span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{min}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0833279999999998em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.824664em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) e a tensão deixa de ser controlada partir de então.</p><p>A verificação de violação e troca de tipo de barra pode ser realizada a cada iteração ou ao final da convergência do cálculo. No PSP-UFU foi implementada a última estratégia, uma vez que separa os conceitos de cálculo e de verificação de limites, tornando mais fácil o desenvolvimento de novos métodos numéricos e limites. Nessa abordagem, ao ajustar o sistema para a nova situação não violadora o cálculo iterativo deve ser retomado até que obtenha novamente a convergência.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="referências"></a>Referências<a aria-hidden="true" tabindex="-1" class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ul><li>MONTICELLI, A. J. Fluxo de Carga em Redes de Energia Elétrica. São Paulo: Edgar Blücher, 1983.</li><li>STEVENSON JR.; WILLIAN, D. Elementos de Análise de Sistemas de Potência. 2ª ed. São Paulo: McGraw-Hill, 1986.</li><li>MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: <a href="https://doi.org/10.1007/978-3-642-13669-6" target="_blank" rel="noopener noreferrer">https://doi.org/10.1007/978-3-642-13669-6</a></li><li>ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley & Sons, New York, 2001. doi: <a href="https://doi.org/10.1002/9781118878286" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/9781118878286</a></li><li>TINNEY, W. F.; HART, C. E. Power Flow Solution by Newton’s Method. IEEE Transaction on Power Apparatus and Systems, v. PAS-86, n. 11, nov. 1967. doi: <a href="https://doi.org/10.1109/TPAS.1967.291823">https://doi.org/10.1109/TPAS.1967.291823</a></li></ul></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/powerFlow.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/mathExpression"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Expressão Matemática</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/fault"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Curto-Circuito »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_TbNY"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#formulação-do-problema-de-fluxo-de-carga" class="table-of-contents__link">Formulação do problema de fluxo de carga</a></li><li><a href="#execução-do-fluxo-de-carga-no-psp-ufu" class="table-of-contents__link">Execução do fluxo de carga no PSP-UFU</a><ul><li><a href="#erros-comuns-na-execução-do-fluxo-de-carga" class="table-of-contents__link">Erros comuns na execução do fluxo de carga</a></li></ul></li><li><a href="#métodos-de-solução-numérica-do-fluxo-de-carga-no-psp-ufu" class="table-of-contents__link">Métodos de solução numérica do fluxo de carga no PSP-UFU</a><ul><li><a href="#gauss-seidel" class="table-of-contents__link">Gauss-Seidel</a></li><li><a href="#newton-raphson" class="table-of-contents__link">Newton-Raphson</a></li></ul></li><li><a href="#controles-e-limites-em-um-problema-de-fluxo-de-carga" class="table-of-contents__link">Controles e limites em um problema de fluxo de carga</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. Built with Docusaurus.</div></div></div></footer></div> +Na formulação do problema a cada barra da rede são associadas quatro variáveis, sendo que duas delas surgem como dados e duas como incógnitas (em uma barra de índice <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>):</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a magnitude da tensão na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">θ_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é o ângulo da tensão na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a injeção líquida de potência ativa na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a injeção líquida de potência reativa na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>.</li></ul><p>Relativo às variáveis que são incógnitas e os dados do sistema, podem ser definidos <strong>três tipos de barras</strong>:</p><ul><li><strong>Barra PQ</strong>: <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são dados, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">θ_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são calculados;</li><li><strong>Barra PV</strong>: <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são dados, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">θ_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são calculados;</li><li><strong>Barra de Referência</strong> <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">θ_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são dados, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são calculados.</li></ul><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>As <strong>barras PQ</strong> geralmente são representadas pelos barramentos de carga, os quais não há geração e controle de tensão.</p><p>As <strong>barras PV</strong> se caracterizam pelo controle de tensão mediante a injeção ou absorção de potência reativa por meio do controle da excitação de uma máquina síncrona.</p><p>A <strong>barra de Referência</strong> (ou de folga, de oscilação) tem como função, assim como o próprio nome diz, servir de referência de tensão e ângulo do sistema. Essa barra é necessariamente geradora, uma vez que ela é responsável pelo equilíbrio do balanço de potência do sistema.</p></div></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O tipo de barra deve ser definido no <a href="/PSP/docs/bus">elemento barramento</a>.</p><p>Note que <strong>o sistema deve possuir somente uma barra de referência</strong>.</p></div></div><p>Como mencionado anteriormente, as equações são não-lineares e a solução analítica não é prática. As soluções dessas equações seguem <strong><a href="https://en.wikipedia.org/wiki/Iterative_method" target="_blank" rel="noopener noreferrer">processos iterativos</a></strong>, em que são atribuídos valores estimados (ou iniciais) para as barras com tensões desconhecidas e, baseado na potência ativa e reativa e módulo da tensão especificados, calcula-se por meio das equações previamente apresentadas as novas tensões complexas em cada nó do sistema.</p><p>Na sequência, esse conjunto de valores para as tensões em cada barra é utilizado para novamente calcular outro grupo de tensões. Cada cálculo de um novo conjunto de tensões é chamado iteração. O processo iterativo é repetido até que as mudanças em todas as barras sejam menores do que um valor pré-estipulado, obtendo assim a convergência.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="execução-do-fluxo-de-carga-no-psp-ufu"></a>Execução do fluxo de carga no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#execução-do-fluxo-de-carga-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Após a construção do diagrama unifilar no <a href="/PSP/docs/powerEditor">editor de potência</a>, a execução do fluxo de carga é realizada no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> clicando no botão <strong>Fluxo de carga</strong>.</p><div><center><img src="/PSP/images/menuSimulationPF.svg" alt="Menu Simulação" title="Menu Simulação"></center></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Caso o fluxo de carga tenha sido executado com sucesso, as setas de potência serão exibidas, a barra de status indicará sucesso na operação e os <a href="/PSP/docs/text">elementos de texto</a> serão atualizados.</p></div></div><p><strong>Outra possibilidade</strong> é a execução por meio do cálculo contínuo, também presente no <a href="/PSP/docs/mainScreen#menu-ribbon">menu Simulação</a> e seu acionamento é realizado co clicar no botão <strong>Habilitar solução</strong>. Com essa opção, os cálculos estáticos selecionados nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a> são automaticamente realizados ao modificar quaisquer parâmetros da rede, como dados elétricos e acionamento dos disjuntores dos elementos (remoção ou inserção).</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Evite construir o circuito com o cálculo contínuo habilitado, uma vez que configurações temporárias podem levar a erros de execução da simulação.</p><p>Para desabilitar o cálculo contínuo clique no botão <strong>Desabilitar solução</strong>.</p></div></div><p>Os resultados do fluxo de carga são exibidos nos <a href="/PSP/docs/text">elementos de texto vinculado</a>, ao posicionar o mouse sobre os elementos e em <a href="/PSP/docs/tabularReport">relatórios tabulares</a>.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="erros-comuns-na-execução-do-fluxo-de-carga"></a>Erros comuns na execução do fluxo de carga<a aria-hidden="true" tabindex="-1" class="hash-link" href="#erros-comuns-na-execução-do-fluxo-de-carga" title="Direct link to heading">#</a></h3><p>A seguir são apresentados os erros mais comuns relacionados ao fluxo de carga.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="a-seguinte-mensagem-de-erro-é-exibida-o-número-máximo-de-iterações-foi-alcançado"></a>A seguinte mensagem de erro é exibida: "O número máximo de iterações foi alcançado"<a aria-hidden="true" tabindex="-1" class="hash-link" href="#a-seguinte-mensagem-de-erro-é-exibida-o-número-máximo-de-iterações-foi-alcançado" title="Direct link to heading">#</a></h4><p>Essa mensagem de erro é exibida quando o método de solução numérica selecionado nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a> atinge o número máximo de iterações inserido. As seguintes situações podem ocasionar esse erro:</p><ul><li><strong>Os parâmetros do circuito estão incorretos</strong>. caso parâmetros muito fora dos valores padrão dos elementos elétricos sejam inseridos, o cálculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente.</li><li><strong>Algum elemento possui parâmetros discrepantes dos demais</strong>. Caso um dos elementos inseridos possua um valor de impedância muito distinto dos demais, como por exemplo uma linha com impedância muito elevada ou uma carga muito pequena, pode levar à divergência do método numérico. Nesse caso, reconsidere a necessidade de representação desses elementos no circuito e alterne entre os métodos numéricos de solução disponíveis.</li><li><strong>O número máximo de iterações está muito baixo</strong>. Alguns circuitos exigem um número maior de iterações, portanto altere o valor do máximo de iterações nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>. Também tente alternar entre os métodos numéricos de solução disponíveis.</li><li><strong>Os parâmetros de simulação estão inadequados</strong>. Caso um parâmetro do método de solução esteja inadequado, como fator de aceleração ou tolerância, o cálculo pode não alcançar a convergência. Altere esses parâmetros nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>.</li></ul><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="os-dados-de-saída-são-exibidos-como-nan-ou-nan"></a>Os dados de saída são exibidos como "NaN" ou "nan"<a aria-hidden="true" tabindex="-1" class="hash-link" href="#os-dados-de-saída-são-exibidos-como-nan-ou-nan" title="Direct link to heading">#</a></h4><p>Isso ocorre devido a erros de operações matemáticas nos cálculos de fluxo de carga. "NaN" significa <em>Not a Number</em>.</p><ul><li><strong>Algum barramento está isolado</strong>. Esse erro é bastante comum e pode ocorrer ao inserir um barramento sem conectá-lo ao sistema ou ao remover os elementos de ramo que conectam uma barra ao sistema. A solução é eliminar essa barra do diagrama.</li><li><strong>Os parâmetros do circuito estão incorretos</strong>. caso parâmetros muito fora dos valores padrão dos elementos elétricos sejam inseridos, o cálculo de fluxo de carga pode divergir. Verifique se os dados foram inseridos corretamente.</li><li><strong>Algum elemento possui parâmetros discrepantes dos demais</strong>. Caso um dos elementos inseridos possua um valor de impedância muito distinto dos demais, como por exemplo uma linha com impedância muito elevada ou uma carga muito pequena, pode levar à divergência do método numérico. Nesse caso, reconsidere a necessidade de representação desses elementos no circuito e alterne entre os métodos numéricos de solução disponíveis.</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="métodos-de-solução-numérica-do-fluxo-de-carga-no-psp-ufu"></a>Métodos de solução numérica do fluxo de carga no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#métodos-de-solução-numérica-do-fluxo-de-carga-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Os métodos implementados no programa para solução do problema de fluxo de carga no PSP-UFU são <strong>Gauss-Seidel (GS)</strong> e <strong>Newton-Raphson (NR)</strong>. Além desses métodos clássicos, um método híbrido pode ser utilizado (definido nas <a href="/PSP/docs/simulationConfig">configurações de simulação</a>), em que é utilizado inicialmente o GS e na sequência o NR, aumentando a chance de convergência do NR.</p><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="gauss-seidel"></a>Gauss-Seidel<a aria-hidden="true" tabindex="-1" class="hash-link" href="#gauss-seidel" title="Direct link to heading">#</a></h3><p>O método de Gauss-Seidel tem sido bastante utilizado nas últimas décadas para solução do problema de fluxo de carga, uma vez que não há a necessidade de fatorar a matrizes, reduzindo o esforço computacional. Atualmente, restrições computacionais são menos problemáticas e outros métodos são normalmente escolhidos, porém o Gauss-Seidel ainda possui valor didático e, visto que o PSP-UFU também possui fins educacionais, optou-se pela implementação desse método.</p><p>Para iniciar as iterações do método são necessários valores iniciais para as tensões (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mn>0</mn></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_i^0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.178854em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.441336em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) que devem ser calculadas. Normalmente para barras do tipo PQ tem-se <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mn>0</mn></msubsup><mo>=</mo><mn>1</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mo>+</mo><mi>j</mi><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\dot{V}_i^0=1{,}0+j0{,}0~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.178854em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.441336em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> e para barras do tipo PV <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mn>0</mn></msubsup><mo>=</mo><msubsup><mi>V</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup><mo>+</mo><mi>j</mi><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\dot{V}_i^0=V_i^{esp}+j0{,}0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.178854em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-2.441336em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">0</span></span></span></span></span> [p.u.], em que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>V</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">V_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span> é o módulo da tensão especificada para a barra PV. A barra de referência tem o módulo e ângulo de tensão fixos e não participam no processo iterativo.</p><p>O método de Gauss-Seidel mostra um <strong>número excessivo de iterações</strong> e, com o intuito reduzi-los, multiplicam-se as correções de tensões por uma constante. Essa operação amplia o valor da correção, trazendo a tensão para mais perto do valor do valor final. Os multiplicadores que realizam essa convergência melhorada são chamados de fatores de aceleração. Para qualquer sistema existem valores ótimos para os fatores de aceleração e uma escolha inadequada pode resultar em uma convergência mais lenta ou torná-la impossível. Normalmente é utilizado um fator de aceleração igual a 1,6, valor definido como padrão no programa. A utilização do fator de aceleração é realizada por meio da equação:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mi>α</mi><mrow><mo fence="true">(</mo><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>−</mo><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mo fence="true">)</mo></mrow><mo>+</mo><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i_{AC}}^{(v + 1)} = \alpha \left( \dot{V}_{i}^{(v + 1)}- \dot{V}_{i_{AC}}^{(v)} \right) + \dot{V}_{i_{AC}}^{(v)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i_{AC}}^{(v + 1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão complexa da iteração atual com fator de aceleração aplicado</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i}^{(v + 1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.27686399999999994em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão complexa da iteração atual</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mover accent="true"><mi>V</mi><mo>˙</mo></mover><msub><mi>i</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\dot{V}_{i_{AC}}^{(v)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4219689999999998em;vertical-align:-0.377169em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">A</span><span class="mord mathdefault mtight" style="margin-right:0.07153em">C</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.377169em"><span></span></span></span></span></span></span></span></span></span></span> é a tensão complexa da iteração anterior com fator de aceleração aplicado</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.0037em">α</span></span></span></span></span> é o fator de aceleração</li></ul><p>O fluxograma abaixo demonstra como o método de Gauss-Seidel foi implementado no PSP-UFU:</p><div><center><img src="/PSP/images/gaussSeidel.svg" alt="Método numérico de Gauss-Seidel para fluxo de carga" title="Método numérico de Gauss-Seidel para fluxo de carga"></center></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="newton-raphson"></a>Newton-Raphson<a aria-hidden="true" tabindex="-1" class="hash-link" href="#newton-raphson" title="Direct link to heading">#</a></h3><p>O método de Newton-Raphson (também conhecido como método de Newton ou Newton-Fourier) para solução do fluxo de carga é descrito em vários livros e artigos. Atualmente é o algoritmo mais utilizado para solução do fluxo de carga. Para casos bem condicionados, esse método geralmente converge em 4 a 5 iterações, porém existe a possibilidade da técnica contornar o ponto da solução sem nunca atingi-la, o que também justifica implementação do método de Gauss-Seidel no <em>software</em>.