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<html xmlns="http://www.w3.org/1999/xhtml"><head><meta charset="utf-8"><meta name="generator" content="pdf2htmlEX"><meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"><link rel="stylesheet" href="https://csdnimg.cn/release/download_crawler_static/css/base.min.css"><link rel="stylesheet" href="https://csdnimg.cn/release/download_crawler_static/css/fancy.min.css"><link rel="stylesheet" href="https://csdnimg.cn/release/download_crawler_static/85190235/raw.css"><script src="https://csdnimg.cn/release/download_crawler_static/js/compatibility.min.js"></script><script src="https://csdnimg.cn/release/download_crawler_static/js/pdf2htmlEX.min.js"></script><script>try{pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({});}catch(e){}</script><title></title></head><body><div id="sidebar" style="display: none"><div id="outline"></div></div><div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://csdnimg.cn/release/download_crawler_static/85190235/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">IEEE<span class="_"> </span>TRANSA<span class="_ _0"></span>CTIONS<span class="_"> </span>ON<span class="_"> </span>POWER<span class="_"> </span>SYSTEMS,<span class="_"> </span>V<span class="_ _0"></span>OL.<span class="_"> </span>32,<span class="_"> </span>NO.<span class="_"> </span>1,<span class="_"> </span>J<span class="_ _1"></span>ANU<span class="_ _0"></span>AR<span class="_ _1"></span>Y<span class="_"> </span>2017<span class="_ _2"> </span><span class="ls1">39</span></div><div class="t m0 x2 h3 y2 ff1 fs1 fc0 sc0 ls2 ws0">A<span class="_"> </span>Distrib<span class="_ _1"></span>utionally<span class="_"> </span>Rob<span class="_ _1"></span>ust<span class="_"> </span>Optimization<span class="_"> </span>Model</div><div class="t m0 x3 h3 y3 ff1 fs1 fc0 sc0 ls2 ws0">for<span class="_"> </span>Unit<span class="_"> </span>Commitment<span class="_"> </span>Considering<span class="_"> </span>Uncertain</div><div class="t m0 x4 h3 y4 ff1 fs1 fc0 sc0 ls3 ws0">W<span class="_ _3"></span>ind<span class="_"> </span>Po<span class="_ _1"></span>wer<span class="_"> </span>Generation</div><div class="t m0 x5 h4 y5 ff1 fs2 fc0 sc0 ls4 ws0">Peng<span class="_"> </span>Xiong,<span class="_"> </span>Panida<span class="_"> </span>Jirutitijaroen<span class="ff2 ls3">,<span class="_"> </span>Senior<span class="_"> </span>Member<span class="_ _4"></span>,<span class="_"> </span>IEEE<span class="ff1 ls5">,<span class="_"> </span>and<span class="_"> </span>Chanan<span class="_"> </span>Singh</span><span class="ls4">,<span class="_"> </span>F<span class="_ _3"></span>ellow<span class="_ _1"></span>,<span class="_"> </span>IEEE</span></span></div><div class="t m0 x6 h5 y6 ff3 fs3 fc0 sc0 ls6 ws0">Abstract<span class="ff4">—This<span class="_ _5"> </span>paper<span class="_ _5"> </span>proposes<span class="_ _5"> </span>a<span class="_ _5"> </span>distrib<span class="_ _1"></span>utionally<span class="_ _5"> </span>robust<span class="_ _5"> </span>opti-</span></div><div class="t m0 x1 h5 y7 ff4 fs3 fc0 sc0 ls3 ws0">mization<span class="_ _6"> </span>model<span class="_ _6"> </span>for<span class="_"> </span>solving<span class="_ _6"> </span>unit<span class="_ _6"> </span>commitment<span class="_ _6"> </span>(UC)<span class="_ _6"> </span>problems<span class="_ _6"> </span>con-</div><div class="t m0 x1 h5 y8 ff4 fs3 fc0 sc0 ls3 ws0">sidering<span class="_ _6"> </span>volatile<span class="_"> </span>wind<span class="_ _6"> </span>power<span class="_ _6"> </span>generation.<span class="_ _6"> </span>The<span class="_ _6"> </span>uncertainty<span class="_ _6"> </span>of<span class="_"> </span>wind</div><div class="t m0 x1 h5 y9 ff4 fs3 fc0 sc0 ls3 ws0">power<span class="_ _5"> </span>is<span class="_ _5"> </span>captured<span class="_ _5"> </span>by<span class="_ _5"> </span>an<span class="_ _5"> </span>ambiguity<span class="_ _5"> </span>set<span class="_ _5"> </span>that<span class="_ _5"> </span>deﬁnes<span class="_ _5"> </span>a<span class="_ _5"> </span>family<span class="_ _5"> </span>of</div><div class="t m0 x1 h5 ya ff4 fs3 fc0 sc0 ls3 ws0">wind<span class="_ _7"> </span>power<span class="_ _7"> </span>distrib<span class="_ _0"></span>utions,<span class="_ _7"> </span>and<span class="_ _7"> </span>the<span class="_ _7"> </span>expected<span class="_ _7"> </span>total<span class="_ _7"> </span>cost<span class="_ _7"> </span>under<span class="_ _7"> </span>the</div><div class="t m0 x1 h5 yb ff4 fs3 fc0 sc0 ls6 ws0">worst-case<span class="_ _7"> </span>distribution<span class="_ _5"> </span>is<span class="_ _5"> </span>minimized.<span class="_ _5"> </span>Compared<span class="_ _7"> </span>with<span class="_ _5"> </span>stochastic</div><div class="t m0 x1 h5 yc ff4 fs3 fc0 sc0 ls3 ws0">programming,<span class="_"> </span>this<span class="_ _6"> </span>method<span class="_"> </span>may<span class="_ _6"> </span>ha<span class="_ _1"></span>ve<span class="_ _6"> </span>less<span class="_"> </span>dependence<span class="_ _6"> </span>on<span class="_"> </span>the<span class="_ _6"> </span>data</div><div class="t m0 x1 h5 yd ff4 fs3 fc0 sc0 ls3 ws0">of<span class="_ _7"> </span>exact<span class="_ _7"> </span>probability<span class="_ _7"> </span>distrib<span class="_ _0"></span>utions.<span class="_ _7"> </span>It<span class="_ _7"> </span>should<span class="_ _7"> </span>also<span class="_ _7"> </span>outperform<span class="_ _7"> </span>the</div><div class="t m0 x1 h5 ye ff4 fs3 fc0 sc0 ls3 ws0">con<span class="_ _1"></span>ventional<span class="_ _6"> </span>rob<span class="_ _1"></span>ust<span class="_ _6"> </span>optimization<span class="_"> </span>methods<span class="_ _6"> </span>because<span class="_ _6"> </span>some<span class="_"> </span>distribu-</div><div class="t m0 x1 h5 yf ff4 fs3 fc0 sc0 ls3 ws0">tion<span class="_ _7"> </span>information<span class="_ _7"> </span>can<span class="_ _7"> </span>be<span class="_ _7"> </span>incorporated<span class="_ _7"> </span>into<span class="_ _7"> </span>the<span class="_ _5"> </span>ambiguity<span class="_ _7"> </span>sets<span class="_ _7"> </span>to</div><div class="t m0 x1 h5 y10 ff4 fs3 fc0 sc0 ls6 ws0">generate<span class="_ _8"> </span>less<span class="_ _8"> </span>conservati<span class="_ _0"></span>ve<span class="_ _6"> </span>results.<span class="_ _8"> </span>In<span class="_ _6"> </span>this<span class="_ _8"> </span>paper<span class="_ _1"></span>,<span class="_ _6"> </span>the<span class="_ _8"> </span>UC<span class="_ _8"> </span>model<span class="_ _8"> </span>is</div><div class="t m0 x1 h5 y11 ff4 fs3 fc0 sc0 ls3 ws0">formulated<span class="_ _6"> </span>based<span class="_ _6"> </span>on<span class="_ _8"> </span>the<span class="_ _6"> </span>typical<span class="_ _8"> </span>two-stage<span class="_ _6"> </span>framework,<span class="_ _8"> </span>where<span class="_ _6"> </span>the</div><div class="t m0 x1 h5 y12 ff4 fs3 fc0 sc0 ls7 ws0">UC<span class="_ _8"> </span>decisions<span class="_ _8"> </span>are<span class="_ _6"> </span>determined<span class="_ _8"> </span>in<span class="_ _8"> </span>a<span class="_ _8"> </span><span class="ff3 ls6">here-and-now<span class="_ _6"> </span></span><span class="ls3">manner<span class="_ _1"></span>,<span class="_ _6"> </span>and<span class="_ _8"> </span>the</span></div><div class="t m0 x1 h5 y13 ff4 fs3 fc0 sc0 ls3 ws0">economic<span class="_"> </span>dispatch<span class="_"> </span>decisions<span class="_ _9"> </span>are<span class="_ _9"> </span>assumed<span class="_"> </span>to<span class="_"> </span>be<span class="_ _9"> </span><span class="ff3 ls6">wait-and-see</span><span class="ls8">,m<span class="_ _a"></span>a<span class="_ _a"></span>d<span class="_ _a"></span>e</span></div><div class="t m0 x1 h5 y14 ff4 fs3 fc0 sc0 ls3 ws0">after<span class="_"> </span>the<span class="_"> </span>observation<span class="_"> </span>of<span class="_"> </span>wind<span class="_ _6"> </span>power<span class="_"> </span>outcomes.<span class="_"> </span>For<span class="_"> </span>computational</div><div class="t m0 x1 h5 y15 ff4 fs3 fc0 sc0 ls3 ws0">tractability<span class="_ _1"></span>,<span class="_"> </span>the<span class="_"> </span>wait-and-see<span class="_"> </span>decisions<span class="_"> </span>ar<span class="_ _1"></span>e<span class="_"> </span>addressed<span class="_"> </span>by<span class="_"> </span>linear<span class="_"> </span>de-</div><div class="t m0 x1 h5 y16 ff4 fs3 fc0 sc0 ls3 ws0">cision<span class="_ _7"> </span>rule<span class="_ _7"> </span>appr<span class="_ _0"></span>oximation,<span class="_ _7"> </span>assuming<span class="_ _7"> </span>that<span class="_ _7"> </span>the<span class="_ _7"> </span>economic<span class="_ _8"> </span>dispatch</div><div class="t m0 x1 h5 y17 ff4 fs3 fc0 sc0 ls7 ws0">decisions<span class="_ _9"> </span>afﬁnely<span class="_ _9"> </span>depend<span class="_ _9"> </span>on<span class="_ _9"> </span>uncertain<span class="_ _9"> </span>parameters<span class="_ _9"> </span>as<span class="_ _9"> </span>well<span class="_ _9"> </span>as<span class="_ _9"> </span>auxil-</div><div class="t m0 x1 h5 y18 ff4 fs3 fc0 sc0 ls3 ws0">iary<span class="_"> </span>random<span class="_ _6"> </span>variables<span class="_"> </span>introduced<span class="_"> </span>to<span class="_ _6"> </span>describe<span class="_"> </span>distributional<span class="_"> </span>char-</div><div class="t m0 x1 h5 y19 ff4 fs3 fc0 sc0 ls3 ws0">acteristics<span class="_ _7"> </span>of<span class="_ _7"> </span>wind<span class="_ _8"> </span>power<span class="_ _7"> </span>generation.<span class="_ _7"> </span>It<span class="_ _7"> </span>is<span class="_ _8"> </span>shown<span class="_ _7"> </span>in<span class="_ _7"> </span>case<span class="_ _8"> </span>studies</div><div class="t m0 x1 h5 y1a ff4 fs3 fc0 sc0 ls3 ws0">that<span class="_ _6"> </span>this<span class="_ _6"> </span>decision<span class="_ _8"> </span>rule<span class="_ _6"> </span>model<span class="_ _8"> </span>tends<span class="_ _6"> </span>to<span class="_ _8"> </span>provide<span class="_ _6"> </span>a<span class="_ _6"> </span>tight<span class="_ _6"> </span>approxima-</div><div class="t m0 x1 h5 y1b ff4 fs3 fc0 sc0 ls3 ws0">tion<span class="_"> </span>to<span class="_ _9"> </span>the<span class="_ _9"> </span>original<span class="_"> </span>tw<span class="_ _0"></span>o-stage<span class="_"> </span>pr<span class="_ _1"></span>oblem,<span class="_"> </span>and<span class="_ _9"> </span>the<span class="_"> </span>perf<span class="_ _1"></span>ormance<span class="_"> </span>of<span class="_"> </span>UC</div><div class="t m0 x1 h5 y1c ff4 fs3 fc0 sc0 ls3 ws0">solutions<span class="_ _8"> </span>may<span class="_ _8"> </span>be<span class="_ _8"> </span>greatly<span class="_ _6"> </span>improved<span class="_ _6"> </span>by<span class="_ _8"> </span>incorporating<span class="_ _8"> </span>information</div><div class="t m0 x1 h5 y1d ff4 fs3 fc0 sc0 ls3 ws0">on<span class="_"> </span>wind<span class="_"> </span>power<span class="_"> </span>distrib<span class="_ _0"></span>utions<span class="_"> </span>into<span class="_"> </span>the<span class="_"> </span>rob<span class="_ _1"></span>ust<span class="_"> </span>model.</div><div class="t m0 x6 h5 y1e ff3 fs3 fc0 sc0 ls3 ws0">Index<span class="_"> </span>T<span class="_ _3"></span>erms<span class="ff4">—Distributionally<span class="_ _9"> </span>robust<span class="_ _9"> </span>optimization,<span class="_"> </span>generalized</span></div><div class="t m0 x1 h5 y1f ff4 fs3 fc0 sc0 ls3 ws0">decision<span class="_"> </span>rule,<span class="_"> </span>Unit<span class="_"> </span>commitment.</div><div class="t m0 x7 h6 y20 ff1 fs4 fc0 sc0 ls3 ws0">N<span class="fs5 ls9">OMENCLA<span class="_ _3"></span>TURE</span></div><div class="t m0 x6 h6 y21 ff1 fs4 fc0 sc0 lsa ws0">Throughout<span class="_ _b"> </span>this<span class="_ _b"> </span>paper<span class="_ _1"></span>,<span class="_ _b"> </span>we<span class="_ _b"> </span>will<span class="_ _b"> </span>use<span class="_ _b"> </span>bold<span class="_ _b"> </span>letters,<span class="_ _b"> </span>to<span class="_ _b"> </span>express<span class="_ _b"> </span>vec-</div><div class="t m0 x1 h6 y22 ff1 fs4 fc0 sc0 lsa ws0">tors<span class="_ _c"> </span>and<span class="_ _c"> </span>matrices,<span class="_ _c"> </span>and<span class="_ _c"> </span>entries<span class="_ _c"> </span>of<span class="_ _c"> </span>vectors<span class="_ _9"> </span>or<span class="_"> </span>matrices<span class="_ _9"> </span>are<span class="_"> </span>denoted</div><div class="t m0 x1 h6 y23 ff1 fs4 fc0 sc0 lsb ws0">by<span class="_ _8"> </span>regular<span class="_ _7"> </span>letters<span class="_ _8"> </span>with<span class="_ _7"> </span>subscripts<span class="_ _8"> </span>indicating<span class="_ _7"> </span>the<span class="_ _8"> </span>corresponding</div><div class="t m0 x1 h6 y24 ff1 fs4 fc0 sc0 lsa ws0">indices.<span class="_ _6"> </span>For<span class="_"> </span>example,<span class="_ _6"> </span><span class="ff5 ls3">C</span></div><div class="t m0 x8 h7 y25 ff5 fs0 fc0 sc0 lsc ws0">mn</div><div class="t m0 x9 h6 y26 ff1 fs4 fc0 sc0 ls3 ws0">is<span class="_ _6"> </span>the<span class="_ _6"> </span>entry<span class="_ _8"> </span>of<span class="_ _6"> </span>matrix<span class="_"> </span><span class="ff5">C</span></div><div class="t m0 xa h8 y27 ff5 fs4 fc0 sc0 ls3 ws0">C</div><div class="t m0 xa h6 y26 ff5 fs4 fc0 sc0 ls3 ws0">C<span class="_"> </span><span class="ff1 lsd">in<span class="_ _6"> </span>the<span class="_ _8"> </span></span>m<span class="ff1 lse">th</span></div><div class="t m0 x1 h6 y28 ff1 fs4 fc0 sc0 lsb ws0">ro<span class="_ _0"></span>w<span class="_"> </span>and<span class="_ _c"> </span>the<span class="_"> </span><span class="ff5 ls3">n</span><span class="lsa">th<span class="_ _c"> </span>column.<span class="_"> </span>The<span class="_ _c"> </span><span class="ff5 ls3">m</span><span class="lse">th<span class="_"> </span>ro<span class="_ _1"></span>w<span class="_"> </span>of<span class="_"> </span>matrix<span class="_ _b"> </span><span class="ff5 ls3">C</span></span></span></div><div class="t m0 xb h8 y29 ff5 fs4 fc0 sc0 ls3 ws0">C</div><div class="t m0 xb h6 y28 ff5 fs4 fc0 sc0 ls3 ws0">C<span class="_ _8"> </span><span class="ff1">is<span class="_"> </span>expressed</span></div><div class="t m0 x1 h6 y2a ff1 fs4 fc0 sc0 lsf ws0">as<span class="_ _c"> </span>a<span class="_ _9"> </span>row<span class="_ _c"> </span>vector<span class="_ _b"> </span><span class="ff5 ls3">C</span></div><div class="t m0 xc h8 y2b ff5 fs4 fc0 sc0 ls3 ws0">C</div><div class="t m0 xc h8 y2a ff5 fs4 fc0 sc0 ls3 ws0">C</div><div class="t m0 xd h7 y2c ff5 fs0 fc0 sc0 ls3 ws0">T</div><div class="t m0 xd h7 y2d ff5 fs0 fc0 sc0 ls3 ws0">m</div><div class="t m0 xe h6 y2e ff1 fs4 fc0 sc0 lsf ws0">,<span class="_ _c"> </span>and<span class="_ _9"> </span>the<span class="_"> </span><span class="ff5 ls3">n</span><span class="lsa">th<span class="_ _9"> </span>column<span class="_"> </span>is<span class="_ _9"> </span>denoted<span class="_"> </span>by<span class="_ _9"> </span>a<span class="_ _c"> </span>column</span></div><div class="t m0 x1 h6 y2f ff1 fs4 fc0 sc0 lsf ws0">vector<span class="_ _9"> </span><span class="ff5 ls3">C</span></div><div class="t m0 xf h8 y30 ff5 fs4 fc0 sc0 ls3 ws0">C</div><div class="t m0 xf h8 y2f ff5 fs4 fc0 sc0 ls3 ws0">C</div><div class="t m0 x10 h7 y31 ff5 fs0 fc0 sc0 ls3 ws0">n</div><div class="t m0 x11 h6 y32 ff1 fs4 fc0 sc0 ls3 ws0">.</div><div class="t m0 x6 h6 y33 ff1 fs4 fc0 sc0 lsb ws0">Deﬁnition<span class="_"> </span>of<span class="_"> </span>each<span class="_"> </span>notation<span class="_"> </span>is<span class="_"> </span>listed<span class="_"> </span>below<span class="_ _3"></span>.</div><div class="t m0 x1 h9 y34 ff2 fs4 fc0 sc0 lsa ws0">A.<span class="_"> </span>Indices<span class="_"> </span>and<span class="_"> </span>Sets<span class="_"> </span>of<span class="_"> </span>the<span class="_"> </span>Unit<span class="_"> </span>Commitment<span class="_"> </span>Model</div><div class="t m0 x1 h6 y35 ff5 fs4 fc0 sc0 lsb ws0">b/<span class="ff6 ls3">B<span class="_ _d"> </span><span class="ff1">Index/set<span class="_"> </span>of<span class="_"> </span>all<span class="_"> </span>b<span class="_ _1"></span>uses.</span></span></div><div class="t m0 x1 h6 y36 ff5 fs4 fc0 sc0 ls10 ws0">f/<span class="_ _3"></span><span class="ff6 ls3">F<span class="_ _e"> </span><span class="ff1">Index/set<span class="_"> </span>of<span class="_"> </span>all<span class="_"> </span>transmission<span class="_"> </span>lines.</span></span></div><div class="t m0 x1 h6 y37 ff5 fs4 fc0 sc0 ls11 ws0">k/<span class="_ _1"></span><span class="ff6 ls3">K<span class="_ _f"> </span><span class="ff1">Index/set<span class="_"> </span>of<span class="_"> </span>all<span class="_"> </span>units.</span></span></div><div class="t m0 x12 ha y38 ff1 fs5 fc0 sc0 ls12 ws0">Manuscript<span class="_ _6"> </span>receiv<span class="_ _0"></span>ed<span class="_ _6"> </span>March<span class="_ _6"> </span>2,<span class="_ _6"> </span>2015;<span class="_ _6"> </span>revised<span class="_ _6"> </span>August<span class="_ _6"> </span>26,<span class="_ _6"> </span>2015<span class="_ _6"> </span>and<span class="_ _6"> </span>January</div><div class="t m0 x1 ha y39 ff1 fs5 fc0 sc0 ls12 ws0">15,<span class="_"> </span>2016;<span class="_ _c"> </span>accepted<span class="_ _c"> </span>March<span class="_"> </span>5,<span class="_ _c"> </span>2016.<span class="_ _c"> </span>Date<span class="_ _c"> </span>of<span class="_ _c"> </span>publication<span class="_ _c"> </span>April<span class="_"> </span>20,<span class="_ _c"> </span>2016;<span class="_ _c"> </span>date<span class="_ _c"> </span>of</div><div class="t m0 x1 ha y3a ff1 fs5 fc0 sc0 ls12 ws0">current<span class="_"> </span>version<span class="_"> </span>December<span class="_"> </span>20,<span class="_"> </span>2016.<span class="_"> </span>P<span class="_ _0"></span>aper<span class="_"> </span>no.<span class="_"> </span>TPWRS-00288-2015.</div><div class="t m0 x12 ha y3b ff1 fs5 fc0 sc0 ls3 ws0">P<span class="_ _3"></span>.<span class="_ _c"> </span>Xiong<span class="_ _6"> </span>and<span class="_ _c"> </span>C.<span class="_ _6"> </span>Singh<span class="_"> </span>are<span class="_ _6"> </span>with<span class="_ _c"> </span>the<span class="_ _6"> </span>Department<span class="_"> </span>of<span class="_ _6"> </span>Electrical<span class="_"> </span>and<span class="_ _6"> </span>Computer</div><div class="t m0 x1 ha y3c ff1 fs5 fc0 sc0 ls3 ws0">Engineering,<span class="_"> </span>T<span class="_ _1"></span>exas<span class="_ _b"> </span>A&M<span class="_"> </span>Uni<span class="_ _0"></span>versity<span class="_ _1"></span>,<span class="_"> </span>Colle<span class="_ _0"></span>ge<span class="_"> </span>Station,<span class="_ _b"> </span>TX<span class="_"> </span>77843<span class="_"> </span>USA<span class="_"> </span>(e-mail:</div><div class="t m0 x1 ha y3d ff1 fs5 fc0 sc0 ls3 ws0">xiongpengnus@gmail.com;<span class="_"> </span>singh@ece.tamu.edu).</div><div class="t m0 x12 ha y3e ff1 fs5 fc0 sc0 ls9 ws0">P<span class="_ _3"></span>.<span class="_ _7"> </span>Jirutitijaroen<span class="_ _8"> </span>is<span class="_ _7"> </span>with<span class="_ _8"> </span>the<span class="_ _7"> </span>Department<span class="_ _8"> </span>of<span class="_ _7"> </span>Electrical<span class="_ _8"> </span>and<span class="_ _7"> </span>Computer<span class="_ _7"> </span>Engi-</div><div class="t m0 x1 ha y3f ff1 fs5 fc0 sc0 ls9 ws0">neering,<span class="_ _c"> </span>National<span class="_ _6"> </span>Univ<span class="_ _1"></span>ersity<span class="_ _6"> </span>of<span class="_ _c"> </span>Singapore,<span class="_ _6"> </span>119077<span class="_ _c"> </span>Singapore<span class="_ _6"> </span>(e-mail:<span class="_ _c"> </span>elejp@</div><div class="t m0 x1 ha y40 ff1 fs5 fc0 sc0 ls3 ws0">nus.edu.sg).</div><div class="t m0 x12 ha y41 ff1 fs5 fc0 sc0 ls3 ws0">Digital<span class="_"> </span>Object<span class="_"> </span>Identiﬁer<span class="_"> </span>10.1109/TPWRS.2016.2544795</div><div class="t m0 x13 hb y42 ff6 fs4 fc0 sc0 ls3 ws0">K</div><div class="t m0 x14 h7 y43 ff5 fs0 fc0 sc0 ls3 ws0">b</div><div class="t m0 x15 h6 y42 ff1 fs4 fc0 sc0 lsa ws0">Set<span class="_"> </span>of<span class="_"> </span>units<span class="_"> </span>located<span class="_"> </span>at<span class="_"> </span>bus<span class="_"> </span><span class="ff5 ls3">b<span class="ff1">.</span></span></div><div class="t m0 x13 h6 y44 ff5 fs4 fc0 sc0 ls13 ws0">r/<span class="_ _0"></span><span class="ff6 ls3">R<span class="_ _e"> </span><span class="ff1 lsa">Index/set<span class="_"> </span>of<span class="_"> </span>all<span class="_"> </span>wind<span class="_"> </span>po<span class="_ _1"></span>wer<span class="_"> </span>sources.</span></span></div><div class="t m0 x13 hb y45 ff6 fs4 fc0 sc0 ls3 ws0">R</div><div class="t m0 x16 h7 y46 ff5 fs0 fc0 sc0 ls3 ws0">b</div><div class="t m0 x15 h6 y47 ff1 fs4 fc0 sc0 lsf ws0">Set<span class="_"> </span>of<span class="_"> </span>wind<span class="_"> </span>farms<span class="_"> </span>located<span class="_"> </span>at<span class="_"> </span>bus<span class="_"> </span><span class="ff5 ls3">b<span class="ff1">.</span></span></div><div class="t m0 x13 h6 y48 ff5 fs4 fc0 sc0 lse ws0">t/<span class="ff6 ls3">T<span class="_ _10"> </span><span class="ff1">Index/set<span class="_"> </span>of<span class="_"> </span>all<span class="_"> </span>time<span class="_"> </span>steps.</span></span></div><div class="t m0 x13 h6 y49 ff7 fs4 fc0 sc0 ls3 ws0">T<span class="_ _11"> </span><span class="ff1 lsa">Feasible<span class="_ _9"> </span>set<span class="_ _9"> </span>of<span class="_ _b"> </span>unit<span class="_ _9"> </span>status<span class="_ _9"> </span>decisions,<span class="_ _9"> </span>deﬁned<span class="_ _9"> </span>by<span class="_ _9"> </span>minimum</span></div><div class="t m0 x15 h6 y4a ff1 fs4 fc0 sc0 lsb ws0">up/do<span class="_ _0"></span>wn<span class="_"> </span>time<span class="_"> </span>constraints.</div><div class="t m0 x13 h6 y4b ff5 fs4 fc0 sc0 ls14 ws0">θ/<span class="_ _0"></span><span class="ff8 ls3">Θ<span class="_ _f"> </span><span class="ff1 lsa">Index/set<span class="_ _c"> </span>of<span class="_ _c"> </span>all<span class="_ _c"> </span>segments<span class="_ _9"> </span>of<span class="_"> </span>the<span class="_ _9"> </span>generation<span class="_"> </span>cost<span class="_ _9"> </span>function.</span></span></div><div class="t m0 x13 h6 y4c ff6 fs4 fc0 sc0 ls15 ws0">|·|<span class="_ _12"> </span><span class="ff1 lsa">Number<span class="_"> </span>of<span class="_"> </span>elements<span class="_"> </span>in<span class="_"> </span>a<span class="_"> </span>set.</span></div><div class="t m0 x13 h9 y4d ff2 fs4 fc0 sc0 lsa ws0">B.<span class="_"> </span>Constants<span class="_"> </span>of<span class="_"> </span>the<span class="_"> </span>Unit<span class="_"> </span>Commitment<span class="_"> </span>Model</div><div class="t m0 x13 h8 y4e ff5 fs4 fc0 sc0 ls3 ws0">a</div><div class="t m0 x17 h7 y4f ff5 fs0 fc0 sc0 ls3 ws0">θ</div><div class="t m0 x17 h7 y50 ff5 fs0 fc0 sc0 ls3 ws0">k</div><div class="t m0 x18 h6 y51 ff1 fs4 fc0 sc0 lsa ws0">Linear<span class="_"> </span>term<span class="_"> </span>of<span class="_ _c"> </span>the<span class="_"> </span><span class="ff5 ls3">θ</span><span class="lsb">th<span class="_"> </span>piecewise<span class="_"> </span>linear<span class="_ _c"> </span>cost<span class="_"> </span>function<span class="_"> </span>for</span></div><div class="t m0 x18 h6 y52 ff1 fs4 fc0 sc0 lsf ws0">generator<span class="_"> </span><span class="ff5 ls3">k<span class="_ _13"></span><span class="ff1">.</span></span></div><div class="t m0 x13 h8 y53 ff5 fs4 fc0 sc0 ls3 ws0">b</div><div class="t m0 x19 h7 y54 ff5 fs0 fc0 sc0 ls3 ws0">θ</div><div class="t m0 x19 h7 y55 ff5 fs0 fc0 sc0 ls3 ws0">k</div><div class="t m0 x18 h6 y56 ff1 fs4 fc0 sc0 lsb ws0">Constant<span class="_ _8"> </span>term<span class="_ _8"> </span>of<span class="_ _8"> </span>the<span class="_ _8"> </span><span class="ff5 ls3">θ<span class="_ _13"></span></span>th<span class="_ _8"> </span>piecewise<span class="_ _6"> </span>linear<span class="_ _8"> </span>cost<span class="_ _7"> </span>function</div><div class="t m0 x18 h6 y57 ff1 fs4 fc0 sc0 lsa ws0">for<span class="_"> </span>generator<span class="_"> </span><span class="ff5 ls3">k<span class="_ _13"></span><span class="ff1">.</span></span></div><div class="t m0 x13 h8 y58 ff5 fs4 fc0 sc0 ls3 ws0">C</div><div class="t m0 x14 h7 y59 ff5 fs0 fc0 sc0 ls3 ws0">s</div><div class="t m0 x14 h7 y5a ff5 fs0 fc0 sc0 ls3 ws0">k</div><div class="t m0 x18 h6 y5b ff1 fs4 fc0 sc0 lsb ws0">Start-up<span class="_"> </span>cost<span class="_"> </span>of<span class="_"> </span>unit<span class="_"> </span><span class="ff5 ls3">k<span class="_ _6"> </span><span class="ff1">(<span class="ff8">$</span><span class="ls16">).</span></span></span></div><div class="t m0 x13 h8 y5c ff5 fs4 fc0 sc0 ls3 ws0">D</div><div class="t m0 x16 h7 y5d ff5 fs0 fc0 sc0 ls17 ws0">bt</div><div class="t m0 x18 h6 y5e ff1 fs4 fc0 sc0 ls18 ws0">Po<span class="_ _0"></span>wer<span class="_"> </span>demand<span class="_"> </span>at<span class="_"> </span>bus<span class="_"> </span><span class="ff5 ls3">b<span class="_ _c"> </span></span><span class="lsb">during<span class="_"> </span>time<span class="_"> </span><span class="ff5 ls3">t<span class="_ _6"> </span></span><span class="lse">(MW).</span></span></div><div class="t m0 x13 h8 y5f ff5 fs4 fc0 sc0 ls3 ws0">F</div><div class="t m0 x1a h7 y60 ff5 fs0 fc0 sc0 ls3 ws0">f</div><div class="t m0 x18 h6 y61 ff1 fs4 fc0 sc0 lsb ws0">Capacity<span class="_"> </span>of<span class="_"> </span>transmission<span class="_"> </span>line<span class="_"> </span><span class="ff5 ls3">f<span class="_ _7"> </span></span><span class="lse">(MW).</span></div><div class="t m0 x13 h8 y62 ff5 fs4 fc0 sc0 ls3 ws0">K</div><div class="t m0 x16 h7 y63 ff5 fs0 fc0 sc0 ls19 ws0">fb</div><div class="t m0 x18 h6 y64 ff1 fs4 fc0 sc0 ls3 ws0">Line<span class="_ _7"> </span>ﬂo<span class="_ _0"></span>w<span class="_ _7"> </span>distribution<span class="_ _8"> </span>factor<span class="_ _7"> </span>for<span class="_ _7"> </span>transmission<span class="_ _7"> </span>line<span class="_ _7"> </span><span class="ff5">f<span class="_ _14"> </span></span><span class="ls18">at</span></div><div class="t m0 x18 h6 y65 ff1 fs4 fc0 sc0 ls1a ws0">bus<span class="_ _6"> </span><span class="ff5 ls3">b<span class="ff1">.