</p><p>A expansão da <a href="https://en.wikipedia.org/wiki/Taylor_series" target="_blank" rel="noopener noreferrer">série de Taylor</a> para uma função de duas ou mais variáveis é a base do método de Newton-Raphson para resolver o problema do fluxo de carga. Fazendo a expansão em série de Taylor, para duas equações e duas incógnitas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mn>1</mn></msub><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">f_1\left( x_{1}, x_{2} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mn>2</mn></msub><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">f_2\left( x_{1}, x_{2} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span></span>), sem listar as derivadas parciais maiores que 1 na forma matricial, obtêm-se:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>K</mi><mn>1</mn></msub><mo>−</mo><msub><mi>f</mi><mn>1</mn></msub><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo separator="true">,</mo><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>K</mi><mn>2</mn></msub><mo>−</mo><msub><mi>f</mi><mn>2</mn></msub><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo separator="true">,</mo><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>f</mi><mn>1</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>f</mi><mn>1</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>f</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>1</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>f</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>x</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} K_1 - f_1\left( x_{1}^{(0)}, x_{2}^{(0)} \right)\\ K_2 - f_2\left( x_{1}^{(0)}, x_{2}^{(0)} \right) \end{bmatrix} = \begin{bmatrix} \displaystyle\frac{\partial f_1}{\partial x_1} & \displaystyle\frac{\partial f_1}{\partial x_2}\\ \displaystyle\frac{\partial f_2}{\partial x_1} & \displaystyle\frac{\partial f_2}{\partial x_2} \end{bmatrix} \begin{bmatrix} \Delta x_{1}^{(0)}\\ \Delta x_{2}^{(0)} \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.6010299999999997em;vertical-align:-1.55002em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05002em"><span style="top:-4.05002em"><span class="pstrut" style="height:3.15em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.15em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.8099900000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.05101em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:4.41488em;vertical-align:-1.95744em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.45744em"><span style="top:-4.45744em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.95744em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.45744em"><span style="top:-4.45744em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.3714399999999998em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.95744em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6548em"><span style="top:-3.6548em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-2.2499999999999996em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.1548000000000005em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size4">]</span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">K</span></span></span></span></span> é o resultado da equação <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">f\left( x_{1}, x_{2} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span></span> é a estimativa iniciais de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">x</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\Delta x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault">x</span></span></span></span></span> é o valor acrescido de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879999999999999em;vertical-align:0em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span></span> que resulta em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">x</span></span></span></span></span>, ou seja: <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mrow><mo fence="true">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow><mo>=</mo><msub><mi>f</mi><mn>1</mn></msub><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mn>2</mn></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">f\left( x_{1}, x_{2} \right) = f_1\left( x_{1}^{(0)} + \Delta x_1, x_{2}^{(0)} + \Delta x_2 \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span></span></span></span></span></li></ul><p>Essa expressão pode ser resumida em:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>K</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>K</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><msup><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>J</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} \Delta K_1^{(0)}\\ \Delta K_2^{(0)} \end{bmatrix} = \begin{bmatrix} J \end{bmatrix}^{(0)} \begin{bmatrix} \Delta x_{1}^{(0)}\\ \Delta x_{2}^{(0)} \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6548em"><span style="top:-3.6548em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-2.2499999999999996em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:-0.07153em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.1548000000000005em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size4">]</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.09618em">J</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.1279000000000001em"><span style="top:-3.3029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6548em"><span style="top:-3.6548em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span><span style="top:-2.2499999999999996em"><span class="pstrut" style="height:3.0448em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.1548000000000005em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size4">]</span></span></span></span></span></span></span></div><p>Em que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>J</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[J]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mopen">[</span><span class="mord mathdefault" style="margin-right:0.09618em">J</span><span class="mclose">]</span></span></span></span></span> é a <strong>Matriz Jacobiana</strong></p><p>Com a equação acima é possível calcular os valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta x_{1}^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.26630799999999993em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta x_{2}^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.26630799999999993em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span></span>. Entretanto, esses valores somados às estimativas iniciais não determinam a solução correta, sendo necessário repetir o processo de determinação das constantes, formação da matriz jacobiana e solução da equação acima, o qual será refeito determinando novas estimativas <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta x_{1}^{(1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.26630799999999993em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msubsup><mi>x</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta x_{2}^{(1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.26630799999999993em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26630799999999993em"><span></span></span></span></span></span></span></span></span></span></span>.</p><p>Esse processo é repetido até que as correções se tornem tão pequenas que satisfaçam uma precisão escolhida.</p><p>Para aplicar o método de Newton-Raphson à solução das equações do fluxo de carga, utiliza-se as equações que representam as potências ativa e reativa injetadas em uma barra.</p><p>Assim como no método de Gauss-Seidel, a barra de referência é omitida da solução iterativa para determinar as tensões, pois a tensão complexa dessa barra é especificada. Como é conhecido o valor da potência ativa injetada (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>P</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">P_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span>) nas barras do tipo PQ e PV, além da potência reativa injetada (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span>) nas barras PQ pode-se definir <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>P</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">P_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{esp}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7823em"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span> como os valores de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.07153em">K</span></span></span></span></span>. Os valores estimados do módulo e ângulo da tensão correspondem aos valores estimados para <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>.</p><p>O jacobiano consiste nas derivadas parciais de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">P_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Q</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Q_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> em relação a cada uma das variáveis das equações injeção de potência líquida na barra <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>. A matriz coluna formada por <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\Delta x_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Delta x_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> corresponde às correções de ângulo (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>θ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\Delta \theta_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) e módulo (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\Delta V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) das tensões de barra.</p><p>Com isso pode-se escrever a equação matricial de um sistema de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">n</span></span></span></span></span> barras, em que a barra número 1 corresponde à barra de referência e as barras de número 2 a n são barras do tipo PQ:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtable rowspacing="0.15999999999999992em" columnalign="right" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>NPQ</mtext><mo>+</mo><mtext>NPV</mtext><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>NPQ</mtext><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext> </mtext></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr></mtable><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>P</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>P</mi><mi>n</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>P</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>θ</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">…</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><msub><mi>Q</mi><mi>n</mi></msub></mrow><mrow><mi mathvariant="normal">∂</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>θ</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>V</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><msub><mi>V</mi><mi>n</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{array}{r} \text{NPQ} + \text{NPV} \begin{cases} ~\\ ~ \end{cases}\\ ~\\ \text{NPQ} \begin{cases} ~\\ ~ \end{cases} \end{array} \begin{bmatrix} \Delta P_2\\ \vdots\\ \Delta P_n\\ \Delta Q_2\\ \vdots\\ \Delta Q_n\\ \end{bmatrix} = \begin{bmatrix} \displaystyle\frac{\partial P_2}{\partial \theta_2} & \dots & \displaystyle\frac{\partial P_2}{\partial \theta_n} & \displaystyle\frac{\partial P_2}{\partial V_2} & \dots & \displaystyle\frac{\partial P_2}{\partial V_n}\\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\ \displaystyle\frac{\partial P_n}{\partial \theta_2} & \dots & \displaystyle\frac{\partial P_n}{\partial \theta_n} & \displaystyle\frac{\partial P_n}{\partial V_2} & \dots & \displaystyle\frac{\partial P_n}{\partial V_n}\\ \displaystyle\frac{\partial Q_2}{\partial \theta_2} & \dots & \displaystyle\frac{\partial Q_2}{\partial \theta_n} & \displaystyle\frac{\partial Q_2}{\partial V_2} & \dots & \displaystyle\frac{\partial Q_2}{\partial V_n}\\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\ \displaystyle\frac{\partial Q_n}{\partial \theta_2} & \dots & \displaystyle\frac{\partial Q_n}{\partial \theta_n} & \displaystyle\frac{\partial Q_n}{\partial V_2} & \dots & \displaystyle\frac{\partial Q_n}{\partial V_n} \end{bmatrix} \begin{bmatrix} \Delta \theta_2\\ \vdots\\ \Delta \theta_2\\ \Delta V_2\\ \vdots\\ \Delta V_n\\ \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:8.520000000000003em;vertical-align:-4.010000000000001em"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.8500299999999994em"><span style="top:-5.850029999999999em"><span class="pstrut" style="height:3.75em"></span><span class="mord"><span class="mord text"><span class="mord">NPQ</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord text"><span class="mord">NPV</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.7599999999999993em"><span class="pstrut" style="height:3.75em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span><span style="top:-1.6499999999999997em"><span class="pstrut" style="height:3.75em"></span><span class="mord"><span class="mord text"><span class="mord">NPQ</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em"><span style="top:-3.69em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span><span style="top:-2.25em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mspace nobreak"> </span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.3500300000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.458970000000001em"><span style="top:0.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-0.9999600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.5959600000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.191960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.787960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.3839600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.9799600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.57596em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.17196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.217950000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.458970000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.9500599999999997em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.510000000000002em"><span style="top:-7.357500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-5.497500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.297500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.0975000000000006em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.2375000000000005em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-0.03749999999999937em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:4.010000000000001em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.458970000000001em"><span style="top:0.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-0.9999600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.5959600000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.191960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.787960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.3839600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.9799600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.57596em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.17196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.217950000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.458970000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.9500599999999997em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:12.61603em;vertical-align:-6.050095em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.5659350000000005em"><span style="top:2.2500749999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:1.1000749999999995em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:0.5040749999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-0.0919250000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-0.6879250000000008em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.2839250000000009em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.879925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.475925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.071925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6679250000000008em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.263925em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.8599250000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.455925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.051925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.647925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-7.243925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-7.324915000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-8.565935000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:6.050095em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.84094em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-6.5049399999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.773499999999999em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.5660599999999993em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.23005999999999965em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:1.5013799999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.65344em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-6.3174399999999995em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:-4.585999999999999em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-2.3785599999999993em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-0.042559999999999654em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:1.6888799999999995em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.84094em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-6.5049399999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.773499999999999em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.5660599999999993em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.23005999999999965em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:1.5013799999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.84094em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-6.5049399999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.773499999999999em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.5660599999999993em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.23005999999999965em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:1.5013799999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.65344em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-6.3174399999999995em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:-4.585999999999999em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-2.3785599999999993em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span><span style="top:-0.042559999999999654em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:1.6888799999999995em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">…</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.52488em"><span style="top:-8.84094em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-6.5049399999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.773499999999999em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.5660599999999993em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-0.23005999999999965em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:1.5013799999999995em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord" style="margin-right:0.05556em">∂</span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8360000000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:6.02488em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.5659350000000005em"><span style="top:2.2500749999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:1.1000749999999995em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:0.5040749999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-0.0919250000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-0.6879250000000008em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.2839250000000009em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.879925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.475925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.071925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6679250000000008em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.263925em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.8599250000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.455925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.051925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.647925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-7.243925000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-7.324915000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-8.565935000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:6.050095em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.458970000000001em"><span style="top:0.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-0.9999600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.5959600000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.191960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.787960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.3839600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.9799600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.57596em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.17196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.217950000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.458970000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.9500599999999997em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.510000000000002em"><span style="top:-7.357500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-5.497500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-4.297500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.0975000000000006em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.2375000000000005em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-0.03749999999999937em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:4.010000000000001em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.458970000000001em"><span style="top:0.