</span></span></div><div class="t m0 x13 h8 y66 ff5 fs4 fc0 sc0 ls3 ws0">P</div><div class="t m0 x14 h7 y67 ff5 fs0 fc0 sc0 ls3 ws0">k</div><div class="t m0 x18 h6 y66 ff1 fs4 fc0 sc0 lsa ws0">Maximum<span class="_"> </span>capacity<span class="_"> </span>of<span class="_"> </span>unit<span class="_"> </span><span class="ff5 ls3">k<span class="_ _6"> </span></span><span class="lse">(MW).</span></div><div class="t m0 x13 h8 y68 ff5 fs4 fc0 sc0 ls3 ws0">P</div><div class="t m0 x14 h7 y69 ff5 fs0 fc0 sc0 ls3 ws0">k</div><div class="t m0 x18 h6 y6a ff1 fs4 fc0 sc0 lsa ws0">Minimum<span class="_"> </span>capacity<span class="_"> </span>of<span class="_"> </span>unit<span class="_"> </span><span class="ff5 ls3">k<span class="_ _6"> </span></span><span class="lse">(MW).</span></div><div class="t m0 x13 h8 y6b ff5 fs4 fc0 sc0 ls1b ws0">RD</div><div class="t m0 x1b h7 y6c ff5 fs0 fc0 sc0 ls3 ws0">k</div><div class="t m0 x18 h6 y6d ff1 fs4 fc0 sc0 lsb ws0">Ramp-do<span class="_ _0"></span>wn<span class="_"> </span>limit<span class="_"> </span>of<span class="_"> </span>unit<span class="_"> </span><span class="ff5 ls3">k<span class="_ _6"> </span></span><span class="lsd">(MW/h).</span></div><div class="t m0 x13 h8 y6e ff5 fs4 fc0 sc0 ls1b ws0">RU</div><div class="t m0 x1c h7 y6f ff5 fs0 fc0 sc0 ls3 ws0">k</div><div class="t m0 x18 h6 y70 ff1 fs4 fc0 sc0 lsb ws0">Ramp-up<span class="_"> </span>limit<span class="_"> </span>of<span class="_"> </span>unit<span class="_"> </span><span class="ff5 ls3">k<span class="_ _13"></span><span class="ff1">.</span></span></div><div class="t m0 x13 h6 y71 ff5 fs4 fc0 sc0 ls3 ws0">V<span class="_ _15"> </span><span class="ff1 lsb">V<span class="_ _4"></span>alue<span class="_ _6"> </span>of<span class="_"> </span>load<span class="_"> </span>loss<span class="_"> </span>(<span class="ff8 ls3">$</span><span class="lsd">/MWh).</span></span></div><div class="t m0 x13 h9 y72 ff2 fs4 fc0 sc0 lsa ws0">C.<span class="_"> </span>Decision<span class="_"> </span>V<span class="_ _3"></span>ariables<span class="_"> </span>of<span class="_"> </span>the<span class="_"> </span>Unit<span class="_"> </span>Commitment<span class="_"> </span>Model</div><div class="t m0 x13 h8 y73 ff5 fs4 fc0 sc0 ls3 ws0">c</div><div class="t m0 x19 h7 y74 ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 x1d h6 y75 ff1 fs4 fc0 sc0 lsa ws0">Generation<span class="_"> </span>cost<span class="_"> </span>of<span class="_"> </span>unit<span class="_"> </span><span class="ff5 ls3">k<span class="_ _6"> </span></span><span class="lsd">in<span class="_"> </span>time<span class="_"> </span><span class="ff5 ls3">t<span class="_ _6"> </span><span class="ff1">(<span class="ff8">$</span><span class="ls16">).</span></span></span></span></div><div class="t m0 x13 h8 y76 ff5 fs4 fc0 sc0 ls3 ws0">p</div><div class="t m0 x17 h7 y77 ff5 fs0 fc0 sc0 ls3 ws0">t</div><div class="t m0 x17 h7 y78 ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 x1d h6 y79 ff1 fs4 fc0 sc0 lsa ws0">Generation<span class="_"> </span>of<span class="_"> </span>thermal<span class="_"> </span>unit<span class="_"> </span><span class="ff5 ls3">k<span class="_ _6"> </span></span><span class="lsb">during<span class="_"> </span>time<span class="_"> </span><span class="ff5 ls3">t<span class="_ _6"> </span></span><span class="lse">(MW).</span></span></div><div class="t m0 x13 h8 y7a ff5 fs4 fc0 sc0 ls3 ws0">p</div><div class="t m0 x17 h7 y7b ff5 fs0 fc0 sc0 ls3 ws0">w</div><div class="t m0 x17 h7 y7c ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 x1d h6 y7d ff1 fs4 fc0 sc0 lsb ws0">Generation<span class="_"> </span>of<span class="_"> </span>wind<span class="_"> </span>farm<span class="_"> </span><span class="ff5 ls3">r<span class="_ _6"> </span></span>during<span class="_"> </span>time<span class="_"> </span><span class="ff5 ls3">t<span class="_ _6"> </span></span><span class="lsd">(MW).</span></div><div class="t m0 x13 h8 y7e ff5 fs4 fc0 sc0 ls3 ws0">l</div><div class="t m0 x1e h7 y7f ff5 fs0 fc0 sc0 ls17 ws0">bt</div><div class="t m0 x1d h6 y7e ff1 fs4 fc0 sc0 lsa ws0">Load<span class="_"> </span>loss<span class="_"> </span>at<span class="_"> </span>bus<span class="_"> </span><span class="ff5 ls3">b<span class="_ _c"> </span></span><span class="lsb">during<span class="_"> </span>time<span class="_"> </span><span class="ff5 ls3">t<span class="_ _6"> </span></span><span class="lse">(MW).</span></span></div><div class="t m0 x13 h8 y80 ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 x17 h7 y81 ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 x1d h6 y82 ff1 fs4 fc0 sc0 lsa ws0">Binary<span class="_ _8"> </span>decision<span class="_ _7"> </span>v<span class="_ _1"></span>ariable<span class="_ _7"> </span>indicating<span class="_ _8"> </span>the<span class="_ _7"> </span>unit<span class="_ _8"> </span>commitment</div><div class="t m0 x1d h6 y83 ff1 fs4 fc0 sc0 lsa ws0">status<span class="_"> </span>of<span class="_"> </span>generator<span class="_"> </span><span class="ff5 ls3">k<span class="_ _6"> </span></span><span class="lsb">during<span class="_"> </span>time<span class="_"> </span><span class="ff5 ls3">t<span class="ff1">.</span></span></span></div><div class="t m0 x13 h8 y84 ff5 fs4 fc0 sc0 ls3 ws0">y</div><div class="t m0 x13 h8 y85 ff5 fs4 fc0 sc0 ls3 ws0">y</div><div class="t m0 x13 h6 y84 ff5 fs4 fc0 sc0 ls3 ws0">y<span class="_ _16"> </span><span class="ff1 lsa">V<span class="_ _4"></span>ector<span class="_ _17"> </span>of<span class="_ _5"> </span>all<span class="_ _5"> </span>second-stage<span class="_ _5"> </span>decisions,<span class="_ _5"> </span>including<span class="_ _5"> </span><span class="ff5 ls3">c</span></span></div><div class="t m0 x1f h7 y86 ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 x20 h6 y87 ff1 fs4 fc0 sc0 ls3 ws0">,<span class="_ _5"> </span><span class="ff5">p</span></div><div class="t m0 x21 h7 y88 ff5 fs0 fc0 sc0 ls3 ws0">t</div><div class="t m0 x21 h7 y89 ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 x22 h6 y87 ff1 fs4 fc0 sc0 ls3 ws0">,</div><div class="t m0 x1d h8 y8a ff5 fs4 fc0 sc0 ls3 ws0">p</div><div class="t m0 x15 h7 y8b ff5 fs0 fc0 sc0 ls3 ws0">w</div><div class="t m0 x15 h7 y8c ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 x23 h6 y8d ff1 fs4 fc0 sc0 ls3 ws0">,<span class="_"> </span><span class="ff5">l</span></div><div class="t m0 x24 h7 y8e ff5 fs0 fc0 sc0 ls17 ws0">bt</div><div class="t m0 x25 h6 y8d ff1 fs4 fc0 sc0 ls18 ws0">,<span class="_"> </span>and<span class="_"> </span><span class="ff5 ls3">z</span></div><div class="t m0 x26 h7 y8e ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 x27 h6 y8d ff1 fs4 fc0 sc0 ls3 ws0">.</div><div class="t m0 x13 h8 y8f ff5 fs4 fc0 sc0 ls3 ws0">z</div><div class="t m0 x19 h7 y90 ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 x1d h6 y91 ff1 fs4 fc0 sc0 lsb ws0">Start-up<span class="_"> </span>cost<span class="_"> </span>of<span class="_"> </span>unit<span class="_"> </span><span class="ff5 ls3">k<span class="_ _6"> </span></span>during<span class="_"> </span>time<span class="_"> </span><span class="ff5 ls3">t<span class="_ _6"> </span><span class="ff1">(<span class="ff8">$</span><span class="ls16">).</span></span></span></div><div class="t m0 x13 h9 y92 ff2 fs4 fc0 sc0 lsa ws0">D.<span class="_"> </span>Notations<span class="_"> </span>for<span class="_"> </span>the<span class="_"> </span>Uncertainty<span class="_"> </span>Model</div><div class="t m0 x13 hc y93 ff7 fs4 fc0 sc0 ls3 ws0">E</div><div class="t m0 x14 hd y94 ff9 fs0 fc0 sc0 ls3 ws0">P</div><div class="t m0 x28 h6 y95 ff1 fs4 fc0 sc0 lsa ws0">Expectations<span class="_"> </span>under<span class="_"> </span>distribution<span class="_"> </span><span class="ff7 ls3">P<span class="_ _18"></span><span class="ff1">.</span></span></div><div class="t m0 x13 h6 y96 ff7 fs4 fc0 sc0 ls3 ws0">F<span class="_ _11"> </span><span class="ff1 lsb">Ambiguity<span class="_"> </span>set<span class="_"> </span>for<span class="_"> </span>distributions<span class="_"> </span>of<span class="_"> </span>random<span class="_"> </span>variables</span></div><div class="t m0 x29 h8 y97 ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x29 h8 y96 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x29 h8 y97 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x29 h6 y96 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff1">.</span></div><div class="t m0 x13 h6 y98 ff7 fs4 fc0 sc0 ls3 ws0">G<span class="_ _19"> </span><span class="ff1 lsb">Extended<span class="_ _b"> </span>form<span class="_ _b"> </span>of<span class="_ _b"> </span>the<span class="_ _b"> </span>ambiguity<span class="_ _b"> </span>set<span class="_ _b"> </span></span>F<span class="_ _18"></span><span class="ff1 ls1e">,t<span class="_ _1a"></span>h<span class="_ _1a"></span>es<span class="_ _1a"></span>e<span class="_ _4"></span>tf<span class="_ _1a"></span>o<span class="_ _1a"></span>rj<span class="_ _1a"></span>o<span class="_ _4"></span>i<span class="_ _1a"></span>n<span class="_ _1a"></span>td<span class="_ _4"></span>i<span class="_ _1a"></span>s<span class="_ _1a"></span>-</span></div><div class="t m0 x28 h6 y99 ff1 fs4 fc0 sc0 lsb ws0">tributions<span class="_ _6"> </span>of<span class="_ _8"> </span>random<span class="_ _7"> </span>v<span class="_ _1"></span>ariables</div><div class="t m0 x2a h8 y9a ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x2a h8 y99 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x2a h8 y9a ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x2a h6 y99 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_"> </span><span class="ff1 lsb">and<span class="_ _8"> </span>auxiliary<span class="_ _8"> </span>variables</span></div><div class="t m0 x28 h8 y9b ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x28 h8 y9c ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 x28 h8 y9b ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 x28 h6 y9c ff5 fs4 fc0 sc0 ls3 ws0">u<span class="ff1">.</span></div><div class="t m0 x13 h6 y9d ff5 fs4 fc0 sc0 lsf ws0">i/<span class="ff6 ls3">I<span class="_ _1b"> </span><span class="ff1 lsb">Index/set<span class="_"> </span>of<span class="_"> </span>functions<span class="_ _6"> </span>describing<span class="_ _6"> </span>the<span class="_ _6"> </span>distribution<span class="_"> </span>of<span class="_ _6"> </span>ran-</span></span></div><div class="t m0 x28 h6 y9e ff1 fs4 fc0 sc0 lsb ws0">dom<span class="_"> </span>v<span class="_ _1"></span>ariables,<span class="_"> </span>or<span class="_"> </span>the<span class="_ _c"> </span>index<span class="_"> </span>of<span class="_ _9"> </span>auxiliary<span class="_"> </span>variables.<span class="_ _c"> </span>In<span class="_"> </span>this</div><div class="t m0 x28 h6 y9f ff1 fs4 fc0 sc0 lsa ws0">case,<span class="_"> </span><span class="ff5 ls3">i<span class="_ _6"> </span><span class="ff8 ls1f">=(<span class="_ _1c"></span><span class="ff5 ls20">j,<span class="_ _b"> </span>r<span class="_ _0"></span>,<span class="_ _b"> </span>t<span class="ff8 ls3">)<span class="ff1 ls18">,<span class="_"> </span>and<span class="_"> </span></span><span class="ff6">I<span class="_ _7"> </span></span>=<span class="_ _8"> </span><span class="ff6 ls21">J×<span class="_ _1a"></span>R<span class="_ _a"></span>×<span class="_ _1a"></span>T<span class="_ _1a"></span><span class="ff1 ls3">.</span></span></span></span></span></span></div><div class="t m0 x2b ha ya0 ff1 fs5 fc0 sc0 ls3 ws0">0885-8950<span class="_"> </span>©<span class="_"> </span>2016<span class="_"> </span>IEEE.<span class="_"> </span>Personal<span class="_"> </span>use<span class="_"> </span>is<span class="_"> </span>permitted,<span class="_"> </span>but<span class="_"> </span>republication/redistrib<span class="_ _1"></span>ution<span class="_ _c"> </span>requires<span class="_"> </span>IEEE<span class="_"> </span>permission.</div><div class="t m0 x2c ha ya1 ff1 fs5 fc0 sc0 ls9 ws0">See<span class="_"> </span>http://www<span class="_ _1"></span>.ieee.org/publications</div><div class="t m0 x2d ha ya2 ff1 fs5 fc0 sc0 ls3 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<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://csdnimg.cn/release/download_crawler_static/85190235/bg2.jpg"><div class="t m0 x2e h2 y1 ff1 fs0 fc0 sc0 ls1 ws0">40<span class="_ _2"> </span><span class="ls0">IEEE<span class="_"> </span>TRANSA<span class="_ _1"></span>CTIONS<span class="_"> </span>ON<span class="_"> </span>POWER<span class="_"> </span>SYSTEMS,<span class="_"> </span>V<span class="_ _0"></span>OL.<span class="_"> </span>32,<span class="_"> </span>NO.<span class="_"> </span>1,<span class="_"> </span>J<span class="_ _1"></span>ANUAR<span class="_ _1"></span>Y<span class="_"> </span>2017</span></div><div class="t m0 x2e h6 ya4 ff5 fs4 fc0 sc0 ls22 ws0">j/<span class="_ _1"></span><span class="ff6 ls3">J<span class="_ _1d"> </span><span class="ff1 lsb">Index/set<span class="_ _7"> </span>of<span class="_ _7"> </span>segments<span class="_ _7"> </span>that<span class="_ _7"> </span>describes<span class="_ _7"> </span>the<span class="_ _5"> </span>distribution<span class="_ _7"> </span>of</span></span></div><div class="t m0 x2f h6 ya5 ff1 fs4 fc0 sc0 lsa ws0">each<span class="_"> </span>random<span class="_"> </span>variable<span class="_"> </span><span class="ff5 ls3">v</span></div><div class="t m0 x30 h7 ya6 ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 x31 h6 ya7 ff1 fs4 fc0 sc0 ls3 ws0">.</div><div class="t m0 x2e h6 ya8 ff7 fs4 fc0 sc0 ls3 ws0">P<span class="_ _11"> </span><span class="ff1 lsb">Distrib<span class="_ _1"></span>ution<span class="_"> </span>of<span class="_"> </span>random<span class="_"> </span>variables</span></div><div class="t m0 x32 h8 ya9 ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x32 h8 ya8 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x32 h8 ya9 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x32 h6 ya8 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff1">.</span></div><div class="t m0 x2e h6 yaa ff7 fs4 fc0 sc0 ls3 ws0">Q<span class="_ _1e"> </span><span class="ff1 lsb">Joint<span class="_ _1f"> </span>distribution<span class="_ _17"> </span>of<span class="_ _1f"> </span>random<span class="_ _1f"> </span>variables</span></div><div class="t m0 x33 h8 yab ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x33 h8 yaa ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x33 h8 yab ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x33 h6 yaa ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _14"> </span><span class="ff1 lsb">and<span class="_ _1f"> </span>auxiliary</span></div><div class="t m0 x2f h6 yac ff1 fs4 fc0 sc0 lsb ws0">v<span class="_ _0"></span>ariables</div><div class="t m0 x34 h8 yad ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 xd h8 yac ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 xd h8 yad ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 xd h6 yac ff5 fs4 fc0 sc0 ls3 ws0">u<span class="ff1">.</span></div><div class="t m0 x2e h6 yae ff5 fs4 fc0 sc0 lsd ws0">s/<span class="ff6 ls3">S<span class="_ _20"> </span><span class="ff1 lsa">Index/set<span class="_ _5"> </span>of<span class="_ _1f"> </span>random<span class="_ _17"> </span>variables.<span class="_ _5"> </span>In<span class="_ _17"> </span>this<span class="_ _1f"> </span>case,<span class="_ _17"> </span></span><span class="ff5">s<span class="_ _6"> </span><span class="ff8 ls23">=(<span class="_ _1c"></span><span class="ff5 ls24">r,<span class="_ _9"> </span>t<span class="_ _13"></span><span class="ff8 ls3">)<span class="ff1">,</span></span></span></span></span></span></div><div class="t m0 x2f h6 yaf ff1 fs4 fc0 sc0 ls18 ws0">and<span class="_"> </span><span class="ff6 ls3">S<span class="_ _7"> </span><span class="ff8">=<span class="_ _6"> </span></span><span class="ls25">R×T<span class="_ _13"></span></span><span class="ff1">.</span></span></div><div class="t m0 x35 h8 yb0 ff8 fs4 fc0 sc0 ls3 ws0">˜<span class="_ _21"></span><span class="ff5">u</span></div><div class="t m0 x6 h7 yb1 ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x2f h6 yb2 ff1 fs4 fc0 sc0 ls3 ws0">Auxiliary<span class="_"> </span>va<span class="_ _0"></span>riable<span class="_"> </span>associated<span class="_"> </span>with<span class="_"> </span>function<span class="_"> </span><span class="ff5">g</span></div><div class="t m0 x36 h7 yb1 ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x37 h8 yb2 ff8 fs4 fc0 sc0 ls3 ws0">(</div><div class="t m0 x2d h8 yb3 ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x38 h8 yb2 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x38 h8 yb3 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x38 h6 yb2 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8">)<span class="ff1">.</span></span></div><div class="t m0 x2e h6 yb4 ff5 fs4 fc0 sc0 ls3 ws0">U<span class="_ _22"> </span><span class="ff1 lsf">Number<span class="_ _9"> </span>of<span class="_"> </span>constraints<span class="_ _b"> </span>used<span class="_"> </span>to<span class="_ _b"> </span>deﬁne<span class="_"> </span>the<span class="_ _9"> </span>extended<span class="_ _9"> </span>uncer-</span></div><div class="t m0 x2f h6 yb5 ff1 fs4 fc0 sc0 ls3 ws0">tainty<span class="_"> </span>set</div><div class="t m0 x39 h8 yb6 ff8 fs4 fc0 sc0 ls3 ws0">¯</div><div class="t m0 x34 h6 yb5 ff6 fs4 fc0 sc0 ls3 ws0">V<span class="_ _18"></span><span class="ff1">.</span></div><div class="t m0 x2e h8 yb7 ff8 fs4 fc0 sc0 ls3 ws0">˜<span class="_ _23"></span><span class="ff5">v</span></div><div class="t m0 x3a h7 yb8 ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 x2f h6 yb9 ff1 fs4 fc0 sc0 lsa ws0">Random<span class="_ _17"> </span>v<span class="_ _0"></span>ariable<span class="_ _17"> </span>indicating<span class="_ _17"> </span>the<span class="_ _5"> </span>forecast<span class="_ _17"> </span>error<span class="_ _17"> </span>of<span class="_ _17"> </span>wind</div><div class="t m0 x2f h6 yba ff1 fs4 fc0 sc0 ls18 ws0">po<span class="_ _0"></span>wer<span class="_"> </span>at<span class="_"> </span>bus<span class="_"> </span><span class="ff5 ls3">r</span><span class="lsb">,<span class="_"> </span>during<span class="_"> </span>time<span class="_"> </span><span class="ff5 ls3">t<span class="_ _6"> </span></span><span class="lsd">(MW).</span></span></div><div class="t m0 x2e h8 ybb ff5 fs4 fc0 sc0 ls3 ws0">V</div><div class="t m0 x3b h7 ybc ff8 fs0 fc0 sc0 ls3 ws0">+</div><div class="t m0 x6 h7 ybd ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 x2f h6 ybe ff1 fs4 fc0 sc0 lsa ws0">Upper<span class="_ _6"> </span>bound<span class="_ _8"> </span>of<span class="_ _8"> </span>wind<span class="_ _8"> </span>forecast<span class="_ _8"> </span>error<span class="_ _7"> </span><span class="ff8 ls3">˜<span class="_ _21"></span><span class="ff5">v</span></span></div><div class="t m0 x3c h7 ybf ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 x33 h6 ybe ff1 fs4 fc0 sc0 ls18 ws0">at<span class="_ _6"> </span>bus<span class="_ _8"> </span><span class="ff5 ls3">r<span class="_ _13"></span></span><span class="lsb">,<span class="_ _6"> </span>during</span></div><div class="t m0 x2f h6 yc0 ff1 fs4 fc0 sc0 lsd ws0">time<span class="_"> </span><span class="ff5 ls3">t<span class="_ _6"> </span></span>(MW).</div><div class="t m0 x2e h8 yc1 ff5 fs4 fc0 sc0 ls3 ws0">V</div><div class="t m0 x3b hf yc2 ff6 fs0 fc0 sc0 ls3 ws0">−</div><div class="t m0 x6 h7 yc3 ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 x2f h6 yc4 ff1 fs4 fc0 sc0 lsa ws0">Lo<span class="_ _0"></span>wer<span class="_ _6"> </span>bound<span class="_ _8"> </span>of<span class="_ _6"> </span>wind<span class="_ _8"> </span>forecast<span class="_ _6"> </span>error<span class="_ _7"> </span><span class="ff8 ls3">˜<span class="_ _23"></span><span class="ff5">v</span></span></div><div class="t m0 x3c h7 yc5 ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 xb h6 yc4 ff1 fs4 fc0 sc0 ls18 ws0">at<span class="_ _6"> </span>bus<span class="_ _6"> </span><span class="ff5 ls3">r<span class="_ _13"></span></span><span class="lsb">,<span class="_ _6"> </span>during</span></div><div class="t m0 x2f h6 yc6 ff1 fs4 fc0 sc0 lsd ws0">time<span class="_"> </span><span class="ff5 ls3">t<span class="_ _6"> </span></span>(MW).</div><div class="t m0 x2e h6 yc7 ff6 fs4 fc0 sc0 ls3 ws0">V<span class="_ _11"> </span><span class="ff1 lsa">Uncertainty<span class="_ _6"> </span>set<span class="_ _6"> </span>of<span class="_ _6"> </span>random<span class="_ _8"> </span>variables</span></div><div class="t m0 x3d h8 yc8 ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x3d h8 yc7 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x3d h8 yc8 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x3d h6 yc7 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff1 lsf">,<span class="_ _6"> </span>deﬁned<span class="_ _6"> </span>by<span class="_ _8"> </span>linear</span></div><div class="t m0 x2f h6 yc9 ff1 fs4 fc0 sc0 ls3 ws0">constraints.</div><div class="t m0 x35 h8 yca ff8 fs4 fc0 sc0 ls3 ws0">¯</div><div class="t m0 x2e h6 ycb ff6 fs4 fc0 sc0 ls3 ws0">V<span class="_ _11"> </span><span class="ff1 lsa">Uncertainty<span class="_ _24"> </span>set<span class="_ _24"> </span>of<span class="_ _24"> </span>random<span class="_ _24"> </span>v<span class="_ _0"></span>ariables</span></div><div class="t m0 x3e h8 ycc ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x3e h8 ycb ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x3e h8 ycc ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x3e h6 ycb ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _25"> </span><span class="ff1 lsb">and<span class="_ _24"> </span>auxiliary</span></div><div class="t m0 x2f h6 ycd ff1 fs4 fc0 sc0 lsb ws0">v<span class="_ _0"></span>ariables</div><div class="t m0 x3f h8 yce ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x39 h8 ycd ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 x3f h8 yce ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 x3f h6 ycd ff5 fs4 fc0 sc0 ls3 ws0">u<span class="ff1 lsa">,<span class="_ _24"> </span>as<span class="_ _24"> </span>an<span class="_ _24"> </span>extended<span class="_ _24"> </span>form<span class="_ _24"> </span>of<span class="_ _24"> </span>the<span class="_ _24"> </span>uncertainty</span></div><div class="t m0 x2f h6 ycf ff1 fs4 fc0 sc0 ls3 ws0">set<span class="_"> </span><span class="ff6">V<span class="_ _18"></span></span>.</div><div class="t m0 x2e h9 yd0 ff2 fs4 fc0 sc0 lsa ws0">E.<span class="_"> </span>Notations<span class="_"> </span>for<span class="_"> </span>the<span class="_"> </span>Decision<span class="_"> </span>Rule</div><div class="t m0 x2e hb yd1 ff6 fs4 fc0 sc0 ls3 ws0">I</div><div class="t m0 x3a h7 yd2 ff5 fs0 fc0 sc0 ls3 ws0">n</div><div class="t m0 x40 h6 yd3 ff1 fs4 fc0 sc0 lsa ws0">Set<span class="_ _5"> </span>of<span class="_ _17"> </span>auxiliary<span class="_ _17"> </span>variables<span class="_ _5"> </span>that<span class="_ _5"> </span>the<span class="_ _17"> </span><span class="ff5 ls3">n</span>th<span class="_ _17"> </span>linear<span class="_ _17"> </span>afﬁne</div><div class="t m0 x40 h6 yd4 ff1 fs4 fc0 sc0 lsb ws0">function<span class="_"> </span><span class="ff5 ls3">y</span></div><div class="t m0 x41 h7 yd5 ff5 fs0 fc0 sc0 ls3 ws0">n</div><div class="t m0 x7 h8 yd6 ff8 fs4 fc0 sc0 ls3 ws0">(<span class="_ _1"></span><span class="ff5">v</span></div><div class="t m0 x8 h8 yd7 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x8 h8 yd6 ff5 fs4 fc0 sc0 ls26 ws0">v,<span class="_ _18"></span>u</div><div class="t m0 x42 h8 yd7 ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 x42 h6 yd6 ff5 fs4 fc0 sc0 ls3 ws0">u<span class="ff8">)<span class="_ _6"> </span><span class="ff1 ls18">depends<span class="_"> </span>on.</span></span></div><div class="t m0 x2e hb yd8 ff6 fs4 fc0 sc0 ls3 ws0">N</div><div class="t m0 x43 h7 yd9 ff5 fs0 fc0 sc0 ls3 ws0">v</div><div class="t m0 x3b h7 yda ff5 fs0 fc0 sc0 ls3 ws0">s</div><div class="t m0 x40 h6 ydb ff1 fs4 fc0 sc0 lsa ws0">Set<span class="_ _1f"> </span>of<span class="_ _14"> </span>linear<span class="_ _14"> </span>decision<span class="_ _14"> </span>rule<span class="_ _14"> </span>functions<span class="_ _14"> </span><span class="ff5 ls3">y</span></div><div class="t m0 x44 h7 ydc ff5 fs0 fc0 sc0 ls3 ws0">n</div><div class="t m0 x45 h8 ydb ff8 fs4 fc0 sc0 ls3 ws0">(<span class="_ _1"></span><span class="ff5">v</span></div><div class="t m0 x38 h8 ydd ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x38 h8 ydb ff5 fs4 fc0 sc0 ls26 ws0">v,<span class="_ _18"></span>u</div><div class="t m0 x46 h8 ydd ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 x46 h6 ydb ff5 fs4 fc0 sc0 ls3 ws0">u<span class="ff8">)<span class="_ _1f"> </span><span class="ff1 lsf">that</span></span></div><div class="t m0 x40 h6 yde ff1 fs4 fc0 sc0 lsf ws0">depend<span class="_"> </span>on<span class="_"> </span>random<span class="_"> </span>variable<span class="_"> </span><span class="ff5 ls3">v</span></div><div class="t m0 x47 h7 ydf ff5 fs0 fc0 sc0 ls3 ws0">s</div><div class="t m0 x48 h6 ye0 ff1 fs4 fc0 sc0 ls3 ws0">.</div><div class="t m0 x2e hb ye1 ff6 fs4 fc0 sc0 ls3 ws0">N</div><div class="t m0 x43 h7 ye2 ff5 fs0 fc0 sc0 ls3 ws0">u</div><div class="t m0 x3b h7 ye3 ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x40 h6 ye4 ff1 fs4 fc0 sc0 lsa ws0">Set<span class="_ _1f"> </span>of<span class="_ _14"> </span>linear<span class="_ _14"> </span>decision<span class="_ _14"> </span>rule<span class="_ _14"> </span>functions<span class="_ _14"> </span><span class="ff5 ls3">y</span></div><div class="t m0 x44 h7 ye5 ff5 fs0 fc0 sc0 ls3 ws0">n</div><div class="t m0 x45 h8 ye4 ff8 fs4 fc0 sc0 ls3 ws0">(<span class="_ _1"></span><span class="ff5">v</span></div><div class="t m0 x38 h8 ye6 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x38 h8 ye4 ff5 fs4 fc0 sc0 ls26 ws0">v,<span class="_ _18"></span>u</div><div class="t m0 x46 h8 ye6 ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 x46 h6 ye4 ff5 fs4 fc0 sc0 ls3 ws0">u<span class="ff8">)<span class="_ _1f"> </span><span class="ff1 lsf">that</span></span></div><div class="t m0 x40 h6 ye7 ff1 fs4 fc0 sc0 lsa ws0">depend<span class="_"> </span>on<span class="_"> </span>auxiliary<span class="_"> </span>variable<span class="_"> </span><span class="ff5 ls3">u</span></div><div class="t m0 x49 h7 ye8 ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x4a h6 ye9 ff1 fs4 fc0 sc0 ls3 ws0">.