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-0.9999600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.5959600000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.191960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.787960000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.3839600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.9799600000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.57596em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.17196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.217950000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.458970000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.9500599999999997em"><span></span></span></span></span></span></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>NPQ</mtext></mrow><annotation encoding="application/x-tex">\text{NPQ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord text"><span class="mord">NPQ</span></span></span></span></span></span> é o número de barras PQ</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>NPV</mtext></mrow><annotation encoding="application/x-tex">\text{NPV}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord text"><span class="mord">NPV</span></span></span></span></span></span> é o número de barras PV</li></ul><p>O processo iterativo se inicia calculando as potências ativas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>P</mi><mi>i</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">P_i^{calc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.107772em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) para as barras PQ e PV e as potências reativas (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{calc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.107772em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) para as barras PQ, ambas as equações utilizando as estimativas iniciais das tensões complexas. Calcula-se, então, as correções de potência (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>P</mi></mrow><annotation encoding="application/x-tex">\Delta P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault" style="margin-right:0.13889em">P</span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>Q</mi></mrow><annotation encoding="application/x-tex">\Delta Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord">Δ</span><span class="mord mathdefault">Q</span></span></span></span></span>):</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>P</mi><mo>=</mo><msubsup><mi>P</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup><mo>−</mo><msubsup><mi>P</mi><mi>i</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow></msubsup><mspace linebreak="newline"></mspace><mi mathvariant="normal">Δ</mi><mi>Q</mi><mo>=</mo><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>e</mi><mi>s</mi><mi>p</mi></mrow></msubsup><mo>−</mo><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>c</mi><mi>a</mi><mi>l</mi><mi>c</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Delta P = P_i^{esp} - P_i^{calc}\\ \Delta Q = Q_i^{esp} - Q_i^{calc}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7822999999999999em"><span style="top:-2.4231360000000004em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.180908em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.146108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999998em"><span style="top:-2.4530000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em"></span><span class="mord">Δ</span><span class="mord mathdefault">Q</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.059164em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7822999999999999em"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.180908em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.146108em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999998em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span></span></span></span></span></div><p>O passo seguinte é a formação da matriz jacobiana. Com isso é possível calcular as correções de módulo e ângulo das tensões de todas as barras (com exceção da barra de referência). Para tanto, no PSP-UFU utiliza-se o método de <a href="https://en.wikipedia.org/wiki/Gaussian_elimination" target="_blank" rel="noopener noreferrer">Eliminação Gaussiana</a> e em sequência a substituição regressiva. Esse procedimento diminui o esforço computacional, uma vez que a inversão da matriz jacobiana em todas as iterações é evitada.</p><p>Com as correções de módulo e ângulo das tensões das barras calculados, aplicam-se as seguintes equações:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>θ</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><msubsup><mi>θ</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msubsup><mi>θ</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mspace linebreak="newline"></mspace><msubsup><mi>V</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><msubsup><mi>V</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">Δ</mi><msubsup><mi>V</mi><mi>i</mi><mrow><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">\theta_i^{(v+1)}=\theta_i^{(v)}+\Delta \theta_i^{(v)}\\ V_i^{(v+1)}=V_i^{(v)}+\Delta V_i^{(v)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">θ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em"></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em"><span style="top:-2.4231360000000004em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight" style="margin-right:0.03588em">v</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em"><span></span></span></span></span></span></span></span></span></span></span></div><p>O processo é então reiniciado e será repetido até que se obtenha a convergência, quando as correções se tornam tão pequenas que satisfaçam uma tolerância pré-estipulada.</p><p>O fluxograma abaixo mostra o método de Newton-Raphson para solução do fluxo de carga implementado.</p><div><center><img src="/PSP/images/newtonRaphson.svg" alt="Método numérico de Newton-Raphson para fluxo de carga" title="Método numérico de Newton-Raphson para fluxo de carga"></center></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="controles-e-limites-em-um-problema-de-fluxo-de-carga"></a>Controles e limites em um problema de fluxo de carga<a aria-hidden="true" tabindex="-1" class="hash-link" href="#controles-e-limites-em-um-problema-de-fluxo-de-carga" title="Direct link to heading">#</a></h2><p>Nas barras de <a href="/PSP/docs/syncGenerator">geração</a> e também naquelas em que se encontra um <a href="/PSP/docs/syncMotor">compensador síncrono</a> conectado, o controle da magnitude da tensão no barramento é realizado por meio do ajuste da corrente de campo das máquinas síncronas, as quais podem operar sobrexcitadas (injetando reativos) ou subexcitadas (absorvendo reativos). Os valores limites de potência reativa que podem ser injetadas ou absorvidas dependem da máquina síncrona em estudo. Esses limites são incluídos no fluxo de carga com a criação de dois novos parâmetros, a potência reativa máxima (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>m</mi><mover accent="true"><mi>a</mi><mo>ˊ</mo></mover><mi>x</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{máx}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.107772em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.69444em"><span style="top:-2.7em"><span class="pstrut" style="height:2.7em"></span><span class="mord mathdefault mtight">a</span></span><span style="top:-2.7em"><span class="pstrut" style="height:2.7em"></span><span class="accent-body" style="left:-0.25em"><span class="mord mtight">ˊ</span></span></span></span></span></span></span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) e potência reativa mínima (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{min}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0833279999999998em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.824664em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>), sendo esses valores a soma dos limites individuais das máquinas em uma mesma barra genérica <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span>.</p><p>A manutenção da potência reativa dentro dos limites é realizada pela troca do tipo de barra, ou seja, as barras violadoras que controlam a tensão (PV), passam a ser barras de carga (PQ), cuja potência é fixada como o limite que seria ultrapassado (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>m</mi><mover accent="true"><mi>a</mi><mo>ˊ</mo></mover><mi>x</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{máx}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.107772em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord accent mtight"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.69444em"><span style="top:-2.7em"><span class="pstrut" style="height:2.7em"></span><span class="mord mathdefault mtight">a</span></span><span style="top:-2.7em"><span class="pstrut" style="height:2.7em"></span><span class="accent-body" style="left:-0.25em"><span class="mord mtight">ˊ</span></span></span></span></span></span></span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>Q</mi><mi>i</mi><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">Q_i^{min}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0833279999999998em;vertical-align:-0.258664em"></span><span class="mord"><span class="mord mathdefault">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.824664em"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em"><span></span></span></span></span></span></span></span></span></span></span>) e a tensão deixa de ser controlada partir de então.</p><p>A verificação de violação e troca de tipo de barra pode ser realizada a cada iteração ou ao final da convergência do cálculo. No PSP-UFU foi implementada a última estratégia, uma vez que separa os conceitos de cálculo e de verificação de limites, tornando mais fácil o desenvolvimento de novos métodos numéricos e limites. Nessa abordagem, ao ajustar o sistema para a nova situação não violadora o cálculo iterativo deve ser retomado até que obtenha novamente a convergência.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="referências"></a>Referências<a aria-hidden="true" tabindex="-1" class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>MONTICELLI, A. J. Fluxo de Carga em Redes de Energia Elétrica. São Paulo: Edgar Blücher, 1983.</li><li>STEVENSON JR.; WILLIAN, D. Elementos de Análise de Sistemas de Potência. 2ª ed. São Paulo: McGraw-Hill, 1986.</li><li>MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: <a href="https://doi.org/10.1007/978-3-642-13669-6" target="_blank" rel="noopener noreferrer">https://doi.org/10.1007/978-3-642-13669-6</a></li><li>ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley & Sons, New York, 2001. doi: <a href="https://doi.org/10.1002/9781118878286" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/9781118878286</a></li><li>TINNEY, W. F.; HART, C. E. Power Flow Solution by Newton’s Method. IEEE Transaction on Power Apparatus and Systems, v. PAS-86, n. 11, nov. 1967. doi: <a href="https://doi.org/10.1109/TPAS.1967.291823">https://doi.org/10.1109/TPAS.1967.291823</a></li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/powerFlow.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/mathExpression"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Expressão Matemática</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/fault"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Curto-Circuito »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_TbNY"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#formulação-do-problema-de-fluxo-de-carga" class="table-of-contents__link">Formulação do problema de fluxo de carga</a></li><li><a href="#execução-do-fluxo-de-carga-no-psp-ufu" class="table-of-contents__link">Execução do fluxo de carga no PSP-UFU</a><ul><li><a href="#erros-comuns-na-execução-do-fluxo-de-carga" class="table-of-contents__link">Erros comuns na execução do fluxo de carga</a></li></ul></li><li><a href="#métodos-de-solução-numérica-do-fluxo-de-carga-no-psp-ufu" class="table-of-contents__link">Métodos de solução numérica do fluxo de carga no PSP-UFU</a><ul><li><a href="#gauss-seidel" class="table-of-contents__link">Gauss-Seidel</a></li><li><a href="#newton-raphson" class="table-of-contents__link">Newton-Raphson</a></li></ul></li><li><a href="#controles-e-limites-em-um-problema-de-fluxo-de-carga" class="table-of-contents__link">Controles e limites em um problema de fluxo de carga</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Limitador de taxa</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Os blocos de limite de taxa têm como objetivo conter a taxa de crescimento ou decrescimento da sua entrada a valores inseridos pelo usuário. A taxa de variação é calculada a partir dos valores de entrada atual e passado e, caso seja violada, será definida uma saída que respeite essas taxas (de crescimento ou decrescimento).</p><p>A taxa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span></span></span></span></span>) é calculada de acordo com a equação:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>=</mo><mfrac><mrow><msub><mi>u</mi><mi>n</mi></msub><mo>−</mo><msub><mi>y</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><mi>h</mi></mfrac></mrow><annotation encoding="application/x-tex">r = \frac{u_n - y_{n-1}}{h}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.9463300000000001em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.26033em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">h</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>u</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">u_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> é a entrada atual do bloco</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">y_{n-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span></span> é o valor de saída anterior do bloco</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em"></span><span class="mord mathdefault">h</span></span></span></span></span> é o passo de integração</li></ul><p>Obtida a taxa, o bloco irá resultar em um valor calculado pelas seguintes condições:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>h</mi><mo>×</mo><msub><mi>R</mi><mrow><mi>c</mi><mi>r</mi><mi>e</mi></mrow></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><mi>r</mi><mo>></mo><msub><mi>R</mi><mrow><mi>c</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>u</mi><mi>n</mi></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><msub><mi>R</mi><mrow><mi>d</mi><mi>e</mi><mi>c</mi></mrow></msub><mo>≤</mo><mi>r</mi><mo>≤</mo><msub><mi>R</mi><mrow><mi>c</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>h</mi><mo>×</mo><msub><mi>R</mi><mrow><mi>d</mi><mi>e</mi><mi>c</mi></mrow></msub><mo separator="true" lspace="0em" rspace="0em">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>se </mtext><mi>r</mi><mo><</mo><msub><mi>R</mi><mrow><mi>d</mi><mi>e</mi><mi>c</mi></mrow></msub></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases} h \times R_{cre}{,} & \text{se } r > R_{cre} \\ u_n{,} & \text{se } R_{dec} \le r \le R_{cre}\\ h \times R_{dec}{,} & \text{se } r < R_{dec} \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.9099999999999997em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35002em"><span style="top:-2.19999em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.19499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-2.20499em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.15001em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.2950099999999996em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.30501em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.60002em"><span class="pstrut" style="height:3.15em"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.8500199999999998em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mpunct">,</span></span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mpunct">,</span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mpunct">,</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em"><span style="top:-4.41em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.97em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-1.5300000000000002em"><span class="pstrut" style="height:3.008em"></span><span class="mord"><span class="mord text"><span class="mord">se </span></span><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>R</mi><mrow><mi>c</mi><mi>r</mi><mi>e</mi></mrow></msub></mrow><annotation encoding="application/x-tex">R_{cre}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>R</mi><mrow><mi>d</mi><mi>e</mi><mi>c</mi></mrow></msub></mrow><annotation encoding="application/x-tex">R_{dec}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">c</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> são as taxas de crescimento e decaimento, respectivamente</li></ul><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="formulário-de-edição-de-dados-do-limitador-de-taxa"></a>Formulário de edição de dados do Limitador de taxa<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-de-dados-do-limitador-de-taxa" title="Direct link to heading">#</a></h2><p>A figura abaixo apresenta o formulário de edição de dados do bloco limitador de taxa.</p><div><center><img src="/PSP/images/rateLimitForm.png" alt="Formulário de edição de dados do bloco limitador de taxa no PSP-UFU" title="Formulário de edição de dados do bloco limitador de taxa no PSP-UFU"></center></div><p>Esse bloco é definido pelo <strong>limite superior (taxa de crescimento máxima) e inferior (taxa de decrescimento máxima)</strong>, inseridos pelo usuário.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Esses blocos são úteis para limitar variações bruscas na entrada que possam comprometer o controle ou limites físicos de modelos de turbina, por exemplo.</p></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/rateLimiter.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" 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Built with Docusaurus.</div></div></div></footer></div> <script src="/PSP/styles.26e2fe91.js"></script> -<script src="/PSP/runtime~main.3ede30cf.js"></script> -<script src="/PSP/main.1c1af766.js"></script> +<script src="/PSP/runtime~main.57a83c0d.js"></script> +<script src="/PSP/main.1dbec03f.js"></script> <script src="/PSP/1.837a9375.js"></script> <script src="/PSP/2.19116c85.js"></script> <script src="/PSP/64.2a8261e0.js"></script> diff --git a/docs/docs/sum/index.html b/docs/docs/sum/index.html index 871543c..48bfedd 100644 --- a/docs/docs/sum/index.html +++ b/docs/docs/sum/index.html @@ -6,8 +6,8 @@ <meta name="generator" content="Docusaurus v2.0.0-alpha.58"> <link rel="search" type="application/opensearchdescription+xml" title="PSP-UFU" href="/PSP/opensearch.xml"><title data-react-helmet="true">Somador | PSP-UFU</title><meta data-react-helmet="true" name="twitter:card" content="summary_large_image"><meta data-react-helmet="true" property="og:title" content="Somador | 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class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_FFX1"><div class="container padding-vert--lg docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Somador</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>Como o próprio nome sugere, os somadores são elementos de controle capazes de somar qualquer quantidade de valores reais.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="formulário-de-edição-de-dados-do-somador"></a>Formulário de edição de dados do somador<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-de-dados-do-somador" title="Direct link to heading">#</a></h2><p>A quantidade de entradas, assim como seus sinais são definidas pelo usuário, como mostra a figura abaixo.</p><div><center><img src="/PSP/images/sumForm.png" alt="Formulário de edição de dados do somador no PSP-UFU" title="Formulário de edição de dados do somador no PSP-UFU"></center></div><p>No seu único campo “<strong>Sinais</strong>” é inserida uma lista de sinais <strong>separados por espaços</strong> contendo os símbolos “+” para uma entrada positiva e “-” para negativa.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>O usuário pode inserir duas ou mais entradas de sinais no bloco.</p></div></div><p>Uma vez inseridas novas entradas, nós correspondentes serão incluídos no elemento gráfico do somador, o qual poderá ser conectado por meio das linhas de conexão.</p><p>De forma semelhante, o usuário poderá excluir entradas de sinais pré-existentes, as quais serão removidas no ícone gráfico pela remoção do nó e da linha (caso exista alguma conectada).</p></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/sum.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div 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Matemática</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulações</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_FFX1"><div class="container padding-vert--lg docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Gerador Síncrono</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Uma máquina de corrente alternada na qual a frequência das tensões geradas e a velocidade da máquina estão em uma proporção constante. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=411-31-08" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="gerador-síncrono-no-psp-ufu"></a>Gerador Síncrono no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#gerador-síncrono-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Os geradores síncronos são a fonte de potência para o cálculo do fluxo de carga e estabilidade, além de serem um dos principais contribuintes para as correntes de falta.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Somente barramentos que possuem esse elemento conectado podem ser considerados <a href="/PSP/docs/powerFlow">barras de referência</a>. <a href="/PSP/docs/powerFlow">Barras do tipo PV</a> devem conter um gerador síncrono ou um <a href="/PSP/docs/syncMotor">motor síncrono (compensador síncrono)</a>.</p></div></div><p>O formulário de dados é dividido em dois, sendo o primeiro responsável pelos dados gerais, de <a href="/PSP/docs/powerFlow">fluxo de carga</a> e de <a href="/PSP/docs/fault">falta</a> e um segundo pelos dados de <a href="/PSP/docs/stability">estabilidade</a> . Esse último também permite o acesso aos controles das máquinas síncronas manipulados pelo editor de controle.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Os dados referentes às <strong>impedâncias de sequência positiva do gerador síncrono</strong> inseridas no <a href="/PSP/docs/syncGenerator#falta">contexto (aba) "Falta"</a> são utilizados <strong>tanto para os estudos de <a href="/PSP/docs/fault">curto-circuito</a> quanto para os estudos de <a href="/PSP/docs/harmonics">harmônicos</a></strong>.</p><p>Esses dados são ignorados nos estudos de <a href="/PSP/docs/powerFlow">fluxo de carga</a> (<em>não são utilizado nesse estudo</em>) e <a href="/PSP/docs/stability">estabilidade</a> (<em>são utilizados dados inseridos em formulário específico</em>).</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="gerador-síncrono-no-fluxo-de-carga"></a>Gerador Síncrono no fluxo de carga<a aria-hidden="true" tabindex="-1" class="hash-link" href="#gerador-síncrono-no-fluxo-de-carga" title="Direct link to heading">#</a></h3><p>O gerador síncrono é a fonte de potência do PSP-UFU no estudo de fluxo de carga. Seu comportamento difere de acordo com o tipo de barra conectada:</p><ul><li><a href="/PSP/docs/powerFlow">Barra de referência</a>: Os dados de potência ativa e reativa inseridos são desprezados, uma vez que esse elemento será utilizado para completar o balanço de potência do estudo de fluxo d carga;</li><li><a href="/PSP/docs/powerFlow">Barra PV</a>: O dado de potência ativa é considerado, porém o dados de potência reativa são desprezados. O valor da potência reativa é utilizado para manter o módulo da tensão constante no barramento conectado;</li><li><a href="/PSP/docs/powerFlow">Barra PQ</a>: Para os geradores conectados a essa barra tanto a potência ativa quanto reativa inseridas são consideradas.</li></ul><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>Caso o limite de potência reativa seja excedido, o programa automaticamente transforma a <a href="/PSP/docs/powerFlow">Barra PV</a> conectada em uma <a href="/PSP/docs/powerFlow">Barra PQ</a>, utilizando o valor limite de potência reativa que seria ultrapassado.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="gerador-síncrono-no-estudo-de-curto-circuito"></a>Gerador Síncrono no estudo de curto-circuito<a aria-hidden="true" tabindex="-1" class="hash-link" href="#gerador-síncrono-no-estudo-de-curto-circuito" title="Direct link to heading">#</a></h3><p>Enquanto os geradores no estudo de fluxo de carga são modelados somente por correntes injetadas nas barras, para o curto-circuito utiliza-se uma tensão atrás de uma impedância. A figura abaixo mostra o caminho da corrente e o circuito equivalente de cada sequência nos geradores.</p><div><center><img src="/PSP/images/sycGeneratorSeq.svg" alt="Caminho das correntes e circuito equivalente: (a) sequência positiva; (b) sequência negativa; (c) sequência zero" title="Caminho das correntes e circuito equivalente: (a) sequência positiva; (b) sequência negativa; (c) sequência zero"></center></div><p>As tensões geradas são somente de sequência positiva, uma vez que o gerador fornece sempre tensões trifásicas equilibradas. Portanto a rede de sequência positiva é composta de uma tensão pré-falta atrás de uma impedância de sequência positiva. As redes de sequência negativa e zero não contêm forças eletromotrizes, porém incluem as impedâncias do gerador de sequência negativa e zero.</p><p>A corrente que circula na impedância <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\overline{z}_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> entre o neutro e a terra é <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">3\dot{I}_{a0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.07019em;vertical-align:-0.15em"></span><span class="mord">3</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>. Pela figura acima (c), observa-se que a queda de tensão de sequência zero do ponto a para terra (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\dot{V}_{a0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.07019em;vertical-align:-0.15em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) é:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub><mo>=</mo><mo>−</mo><mn>3</mn><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>n</mi></msub><mo>−</mo><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>a</mi><mn>0</mn></mrow></msub><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>g</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\dot{V}_{a0} = -3\dot{I}_{a0}\overline{z}_n - \dot{I}_{a0}\overline{z}_{g0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.07019em;vertical-align:-0.15em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.07019em;vertical-align:-0.15em"></span><span class="mord">−</span><span class="mord">3</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.206298em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">g</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span></div><p>A rede de sequência zero, que é um circuito monofásico pelo qual se supõe que circule apenas corrente de sequência zero e deve, portanto, ter uma impedância definida pela seguinte equação:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mn>0</mn></msub><mo>=</mo><mn>3</mn><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>n</mi></msub><mo>+</mo><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mrow><mi>g</mi><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\overline{z}_{0} = 3\overline{z}_n + \overline{z}_{g0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.916668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">g</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>Caso o gerador não seja aterrado, não circulará corrente de sequência zero por ele. Nesse caso, dependendo da conexão do transformador próximo ao gerador sem aterramento, a seguinte mensagem de erro pode ser exibida:</p><p><center>"<i>Falha ao inverter a matriz admitância de sequência zero</i>"</center></p><p>Isso ocorre porque a matriz admitância de sequência zero é singular. Para contornar esse problema escolha <strong>uma</strong> das duas soluções abaixo:</p><ol><li>Marque a opção "Neutro aterrado" e insira um alto valor de reatância de aterramento (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mn>9999</mn><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">j9999~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, por exemplo);</li><li><strong>Ou</strong>, na barra do gerador, insira um <a href="/PSP/docs/inductor">reator</a> de baixo valor de potência reativa (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mtext> </mtext><mi>v</mi><mi>a</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">1,0~var</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault" style="margin-right:0.03588em">v</span><span class="mord mathdefault">a</span><span class="mord mathdefault" style="margin-right:0.02778em">r</span></span></span></span></span>, por exemplo).</li></ol></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="gerador-síncrono-no-estudo-de-estabilidade"></a>Gerador Síncrono no estudo de estabilidade<a aria-hidden="true" tabindex="-1" class="hash-link" href="#gerador-síncrono-no-estudo-de-estabilidade" title="Direct link to heading">#</a></h3><p>A relação de valores observados em testes adequados (definidos na <a href="https://doi.org/10.1109/IEEESTD.2020.9050934">IEEE Std. 115-2019</a>), denominados parâmetros padrões, são utilizados para modelar a máquina síncrona no estudo de estabilidade do PSP-UFU.</p><p>Os parâmetros da máquina síncrona que influenciam rapidamente no decaimento de valores são chamados <strong>subtransitórios</strong> (indicados por <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow></mrow><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msup></mrow><annotation encoding="application/x-tex">''</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span>), aqueles que influenciam mais lentamente são chamados <strong>transitórios</strong> (indicados por <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span></span>) e, finalmente, aqueles que influenciam continuamente são chamados de parâmetros <strong>síncronos</strong> (sem indicação de sobrescrito).</p><p>Um conjunto de equações algébrico-diferenciais determinam o comportamento da máquina síncrona no estudo de estabilidade:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><msub><mi>V</mi><mi>q</mi></msub><mo>=</mo><msub><mi>r</mi><mi>a</mi></msub><msub><mi>I</mi><mi>q</mi></msub><mo>−</mo><msubsup><mi>x</mi><mrow><mi>d</mi><mi>s</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mi>I</mi><mi>d</mi><mspace linebreak="newline"></mspace><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><msub><mi>V</mi><mi>d</mi></msub><mo>=</mo><msub><mi>r</mi><mi>a</mi></msub><msub><mi>I</mi><mi>d</mi></msub><mo>−</mo><msubsup><mi>x</mi><mrow><mi>q</mi><mi>s</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mi>I</mi><mi>q</mi><mspace linebreak="newline"></mspace><msubsup><mi>E</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>−</mo><msub><mi>V</mi><mi>q</mi></msub><mo>=</mo><msub><mi>r</mi><mi>a</mi></msub><msub><mi>I</mi><mi>q</mi></msub><mo>−</mo><msubsup><mi>x</mi><mrow><mi>d</mi><mi>s</mi></mrow><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mi>I</mi><mi>d</mi><mspace linebreak="newline"></mspace><msubsup><mi>E</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>−</mo><msub><mi>V</mi><mi>d</mi></msub><mo>=</mo><msub><mi>r</mi><mi>a</mi></msub><msub><mi>I</mi><mi>d</mi></msub><mo>−</mo><msubsup><mi>x</mi><mrow><mi>q</mi><mi>s</mi></mrow><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mi>I</mi><mi>q</mi><mspace linebreak="newline"></mspace><mtext> </mtext><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>V</mi><mrow><mi>f</mi><mi>d</mi></mrow></msub><mo>+</mo><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>d</mi></msub><mo>−</mo><msubsup><mi>x</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>d</mi></msub><mo>−</mo><msub><mi>s</mi><mi>d</mi></msub><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><msubsup><mi>T</mi><mrow><mi>d</mi><mn>0</mn></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mfrac><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>−</mo><mrow><mo fence="true">(</mo><msub><mi>x</mi><mi>q</mi></msub><mo>−</mo><msubsup><mi>x</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>q</mi></msub><mo>−</mo><msub><mi>s</mi><mi>q</mi></msub><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><msubsup><mi>T</mi><mrow><mi>q</mi><mn>0</mn></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mfrac><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>s</mi><mi>d</mi></msub><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>+</mo><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><msubsup><mi>x</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>d</mi></msub><mo>−</mo><msub><mi>s</mi><mi>d</mi></msub><msubsup><mi>E</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><msubsup><mi>T</mi><mrow><mi>d</mi><mn>0</mn></mrow><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mfrac><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>s</mi><mi>d</mi></msub><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>+</mo><mrow><mo fence="true">(</mo><msubsup><mi>x</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>−</mo><msubsup><mi>x</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo fence="true">)</mo></mrow><msub><mi>I</mi><mi>q</mi></msub><mo>−</mo><msub><mi>s</mi><mi>d</mi></msub><msubsup><mi>E</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><msubsup><mi>T</mi><mrow><mi>q</mi><mn>0</mn></mrow><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mfrac><mspace linebreak="newline"></mspace><mtext> </mtext><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><mi>ω</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><msub><mi>ω</mi><mi>r</mi></msub><mrow><mn>2</mn><mi>H</mi></mrow></mfrac><mrow><mo fence="true">[</mo><msub><mi>P</mi><mi>m</mi></msub><mo>−</mo><msub><mi>P</mi><mi>e</mi></msub><mo>−</mo><msub><mi>D</mi><mi>a</mi></msub><mrow><mo fence="true">(</mo><mi>ω</mi><mo>−</mo><msub><mi>ω</mi><mi>r</mi></msub><mo fence="true">)</mo></mrow><mo fence="true">]</mo></mrow><mspace linebreak="newline"></mspace><mfrac><mrow><mi>d</mi><mi>δ</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><msub><mi mathvariant="normal">Ω</mi><mi>b</mi></msub><mrow><mo fence="true">(</mo><mi>ω</mi><mo>−</mo><msub><mi>ω</mi><mi>r</mi></msub><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">E_{q}' - V_q = r_aI_q - x_{ds}'Id\\ E_{d}' - V_d = r_aI_d - x_{qs}'Iq\\ E_{q}'' - V_q = r_aI_q - x_{ds}''Id\\ E_{d}'' - V_d = r_aI_d - x_{qs}''Iq\\ ~\\ \frac{dE_{q}'}{dt} = \frac{V_{fd} + \left( x_d - x_{d}' \right)I_d - s_d E_{q}'}{T_{d0}'}\\ \frac{dE_{d}'}{dt} = \frac{- \left( x_q - x_{q}' \right)I_q - s_q E_{d}'}{T_{q0}'}\\ \frac{dE_{q}''}{dt} = \frac{s_dE_{q}' + \left( x_{d}' - x_{d}'' \right)I_d - s_d E_{q}''}{T_{d0}''}\\ \frac{dE_{d}''}{dt} = \frac{s_dE_{d}' + \left( x_{q}' - x_{q}'' \right)I_q - s_d E_{d}''}{T_{q0}''}\\ ~\\ \frac{d\omega}{dt} = \frac{\omega_r}{2H} \left[ P_m - P_e - D_a \left( \omega - \omega_r \right) \right]\\ \frac{d\delta}{dt} = \Omega_b\left( \omega - \omega_r \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.185em;vertical-align:-0.383108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="mord mathdefault">d</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.185em;vertical-align:-0.383108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="mord mathdefault" style="margin-right:0.03588em">q</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.185em;vertical-align:-0.383108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="mord mathdefault">d</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.048892em;vertical-align:-0.247em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.185em;vertical-align:-0.383108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span><span class="mord mathdefault mtight">s</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="mord mathdefault" style="margin-right:0.03588em">q</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0em;vertical-align:0em"></span><span class="mspace nobreak"> </span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.211em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.525em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.512308em;vertical-align:-0.9873080000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.525em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.398692em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.0448em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.30130799999999996em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em">f</span><span class="mord mathdefault mtight">d</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9873080000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.114892em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.7115240000000003em;vertical-align:-1.088416em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6231080000000002em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.4336920000000006em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4024159999999999em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">−</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.088416em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.211em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.525em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.512308em;vertical-align:-0.9873080000000001em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.525em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.398692em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.0448em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.30130799999999996em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9873080000000001em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.114892em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.428892em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.7115240000000003em;vertical-align:-1.088416em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6231080000000002em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.733692em"><span style="top:-2.4336920000000006em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4024159999999999em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.773108em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.07847em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.088416em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0em;vertical-align:0em"></span><span class="mspace nobreak"> </span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1.7935599999999998em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">2</span><span class="mord mathdefault" style="margin-right:0.08125em">H</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">e</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">a</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span><span class="mclose delimcenter" style="top:0em">]</span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault">t</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.03785em">δ</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord"><span class="mord">Ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span></span></div><p>As duas últimas equações diferenciais são as equações mecânicas da máquina; e as demais são equações elétricas (consulte essa <a href="https://dx.doi.org/10.14393/ufu.te.2019.2444">tese</a> para maiores detalhes acerca dos parâmetros dessas equações).</p><p>Utilizando as equações transitórias e subtransitórias podem-se definir cinco modelos de distintas complexidades.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O modelo da máquina síncrona é <strong>selecionado automaticamente</strong> de acordo com os dados fornecidos ao programa.</p></div></div><p>Na sequência são apresentados tais modelos, incluídos os efeitos da saturação magnética, em conjunto com seus diagramas de blocos:</p><ul><li><strong>Modelo 1</strong>: Corresponde a uma tensão constante atrás de uma reatância transitória de eixo direto (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><annotation encoding="application/x-tex">x_{d}'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.035em;vertical-align:-0.2831079999999999em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span></span>), não exigindo equações diferenciais;</li><li><strong>Modelo 2</strong>: São representados os efeitos transitórios de eixo direto, sendo necessária a solução de uma equação diferencial (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}'}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>), cujo diagrama de blocos é apresentado na figura abaixo:</li></ul><div><center><img src="/PSP/images/model2SyncGenerator.svg" alt="Diagrama de blocos do Modelo 2 das máquinas síncronas" title="Diagrama de blocos do Modelo 2 das máquinas síncronas"></center></div><ul><li><strong>Modelo 3</strong>: São representados os efeitos transitórios de eixo direto e em quadratura, exigindo duas equações diferenciais (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}'}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{d}'}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.43468em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.08968em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.5102em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>), cujo diagrama de blocos é apresentado na figura abaixo:</li></ul><div><center><img src="/PSP/images/model3SyncGenerator.svg" alt="Diagrama de blocos do Modelo 3 das máquinas síncronas" title="Diagrama de blocos do Modelo 3 das máquinas síncronas"></center></div><ul><li><strong>Modelo 4</strong>: São representados os efeitos subtransitórios de eixo direto e em quadratura, sendo necessária a solução de três equações diferenciais (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}'}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}''}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{d}''}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.43468em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.08968em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.5102em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>), cujo diagrama de blocos é apresentado na figura abaixo:</li></ul><div><center><img src="/PSP/images/model4SyncGenerator.svg" alt="Diagrama de blocos do Modelo 4 das máquinas síncronas" title="Diagrama de blocos do Modelo 4 das máquinas síncronas"></center></div><ul><li><strong>Modelo 5</strong>: São representados os efeitos subtransitórios de eixo direto e em quadratura, sendo necessária a solução de quatro equações diferenciais (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}'}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{d}'}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.43468em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.08968em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.5102em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>q</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{q}''}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.5319em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1869em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.60742em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.42488571428571426em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>d</mi><msubsup><mi>E</mi><mi>d</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{dE_{d}''}{dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.43468em;vertical-align:-0.