</div><div class="t m0 x2e hb yea ff6 fs4 fc0 sc0 ls3 ws0">S</div><div class="t m0 x6 h7 yeb ff5 fs0 fc0 sc0 ls3 ws0">n</div><div class="t m0 x40 h6 yec ff1 fs4 fc0 sc0 lsa ws0">Set<span class="_ _14"> </span>of<span class="_ _14"> </span>random<span class="_ _14"> </span>variables<span class="_ _1f"> </span>that<span class="_ _14"> </span>the<span class="_ _14"> </span><span class="ff5 ls3">n</span>th<span class="_ _14"> </span>linear<span class="_ _14"> </span>afﬁne</div><div class="t m0 x40 h6 yed ff1 fs4 fc0 sc0 lsb ws0">function<span class="_"> </span><span class="ff5 ls3">y</span></div><div class="t m0 x41 h7 yee ff5 fs0 fc0 sc0 ls3 ws0">n</div><div class="t m0 x7 h8 yef ff8 fs4 fc0 sc0 ls3 ws0">(<span class="_ _1"></span><span class="ff5">v</span></div><div class="t m0 x8 h8 yf0 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x8 h8 yef ff5 fs4 fc0 sc0 ls26 ws0">v,<span class="_ _18"></span>u</div><div class="t m0 x42 h8 yf0 ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 x42 h6 yef ff5 fs4 fc0 sc0 ls3 ws0">u<span class="ff8">)<span class="_ _6"> </span><span class="ff1 ls18">depends<span class="_"> </span>on.</span></span></div><div class="t m0 x2e h8 yf1 ff5 fs4 fc0 sc0 ls3 ws0">y</div><div class="t m0 x2e h8 yf2 ff5 fs4 fc0 sc0 ls3 ws0">y</div><div class="t m0 x2e h8 yf1 ff5 fs4 fc0 sc0 ls3 ws0">y<span class="_ _13"></span><span class="ff8">(<span class="_ _1"></span><span class="ff5">v</span></span></div><div class="t m0 x4b h8 yf2 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x4b h6 yf1 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8">)<span class="_ _15"> </span><span class="ff1 lsa">Optimal<span class="_"> </span>recourse<span class="_ _b"> </span>decision<span class="_"> </span>under<span class="_ _9"> </span>wind<span class="_ _9"> </span>power<span class="_ _9"> </span>realiza-</span></span></div><div class="t m0 x40 h6 yf3 ff1 fs4 fc0 sc0 ls3 ws0">tion<span class="_ _9"> </span><span class="ff5">v</span></div><div class="t m0 x4c h8 yf4 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x4c h6 yf3 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff1">.</span></div><div class="t m0 x2e h8 yf5 ff5 fs4 fc0 sc0 ls3 ws0">y</div><div class="t m0 x3a h7 yf6 ff5 fs0 fc0 sc0 ls3 ws0">n</div><div class="t m0 x4d h8 yf7 ff8 fs4 fc0 sc0 ls3 ws0">(<span class="_ _1"></span><span class="ff5">v</span></div><div class="t m0 x4e h8 yf8 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x4e h8 yf7 ff5 fs4 fc0 sc0 ls26 ws0">v,<span class="_ _18"></span>u</div><div class="t m0 x4f h8 yf8 ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 x4f h6 yf7 ff5 fs4 fc0 sc0 ls3 ws0">u<span class="ff8">)<span class="_ _1d"> </span></span>n<span class="ff1 lsa">th<span class="_ _b"> </span>recourse<span class="_ _b"> </span>decision<span class="_ _9"> </span>approximated<span class="_ _9"> </span>by<span class="_ _b"> </span>a<span class="_ _9"> </span>linear<span class="_ _b"> </span>afﬁne</span></div><div class="t m0 x40 h6 yf9 ff1 fs4 fc0 sc0 lsa ws0">function<span class="_ _b"> </span>of<span class="_ _b"> </span>random<span class="_ _26"></span>v<span class="_ _0"></span>ariables<span class="_ _18"></span><span class="ff5 ls3">v</span></div><div class="t m0 x50 h8 yfa ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x50 h6 yf9 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _9"> </span><span class="ff1 lsb">and<span class="_ _26"> </span>auxiliary<span class="_ _b"> </span>variables</span></div><div class="t m0 x40 h8 yfb ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 x40 h8 yfc ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 x40 h6 yfb ff5 fs4 fc0 sc0 ls3 ws0">u<span class="ff1">.</span></div><div class="t m0 x2e h8 yfd ff5 fs4 fc0 sc0 ls3 ws0">y</div><div class="t m0 x3a h7 yfe ff8 fs0 fc0 sc0 ls3 ws0">0</div><div class="t m0 x3a h7 yff ff5 fs0 fc0 sc0 ls3 ws0">n</div><div class="t m0 x40 h6 y100 ff1 fs4 fc0 sc0 lsa ws0">Constant<span class="_ _7"> </span>term<span class="_ _7"> </span>of<span class="_ _8"> </span>the<span class="_ _7"> </span>linear<span class="_ _7"> </span>afﬁne<span class="_ _8"> </span>function<span class="_ _7"> </span><span class="ff5 ls3">y</span></div><div class="t m0 x51 h7 y101 ff5 fs0 fc0 sc0 ls3 ws0">n</div><div class="t m0 x52 h8 y100 ff8 fs4 fc0 sc0 ls3 ws0">(<span class="_ _1"></span><span class="ff5">v</span></div><div class="t m0 x53 h8 y102 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x53 h8 y100 ff5 fs4 fc0 sc0 ls26 ws0">v,<span class="_ _18"></span>u</div><div class="t m0 x54 h8 y102 ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 x54 h8 y100 ff5 fs4 fc0 sc0 ls3 ws0">u<span class="ff8">)</span></div><div class="t m0 x40 h6 y103 ff1 fs4 fc0 sc0 lsa ws0">that<span class="_"> </span>approximates<span class="_"> </span>the<span class="_"> </span><span class="ff5 ls3">n</span><span class="lsb">th<span class="_"> </span>recourse<span class="_"> </span>decision<span class="_"> </span><span class="ff5 ls3">y</span></span></div><div class="t m0 x55 h7 y104 ff5 fs0 fc0 sc0 ls3 ws0">n</div><div class="t m0 x56 h8 y105 ff8 fs4 fc0 sc0 ls3 ws0">(<span class="_ _1"></span><span class="ff5">v</span></div><div class="t m0 x57 h8 y106 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x57 h6 y105 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8">)<span class="ff1">.</span></span></div><div class="t m0 x2e h8 y107 ff5 fs4 fc0 sc0 ls3 ws0">y</div><div class="t m0 x3a h7 y108 ff5 fs0 fc0 sc0 ls3 ws0">u</div><div class="t m0 x3a h7 y109 ff5 fs0 fc0 sc0 ls27 ws0">ni</div><div class="t m0 x40 h6 y10a ff1 fs4 fc0 sc0 lsa ws0">Coefﬁcient<span class="_ _8"> </span>of<span class="_ _7"> </span>the<span class="_ _8"> </span>auxiliary<span class="_ _7"> </span>v<span class="_ _0"></span>ariable<span class="_ _7"> </span><span class="ff5 ls3">u</span></div><div class="t m0 x58 h7 y10b ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x59 h6 y10a ff1 fs4 fc0 sc0 lsb ws0">in<span class="_ _7"> </span>the<span class="_ _8"> </span>linear</div><div class="t m0 x40 h6 y10c ff1 fs4 fc0 sc0 lsa ws0">afﬁne<span class="_ _17"> </span>function<span class="_ _1f"> </span><span class="ff5 ls3">y</span></div><div class="t m0 x5a h7 y10d ff5 fs0 fc0 sc0 ls3 ws0">n</div><div class="t m0 x5b h8 y10e ff8 fs4 fc0 sc0 ls3 ws0">(<span class="_ _1"></span><span class="ff5">v</span></div><div class="t m0 x30 h8 y10f ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x30 h8 y10e ff5 fs4 fc0 sc0 ls26 ws0">v,<span class="_ _18"></span>u</div><div class="t m0 x5c h8 y10f ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 x5c h6 y10e ff5 fs4 fc0 sc0 ls3 ws0">u<span class="ff8">)<span class="_ _1f"> </span><span class="ff1 lsa">that<span class="_ _1f"> </span>approximates<span class="_ _1f"> </span>the<span class="_ _1f"> </span></span></span>n<span class="ff1 lse">th</span></div><div class="t m0 x40 h6 y110 ff1 fs4 fc0 sc0 lsb ws0">recourse<span class="_"> </span>decision<span class="_"> </span><span class="ff5 ls3">y</span></div><div class="t m0 x5d h7 y111 ff5 fs0 fc0 sc0 ls3 ws0">n</div><div class="t m0 x5e h8 y112 ff8 fs4 fc0 sc0 ls3 ws0">(<span class="_ _1"></span><span class="ff5">v</span></div><div class="t m0 x5f h8 y113 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x5f h6 y112 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8">)<span class="ff1">.</span></span></div><div class="t m0 x2e h8 y114 ff5 fs4 fc0 sc0 ls3 ws0">y</div><div class="t m0 x3a h7 y115 ff5 fs0 fc0 sc0 ls3 ws0">v</div><div class="t m0 x3a h7 y116 ff5 fs0 fc0 sc0 ls27 ws0">ns</div><div class="t m0 x40 h6 y117 ff1 fs4 fc0 sc0 lsf ws0">Coefﬁcient<span class="_ _7"> </span>of<span class="_ _17"> </span>the<span class="_ _17"> </span>random<span class="_ _5"> </span>variable<span class="_ _5"> </span><span class="ff5 ls3">v</span></div><div class="t m0 x60 h7 y118 ff5 fs0 fc0 sc0 ls3 ws0">s</div><div class="t m0 x61 h6 y117 ff1 fs4 fc0 sc0 lsb ws0">in<span class="_ _5"> </span>the<span class="_ _17"> </span>linear</div><div class="t m0 x40 h6 y119 ff1 fs4 fc0 sc0 lsa ws0">afﬁne<span class="_ _17"> </span>function<span class="_ _1f"> </span><span class="ff5 ls3">y</span></div><div class="t m0 x5a h7 y11a ff5 fs0 fc0 sc0 ls3 ws0">n</div><div class="t m0 x5b h8 y11b ff8 fs4 fc0 sc0 ls3 ws0">(<span class="_ _1"></span><span class="ff5">v</span></div><div class="t m0 x30 h8 y11c ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x30 h8 y11b ff5 fs4 fc0 sc0 ls26 ws0">v,<span class="_ _18"></span>u</div><div class="t m0 x5c h8 y11c ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 x5c h6 y11b ff5 fs4 fc0 sc0 ls3 ws0">u<span class="ff8">)<span class="_ _1f"> </span><span class="ff1 lsa">that<span class="_ _1f"> </span>approximates<span class="_ _1f"> </span>the<span class="_ _1f"> </span></span></span>n<span class="ff1 lse">th</span></div><div class="t m0 x40 h6 y11d ff1 fs4 fc0 sc0 lsb ws0">recourse<span class="_"> </span>decision<span class="_"> </span>variable<span class="_"> </span><span class="ff5 ls3">y</span></div><div class="t m0 x62 h7 y11e ff5 fs0 fc0 sc0 ls3 ws0">n</div><div class="t m0 x63 h8 y11f ff8 fs4 fc0 sc0 ls3 ws0">(<span class="_ _1"></span><span class="ff5">v</span></div><div class="t m0 x48 h8 y120 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x48 h6 y11f ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8">)<span class="ff1">.</span></span></div><div class="t m0 x2e h9 y121 ff2 fs4 fc0 sc0 lsa ws0">F<span class="_ _4"></span>.<span class="_"> </span>Others</div><div class="t m0 x2e hb y122 ff6 fs4 fc0 sc0 ls3 ws0">M</div><div class="t m0 x64 h7 y123 ff8 fs0 fc0 sc0 ls3 ws0">1</div><div class="t m0 x65 h6 y124 ff1 fs4 fc0 sc0 lsb ws0">Set<span class="_"> </span>of<span class="_"> </span>all<span class="_"> </span>ﬁrst-stage<span class="_"> </span>constraints.</div><div class="t m0 x2e hb y125 ff6 fs4 fc0 sc0 ls3 ws0">M</div><div class="t m0 x64 h7 y126 ff8 fs0 fc0 sc0 ls3 ws0">2</div><div class="t m0 x65 h6 y127 ff1 fs4 fc0 sc0 lsb ws0">Set<span class="_"> </span>of<span class="_"> </span>all<span class="_"> </span>second-stage<span class="_"> </span>constraints.</div><div class="t m0 x2e hb y128 ff6 fs4 fc0 sc0 ls3 ws0">N</div><div class="t m0 x43 h7 y129 ff8 fs0 fc0 sc0 ls3 ws0">1</div><div class="t m0 x65 h6 y12a ff1 fs4 fc0 sc0 lsb ws0">Set<span class="_"> </span>of<span class="_"> </span>all<span class="_"> </span>ﬁrst-stage<span class="_"> </span>decision<span class="_"> </span>variables.</div><div class="t m0 x2e hb y12b ff6 fs4 fc0 sc0 ls3 ws0">N</div><div class="t m0 x43 h7 y12c ff8 fs0 fc0 sc0 ls3 ws0">2</div><div class="t m0 x65 h6 y12d ff1 fs4 fc0 sc0 lsa ws0">Set<span class="_"> </span>of<span class="_"> </span>all<span class="_"> </span>second-stage<span class="_"> </span>decision<span class="_"> </span>variables.</div><div class="t m0 x2e h8 y12e ff5 fs4 fc0 sc0 ls3 ws0">Q<span class="ff8">(<span class="_ _1"></span><span class="ff5">x</span></span></div><div class="t m0 x64 h8 y12f ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 x64 h8 y12e ff5 fs4 fc0 sc0 ls28 ws0">x,<span class="_ _18"></span>v</div><div class="t m0 x66 h8 y12f ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x66 h6 y12e ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8">)<span class="_ _27"> </span><span class="ff1 lsb">Economic<span class="_"> </span>dispatch<span class="_"> </span>cost<span class="_"> </span>associated<span class="_"> </span>with<span class="_"> </span>the<span class="_"> </span>UC<span class="_"> </span>deci-</span></span></div><div class="t m0 x65 h6 y130 ff1 fs4 fc0 sc0 lsb ws0">sion<span class="_ _9"> </span><span class="ff5 ls3">x</span></div><div class="t m0 x67 h8 y131 ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 x67 h6 y130 ff5 fs4 fc0 sc0 ls3 ws0">x<span class="ff1 lsf">,<span class="_"> </span>under<span class="_"> </span>wind<span class="_"> </span>power<span class="_"> </span>outcome<span class="_ _9"> </span></span>v</div><div class="t m0 x68 h8 y131 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x68 h6 y130 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff1">.</span></div><div class="t m0 x2e hb y132 ff6 fs4 fc0 sc0 ls3 ws0">Q<span class="ff8">(<span class="_ _1"></span><span class="ff5">x</span></span></div><div class="t m0 x64 h8 y133 ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 x64 h6 y132 ff5 fs4 fc0 sc0 ls3 ws0">x<span class="ff8">)<span class="_ _28"> </span><span class="ff1 lsb">Expected<span class="_"> </span>recourse<span class="_ _9"> </span>cost<span class="_"> </span>under<span class="_"> </span>the<span class="_ _9"> </span>worst-case<span class="_"> </span>distrib<span class="_ _1"></span>u-</span></span></div><div class="t m0 x65 h6 y134 ff1 fs4 fc0 sc0 lsa ws0">tion.</div><div class="t m0 x69 h8 y135 ff8 fs4 fc0 sc0 ls3 ws0">¯</div><div class="t m0 x2e hb y136 ff6 fs4 fc0 sc0 ls3 ws0">Q<span class="ff8">(<span class="_ _1"></span><span class="ff5">x</span></span></div><div class="t m0 x64 h8 y137 ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 x64 h6 y136 ff5 fs4 fc0 sc0 ls3 ws0">x<span class="ff8">)<span class="_ _28"> </span><span class="ff1 lsa">Approximated<span class="_ _25"> </span>formulation<span class="_ _25"> </span>as<span class="_ _25"> </span>an<span class="_ _24"> </span>upper<span class="_ _25"> </span>bound<span class="_ _25"> </span>of</span></span></div><div class="t m0 x65 hb y138 ff6 fs4 fc0 sc0 ls3 ws0">Q<span class="ff8">(<span class="_ _1"></span><span class="ff5">x</span></span></div><div class="t m0 x6a h8 y139 ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 x6a h6 y138 ff5 fs4 fc0 sc0 ls3 ws0">x<span class="ff8">)<span class="ff1">.</span></span></div><div class="t m0 x6b h6 y13a ff1 fs4 fc0 sc0 ls16 ws0">I.<span class="_ _24"> </span>I</div><div class="t m0 x6c ha y13b ff1 fs5 fc0 sc0 ls29 ws0">NTR<span class="_ _1"></span>ODUCTION</div><div class="t m0 x19 h10 y13c ff4 fs7 fc0 sc0 ls3 ws0">T</div><div class="t m0 x6d h6 y13d ff1 fs4 fc0 sc0 lsa ws0">HE<span class="_ _1f"> </span>rapid<span class="_ _14"> </span>growth<span class="_ _1f"> </span>of<span class="_ _1f"> </span>wind<span class="_ _14"> </span>power<span class="_ _17"> </span>penetration<span class="_ _14"> </span>in<span class="_ _14"> </span>energy</div><div class="t m0 x6d h6 y13e ff1 fs4 fc0 sc0 lsb ws0">systems<span class="_ _26"> </span>has<span class="_ _b"> </span>greatly<span class="_ _b"> </span>reduced<span class="_ _26"> </span>the<span class="_ _b"> </span>consumption<span class="_ _b"> </span>of<span class="_ _26"> </span>fossil<span class="_ _b"> </span>fuels</div><div class="t m0 x19 h6 y13f ff1 fs4 fc0 sc0 lsa ws0">and<span class="_ _b"> </span>the<span class="_ _26"> </span>emission<span class="_ _b"> </span>of<span class="_ _b"> </span>greenhouse<span class="_ _b"> </span>gases.<span class="_ _b"> </span>It<span class="_ _b"> </span>is<span class="_ _b"> </span>also<span class="_ _b"> </span>changing<span class="_ _b"> </span>the<span class="_ _b"> </span>way</div><div class="t m0 x19 h6 y140 ff1 fs4 fc0 sc0 lsf ws0">po<span class="_ _0"></span>wer<span class="_"> </span>systems<span class="_ _6"> </span>are<span class="_ _6"> </span>operated,<span class="_ _6"> </span>because<span class="_ _6"> </span>the<span class="_ _6"> </span>output<span class="_ _6"> </span>of<span class="_ _6"> </span>wind<span class="_ _6"> </span>power</div><div class="t m0 x19 h6 y141 ff1 fs4 fc0 sc0 lsa ws0">is<span class="_ _c"> </span>highly<span class="_"> </span>uncertain<span class="_ _9"> </span>and<span class="_"> </span>intermittent,<span class="_ _9"> </span>and<span class="_"> </span>it<span class="_ _c"> </span>cannot<span class="_"> </span>be<span class="_ _9"> </span>dispatched</div><div class="t m0 x19 h6 y142 ff1 fs4 fc0 sc0 lsb ws0">in<span class="_ _1f"> </span>the<span class="_ _14"> </span>classical<span class="_ _14"> </span>sense<span class="_ _14"> </span>[1].<span class="_ _14"> </span>New<span class="_ _1f"> </span>methods<span class="_ _14"> </span>are<span class="_ _14"> </span>thus<span class="_ _14"> </span>needed<span class="_ _14"> </span>to</div><div class="t m0 x19 h6 y143 ff1 fs4 fc0 sc0 lsa ws0">model<span class="_ _17"> </span>uncertain<span class="_ _1f"> </span>wind<span class="_ _1f"> </span>power<span class="_ _5"> </span>generation<span class="_ _1f"> </span>in<span class="_ _1f"> </span>unit<span class="_ _17"> </span>commitment</div><div class="t m0 x19 h6 y144 ff1 fs4 fc0 sc0 lsb ws0">(UC)<span class="_"> </span>problems.</div><div class="t m0 x1c h6 y145 ff1 fs4 fc0 sc0 lsa ws0">Among<span class="_ _17"> </span>all<span class="_ _17"> </span>uncertainty<span class="_ _17"> </span>models<span class="_ _1f"> </span>for<span class="_ _17"> </span>UC<span class="_ _17"> </span>and<span class="_ _17"> </span>dispatch<span class="_ _1f"> </span>under</div><div class="t m0 x19 h6 y146 ff1 fs4 fc0 sc0 lsa ws0">wind<span class="_ _7"> </span>power<span class="_ _7"> </span>generation,<span class="_ _5"> </span>stochastic<span class="_ _7"> </span>programming<span class="_ _5"> </span>[2]<span class="_ _7"> </span>has<span class="_ _5"> </span>been</div><div class="t m0 x19 h6 y147 ff1 fs4 fc0 sc0 lsb ws0">most<span class="_"> </span>e<span class="_ _1"></span>xtensiv<span class="_ _0"></span>ely<span class="_"> </span>e<span class="_ _1"></span>xplored<span class="_"> </span>during<span class="_ _c"> </span>the<span class="_"> </span>last<span class="_ _9"> </span>decade<span class="_"> </span>[3]–[7].<span class="_ _c"> </span>These</div><div class="t m0 x19 h6 y148 ff1 fs4 fc0 sc0 lsa ws0">studies<span class="_ _7"> </span>ha<span class="_ _1"></span>ve<span class="_ _7"> </span>sho<span class="_ _1"></span>wn<span class="_ _7"> </span>that<span class="_ _7"> </span>the<span class="_ _8"> </span>stochastic<span class="_ _7"> </span>model<span class="_ _7"> </span>can<span class="_ _8"> </span>improve<span class="_ _8"> </span>the</div><div class="t m0 x19 h6 y149 ff1 fs4 fc0 sc0 lsa ws0">expected<span class="_ _9"> </span>performance<span class="_"> </span>of<span class="_ _c"> </span>UC<span class="_ _c"> </span>decisions<span class="_"> </span>under<span class="_ _9"> </span>system<span class="_"> </span>uncertain-</div><div class="t m0 x19 h6 y14a ff1 fs4 fc0 sc0 lsb ws0">ties.<span class="_ _7"> </span>Ho<span class="_ _1"></span>wever<span class="_ _1"></span>,<span class="_ _7"> </span>stochastic<span class="_ _8"> </span>programming<span class="_ _7"> </span>methods<span class="_ _7"> </span>are<span class="_ _7"> </span>computa-</div><div class="t m0 x19 h6 y14b ff1 fs4 fc0 sc0 lsb ws0">tionally<span class="_ _b"> </span>challenging<span class="_ _9"> </span>due<span class="_ _9"> </span>to<span class="_ _b"> </span>the<span class="_ _9"> </span>scenario<span class="_ _9"> </span>representation<span class="_ _b"> </span>of<span class="_ _9"> </span>system</div><div class="t m0 x19 h6 y14c ff1 fs4 fc0 sc0 lsb ws0">uncertainties.<span class="_ _8"> </span>This<span class="_ _7"> </span>dif<span class="_ _1"></span>ﬁculty<span class="_ _7"> </span>is<span class="_ _8"> </span>usually<span class="_ _8"> </span>addressed<span class="_ _7"> </span>by<span class="_ _8"> </span>advanced</div><div class="t m0 x19 h6 y14d ff1 fs4 fc0 sc0 lsb ws0">scenario<span class="_ _8"> </span>selection<span class="_ _8"> </span>algorithms<span class="_ _7"> </span>[6],<span class="_ _6"> </span>[8]<span class="_ _7"> </span>and<span class="_ _6"> </span>decomposition<span class="_ _7"> </span>tech-</div><div class="t m0 x19 h6 y14e ff1 fs4 fc0 sc0 ls3 ws0">niques<span class="_"> </span>[6],<span class="_"> </span>[9].</div><div class="t m0 x1c h6 y14f ff1 fs4 fc0 sc0 lsa ws0">Another<span class="_ _c"> </span>tool<span class="_ _9"> </span>for<span class="_"> </span>optimization<span class="_ _9"> </span>in<span class="_ _c"> </span>the<span class="_ _c"> </span>presence<span class="_ _c"> </span>of<span class="_ _9"> </span>uncertainties</div><div class="t m0 x19 h6 y150 ff1 fs4 fc0 sc0 lsb ws0">is<span class="_ _8"> </span>robust<span class="_ _6"> </span>optimization<span class="_ _8"> </span>[10].<span class="_ _8"> </span>This<span class="_ _8"> </span>method<span class="_ _8"> </span>minimizes<span class="_ _8"> </span>the<span class="_ _7"> </span>w<span class="_ _0"></span>orst-</div><div class="t m0 x19 h6 y151 ff1 fs4 fc0 sc0 lsb ws0">case<span class="_ _6"> </span>total<span class="_ _8"> </span>cost<span class="_ _8"> </span>ov<span class="_ _0"></span>er<span class="_ _6"> </span>all<span class="_ _8"> </span>possible<span class="_ _8"> </span>realizations<span class="_ _6"> </span>within<span class="_ _8"> </span>a<span class="_ _8"> </span>determin-</div><div class="t m0 x19 h6 y152 ff1 fs4 fc0 sc0 lsb ws0">istic<span class="_ _c"> </span>uncertainty<span class="_"> </span>set,<span class="_ _9"> </span>so<span class="_"> </span>decisions<span class="_ _c"> </span>can<span class="_"> </span>be<span class="_ _9"> </span>made<span class="_"> </span>without<span class="_ _c"> </span>knowing</div><div class="t m0 x19 h6 y153 ff1 fs4 fc0 sc0 lsb ws0">the<span class="_ _9"> </span>exact<span class="_ _9"> </span>distribution<span class="_ _9"> </span>of<span class="_ _9"> </span>random<span class="_ _c"> </span>parameters.<span class="_ _9"> </span>Recently<span class="_ _9"> </span>more<span class="_ _c"> </span>em-</div><div class="t m0 x19 h6 y154 ff1 fs4 fc0 sc0 lsa ws0">phasis<span class="_ _9"> </span>has<span class="_ _c"> </span>been<span class="_ _9"> </span>placed<span class="_ _c"> </span>on<span class="_ _9"> </span>the<span class="_"> </span>application<span class="_ _b"> </span>of<span class="_ _c"> </span>robust<span class="_ _9"> </span>optimization</div><div class="t m0 x19 h6 y155 ff1 fs4 fc0 sc0 lsb ws0">in<span class="_ _8"> </span>solving<span class="_ _7"> </span>UC<span class="_ _8"> </span>problems.<span class="_ _8"> </span>In<span class="_ _7"> </span>[11]<span class="_ _6"> </span>and<span class="_ _7"> </span>[12],<span class="_ _8"> </span>for<span class="_ _8"> </span>instance,<span class="_ _7"> </span>robust</div><div class="t m0 x19 h6 y156 ff1 fs4 fc0 sc0 lsa ws0">optimization<span class="_"> </span>is<span class="_ _9"> </span>used<span class="_"> </span>to<span class="_ _c"> </span>optimize<span class="_"> </span>UC<span class="_ _9"> </span>schedules<span class="_"> </span>under<span class="_"> </span>the<span class="_ _9"> </span>worst-</div><div class="t m0 x19 h6 y157 ff1 fs4 fc0 sc0 lsa ws0">case<span class="_ _8"> </span>unit<span class="_ _8"> </span>outage<span class="_ _7"> </span>contingencies.<span class="_ _8"> </span>Zhao<span class="_ _8"> </span><span class="ff2 lsf">et<span class="_ _8"> </span>al.<span class="_ _7"> </span></span>[13]–[18]<span class="_ _6"> </span>modeled</div><div class="t m0 x19 h6 y158 ff1 fs4 fc0 sc0 lsa ws0">the<span class="_"> </span>uncertain<span class="_ _9"> </span>demand<span class="_"> </span>or<span class="_ _c"> </span>renew<span class="_ _1"></span>able<span class="_"> </span>energy<span class="_ _9"> </span>by<span class="_"> </span>the<span class="_"> </span>adjustable<span class="_ _9"> </span>ro-</div><div class="t m0 x19 h6 y159 ff1 fs4 fc0 sc0 lsb ws0">bust<span class="_"> </span>optimization<span class="_"> </span>frame<span class="_ _1"></span>work,<span class="_"> </span>and<span class="_"> </span>Xiong<span class="_"> </span>and<span class="_"> </span>Jirutitijaroen<span class="_"> </span>[19]</div><div class="t m0 x19 h6 y15a ff1 fs4 fc0 sc0 lsa ws0">proposed<span class="_ _c"> </span>a<span class="_"> </span>rob<span class="_ _1"></span>ust<span class="_"> </span>optimization<span class="_ _9"> </span>formulation<span class="_"> </span>to<span class="_ _c"> </span>address<span class="_ _c"> </span>the<span class="_"> </span>com-</div><div class="t m0 x19 h6 y15b ff1 fs4 fc0 sc0 lsa ws0">bined<span class="_ _9"> </span>impact<span class="_"> </span>of<span class="_ _b"> </span>unit<span class="_"> </span>outages<span class="_ _b"> </span>and<span class="_"> </span>load<span class="_ _9"> </span>uncertainty<span class="_ _1"></span>.<span class="_ _9"> </span>Despite<span class="_ _c"> </span>these</div><div class="t m0 x19 h6 y15c ff1 fs4 fc0 sc0 ls3 ws0">efforts,<span class="_ _6"> </span>it<span class="_ _8"> </span>is<span class="_ _7"> </span>dif<span class="_ _1"></span>ﬁcult<span class="_ _7"> </span>to<span class="_ _6"> </span>properly<span class="_ _7"> </span>incorporate<span class="_ _6"> </span>distribution<span class="_ _8"> </span>infor-</div><div class="t m0 x19 h6 y15d ff1 fs4 fc0 sc0 lsb ws0">mation<span class="_ _8"> </span>into<span class="_ _8"> </span>the<span class="_ _7"> </span>rob<span class="_ _1"></span>ust<span class="_ _7"> </span>optimization<span class="_ _6"> </span>model,<span class="_ _7"> </span>and<span class="_ _6"> </span>the<span class="_ _7"> </span>worst-case</div><div class="t m0 x19 h6 y15e ff1 fs4 fc0 sc0 lsb ws0">realization<span class="_ _5"> </span>is<span class="_ _5"> </span>sometimes<span class="_ _5"> </span>too<span class="_ _5"> </span>pessimistic<span class="_ _5"> </span>for<span class="_ _5"> </span>modeling<span class="_ _5"> </span>system</div><div class="t m0 x19 h6 y15f ff1 fs4 fc0 sc0 ls3 ws0">uncertainties,<span class="_ _5"> </span>resulting<span class="_ _5"> </span>in<span class="_ _5"> </span>ov<span class="_ _1"></span>er-conservati<span class="_ _1"></span>ve<span class="_ _5"> </span>solutions.<span class="_ _5"> </span>This<span class="_ _5"> </span>is</div><div class="t m0 x19 h6 y160 ff1 fs4 fc0 sc0 lsb ws0">why<span class="_ _7"> </span>efforts<span class="_ _5"> </span>are<span class="_ _5"> </span>made<span class="_ _5"> </span>in<span class="_ _5"> </span>[20]<span class="_ _5"> </span>to<span class="_ _5"> </span>combine<span class="_ _5"> </span>robust<span class="_ _7"> </span>optimization</div><div class="t m0 x19 h6 y161 ff1 fs4 fc0 sc0 lsb ws0">with<span class="_"> </span>a<span class="_ _8"> </span>stochastic<span class="_ _6"> </span>programming<span class="_ _6"> </span>formulation,<span class="_ _8"> </span>in<span class="_"> </span>order<span class="_ _8"> </span>to<span class="_ _6"> </span>exploit</div><div class="t m0 x19 h6 y162 ff1 fs4 fc0 sc0 lsf ws0">the<span class="_"> </span>advantages<span class="_"> </span>of<span class="_"> </span>both<span class="_"> </span>methods.