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.08968em"><span style="top:-2.6550000000000002em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">t</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.5102em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8278285714285715em"><span style="top:-2.214em;margin-left:-0.05764em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-2.931em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>), cujo diagrama de blocos é apresentado na figura abaixo:</li></ul><div><center><img src="/PSP/images/model5SyncGenerator.svg" alt="Diagrama de blocos do Modelo 4 das máquinas síncronas" title="Diagrama de blocos do Modelo 4 das máquinas síncronas"></center></div><div class="admonition admonition-note alert alert--secondary"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M6.3 5.69a.942.942 0 0 1-.28-.7c0-.28.09-.52.28-.7.19-.18.42-.28.7-.28.28 0 .52.09.7.28.18.19.28.42.28.7 0 .28-.09.52-.28.7a1 1 0 0 1-.7.3c-.28 0-.52-.11-.7-.3zM8 7.99c-.02-.25-.11-.48-.31-.69-.2-.19-.42-.3-.69-.31H6c-.27.02-.48.13-.69.31-.2.2-.3.44-.31.69h1v3c.02.27.11.5.31.69.2.2.42.31.69.31h1c.27 0 .48-.11.69-.31.2-.19.3-.42.31-.69H8V7.98v.01zM7 2.3c-3.14 0-5.7 2.54-5.7 5.68 0 3.14 2.56 5.7 5.7 5.7s5.7-2.55 5.7-5.7c0-3.15-2.56-5.69-5.7-5.69v.01zM7 .98c3.86 0 7 3.14 7 7s-3.14 7-7 7-7-3.12-7-7 3.14-7 7-7z"></path></svg></span>Nota</h5></div><div class="admonition-content"><p>Em todos os modelos as equações diferenciais mecânicas são solucionadas.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="saturação"></a>Saturação<a aria-hidden="true" tabindex="-1" class="hash-link" href="#saturação" title="Direct link to heading">#</a></h4><p>Para representar matematicamente o efeito da saturação nas equações das máquinas síncronas são introduzidos “fatores de saturação” que modificam as impedâncias do circuito equivalente, os quais dependem de uma reatância de dispersão efetiva, chamada de reatância de Potier (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>p</mi></msub></mrow><annotation encoding="application/x-tex">x_p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>).</p><p>Tal reatância pode ser obtida por meio de ensaios (utilizando curvas de saturação de circuito aberto e carga de fator de potência zero) ou estimadas de forma aproximada por outros parâmetros da máquina. A reatância de dispersão (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>l</mi></msub></mrow><annotation encoding="application/x-tex">x_l</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>), aqui substituída de forma aproximada por <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>p</mi></msub></mrow><annotation encoding="application/x-tex">x_p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>, representa a parcela da reatância da máquina originados do fluxo magnético que percorrem o ar na maioria de seu caminho e, portanto, é independente da saturação.</p><p>O método implementado no programa permite reproduzir a saturação em ambos os eixos (direto e em quadratura), diferindo entre si devido à divergência no tamanho do entreferro. É assumido que a soma vetorial das duas componentes do fluxo magnético saturado está em fase com a f.m.m. e proporcional à Tensão de Potier (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>E</mi><mi>p</mi></msub></mrow><annotation encoding="application/x-tex">E_p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>, a qual é a tensão atrás da reatância de Potier).</p><p>Para isso, são utilizados internamente dois fatores de saturação, sendo um no eixo direto (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>d</mi></msub></mrow><annotation encoding="application/x-tex">s_d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>) e outro no eixo em quadratura (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>q</mi></msub></mrow><annotation encoding="application/x-tex">s_q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span></span>). Esses fatores de saturação são <strong>automaticamente calculados</strong> a cada passo de integração e dependem da curva de saturação da máquina definida pelo <a href="/PSP/docs/syncGenerator#fator-de-satura%C3%A7%C3%A3o">fator de saturação inserido no formulário de edição de dados</a>.</p><p>Portanto as reatâncias saturadas, que devem ser inseridas nas equações algébricas da máquina, são definidas pelas seguintes equações:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mrow><mi>d</mi><mi>s</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mi>d</mi></msub><mo>−</mo><msub><mi>x</mi><mi>p</mi></msub></mrow><mrow><msub><mi>s</mi><mi>d</mi></msub><mo>+</mo><msub><mi>x</mi><mi>p</mi></msub></mrow></mfrac><mspace linebreak="newline"></mspace><msub><mi>x</mi><mrow><mi>q</mi><mi>s</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>x</mi><mi>q</mi></msub><mo>−</mo><msub><mi>x</mi><mi>p</mi></msub></mrow><mrow><msub><mi>s</mi><mi>q</mi></msub><mo>+</mo><msub><mi>x</mi><mi>p</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">x_{ds}=\frac{x_d-x_p}{s_d +x_p}\\ x_{qs}=\frac{x_q-x_p}{s_q +x_p}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.232438em;vertical-align:-0.972108em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.26033em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.972108em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span><span class="mord mathdefault mtight">s</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.232438em;vertical-align:-0.972108em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.26033em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em">q</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.972108em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Essas equações também são utilizadas para as reatâncias transitórias e subtransitórias, visto que o valor da reatância de Potier (ou de dispersão) não é alterada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="barramento-infinito"></a>Barramento infinito<a aria-hidden="true" tabindex="-1" class="hash-link" href="#barramento-infinito" title="Direct link to heading">#</a></h4><p>Algumas referências incluem um modelo sem equações diferenciais, em que a máquina é somente representada por uma tensão constante atrás de uma reatância transitória de eixo direto. Tal é utilizado na representação de um <strong>barramento infinito</strong>, o qual é normalmente constituído de um subsistema muito maior àquele simulado.</p><p>No PSP-UFU a <strong>representação de um barramento infinito</strong> pode ser obtido por meio da utilização de uma máquina representada pelo Modelo 1 cujo valor da constante de inércia (H) é infinito ou muito grande (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>9999</mn><mtext> </mtext><mi>s</mi></mrow><annotation encoding="application/x-tex">9999~s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mord">9</span><span class="mspace nobreak"> </span><span class="mord mathdefault">s</span></span></span></span></span>, por exemplo) em relação às demais máquinas do sistema, e o valor de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mi>d</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow><annotation encoding="application/x-tex">x_{d}'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.035em;vertical-align:-0.2831079999999999em"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2831079999999999em"><span></span></span></span></span></span></span></span></span></span></span> deve ser um valor muito pequeno (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">10^{-3}~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.008548em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span>, por exemplo).</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="centro-de-inércia"></a>Centro de inércia<a aria-hidden="true" tabindex="-1" class="hash-link" href="#centro-de-inércia" title="Direct link to heading">#</a></h4><p>Normalmente utiliza-se a velocidade de referência como sendo a síncrona e portanto, nesse caso, <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ω</mi><mi>r</mi></msub><mo>=</mo><msub><mi>ω</mi><mi>b</mi></msub><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mtext> </mtext><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">\omega_r = \omega_b = 1,0~p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord">0</span><span class="mspace nobreak"> </span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> Essa abordagem, adotada por vários livros de estabilidade, considera como referência uma máquina fictícia girando sempre na velocidade síncrona independente das perturbações aplicadas no sistema. No PSP-UFU foi implementado o conceito de centro de inércia (COI, do inglês, <em>Center of Inertia</em>), que constitui uma soma ponderada das velocidades das máquinas presentes no sistema:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ω</mi><mi>r</mi></msub><mo>=</mo><mfrac><mrow><mo fence="true">(</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>H</mi><mi>i</mi></msub><msub><mi>ω</mi><mi>i</mi></msub><mo fence="true">)</mo></mrow><mrow><mo fence="true">(</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>H</mi><mi>i</mi></msub><mo fence="true">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\omega_r=\frac{\left( \sum_{i=1}^{n} H_i \omega_i \right)}{\left( \sum_{i=1}^{n} H_i \right)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.488004em;vertical-align:-0.994002em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.494002em"><span style="top:-2.305708em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.08125em">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6897100000000003em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.08125em">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.994002em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>Em que: <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">n</span></span></span></span></span> é o número de máquinas síncronas <strong>conectadas</strong> no sistema.</p><p>A aplicação do COI resulta em dados de saída, como o ângulo do rotor, mais fáceis de serem analisados. Na implementação realizada no programa a utilização ou não desse recurso é opcional e pode ser <a href="/PSP/docs/simulationConfig">definida pelo usuário</a>.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="formulário-de-edição-dos-geradores-síncronos"></a>Formulário de edição dos geradores síncronos<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-dos-geradores-síncronos" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos geradores síncronos:</p><div><center><img src="/PSP/images/syncGeneratorForm.png" alt="Formulário dos geradores síncronos no PSP-UFU" title="Formulário dos geradores síncronos no PSP-UFU"></center></div><p>Um segundo pelos dados de estabilidade, como mostra a figura abaixo, acessado ao clicar no botão "Estabilidade" do formulário principal. Nele é possível também acessar aos controles das máquinas síncronas manipulados pelo <a href="/PSP/docs/controlEditor">editor de controle</a>.</p><div><center><img src="/PSP/images/syncGeneratorStabForm.png" alt="Formulário de estabilidade dos geradores síncronos no PSP-UFU" title="Formulário de estabilidade dos geradores síncronos no PSP-UFU"></center></div><p>No formulário de estabilidade pode ser observado o botão "Chaveamento" na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do gerador durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/images/syncGeneratorSw.png" alt="Formulário de chaveamento do gerador síncrono" title="Formulário de chaveamento do gerador síncrono"></center></div><div><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_3zTA tabs__item--active">Geral</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_3zTA">Falta</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_3zTA">Botão Estabilidade</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_3zTA">Botão Chaveamento</li></ul><div role="tabpanel" class="margin-vert--md"><div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="nome"></a>Nome<a aria-hidden="true" tabindex="-1" class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="potência-nominal"></a>Potência nominal<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potência-nominal" title="Direct link to heading">#</a></h4><p>Potência nominal do gerador, inserida em MVA, kVA ou VA.</p><p>Esse campo é especialmente importante caso a opção "Utilizar a potência nominal como base" esteja marcada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="potências-ativa-e-reativa"></a>Potências ativa e reativa<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potências-ativa-e-reativa" title="Direct link to heading">#</a></h4><p>Potências ativa (inserida em W, kW, MW ou p.u.) e reativa (inserida em var, kvar, Mvar ou p.u.) do gerador.</p><p>Caso a barra conectada seja PV o valor de potência reativa será ignorado e caso seja de referência ambos os valores inseridos serão desprezados.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Caso mais de um gerador esteja conectado na mesma barra, os valores de potência reativa (nas barras de referência e PV) e ativa (nas barras de referência) são igualmente distribuídas, respeitando os limites individuais de potência reativa.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="potências-reativas-máxima-e-mínima"></a>Potências reativas máxima e mínima<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potências-reativas-máxima-e-mínima" title="Direct link to heading">#</a></h4><p>Limites de potência reativa máxima e mínima do gerador para controle de tensão em barras PV. Caso esses valores sejam ultrapassados, o reativo gerado pela unidade será limitado ao valor inserido e a barra conectada será transformada em PQ, não controlando a tensão estabelecida.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="utilizar-potência-nominal-como-base"></a>Utilizar potência nominal como base<a aria-hidden="true" tabindex="-1" class="hash-link" href="#utilizar-potência-nominal-como-base" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar a potência nominal do gerador como base para a conversão das unidades, inclusive aqueles no formulário de estabilidade, caso contrário será usada a <a href="/PSP/docs/simulationConfig">potência base do sistema</a>.</p></div></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="acesso-aos-controles-da-máquina-síncrona"></a>Acesso aos controles da máquina síncrona<a aria-hidden="true" tabindex="-1" class="hash-link" href="#acesso-aos-controles-da-máquina-síncrona" title="Direct link to heading">#</a></h2><p>Como já <a href="/PSP/docs/syncGenerator#utilizar-avr-e-regulador-de-velocidade">mencionado anteriormente</a>, os reguladores de velocidade e tensão da máquina síncrona podem ser acionados ou inibidos por meio das caixas de seleção "<a href="/PSP/docs/syncGenerator#utilizar-avr-e-regulador-de-velocidade">Utilizar AVR e regulador de velocidade</a>". Ambas as opções irão acessar o <a href="/PSP/docs/controlEditor">editor de controles</a>.</p><p>O acesso aos controles do <strong>AVR</strong> poderão então ser criados e manipulados ao clicar no botão "Editar AVR", assim como o <strong>Regulador de Velocidade</strong> é acessado no botão "Editar regulador de velocidade".</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>No PSP-UFU a opção de editar o <strong>AVR</strong> engloba mais que somente o controle de tensão da máquina. Nele <strong>deve</strong> ser inserida a malha de controle da máquina assim como a <strong>excitatriz da máquina síncrona</strong>. Outras estratégias de controle (opcionais), como PSS (<em>Power System Stabilizer</em>) e/ou controles de sobre e sub excitação, são também implementadas em conjunto.</p></div></div><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Assim como no AVR, o <strong>Regulador de Velocidade</strong> engloba mais que a regulação primária da máquina. Nessa opção <strong>deve</strong> ser inserida ao menos a malha de controle da regulação primária de velocidade, assim como o <strong>modelo da turbina</strong>. Estratégias opcionais de controle da velocidade também são inseridas nessa opção.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="referências"></a>Referências<a aria-hidden="true" tabindex="-1" class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. doi: <a href="https://doi.org/10.1007/978-3-642-13669-6" target="_blank" rel="noopener noreferrer">https://doi.org/10.1007/978-3-642-13669-6</a></li><li>ARRILLAGA, J.; WATSON, N. R. Computer Modelling of Electrical Power Systems. Wiley & Sons, New York, 2001. doi: <a href="https://doi.org/10.1002/9781118878286" target="_blank" rel="noopener noreferrer">https://doi.org/10.1002/9781118878286</a></li><li>KUNDUR, P. Power System Stability and Control. McGraw-Hill, New York, 1994.</li><li>DOMMEL, H. W.; SATO, N. Fast Transient Stability Solutions. IEEE Transactions on Power Aparatus and Systems, v. PAS-91, n. 4, jul 1972, p. 1643-1650. doi: <a href="https://doi.org/10.1109/TPAS.1972.293341">https://doi.org/10.1109/TPAS.1972.293341</a></li><li>IEEE Std 1110-2002 IEEE Guide for Synchronous Generator Modeling Practices and Applications in Power System Stability Analyses. IEEE, New York, nov. 2003. doi: <a href="https://doi.org/10.1109/IEEESTD.2003.94408">https://doi.org/10.1109/IEEESTD.2003.94408</a></li><li>KIMBARK, E. W. Power System Stability: Volume III – Synchronous Machine. New York: Wiley-IEEE Press, 1995.</li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/syncGenerator.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/harmSource"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Fonte de Corrente Harmônica</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/syncMotor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Compensador Síncrono »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_TbNY"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#gerador-síncrono-no-psp-ufu" class="table-of-contents__link">Gerador Síncrono no PSP-UFU</a><ul><li><a href="#gerador-síncrono-no-fluxo-de-carga" class="table-of-contents__link">Gerador Síncrono no fluxo de carga</a></li><li><a href="#gerador-síncrono-no-estudo-de-curto-circuito" class="table-of-contents__link">Gerador Síncrono no estudo de curto-circuito</a></li><li><a href="#gerador-síncrono-no-estudo-de-estabilidade" class="table-of-contents__link">Gerador Síncrono no estudo de estabilidade</a></li></ul></li><li><a href="#formulário-de-edição-dos-geradores-síncronos" class="table-of-contents__link">Formulário de edição dos geradores síncronos</a></li><li><a href="#acesso-aos-controles-da-máquina-síncrona" class="table-of-contents__link">Acesso aos controles da máquina síncrona</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_FFX1"><div class="container padding-vert--lg docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Compensador Síncrono</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Uma máquina síncrona operando sem carga mecânica e fornecendo ou absorvendo energia reativa. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=411-34-03" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="compensador-síncrono-no-psp-ufu"></a>Compensador Síncrono no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#compensador-síncrono-no-psp-ufu" title="Direct link to heading">#</a></h2><p>O compensador síncrono nada mais é que um motor síncrono operando sem carga em seu eixo com o intuito de fornecer ou absorver potência reativa na rede. Em relação ao <a href="/PSP/docs/powerFlow">fluxo de carga</a>, no PSP-UFU esse elemento tem comportamento idêntico ao <a href="/PSP/docs/syncGenerator">gerador síncrono</a>, porém a sua potência ativa é inserida nos algoritmos de solução com sinal negativo.</p><p>Sua parametrização para o estudo de <a href="/PSP/docs/powerFlow">fluxo de carga</a>, <a href="/PSP/docs/fault">falta</a> e <a href="/PSP/docs/harmonics">harmônicos</a> é idêntica ao <a href="/PSP/docs/syncGenerator">gerador síncrono</a>.</p><div class="admonition admonition-warning alert alert--danger"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M5.05.31c.81 2.17.41 3.38-.52 4.31C3.55 5.67 1.98 6.45.9 7.98c-1.45 2.05-1.7 6.53 3.53 7.7-2.2-1.16-2.67-4.52-.3-6.61-.61 2.03.53 3.33 1.94 2.86 1.39-.47 2.3.53 2.27 1.67-.02.78-.31 1.44-1.13 1.81 3.42-.59 4.78-3.42 4.78-5.56 0-2.84-2.53-3.22-1.25-5.61-1.52.13-2.03 1.13-1.89 2.75.09 1.08-1.02 1.8-1.86 1.33-.67-.41-.66-1.19-.06-1.78C8.18 5.31 8.68 2.45 5.05.32L5.03.3l.02.01z"></path></svg></span>Cuidado!</h5></div><div class="admonition-content"><p>O compensador síncrono <strong>não foi implementado no estudo de <a href="/PSP/docs/stability">estabilidade</a></strong> e sua presença no circuito pode causar erros durante os cálculos.</p><p>Não inclua esse elemento para os estudos de <a href="/PSP/docs/stability">estabilidade</a>. Versões futuras do PSP-UFU irão contemplar esse elemento para esse estudo.</p></div></div><div class="admonition admonition-tip alert alert--success"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="12" height="16" viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>Dica</h5></div><div class="admonition-content"><p>É possível utilizar um <a href="/PSP/docs/syncGenerator">gerador síncrono</a> com potência ativa <em>nula</em> ou <em>negativa</em> para representar o compensador síncrono nos estudos de <a href="/PSP/docs/stability">estabilidade</a>.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="formulário-de-edição-dos-compensadores-síncronos"></a>Formulário de edição dos compensadores síncronos<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-dos-compensadores-síncronos" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos compensadores síncronos:</p><div><center><img src="/PSP/images/syncMotorForm.png" alt="Formulário dos compensadores síncronos no PSP-UFU" title="Formulário dos compensadores síncronos no PSP-UFU"></center></div><div><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_3zTA tabs__item--active">Geral</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_3zTA">Falta</li></ul><div role="tabpanel" class="margin-vert--md"><div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="nome"></a>Nome<a aria-hidden="true" tabindex="-1" class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="potência-nominal"></a>Potência nominal<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potência-nominal" title="Direct link to heading">#</a></h4><p>Potência nominal do compensador, inserida em MVA, kVA ou VA.</p><p>Esse campo é especialmente importante caso a opção "Utilizar a potência nominal como base" esteja marcada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="potências-ativa-e-reativa"></a>Potências ativa e reativa<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potências-ativa-e-reativa" title="Direct link to heading">#</a></h4><p>Potências ativa (inserida em W, kW, MW ou p.u.) e reativa (inserida em var, kvar, Mvar ou p.u.) do compensador.</p><p>Caso a barra conectada seja PV o valor de potência reativa será ignorado e caso seja de referência ambos os valores inseridos serão desprezados.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Caso mais de um compensador esteja conectado na mesma barra, os valores de potência reativa (nas barras de referência e PV) e ativa (nas barras de referência) são igualmente distribuídas, respeitando os limites individuais de potência reativa.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="potências-reativas-máxima-e-mínima"></a>Potências reativas máxima e mínima<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potências-reativas-máxima-e-mínima" title="Direct link to heading">#</a></h4><p>Limites de potência reativa máxima e mínima do compensador para controle de tensão em barras PV. Caso esses valores sejam ultrapassados, o reativo gerado pela unidade será limitado ao valor inserido e a barra conectada será transformada em PQ, não controlando a tensão estabelecida.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="utilizar-potência-nominal-como-base"></a>Utilizar potência nominal como base<a aria-hidden="true" tabindex="-1" class="hash-link" href="#utilizar-potência-nominal-como-base" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar a potência nominal do compensador como base para a conversão das unidades, inclusive aqueles no formulário de estabilidade, caso contrário será usada a <a href="/PSP/docs/simulationConfig">potência base do sistema</a>.</p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/syncMotor.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/syncGenerator"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Gerador Síncrono</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/indMotor"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Motor de Indução »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_TbNY"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#compensador-síncrono-no-psp-ufu" class="table-of-contents__link">Compensador Síncrono no PSP-UFU</a></li><li><a href="#formulário-de-edição-dos-compensadores-síncronos" class="table-of-contents__link">Formulário de edição dos compensadores síncronos</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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href="/PSP/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulações</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_FFX1"><div class="container padding-vert--lg docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Função Transferência</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><p>A Função Transferência é um importante bloco elementar e essencial em um sistema de controle, visto que esse componente possibilita a representação de sistemas dinâmicos por meio de equações algébricas no domínio da frequência.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>Nesses blocos é utilizado a letra “<strong>s</strong>” para representação no domínio da frequência, pois essa é a terminologia mais utilizada em sistemas de controle.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="função-transferência-no-psp-ufu"></a>Função Transferência no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#função-transferência-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Após a inserção da função transferência, o programa a transforma na representação em espaço de estado, visto que sua solução é sistemática tornando-a, assim, mais eficiente computacionalmente.</p><p>Dada uma função transferência genérica abaixo:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi>Y</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>U</mi><mo stretchy="false">(</mo><mi mathvariant="bold">s</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>b</mi><mn>0</mn></msub><msup><mi mathvariant="bold">s</mi><mi>k</mi></msup><mo>+</mo><msub><mi>b</mi><mn>1</mn></msub><msup><mi mathvariant="bold">s</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>b</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mi mathvariant="bold">s</mi><mo>+</mo><msub><mi>b</mi><mi>k</mi></msub></mrow><mrow><msup><mi mathvariant="bold">s</mi><mi>k</mi></msup><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><msup><mi mathvariant="bold">s</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>a</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mi mathvariant="bold">s</mi><mo>+</mo><msub><mi>a</mi><mi>k</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{Y(\bold{s})}{U(\bold{s})} = \frac{b_0 \bold{s}^k + b_1 \bold{s}^{k-1} + \cdots + b_{k-1} \bold{s} + b_k}{\bold{s}^k + a_1 \bold{s}^{k-1} + \cdots + a_{k-1} \bold{s} + a_k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em">U</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span></span><span class="mclose">)</span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">s</span></span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.420439em;vertical-align:-0.894331em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.526108em"><span style="top:-2.3139999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7751079999999999em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7751079999999999em"><span style="top:-2.989em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">s</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">s</span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.894331em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></div><p>A transformação em espaço de estado na <em>forma canônica controlável</em> ficará da seguinte forma:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="bold">s</mi><msub><mi>x</mi><mn>1</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="bold">s</mi><msub><mi>x</mi><mn>2</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="bold">s</mi><msub><mi>x</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="bold">s</mi><msub><mi>x</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>a</mi><mi>k</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>a</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>a</mi><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msub><mi>a</mi><mn>1</mn></msub></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>1</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>2</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mi>k</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>+</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><msub><mi>u</mi><mi>n</mi></msub><mspace linebreak="newline"></mspace><msub><mi>y</mi><mi>n</mi></msub><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo fence="true">(</mo><msub><mi>b</mi><mi>n</mi></msub><mo>−</mo><msub><mi>a</mi><mi>n</mi></msub><msub><mi>b</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo fence="true">(</mo><msub><mi>b</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>b</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo fence="true">(</mo><msub><mi>b</mi><mn>1</mn></msub><mo>−</mo><msub><mi>a</mi><mn>1</mn></msub><msub><mi>b</mi><mn>0</mn></msub><mo fence="true">)</mo></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>1</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>2</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mi>k</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>+</mo><msub><mi>b</mi><mn>0</mn></msub><msub><mi>u</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\begin{bmatrix} \bold{s}x_1\\ \bold{s}x_2\\ \vdots\\ \bold{s}x_{k-1}\\ \bold{s}x_k \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 & \cdots & 0\\ 0 & 0 & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & 1\\ -a_k & -a_{k-1} & -a_{k-2} & \cdots & -a_1 \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_{k-1}\\ x_k \end{bmatrix} + \begin{bmatrix} 0\\ 0\\ \vdots\\ 0\\ 1 \end{bmatrix} u_n\\ y_n = \begin{bmatrix} \left( b_n - a_n b_0 \right) & \left( b_{n-1} - a_{n-1} b_0 \right) & \cdots & \left( b_1 - a_1 b_0 \right) \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_k \end{bmatrix} + b_0 u_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6.66em;vertical-align:-3.079999999999999em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathbf">s</span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:6.66em;vertical-align:-3.079999999999999em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.240000000000001em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-5.04em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-3.18em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:-1.9800000000000009em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-0.7800000000000009em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:6.66em;vertical-align:-3.079999999999999em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.3675em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-2.167500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-0.9675000000000009em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.079999999999999em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.555985em"><span style="top:-0.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-1.8999750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.4959750000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.0919750000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6879750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.283975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.314965000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.555985000000001em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.050045em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:5.459999999999999em;vertical-align:-2.4799999999999995em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em">(</span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size1">]</span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953995em"><span style="top:-1.3499850000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.4999850000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.0959850000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.6919850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.712975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.953995em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em"><span style="top:-5.8275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-4.6275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.7674999999999996em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em"></span></span></span></span><span style="top:-1.5675000000000006em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953995em"><span style="top:-1.3499850000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.4999850000000006em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.0959850000000007em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.6919850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.712975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.953995em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></div><p>A equação diferencial matricial é resolvida pelo método <strong>Trapezoidal Implícito</strong>. Um processo iterativo é realizado até que o erro entre os cálculos se torne menor que uma tolerância pré-estipulada e caso as iterações se tornem excessivas, ultrapassando um número máximo definido pelo usuário, o processo é interrompido com erro. Com os valores do vetor de estado (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em"></span><span class="mord mathdefault">x</span></span></span></span></span>) calculados obtém-se o valor da saída do bloco (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">y_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>).</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="formulário-de-edição-de-dados-da-função-transferência"></a>Formulário de edição de dados da Função Transferência<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-de-dados-da-função-transferência" title="Direct link to heading">#</a></h2><p>A figura abaixo apresenta o formulário de edição de dados da função transferência.</p><div><center><img src="/PSP/images/transferFunctionForm.png" alt="Formulário de edição de dados da função transferência no PSP-UFU" title="Formulário de edição de dados da função transferência no PSP-UFU"></center></div><p>A função transferência é definida pelos coeficientes do numerador e denominador, <strong>separados por espaço</strong>.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>Os elementos não presentes são representados como coeficientes de valor zero.</p><p>Por exemplo, um parâmetro inserido igual a “<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>5</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{matrix} 1 & 0{,}5 & 0 & 2 \end{matrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2000000000000002em;vertical-align:-0.35000000000000003em"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">5</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8500000000000001em"><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.35000000000000003em"><span></span></span></span></span></span></span></span></span></span></span></span>” irá gerar: “<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>s</mi><mn>3</mn></msup><mo>+</mo><mn>0</mn><mo separator="true" lspace="0em" rspace="0em">,</mo><mn>5</mn><msup><mi>s</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">s^3+0{,}5s^2+2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.897438em;vertical-align:-0.08333em"></span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1.008548em;vertical-align:-0.19444em"></span><span class="mord">0</span><span class="mord"><span class="mpunct">,</span></span><span class="mord">5</span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em"><span style="top:-3.063em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">2</span></span></span></span></span>”.</p></div></div><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="referências"></a>Referências<a aria-hidden="true" tabindex="-1" class="hash-link" href="#referências" title="Direct link to heading">#</a></h2><ol><li>OGATA, K. Modern Control Engineering. Prentice Hall Inc., New Jersey, 2004.</li></ol></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/transferFunction.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog list page navigation"><div class="pagination-nav__item"><a class="pagination-nav__link" href="/PSP/docs/io"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">« Entrada / Saída</div></a></div><div class="pagination-nav__item pagination-nav__item--next"><a class="pagination-nav__link" href="/PSP/docs/sum"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Somador »</div></a></div></nav></div></div></div><div class="col col--3"><div class="tableOfContents_TbNY"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#função-transferência-no-psp-ufu" class="table-of-contents__link">Função Transferência no PSP-UFU</a></li><li><a href="#formulário-de-edição-de-dados-da-função-transferência" class="table-of-contents__link">Formulário de edição de dados da Função Transferência</a></li><li><a href="#referências" class="table-of-contents__link">Referências</a></li></ul></div></div></div></div></main></div></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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taxa</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/const">Constante</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/gain">Ganho</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/mathExpression">Expressão Matemática</a></li></ul></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Simulações</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/powerFlow">Fluxo de Potência</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/fault">Curto-Circuito</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/harmonics">Harmônicos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/stability">Estabilidade</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/simulationConfig">Configurações da Simulação</a></li></ul></li><li class="menu__list-item menu__list-item--collapsed"><a class="menu__link menu__link--sublist" href="#!">Visualização dos Dados</a><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/text">Texto Vinculado</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/tabularReport">Relatórios Tabulares</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/graphViewer">Visualizador de Gráficos</a></li><li class="menu__list-item"><a class="menu__link" tabindex="-1" href="/PSP/docs/heatmap">Mapa de Tensão</a></li></ul></li></ul></div></div></div><main class="docMainContainer_FFX1"><div class="container padding-vert--lg docItemWrapper_1cc7"><div class="row"><div class="col docItemCol_2GOA"><div class="docItemContainer_2cwg"><article><header><h1 class="docTitle_1vWb">Transformador</h1></header><div class="markdown"><link rel="stylesheet" href="/PSP/katex/katex.min.css"><blockquote><p>Aparelho estático com dois ou mais enrolamentos que, por indução eletromagnética, transforma um sistema de tensão e corrente alternada em outro sistema de tensão e corrente geralmente de valores diferentes e na mesma frequência com o objetivo de transmitir energia elétrica. <a href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=421-01-01" target="_blank" rel="noopener noreferrer"><em>tradução livre</em> - IEC 60050</a>.