</div><div class="t m0 x1c h6 y163 ff1 fs4 fc0 sc0 lsb ws0">As<span class="_ _14"> </span>an<span class="_ _14"> </span>intermediate<span class="_ _29"> </span>approach,<span class="_ _14"> </span>distributionally<span class="_ _14"> </span>robust<span class="_ _1f"> </span>opti-</div><div class="t m0 x19 h6 y164 ff1 fs4 fc0 sc0 lsb ws0">mization<span class="_ _5"> </span>can<span class="_ _17"> </span>be<span class="_ _17"> </span>used<span class="_ _5"> </span>to<span class="_ _17"> </span>address<span class="_ _17"> </span>the<span class="_ _17"> </span>limitations<span class="_ _5"> </span>of<span class="_ _17"> </span>stochastic</div><div class="t m0 x19 h6 y165 ff1 fs4 fc0 sc0 lsb ws0">programming<span class="_ _6"> </span>and<span class="_ _8"> </span>robust<span class="_ _6"> </span>optimization.<span class="_ _8"> </span>This<span class="_ _8"> </span>idea<span class="_ _8"> </span>was<span class="_ _6"> </span>ﬁrst<span class="_ _8"> </span>pro-</div><div class="t m0 x19 h6 y166 ff1 fs4 fc0 sc0 lsf ws0">posed<span class="_"> </span>by<span class="_"> </span>Scarf<span class="_ _c"> </span>in<span class="_"> </span>1958<span class="_"> </span>[21],<span class="_ _c"> </span>and<span class="_"> </span>much<span class="_"> </span>progress<span class="_ _c"> </span>has<span class="_"> </span>been<span class="_"> </span>made</div><div class="t m0 x19 h6 y167 ff1 fs4 fc0 sc0 lsb ws0">in<span class="_"> </span>recent<span class="_ _c"> </span>years<span class="_"> </span>[22]–[26].<span class="_ _9"> </span>This<span class="_"> </span>model<span class="_"> </span>minimizes<span class="_ _c"> </span>the<span class="_"> </span>w<span class="_ _0"></span>orst-case</div><div class="t m0 x19 h6 y168 ff1 fs4 fc0 sc0 lsb ws0">expected<span class="_ _8"> </span>cost<span class="_ _7"> </span>ov<span class="_ _1"></span>er<span class="_ _7"> </span>a<span class="_ _7"> </span>set<span class="_ _8"> </span>of<span class="_ _7"> </span>probability<span class="_ _8"> </span>distributions<span class="_ _7"> </span>character<span class="_ _1"></span>-</div><div class="t m0 x19 h6 y169 ff1 fs4 fc0 sc0 lsb ws0">ized<span class="_ _6"> </span>by<span class="_ _6"> </span>statistical<span class="_ _8"> </span>properties,<span class="_ _6"> </span>such<span class="_ _8"> </span>as<span class="_ _6"> </span>the<span class="_ _8"> </span>support<span class="_ _6"> </span>and<span class="_ _8"> </span>moments</div><div class="t m0 x19 h6 y16a ff1 fs4 fc0 sc0 lsa ws0">of<span class="_ _6"> </span>uncertain<span class="_ _8"> </span>variables,<span class="_ _6"> </span>rather<span class="_ _8"> </span>than<span class="_ _8"> </span>assuming<span class="_ _8"> </span>a<span class="_ _8"> </span>single<span class="_ _6"> </span>candidate</div><div class="t m0 x19 h6 y16b ff1 fs4 fc0 sc0 lsb ws0">distribution,<span class="_ _c"> </span>so<span class="_"> </span>it<span class="_ _9"> </span>is<span class="_"> </span>robust<span class="_ _c"> </span>against<span class="_"> </span>the<span class="_ _c"> </span>incompleteness<span class="_"> </span>and<span class="_ _c"> </span>inac-</div><div class="t m0 x19 h6 y16c ff1 fs4 fc0 sc0 lsb ws0">curacy<span class="_"> </span>of<span class="_"> </span>probability<span class="_"> </span>data.<span class="_"> </span>Distribution<span class="_"> </span>information<span class="_"> </span>in<span class="_"> </span>terms<span class="_"> </span>of</div><div class="t m0 x19 h6 y16d ff1 fs4 fc0 sc0 lsa ws0">such<span class="_"> </span>properties<span class="_"> </span>can<span class="_"> </span>be<span class="_"> </span>incorporated<span class="_"> </span>into<span class="_"> </span>the<span class="_"> </span>optimization<span class="_"> </span>prob-</div><div class="t m0 x19 h6 y16e ff1 fs4 fc0 sc0 lsb ws0">lem,<span class="_ _8"> </span>yielding<span class="_ _7"> </span>less<span class="_ _8"> </span>conservati<span class="_ _1"></span>ve<span class="_ _7"> </span>solutions<span class="_ _8"> </span>than<span class="_ _7"> </span>the<span class="_ _8"> </span>distribution-</div><div class="t m0 x19 h6 y16f ff1 fs4 fc0 sc0 lsb ws0">free<span class="_"> </span>optimization.</div><div class="t m0 x1c h6 y170 ff1 fs4 fc0 sc0 lsb ws0">This<span class="_ _7"> </span>paper<span class="_ _7"> </span>presents<span class="_ _7"> </span>a<span class="_ _7"> </span>new<span class="_ _7"> </span>two-stage<span class="_ _7"> </span>distrib<span class="_ _1"></span>utionally<span class="_ _7"> </span>robust</div><div class="t m0 x19 h6 y171 ff1 fs4 fc0 sc0 lsa ws0">optimization<span class="_ _2a"> </span>formulation<span class="_ _2a"> </span>for<span class="_ _2a"> </span>UC<span class="_ _2a"> </span>scheduling<span class="_ _2a"> </span>under<span class="_ _2a"> </span>wind</div><div class="t m0 x19 h6 y172 ff1 fs4 fc0 sc0 lsa ws0">uncertainty<span class="_ _1"></span>.<span class="_ _8"> </span>It<span class="_ _8"> </span>is<span class="_ _8"> </span>assumed<span class="_ _7"> </span>that<span class="_ _8"> </span>the<span class="_ _8"> </span>here-and-now<span class="_ _8"> </span>UC<span class="_ _8"> </span>decisions</div><div class="t m0 x19 h6 y173 ff1 fs4 fc0 sc0 lsa ws0">are<span class="_ _5"> </span>made<span class="_ _5"> </span>before<span class="_ _17"> </span>the<span class="_ _5"> </span>wind<span class="_ _5"> </span>power<span class="_ _5"> </span>generation<span class="_ _5"> </span>is<span class="_ _5"> </span>observed,<span class="_ _5"> </span>and</div><div class="t m0 xf he ya3 ffa fs6 fc1 sc0 ls3 ws0">Authorized licensed use limited to: SICHUAN UNIVERSITY. Downloaded on January 05,2021 at 09:02:32 UTC from IEEE Xplore. Restrictions apply. </div></div><div class="pi" data-data='{"ctm":[1.616162,0.000000,0.000000,1.616162,0.000000,0.000000]}'></div></div>

<div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://csdnimg.cn/release/download_crawler_static/85190235/bg3.jpg"><div class="t m0 x1 h2 y174 ff1 fs0 fc0 sc0 ls2a ws0">XIONG<span class="_"> </span><span class="ff2">et<span class="_"> </span>al.</span><span class="ls2b">:<span class="_"> </span>DISTRIBUTION<span class="_ _0"></span>ALL<span class="_ _3"></span>Y<span class="_ _9"> </span>ROBUST<span class="_"> </span>OPTIMIZA<span class="_ _3"></span>TION<span class="_"> </span>MODEL<span class="_"> </span>FOR<span class="_"> </span>UC<span class="_"> </span>CONSIDERING<span class="_"> </span>UNCER<span class="_ _1"></span>T<span class="_ _1"></span>AIN<span class="_"> </span>WIND<span class="_"> </span>POWER<span class="_"> </span>GENERA<span class="_ _3"></span>TION<span class="_ _2b"> </span>41</span></div><div class="t m0 x1 h6 y13b ff1 fs4 fc0 sc0 lsb ws0">wait-and-see<span class="_ _2c"> </span>economic<span class="_ _2c"> </span>dispatch<span class="_ _2c"> </span>decisions<span class="_ _2a"> </span>are<span class="_ _2c"> </span>determined</div><div class="t m0 x1 h6 y175 ff1 fs4 fc0 sc0 lsa ws0">after<span class="_ _7"> </span>the<span class="_ _7"> </span>observation<span class="_ _7"> </span>of<span class="_ _7"> </span>wind<span class="_ _7"> </span>outcomes.<span class="_ _5"> </span>The<span class="_ _7"> </span>objectiv<span class="_ _0"></span>e<span class="_ _7"> </span>of<span class="_ _7"> </span>this</div><div class="t m0 x1 h6 y176 ff1 fs4 fc0 sc0 lsb ws0">UC<span class="_ _5"> </span>formulation<span class="_ _17"> </span>is<span class="_ _5"> </span>to<span class="_ _17"> </span>minimize<span class="_ _5"> </span>the<span class="_ _17"> </span>worst-case<span class="_ _5"> </span>expected<span class="_ _5"> </span>total</div><div class="t m0 x1 h6 y177 ff1 fs4 fc0 sc0 lsa ws0">cost<span class="_ _7"> </span>over<span class="_ _7"> </span>a<span class="_ _5"> </span>family<span class="_ _5"> </span>of<span class="_ _7"> </span>wind<span class="_ _5"> </span>power<span class="_ _7"> </span>distributions<span class="_ _7"> </span>deﬁned<span class="_ _5"> </span>by<span class="_ _5"> </span>an</div><div class="t m0 x1 h6 y178 ff1 fs4 fc0 sc0 lsb ws0">ambiguity<span class="_"> </span>set.<span class="_"> </span>Unlike<span class="_"> </span>pre<span class="_ _1"></span>vious<span class="_"> </span>studies<span class="_"> </span>applying<span class="_"> </span>decomposition</div><div class="t m0 x1 h6 y179 ff1 fs4 fc0 sc0 lsb ws0">algorithms<span class="_ _c"> </span>to<span class="_ _9"> </span>solve<span class="_ _c"> </span>the<span class="_ _c"> </span>robust<span class="_ _9"> </span>counterpart,<span class="_ _c"> </span>a<span class="_ _c"> </span>linear<span class="_ _c"> </span>decision<span class="_ _9"> </span>rule</div><div class="t m0 x1 h6 y17a ff1 fs4 fc0 sc0 lsb ws0">model<span class="_ _9"> </span>is<span class="_"> </span>utilized<span class="_ _9"> </span>in<span class="_ _9"> </span>this<span class="_"> </span>paper<span class="_ _9"> </span>to<span class="_ _9"> </span>provide<span class="_ _c"> </span>a<span class="_ _9"> </span>slightly<span class="_"> </span>conserv<span class="_ _1"></span>ativ<span class="_ _1"></span>e,</div><div class="t m0 x1 h6 y17b ff1 fs4 fc0 sc0 lsb ws0">yet<span class="_ _17"> </span>tractable<span class="_ _17"> </span>solution<span class="_ _17"> </span>to<span class="_ _17"> </span>the<span class="_ _1f"> </span>proposed<span class="_ _17"> </span>two-stage<span class="_ _17"> </span>formulation.</div><div class="t m0 x1 h6 y17c ff1 fs4 fc0 sc0 lsf ws0">This<span class="_ _8"> </span>method<span class="_ _8"> </span>has<span class="_ _8"> </span>not<span class="_ _8"> </span>been<span class="_ _8"> </span>applied<span class="_ _8"> </span>in<span class="_ _8"> </span>the<span class="_ _8"> </span>area<span class="_ _8"> </span>of<span class="_ _8"> </span>power<span class="_ _6"> </span>system</div><div class="t m0 x1 h6 y17d ff1 fs4 fc0 sc0 lsb ws0">operation,<span class="_ _17"> </span>until<span class="_ _1f"> </span>recently<span class="_ _17"> </span>in<span class="_ _1f"> </span>references<span class="_ _17"> </span>[17]<span class="_ _1f"> </span>and<span class="_ _17"> </span>[18].<span class="_ _17"> </span>Similar</div><div class="t m0 x1 h6 y17e ff1 fs4 fc0 sc0 lsa ws0">assumptions<span class="_ _9"> </span>also<span class="_ _9"> </span>appear<span class="_"> </span>in<span class="_ _b"> </span>[27]<span class="_ _9"> </span>to<span class="_ _c"> </span>model<span class="_ _9"> </span>generation<span class="_ _c"> </span>re-dispatch</div><div class="t m0 x1 h6 y17f ff1 fs4 fc0 sc0 lsa ws0">in<span class="_"> </span>a<span class="_"> </span>robust<span class="_"> </span>chance-constrained<span class="_"> </span>po<span class="_ _1"></span>wer<span class="_"> </span>ﬂow<span class="_ _3"></span>.</div><div class="t m0 x6 h6 y180 ff1 fs4 fc0 sc0 lsa ws0">The<span class="_"> </span>remaining<span class="_ _c"> </span>part<span class="_"> </span>of<span class="_ _c"> </span>this<span class="_"> </span>paper<span class="_ _c"> </span>is<span class="_"> </span>or<span class="_ _1"></span>ganized<span class="_"> </span>as<span class="_"> </span>follo<span class="_ _1"></span>ws.<span class="_"> </span>The</div><div class="t m0 x1 h6 y181 ff1 fs4 fc0 sc0 lsb ws0">distributionally<span class="_"> </span>robust<span class="_"> </span>UC<span class="_"> </span>formulation<span class="_ _6"> </span>for<span class="_ _6"> </span>wind<span class="_ _6"> </span>power<span class="_"> </span>genera-</div><div class="t m0 x1 h6 y182 ff1 fs4 fc0 sc0 lsb ws0">tion,<span class="_ _7"> </span>as<span class="_ _8"> </span>well<span class="_ _7"> </span>as<span class="_ _7"> </span>the<span class="_ _7"> </span>linear<span class="_ _8"> </span>decision<span class="_ _7"> </span>rule<span class="_ _7"> </span>model,<span class="_ _7"> </span>is<span class="_ _8"> </span>discussed<span class="_ _7"> </span>in</div><div class="t m0 x1 h6 y183 ff1 fs4 fc0 sc0 lsa ws0">the<span class="_ _5"> </span>next<span class="_ _7"> </span>section.<span class="_ _5"> </span>Case<span class="_ _5"> </span>studies<span class="_ _5"> </span>conducted<span class="_ _5"> </span>on<span class="_ _5"> </span>two<span class="_ _5"> </span>test<span class="_ _5"> </span>systems</div><div class="t m0 x1 h6 y184 ff1 fs4 fc0 sc0 lsa ws0">are<span class="_ _5"> </span>then<span class="_ _17"> </span>presented<span class="_ _17"> </span>in<span class="_ _17"> </span>Section<span class="_ _17"> </span>II,<span class="_ _17"> </span>followed<span class="_ _5"> </span>by<span class="_ _17"> </span>the<span class="_ _17"> </span>concluding</div><div class="t m0 x1 h6 y185 ff1 fs4 fc0 sc0 lsa ws0">remarks<span class="_"> </span>that<span class="_"> </span>summarize<span class="_"> </span>our<span class="_"> </span>work.</div><div class="t m0 xf h6 y186 ff1 fs4 fc0 sc0 ls28 ws0">II.