</p></blockquote><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="transformador-no-psp-ufu"></a>Transformador no PSP-UFU<a aria-hidden="true" tabindex="-1" class="hash-link" href="#transformador-no-psp-ufu" title="Direct link to heading">#</a></h2><p>Transformadores com quaisquer conexões e defasagens entre as tensões primárias e secundárias podem ser inseridos no PSP-UFU. Para isso, os transformadores devem ser modelados representando seu tape (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em"></span><span class="mord mathdefault">t</span></span></span></span></span>) e defasagem (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">ϕ</span></span></span></span></span>), utilizando um transformador ideal de relação de transformação <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover><mo>:</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\overline{t}:1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.81508em;vertical-align:0em"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em"></span><span class="mord">1</span></span></span></span></span> em série com sua impedância (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><mo>=</mo><msub><mi>r</mi><mi>T</mi></msub><mo>+</mo><mi>j</mi><msub><mi>x</mi><mi>T</mi></msub></mrow><annotation encoding="application/x-tex">\overline{y}_T = r_T + jx_T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8747em;vertical-align:-0.24414em"></span><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>), em que <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{t}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.81508em;vertical-align:0em"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span></span></span> é um número complexo (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mi mathvariant="normal">∠</mi><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">t\angle\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">t</span><span class="mord">∠</span><span class="mord mathdefault">ϕ</span></span></span></span></span>)</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção</h5></div><div class="admonition-content"><p>O modelo utilizado no PSP-UFU <strong>não considera</strong> o ramo magnetizante ou impedâncias mútuas entre as fases.</p></div></div><p>A figura abaixo mostra o modelo do transformador implementado no PSP-UFU:</p><div><center><img src="/PSP/images/transformerModel.svg" alt="Modelo PI da linha" title="Modelo PI da linha"></center></div><p>Em que:</p><ul><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">r</mi><mi mathvariant="bold">T</mi></msub></mrow><annotation encoding="application/x-tex">\bold{r_T}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.59444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord"><span class="mord mathbf">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33027699999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></span> é a resistência total do transformador;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold">x</mi><mi mathvariant="bold">T</mi></msub></mrow><annotation encoding="application/x-tex">\bold{x_T}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.59444em;vertical-align:-0.15em"></span><span class="mord"><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33027699999999993em"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathbf mtight">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></span> é a reatância indutiva total do transformador;</li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi mathvariant="bold">t</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\bold{\overline{t}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8349200000000001em;vertical-align:0em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8349200000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathbf">t</span></span></span><span style="top:-3.7549200000000003em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span></span></span></span> é a o tape complexo do transformador.</li></ul><p>A relação entre as tensões e correntes das barras genéricas <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em"></span><span class="mord mathdefault">i</span></span></span></span></span> e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em"></span><span class="mord mathdefault" style="margin-right:0.05724em">j</span></span></span></span></span> é obtida pela equação matricial:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>i</mi><mi>j</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>I</mi><mo>˙</mo></mover><mrow><mi>j</mi><mi>i</mi></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><msup><mi>t</mi><mn>2</mn></msup></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><msup><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover><mo>∗</mo></msup></mfrac></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover></mfrac></mstyle></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>×</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>i</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mi>j</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} \dot{I}_{ij}\\ \dot{I}_{ji} \end{bmatrix} = \begin{bmatrix} \displaystyle \frac{\overline{y}_T}{t^2} & -\displaystyle \frac{\overline{y}_T}{\overline{t}^*}\\ -\displaystyle \frac{\overline{y}_T}{\overline{t}} & \overline{y}_T \end{bmatrix} \times \begin{bmatrix} \dot{V}_{i}\\ \dot{V}_{j} \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.5603800000000003em;vertical-align:-1.0301900000000004em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.53019em"><span style="top:-3.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-2.3298099999999997em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07847em">I</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.027780000000000013em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.0301900000000004em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:4.202030000000001em;vertical-align:-1.850025em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3019380000000007em"><span style="top:-4.301938em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.2107020000000004em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.29492em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.70508em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.8019379999999998em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3019380000000007em"><span style="top:-4.301938em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.216324em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.893676em"><span style="top:-3.26798em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.783676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.2107020000000004em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.8019379999999998em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.5603800000000003em;vertical-align:-1.0301900000000004em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.53019em"><span style="top:-3.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.3298099999999997em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9201900000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">V</span></span></span><span style="top:-3.25233em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-0.13889em"><span class="mord">˙</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.0301900000000004em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">]</span></span></span></span></span></span></span></div><p>Assim como nas <a href="/PSP/docs/line">linhas</a>, o transformador de dois enrolamentos deve ser inserido entre barras, porém não há a necessidade das tensões nominais desses barramentos serem idênticas.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>A primeira barra selecionada será o lado <strong>primário</strong>, o qual é indicado por um círculo. Para alterar o lado primário após a inserção basta desconectar os nós dos transformadores e reconectá-los alternando as barras, utilizando as <a href="/PSP/docs/cadTools#conex%C3%A3o-e-desconex%C3%A3o-de-elemetos-na-barra">ferramentas CAD</a>.</p></div></div><h3><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="transformadores-no-estudo-de-curto-circuito"></a>Transformadores no estudo de curto-circuito<a aria-hidden="true" tabindex="-1" class="hash-link" href="#transformadores-no-estudo-de-curto-circuito" title="Direct link to heading">#</a></h3><p>Da mesma forma que no estudo de fluxo de carga, a representação dos elementos do sistema para o estudo de curto-circuito é realizada por meio de circuitos equivalentes inseridos na matriz admitância de barras. Nas <a href="/PSP/docs/fault">faltas desbalanceadas</a> é necessário formar três matrizes admitância de sequência: positiva, negativa e zero.</p><p>A matriz admitância de sequência positiva é construída da mesma forma que os estudos de fluxo de carga. Na sequência negativa o ângulo de defasagem entre tensões primária e secundária (<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em"></span><span class="mord mathdefault">ϕ</span></span></span></span></span>) deve ser invertido, uma vez que os fasores da sequência negativa <strong>são deslocados na direção oposta</strong>. Com isso, a matriz que representa as admitâncias da equação anterior deve ser substituída pela seguinte expressão:</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>Y</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>i</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>Y</mi><mo stretchy="true">‾</mo></mover><mrow><mi>i</mi><mi>j</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>Y</mi><mo stretchy="true">‾</mo></mover><mrow><mi>j</mi><mi>i</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>Y</mi><mo stretchy="true">‾</mo></mover><mrow><mi>j</mi><mi>j</mi></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><msup><mi>t</mi><mn>2</mn></msup></mfrac></mstyle></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover></mfrac></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub><msup><mover accent="true"><mi>t</mi><mo stretchy="true">‾</mo></mover><mo>∗</mo></msup></mfrac></mstyle></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mover accent="true"><mi>y</mi><mo stretchy="true">‾</mo></mover><mi>T</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} \overline{Y}_{ii} & \overline{Y}_{ij}\\ \overline{Y}_{ji} & \overline{Y}_{jj} \end{bmatrix} = \begin{bmatrix} \displaystyle \frac{\overline{y}_T}{t^2} & -\displaystyle \frac{\overline{y}_T}{\overline{t}}\\ -\displaystyle \frac{\overline{y}_T}{\overline{t}^*} & \overline{y}_T \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4866600000000005em;vertical-align:-0.9933300000000005em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.49333em"><span style="top:-3.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833300000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span><span style="top:-3.80333em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.3666699999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833300000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span><span style="top:-3.80333em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9933300000000005em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.49333em"><span style="top:-3.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833300000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span><span style="top:-3.80333em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span><span style="top:-2.3666699999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833300000000001em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span><span style="top:-3.80333em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span><span class="mord mathdefault mtight" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9933300000000005em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:4.202030000000001em;vertical-align:-1.850025em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3019380000000003em"><span style="top:-4.301938em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.314em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em"><span style="top:-2.9890000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.2892979999999996em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.216324em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.893676em"><span style="top:-3.26798em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.783676em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.8019380000000003em"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em"></span><span class="arraycolsep" style="width:0.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3019380000000003em"><span style="top:-4.301938em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.30756em"><span style="top:-2.29492em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.81508em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault">t</span></span></span><span style="top:-3.73508em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6769999999999996em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.70508em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.2892979999999996em"><span class="pstrut" style="height:3.30756em"></span><span class="mord"><span class="mord"><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em">y</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em"><span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.23419099999999995em"><span style="top:-2.4558600000000004em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em">T</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24414em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.8019380000000003em"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.352005em"><span style="top:-1.9499950000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-3.0999950000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.1109850000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.352005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em"><span></span></span></span></span></span></span></span></span></span></span></span></div><p>A impedância de sequência negativa nos transformadores deve ser tratada de maneira específica devido aos diferentes tipos de conexão. Na figura abaixo são mostrados os circuitos equivalentes para cada tipo de conexão de transformadores de dois enrolamentos. As setas indicam os caminhos possíveis para circulação da corrente de sequência zero.</p><div><center><img src="/PSP/images/seqZeroTrafo.svg" alt="Circuitos equivalentes de sequência zero dos transformadores" title="Circuitos equivalentes de sequência zero dos transformadores"></center></div><p>Para os transformadores com conexão estrela aterrado, se a ligação do neutro para a terra apresentar uma impedância <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\overline{z}_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78056em;vertical-align:-0.15em"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>, o circuito equivalente de sequência zero deve ter impedância de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><msub><mover accent="true"><mi>z</mi><mo stretchy="true">‾</mo></mover><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">3\overline{z}_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79444em;vertical-align:-0.15em"></span><span class="mord">3</span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.63056em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em">z</span></span></span><span style="top:-3.55056em"><span class="pstrut" style="height:3em"></span><span class="overline-line" style="border-bottom-width:0.04em"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em"><span style="top:-2.5500000000000003em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> em série com a resistência e reatância equivalentes do transformador.</p><h2><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="formulário-de-edição-dos-transformadores"></a>Formulário de edição dos transformadores<a aria-hidden="true" tabindex="-1" class="hash-link" href="#formulário-de-edição-dos-transformadores" title="Direct link to heading">#</a></h2><p>A imagem abaixo apresenta o formulário de inserção/alteração de dados dos transformadores:</p><div><center><img src="/PSP/images/trafoForm.png" alt="Formulário dos transformadores no PSP-UFU" title="Formulário dos transformadores no PSP-UFU"></center></div><p>Esse formulário é subdividido em dois contextos distintos:</p><ul><li><strong>Geral</strong>: no qual são inseridas informações gerais dos transformadores, informações do fluxo de carga, sua conexão e defasagem;</li><li><strong>Falta</strong>: local onde as impedâncias de sequência zero e impedâncias de aterramento são inseridas.</li></ul><p>Além desses dois contextos, pode ser observado o botão "Estabilidade" na parte inferior esquerda do formulário. Esse formulário, comum a vários outros elementos, permite a inserção e/ou remoção do transformador durante o estudo de <a href="/PSP/docs/stability">estabilidade</a>.</p><div><center><img src="/PSP/images/trafoSw.png" alt="Formulário de chaveamento do transformador" title="Formulário de chaveamento do transformador"></center></div><div><ul role="tablist" aria-orientation="horizontal" class="tabs"><li role="tab" tabindex="0" aria-selected="true" class="tabs__item tabItem_3zTA tabs__item--active">Geral</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_3zTA">Falta</li><li role="tab" tabindex="0" aria-selected="false" class="tabs__item tabItem_3zTA">Botão Estabilidade</li></ul><div role="tabpanel" class="margin-vert--md"><div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="nome"></a>Nome<a aria-hidden="true" tabindex="-1" class="hash-link" href="#nome" title="Direct link to heading">#</a></h4><p>Identificação do elemento elétrico. Podem ser inseridos quaisquer números de caracteres no padrão <a href="https://pt.wikipedia.org/wiki/Unicode" target="_blank" rel="noopener noreferrer">Unicode</a>.</p><p>Todos os componentes de potência do PSP-UFU possuem esse campo.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="tensão-nominal"></a>Tensão nominal<a aria-hidden="true" tabindex="-1" class="hash-link" href="#tensão-nominal" title="Direct link to heading">#</a></h4><p>Campo de informação <em>não editável</em> que apresenta a tensão primária e secundária do transformador. Para alterar esse campo é necessário editar o campo correspondente dos <a href="/PSP/docs/bus#geral">barramento</a> conectados.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="tensão-base"></a>Tensão base<a aria-hidden="true" tabindex="-1" class="hash-link" href="#tensão-base" title="Direct link to heading">#</a></h4><p>Tensão utilizada para converter os parâmetros de impedância inseridos em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Ω</span></span></span></span></span> para <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> As tensões selecionadas por uma caixa de escolha são editadas nos formulários dos <a href="/PSP/docs/bus#geral">barramentos</a> conectados.</p><div class="admonition admonition-info alert alert--info"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="14" height="16" viewBox="0 0 14 16"><path fill-rule="evenodd" d="M7 2.3c3.14 0 5.7 2.56 5.7 5.7s-2.56 5.7-5.7 5.7A5.71 5.71 0 0 1 1.3 8c0-3.14 2.56-5.7 5.7-5.7zM7 1C3.14 1 0 4.14 0 8s3.14 7 7 7 7-3.14 7-7-3.14-7-7-7zm1 3H6v5h2V4zm0 6H6v2h2v-2z"></path></svg></span>Informação</h5></div><div class="admonition-content"><p>A tensão base é ignorada caso os dados inseridos estejam em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="potência-nominal"></a>Potência nominal<a aria-hidden="true" tabindex="-1" class="hash-link" href="#potência-nominal" title="Direct link to heading">#</a></h4><p>Potência nominal do transformador, inserida em MVA, kVA ou VA.</p><p>Esse campo é especialmente importante caso a opção "Utilizar a potência nominal como base" esteja marcada.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="impedância-do-transformador"></a>Impedância do transformador<a aria-hidden="true" tabindex="-1" class="hash-link" href="#impedância-do-transformador" title="Direct link to heading">#</a></h4><p>Impedância série do transformador (resistência e reatância indutiva), inserida em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span> ou <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Ω</span></span></span></span></span>. Caso as impedâncias sejam inseridas em <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Ω</span></span></span></span></span> será utilizada a tensão base selecionada para conversão para <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mi mathvariant="normal">.</mi><mi>u</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">p.u.</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em"></span><span class="mord mathdefault">p</span><span class="mord">.</span><span class="mord mathdefault">u</span><span class="mord">.</span></span></span></span></span></p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="conexão"></a>Conexão<a aria-hidden="true" tabindex="-1" class="hash-link" href="#conexão" title="Direct link to heading">#</a></h4><p>Conexão do transformador utilizada para cálculo das correntes de sequência zero nas faltas desbalanceadas com presença de terra (fase-terra e fase-fase-terra).</p><p>As seguintes conexões estão disponíveis:</p><ol><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub><mo>−</mo><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Y_{aterrado}-Y_{aterrado}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>−</mo><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Y-Y_{aterrado}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub><mo>−</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y_{aterrado}-Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>−</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y-Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mo>−</mo><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta-Y_{aterrado}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord">Δ</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mo>−</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\Delta-Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord">Δ</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mrow><mi>a</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>r</mi><mi>a</mi><mi>d</mi><mi>o</mi></mrow></msub><mo>−</mo><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">Y_{aterrado}-\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em"><span style="top:-2.5500000000000003em;margin-left:-0.22222em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight" style="margin-right:0.02778em">r</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight">o</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>−</mo><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">Y-\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span></span></span></span></span></li><li><span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mo>−</mo><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">\Delta-\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord">Δ</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span></span></span></span></span></li></ol><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="tap"></a>TAP<a aria-hidden="true" tabindex="-1" class="hash-link" href="#tap" title="Direct link to heading">#</a></h4><p>Tape do transformador <strong>em relação ao primário</strong>.</p><div class="admonition admonition-caution alert alert--warning"><div class="admonition-heading"><h5><span class="admonition-icon"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg></span>Atenção!</h5></div><div class="admonition-content"><p>O tape no PSP-UFU deve ser inserido pela relação entre a tensão primária nominal e a tensão de tape <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mtext>TAP</mtext><mo>=</mo><mfrac><msub><mi>V</mi><mrow><mi>n</mi><mi>o</mi><mi>m</mi><mi>i</mi><mi>n</mi><mi>a</mi><mi>l</mi></mrow></msub><msub><mi>V</mi><mrow><mi>t</mi><mi>a</mi><mi>p</mi><mi>e</mi></mrow></msub></mfrac><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\left(\text{TAP} = \frac{V_{nominal}}{V_{tape}}\right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em"></span><span class="minner"><span class="mopen delimcenter" style="top:0em"><span class="delimsizing size2">(</span></span><span class="mord text"><span class="mord">TAP</span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.894191em"><span style="top:-2.655em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.29634285714285713em"><span style="top:-2.357em;margin-left:-0.22222em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">t</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">p</span><span class="mord mathdefault mtight">e</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.41586em"><span class="pstrut" style="height:3em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.22222em">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.3487714285714287em;margin-left:-0.22222em;margin-right:0.07142857142857144em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight" style="margin-right:0.01968em">l</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5423199999999999em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em"><span class="delimsizing size2">)</span></span></span></span></span></span></span>.</p></div></div><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="defasagem"></a>Defasagem<a aria-hidden="true" tabindex="-1" class="hash-link" href="#defasagem" title="Direct link to heading">#</a></h4><p>Defasagem do transformador em relação ao primário, em graus. Esse valor independe do tipo de conexão, portanto para conexões diferentes de <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>−</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y-Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord mathdefault" style="margin-right:0.22222em">Y</span></span></span></span></span>(aterrado ou não) e <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mo>−</mo><mi mathvariant="normal">Δ</mi></mrow><annotation encoding="application/x-tex">\Delta-\Delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em"></span><span class="mord">Δ</span><span class="mspace" style="margin-right:0.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em"></span><span class="mord">Δ</span></span></span></span></span>, deve-se inserir o ângulo de defasagem correto.</p><h4><a aria-hidden="true" tabindex="-1" class="anchor enhancedAnchor_ZqCz" id="utilizar-potência-nominal-como-base"></a>Utilizar potência nominal como base<a aria-hidden="true" tabindex="-1" class="hash-link" href="#utilizar-potência-nominal-como-base" title="Direct link to heading">#</a></h4><p>Caso essa opção seja marcada, o programa irá utilizar a potência nominal do transformador como base para a conversão das unidades, caso contrário será usada a <a href="/PSP/docs/simulationConfig">potência base do sistema</a>.</p></div></div></div></div></article><div class="margin-vert--xl"><div class="row"><div class="col"><a href="https://github.com/Thales1330/PSP/tree/master/docusaurus/docs/transformer.md" target="_blank" rel="noreferrer noopener"><svg fill="currentColor" height="1.2em" width="1.2em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 40 40" style="margin-right:0.3em;vertical-align:sub"><g><path d="m34.5 11.7l-3 3.1-6.3-6.3 3.1-3q0.5-0.5 1.2-0.5t1.1 0.5l3.9 3.9q0.5 0.4 0.5 1.1t-0.5 1.2z m-29.5 17.1l18.4-18.5 6.3 6.3-18.4 18.4h-6.3v-6.2z"></path></g></svg>Edit this page</a></div></div></div><div class="margin-vert--lg"><nav class="pagination-nav" aria-label="Blog 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style="font-style:normal;font-weight:normal;font-size:40px;line-height:0;font-family:sans-serif;fill:#ff2600;fill-opacity:1;stroke:none" + x="432.72137" + y="-28.084957" + id="text10489-5-0"><tspan + id="tspan8831" + sodipodi:role="line" + x="432.72137" + y="-28.084957" + style="font-size:21.3333px;line-height:1.05;text-align:end;text-anchor:end;fill:#ff2600;fill-opacity:1">Calcula o THD em todas as barras causadas</tspan><tspan + id="tspan8843" + sodipodi:role="line" + x="432.72137" + y="-5.6849937" + style="font-size:21.3333px;line-height:1.05;text-align:end;text-anchor:end;fill:#ff2600;fill-opacity:1">pelas fontes de correntes harmônicas</tspan></text> + </g> +</svg> diff --git a/docs/index.html b/docs/index.html index 2d921c6..7e2a5e0 100644 --- a/docs/index.html +++ b/docs/index.html @@ -6,8 +6,8 @@ <meta name="generator" content="Docusaurus v2.0.0-alpha.58"> <link rel="search" type="application/opensearchdescription+xml" title="PSP-UFU" href="/PSP/opensearch.xml"><title data-react-helmet="true">PSP-UFU | PSP-UFU</title><meta data-react-helmet="true" property="og:title" content="PSP-UFU | PSP-UFU"><meta data-react-helmet="true" name="description" content="Power System Platform"><meta data-react-helmet="true" property="og:description" content="Power System Platform"><meta data-react-helmet="true" name="twitter:card" content="summary_large_image"><link data-react-helmet="true" rel="shortcut icon" href="/PSP/img/favicon.ico"><link rel="stylesheet" href="/PSP/styles.68586bc0.css"> <link rel="preload" href="/PSP/styles.26e2fe91.js" as="script"> -<link rel="preload" href="/PSP/runtime~main.3ede30cf.js" as="script"> -<link rel="preload" href="/PSP/main.1c1af766.js" as="script"> +<link rel="preload" href="/PSP/runtime~main.57a83c0d.js" as="script"> +<link rel="preload" href="/PSP/main.1dbec03f.js" as="script"> <link rel="preload" href="/PSP/1.837a9375.js" as="script"> <link rel="preload" href="/PSP/2.19116c85.js" as="script"> <link rel="preload" href="/PSP/ece76bd1.472a390a.js" as="script"> @@ -16,8 +16,8 @@ <script>!function(){function t(t){document.documentElement.setAttribute("data-theme",t)}var e=window.matchMedia("(prefers-color-scheme: dark)"),n=function(){var t=null;try{t=localStorage.getItem("theme")}catch(t){}return t}();null!==n?t(n):e.matches&&t("dark")}()</script><div id="__docusaurus"> <nav class="navbar navbar--light navbar--fixed-top"><div class="navbar__inner"><div class="navbar__items"><div aria-label="Navigation bar toggle" class="navbar__toggle" role="button" tabindex="0"><svg xmlns="http://www.w3.org/2000/svg" width="30" height="30" viewBox="0 0 30 30" role="img" focusable="false"><title>Menu</title><path stroke="currentColor" stroke-linecap="round" stroke-miterlimit="10" stroke-width="2" d="M4 7h22M4 15h22M4 23h22"></path></svg></div><a class="navbar__brand" href="/PSP/"><img class="navbar__logo" src="/PSP/img/logo.svg" alt="PSP-UFU logo"><strong class="navbar__title">PSP-UFU</strong></a><a class="navbar__item navbar__link" href="/PSP/docs/">User Guide</a><a class="navbar__item navbar__link" href="/PSP/blog">Changelog</a></div><div class="navbar__items navbar__items--right"><a href="https://github.com/Thales1330/PSP" target="_blank" rel="noopener noreferrer" class="navbar__item navbar__link">PSP-UFU Source Code</a><div class="react-toggle react-toggle--disabled displayOnlyInLargeViewport_1gtM"><div class="react-toggle-track"><div class="react-toggle-track-check"><span class="toggle_keGJ moon_1gwN"></span></div><div class="react-toggle-track-x"><span class="toggle_keGJ sun_3CPA"></span></div></div><div class="react-toggle-thumb"></div><input type="checkbox" disabled="" aria-label="Dark mode toggle" class="react-toggle-screenreader-only"></div><div class="navbar__search"><div class="searchWrapper_-F3-"><span aria-label="expand searchbar" role="button" class="searchIconButton_1Dnz" tabindex="0"></span><input type="search" id="search_input_react" placeholder="Search" aria-label="Search" class="navbar__search-input searchInput_2eto"></div></div></div></div><div role="presentation" class="navbar-sidebar__backdrop"></div><div class="navbar-sidebar"><div class="navbar-sidebar__brand"><a class="navbar__brand" href="/PSP/"><img class="navbar__logo" src="/PSP/img/logo.svg" alt="PSP-UFU logo"><strong class="navbar__title">PSP-UFU</strong></a></div><div class="navbar-sidebar__items"><div class="menu"><ul class="menu__list"><li class="menu__list-item"><a class="menu__link" href="/PSP/docs/">User Guide</a></li><li class="menu__list-item"><a class="menu__link" href="/PSP/blog">Changelog</a></li><li class="menu__list-item"><a href="https://github.com/Thales1330/PSP" target="_blank" rel="noopener noreferrer" class="menu__link">PSP-UFU Source Code</a></li></ul></div></div></div></nav><div class="main-wrapper"><header class="hero hero--primary heroBanner_2Ftp"><div class="container"><h1 class="hero__title">PSP-UFU</h1><p class="hero__subtitle">Power System Platform of Federal University of Uberlândia</p><p><center><iframe src="https://ghbtns.com/github-btn.html?user=thales1330&repo=psp&type=star&count=true&size=large" frameborder="0" scrolling="0" width="130" height="30" title="GitHub"></iframe></center></p><div class="buttons_1Wc3"><a href="https://github.com/Thales1330/PSP/releases/latest" target="_blank" rel="noopener noreferrer" class="button button--outline button--secondary button--lg" to="https://github.com/Thales1330/PSP/releases/latest">Download</a></div><div class="buttons_1Wc3"><a class="button button--outline button--secondary button--lg" href="/PSP/docs/">User Guide</a></div><div class="buttons_1Wc3"><a href="https://minhaskamal.github.io/DownGit/#/home?url=https://github.com/Thales1330/PSP/tree/master/Project/data/samples" target="_blank" rel="noopener noreferrer" class="button button--outline button--secondary button--lg" to="https://minhaskamal.github.io/DownGit/#/home?url=https://github.com/Thales1330/PSP/tree/master/Project/data/samples">Samples</a></div></div></header><main><section class="features_P2SU"><div class="container"><div class="row"><div class="col col--4"><div class="text--center"><img class="featureImage_3Xqx" src="/PSP/img/complete-platform.svg" alt="[object Object]"></div><h3>Complete Platform</h3><p>PSP-UFU is a <b>cross-platform</b>, <b>multilingual</b>, <b>Free and Open-Source Software (FOSS)</b> with <b>advanced GUI (Graphical User Interface)</b> features and <b>CAD (Computer-Aided Design) tools</b> for electrical power system studies.</p></div><div class="col col--4"><div class="text--center"><img class="featureImage_3Xqx" src="/PSP/img/advanced-cad.svg" alt="[object Object]"></div><h3>Advanced CAD Features</h3><p>The software allows the construction of any electric <b>transmission network</b> and <b>control systems</b> through the deployment of visual elements.</p></div><div class="col col--4"><div class="text--center"><img class="featureImage_3Xqx" src="/PSP/img/visualization.svg" alt="[object Object]"></div><h3>Easy Visualization</h3><p>For the visualization of results, the program offers linked text elements in the main screen, voltage heatmap display, and also table and graph editors.</p></div><div class="col col--4"><div class="text--center"><img class="featureImage_3Xqx" src="/PSP/img/application.svg" alt="[object Object]"></div><h3>Application</h3><p>The PSP-UFU aims to provide efficient computer simulation tools for <b>research</b> and <b>education</b> purposes, in addition to <b>industrial applications</b> in electrical power systems.</p></div></div></div></section><div class="container"><h2>Studies Performed</h2></div><section class="features_P2SU"><div class="container"><div class="row"><div class="col col--4"><h3>Power Flow</h3><ul><li>Newton-Raphson</li><li>Gauss-Seidel</li><li>Hybrid Newton-Gauss</li><li>Three-phase induction motors included</li></ul></div><div class="col col--4"><h3>Short-Circuit calculation</h3><ul><li>Balanced</li><li>Unbalanced</li><li>Short-Circuit power in all system buses</li></ul></div><div class="col col--4"><h3>Harmonics</h3><ul><li>Harmonic voltages and THD (Total Harmonic Distortion) calculation</li><li>Frequency scan</li></ul></div><div class="col col--4"><h3>Transient and Dynamic Stability</h3><ul><li>Several synchronous machine models automatically selected</li><li>Three-phase induction motors</li><li>User-defined machine controls, exciters and prime moves created using block diagrams (Exciters, AVR, PSS, Hydro and Thermal turbines, Speed Governor, etc.)</li></ul></div></div></div></section><div class="container"><h2>Published Papers</h2>Further details can be found in the published papers:<blockquote cite="https://doi.org/10.1002/2050-7038.12185">Oliveira, T. L., Guimarães, G. C., & Silva, L. R. C. (2019). PSP-UFU: An open-source, graphical, and multiplatform software for power system studies. <i>International Transactions on Electrical Energy Systems</i>, e12185. doi: <a href="https://doi.org/10.1002/2050-7038.12185">10.1002/2050-7038.12185</a></blockquote><blockquote cite="https://doi.org/10.1177/0020720919879058">Oliveira, T. L., Guimarães, G. C., Silva, L. R., & Rezende, J. O. (2019). Power system education and research applications using free and open-source, graphical and multiplatform PSP-UFU software. <i>The International Journal of Electrical Engineering & Education</i>, 0020720919879058. doi: <a href="https://doi.org/10.1177/0020720919879058">10.1177/0020720919879058</a></blockquote></div><div class="container"><h2>Code Documentation</h2><p>All detailed descriptions of the source-code can be found at <b><a href="https://thales1330.github.io/PSP/doxygen/html/index.html">Online Documentation</a></b>, generated by <a href="http://www.doxygen.org/">Doxygen</a>.</p></div><div class="container"><h2>Screenshots</h2><center><iframe src="gallery.html" style="position:relative;width:100%;height:56.25vw;display:block;overflow:hidden;border:none">Your browser doesn’t support the this feature.</iframe></center></div></main></div><footer class="footer footer--dark"><div class="container"><div class="row footer__links"><div class="col footer__col"><h4 class="footer__title">Guide</h4><ul class="footer__items"><li class="footer__item"><a class="footer__link-item" href="/PSP/docs/">PSP-UFU Guide</a></li><li class="footer__item"><a href="https://thales1330.github.io/PSP/doxygen/html/index.html" target="_blank" rel="noopener noreferrer" class="footer__link-item" to="https://thales1330.github.io/PSP/doxygen/html/index.html">Code Documentation</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Community</h4><ul class="footer__items"><li class="footer__item"><a href="https://github.com/Thales1330/PSP/issues" target="_blank" rel="noopener noreferrer" class="footer__link-item">Ask a question</a></li><li class="footer__item"><a href="https://twitter.com/PspUfu" target="_blank" rel="noopener noreferrer" class="footer__link-item">Twitter</a></li></ul></div><div class="col footer__col"><h4 class="footer__title">Site Image Credit</h4><ul class="footer__items"><li class="footer__item"><a href="http://www.freepik.com/" target="_blank" rel="noopener noreferrer" class="footer__link-item">Images designed by upklyak / slidesgo / macrovector / Freepik</a></li></ul></div></div><div class="text--center"><div>Copyright © 2020 Thales Lima Oliveira. 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t={pattern:/{\$(?:{(?:{[^{}]+}|[^{}]+)}|[^{}])+}|(^|[^\\{])\$+(?:\w+(?:\[.+?]|->\w+)*)/,lookbehind:!0,inside:{rest:e.languages.php}};e.languages.insertBefore("php","string",{"nowdoc-string":{pattern:/<<<'([^']+)'(?:\r\n?|\n)(?:.*(?:\r\n?|\n))*?\1;/,greedy:!0,alias:"string",inside:{delimiter:{pattern:/^<<<'[^']+'|[a-z_]\w*;$/i,alias:"symbol",inside:{punctuation:/^<<<'?|[';]$/}}}},"heredoc-string":{pattern:/<<<(?:"([^"]+)"(?:\r\n?|\n)(?:.*(?:\r\n?|\n))*?\1;|([a-z_]\w*)(?:\r\n?|\n)(?:.*(?:\r\n?|\n))*?\2;)/i,greedy:!0,alias:"string",inside:{delimiter:{pattern:/^<<<(?:"[^"]+"|[a-z_]\w*)|[a-z_]\w*;$/i,alias:"symbol",inside:{punctuation:/^<<<"?|[";]$/}},interpolation:t}},"single-quoted-string":{pattern:/'(?:\\[\s\S]|[^\\'])*'/,greedy:!0,alias:"string"},"double-quoted-string":{pattern:/"(?:\\[\s\S]|[^\\"])*"/,greedy:!0,alias:"string",inside:{interpolation:t}}}),delete 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t={pattern:/{\$(?:{(?:{[^{}]+}|[^{}]+)}|[^{}])+}|(^|[^\\{])\$+(?:\w+(?:\[.+?]|->\w+)*)/,lookbehind:!0,inside:{rest:e.languages.php}};e.languages.insertBefore("php","string",{"nowdoc-string":{pattern:/<<<'([^']+)'(?:\r\n?|\n)(?:.*(?:\r\n?|\n))*?\1;/,greedy:!0,alias:"string",inside:{delimiter:{pattern:/^<<<'[^']+'|[a-z_]\w*;$/i,alias:"symbol",inside:{punctuation:/^<<<'?|[';]$/}}}},"heredoc-string":{pattern:/<<<(?:"([^"]+)"(?:\r\n?|\n)(?:.*(?:\r\n?|\n))*?\1;|([a-z_]\w*)(?:\r\n?|\n)(?:.*(?:\r\n?|\n))*?\2;)/i,greedy:!0,alias:"string",inside:{delimiter:{pattern:/^<<<(?:"[^"]+"|[a-z_]\w*)|[a-z_]\w*;$/i,alias:"symbol",inside:{punctuation:/^<<<"?|[";]$/}},interpolation:t}},"single-quoted-string":{pattern:/'(?:\\[\s\S]|[^\\'])*'/,greedy:!0,alias:"string"},"double-quoted-string":{pattern:/"(?:\\[\s\S]|[^\\"])*"/,greedy:!0,alias:"string",inside:{interpolation:t}}}),delete 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