<span class="_ _24"> </span>A<span class="_"> </span>D</div><div class="t m0 x5 h6 y187 ff1 fs5 fc0 sc0 ls9 ws0">ISTRIBUTION<span class="_ _1"></span>ALL<span class="_ _1"></span>Y<span class="_ _6"> </span><span class="fs4 ls3">R<span class="_ _1"></span><span class="fs5 ls2c">OBUST<span class="_ _6"> </span><span class="fs4 ls3">O</span><span class="ls9">PTIMIZA<span class="_ _3"></span>TION</span></span></span></div><div class="t m0 x4b h6 y188 ff1 fs4 fc0 sc0 ls3 ws0">F<span class="fs5 ls9">RAMEWORK<span class="_ _c"> </span>FOR<span class="_ _6"> </span></span><span class="lsa">UC<span class="_"> </span>W</span><span class="fs5">ITH<span class="_ _6"> </span></span>W<span class="fs5 ls29">IND<span class="_ _c"> </span></span>P<span class="fs5 ls2d">OW<span class="_ _13"></span>E<span class="_ _13"></span>R<span class="_ _6"> </span></span>G<span class="fs5 ls9">ENERA<span class="_ _1"></span>TION</span></div><div class="t m0 x6 h6 y189 ff1 fs4 fc0 sc0 ls3 ws0">This<span class="_ _5"> </span>section<span class="_ _5"> </span>presents<span class="_ _5"> </span>a<span class="_ _5"> </span>distributionally<span class="_ _5"> </span>robust<span class="_ _7"> </span>optimization</div><div class="t m0 x1 h6 y18a ff1 fs4 fc0 sc0 lsa ws0">formulation<span class="_ _8"> </span>for<span class="_ _8"> </span>day-ahead<span class="_ _7"> </span>UC<span class="_ _6"> </span>optimization<span class="_ _7"> </span>in<span class="_ _8"> </span>the<span class="_ _8"> </span>presence<span class="_ _8"> </span>of</div><div class="t m0 x1 h6 y18b ff1 fs4 fc0 sc0 lsa ws0">system<span class="_"> </span>uncertainties.<span class="_"> </span>W<span class="_ _1"></span>e<span class="_"> </span>mainly<span class="_"> </span>focus<span class="_ _6"> </span>on<span class="_ _6"> </span>modeling<span class="_ _6"> </span>the<span class="_"> </span>impact</div><div class="t m0 x1 h6 y18c ff1 fs4 fc0 sc0 lsa ws0">of<span class="_ _9"> </span>uncertain<span class="_"> </span>wind<span class="_ _9"> </span>po<span class="_ _0"></span>wer<span class="_ _9"> </span>in<span class="_"> </span>UC<span class="_ _9"> </span>scheduling,<span class="_ _9"> </span>because<span class="_"> </span>the<span class="_ _9"> </span>forecast</div><div class="t m0 x1 h6 y18d ff1 fs4 fc0 sc0 lsa ws0">error<span class="_ _1f"> </span>of<span class="_ _14"> </span>wind<span class="_ _14"> </span>power<span class="_ _1f"> </span>generation<span class="_ _14"> </span>is<span class="_ _14"> </span>much<span class="_ _14"> </span>larger<span class="_ _1f"> </span>than<span class="_ _14"> </span>that<span class="_ _14"> </span>of</div><div class="t m0 x1 h6 y18e ff1 fs4 fc0 sc0 lsb ws0">load<span class="_ _17"> </span>[28].<span class="_ _17"> </span>A<span class="_ _1f"> </span>two-stage<span class="_ _17"> </span>UC<span class="_ _17"> </span>formulation<span class="_ _1f"> </span>for<span class="_ _17"> </span>UC<span class="_ _17"> </span>optimization</div><div class="t m0 x1 h6 y18f ff1 fs4 fc0 sc0 lsb ws0">under<span class="_ _9"> </span>volatile<span class="_ _9"> </span>wind<span class="_ _9"> </span>energy<span class="_ _9"> </span>is<span class="_ _9"> </span>presented<span class="_ _c"> </span>in<span class="_ _9"> </span>the<span class="_ _c"> </span>follo<span class="_ _0"></span>wing<span class="_ _9"> </span>section.</div><div class="t m0 x1 h6 y190 ff1 fs4 fc0 sc0 lsb ws0">In<span class="_ _8"> </span>this<span class="_ _7"> </span>formulation,<span class="_ _8"> </span>the<span class="_ _7"> </span>uncertainty<span class="_ _8"> </span>of<span class="_ _8"> </span>wind<span class="_ _7"> </span>po<span class="_ _0"></span>wer<span class="_ _8"> </span>is<span class="_ _7"> </span>captured</div><div class="t m0 x1 h6 y191 ff1 fs4 fc0 sc0 ls18 ws0">by<span class="_ _5"> </span>an<span class="_ _5"> </span><span class="ff2 lsb">ambiguity<span class="_ _5"> </span>set<span class="ff1">,<span class="_ _5"> </span>which<span class="_ _5"> </span>is<span class="_ _5"> </span>discussed<span class="_ _5"> </span>in<span class="_ _5"> </span>Section<span class="_ _5"> </span>II-B.<span class="_ _17"> </span>The</span></span></div><div class="t m0 x1 h6 y192 ff1 fs4 fc0 sc0 lsb ws0">last<span class="_"> </span>section<span class="_ _8"> </span>presents<span class="_"> </span>the<span class="_ _8"> </span>deriv<span class="_ _1"></span>ation<span class="_ _6"> </span>of<span class="_ _6"> </span>a<span class="_ _6"> </span>tractable<span class="_ _8"> </span>mixed-inte<span class="_ _1"></span>ger</div><div class="t m0 x1 h6 y193 ff1 fs4 fc0 sc0 lsa ws0">linear<span class="_ _5"> </span>programming<span class="_ _5"> </span>formulation<span class="_ _17"> </span>based<span class="_ _5"> </span>on<span class="_ _17"> </span>the<span class="_ _5"> </span>linear<span class="_ _17"> </span>decision</div><div class="t m0 x1 h6 y194 ff1 fs4 fc0 sc0 lsb ws0">rule<span class="_"> </span>approximation.</div><div class="t m0 x1 h9 y195 ff2 fs4 fc0 sc0 lsa ws0">A.<span class="_ _25"> </span>UC<span class="_"> </span>Under<span class="_"> </span>W<span class="_ _1"></span>ind<span class="_"> </span>P<span class="_ _3"></span>ower<span class="_"> </span>Uncertainty</div><div class="t m0 x6 h6 y196 ff1 fs4 fc0 sc0 lsa ws0">In<span class="_ _8"> </span>this<span class="_ _8"> </span>paper,<span class="_ _6"> </span>the<span class="_ _7"> </span>uncertain<span class="_ _8"> </span>wind<span class="_ _8"> </span>power<span class="_ _6"> </span>generation,<span class="_ _7"> </span>denoted</div><div class="t m0 x1 h6 y197 ff1 fs4 fc0 sc0 ls18 ws0">by<span class="_ _1f"> </span><span class="ff8 ls3">˜<span class="_ _2d"></span><span class="ff5">w</span></span></div><div class="t m0 x6e h7 y198 ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 xf h6 y199 ff1 fs4 fc0 sc0 lsa ws0">,<span class="_"> </span>is<span class="_"> </span>e<span class="_ _1"></span>xpressed<span class="_"> </span>as<span class="_"> </span>the<span class="_"> </span>sum<span class="_ _c"> </span>of<span class="_"> </span>the<span class="_"> </span>e<span class="_ _1"></span>xpected<span class="_"> </span>value<span class="_ _1f"> </span><span class="ff8 ls3">¯<span class="_ _2d"></span><span class="ff5">w</span></span></div><div class="t m0 x6f h7 y198 ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 x70 h6 y199 ff1 fs4 fc0 sc0 ls18 ws0">and<span class="_"> </span>a</div><div class="t m0 x1 h6 y19a ff1 fs4 fc0 sc0 lsb ws0">random<span class="_"> </span>term<span class="_"> </span>for<span class="_"> </span>forecast<span class="_"> </span>error<span class="_ _6"> </span><span class="ff8 ls3">˜<span class="_ _23"></span><span class="ff5">v</span></span></div><div class="t m0 x5e h7 y19b ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 x71 h6 y19c ff1 fs4 fc0 sc0 ls18 ws0">,<span class="_"> </span>giv<span class="_ _1"></span>en<span class="_"> </span>as</div><div class="t m0 x72 h8 y19d ff8 fs4 fc0 sc0 ls3 ws0">˜<span class="_ _2d"></span><span class="ff5">w</span></div><div class="t m0 x5 h7 y19e ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 xd h8 y19f ff8 fs4 fc0 sc0 ls2e ws0">=¯<span class="_ _2e"></span><span class="ff5 ls3">w</span></div><div class="t m0 x73 h7 y19e ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 x8 h8 y19f ff8 fs4 fc0 sc0 ls2f ws0">+˜<span class="_ _2f"></span><span class="ff5 ls3">v</span></div><div class="t m0 x9 h7 y19e ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 x74 h6 y19f ff6 fs4 fc0 sc0 ls3 ws0">∀<span class="ff5">r<span class="_ _8"> </span></span><span class="ls23">∈R<span class="_ _2a"> </span>∀<span class="_ _1c"></span><span class="ff5 ls3">t<span class="_ _6"> </span><span class="ff6 ls1f">∈T<span class="_ _30"> </span><span class="ff1 lsb">(1)</span></span></span></span></div><div class="t m0 x1 h6 y1a0 ff1 fs4 fc0 sc0 lsa ws0">where<span class="_ _7"> </span><span class="ff6 ls3">R<span class="_ _8"> </span></span><span class="lsf">denotes<span class="_ _7"> </span>the<span class="_ _8"> </span>set<span class="_ _7"> </span>of<span class="_ _8"> </span>all<span class="_ _7"> </span>wind<span class="_ _7"> </span>ener<span class="_ _1"></span>gy<span class="_ _7"> </span>sources,<span class="_ _7"> </span>and<span class="_ _8"> </span><span class="ff6 ls3">T<span class="_ _25"> </span></span><span class="lse">is</span></span></div><div class="t m0 x1 h6 y1a1 ff1 fs4 fc0 sc0 lsa ws0">the<span class="_"> </span>set<span class="_"> </span>of<span class="_"> </span>all<span class="_"> </span>time<span class="_"> </span>steps<span class="_"> </span>considered<span class="_"> </span>in<span class="_"> </span>the<span class="_"> </span>operation<span class="_"> </span>horizon.</div><div class="t m0 x6 h6 y1a2 ff1 fs4 fc0 sc0 lsa ws0">The<span class="_ _8"> </span>proposed<span class="_ _8"> </span>UC<span class="_ _8"> </span>model<span class="_ _7"> </span>can<span class="_ _6"> </span>be<span class="_ _7"> </span>e<span class="_ _1"></span>xpressed<span class="_ _7"> </span>as<span class="_ _6"> </span>a<span class="_ _7"> </span>typical<span class="_ _6"> </span>two-</div><div class="t m0 x1 h6 y1a3 ff1 fs4 fc0 sc0 lsb ws0">stage<span class="_ _7"> </span>formulation.<span class="_ _5"> </span>The<span class="_ _7"> </span>ﬁrst-stage<span class="_ _5"> </span>problem<span class="_ _7"> </span>(2)–(4)<span class="_ _5"> </span>determines</div><div class="t m0 x1 h6 y1a4 ff1 fs4 fc0 sc0 lsa ws0">the<span class="_ _7"> </span>here-and-no<span class="_ _1"></span>w<span class="_ _7"> </span>UC<span class="_ _7"> </span>status,<span class="_ _7"> </span>denoted<span class="_ _8"> </span>by<span class="_ _7"> </span>binary<span class="_ _7"> </span>decision<span class="_ _7"> </span>v<span class="_ _1"></span>ari-</div><div class="t m0 x1 h6 y1a5 ff1 fs4 fc0 sc0 lsf ws0">ables<span class="_ _9"> </span><span class="ff5 ls3">x</span></div><div class="t m0 x75 h8 y1a6 ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 x75 h6 y1a5 ff5 fs4 fc0 sc0 ls3 ws0">x<span class="ff1 lsa">,<span class="_"> </span>before<span class="_"> </span>the<span class="_"> </span>realization<span class="_"> </span>of<span class="_"> </span>future<span class="_"> </span>wind<span class="_"> </span>power<span class="_"> </span>outcomes</span></div><div class="t m0 x2b h8 y1a7 ff8 fs4 fc0 sc0 lsa ws0">min<span class="_ _9"> </span>sup</div><div class="t m0 x76 hf y1a8 ff9 fs0 fc0 sc0 ls3 ws0">P<span class="_ _18"> </span><span class="ff6">∈</span>F</div><div class="t m0 x77 hc y1a9 ff7 fs4 fc0 sc0 ls3 ws0">E</div><div class="t m0 x5d hd y1aa ff9 fs0 fc0 sc0 ls3 ws0">P</div><div class="t m0 x31 hb y1a9 ff6 fs4 fc0 sc0 ls3 ws0">{<span class="ff5">Q<span class="ff8">(<span class="_ _1"></span><span class="ff5">x</span></span></span></div><div class="t m0 x78 h8 y1ab ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 x78 h8 y1a9 ff5 fs4 fc0 sc0 ls28 ws0">x,</div><div class="t m0 x79 h8 y1ab ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x62 h8 y1a9 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x62 h8 y1ab ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x79 h6 y1a9 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8">)<span class="ff6">}<span class="_ _31"> </span><span class="ff1">(2)</span></span></span></div><div class="t m0 x7a h6 y1ac ff1 fs4 fc0 sc0 ls28 ws0">s.t.<span class="_ _6"> </span><span class="ff5 ls3">x</span></div><div class="t m0 x7b h8 y1ad ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 x7b h6 y1ac ff5 fs4 fc0 sc0 ls3 ws0">x<span class="_ _6"> </span><span class="ff6">∈<span class="_ _6"> </span><span class="ff7">T<span class="_ _26"> </span></span></span>,<span class="_ _32"> </span><span class="ff1 lsb">(3)</span></div><div class="t m0 x7b h8 y1ae ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 x7b h8 y1af ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 x7c hb y1ae ff5 fs4 fc0 sc0 ls3 ws0">x<span class="_ _6"> </span><span class="ff6 ls23">∈{<span class="_ _1c"></span><span class="ff8 ls3">0<span class="ff5">,<span class="_ _b"> </span></span>1<span class="ff6">}</span></span></span></div><div class="t m0 x7d hf y1b0 ff6 fs0 fc0 sc0 ls2b ws0">|K|×|T<span class="_ _9"> </span>|</div><div class="t m0 x7e h6 y1b1 ff1 fs4 fc0 sc0 ls3 ws0">(4)</div><div class="t m0 x1 h6 y1b2 ff1 fs4 fc0 sc0 lsa ws0">where<span class="_"> </span><span class="ff6 ls3">K<span class="_ _6"> </span></span>is<span class="_"> </span>the<span class="_"> </span>set<span class="_"> </span>of<span class="_"> </span>all<span class="_"> </span>generators.<span class="_"> </span>The<span class="_"> </span>set<span class="_"> </span><span class="ff7 ls3">T<span class="_ _5"> </span></span><span class="lsb">in<span class="_"> </span>(3)<span class="_"> </span>represents</span></div><div class="t m0 x1 h6 y1b3 ff1 fs4 fc0 sc0 lsb ws0">the<span class="_ _7"> </span>feasibility<span class="_ _8"> </span>region<span class="_ _7"> </span>of<span class="_ _6"> </span><span class="ff5 ls3">x</span></div><div class="t m0 x76 h8 y1b4 ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 x76 h6 y1b3 ff5 fs4 fc0 sc0 ls3 ws0">x<span class="_"> </span><span class="ff1 lsa">deﬁned<span class="_ _7"> </span>by<span class="_ _8"> </span>minimum<span class="_ _7"> </span>up/down<span class="_ _8"> </span>time</span></div><div class="t m0 x1 h6 y1b5 ff1 fs4 fc0 sc0 lsb ws0">constraints.<span class="_ _b"> </span>The<span class="_ _9"> </span>objective<span class="_ _b"> </span>function<span class="_ _9"> </span>minimizes<span class="_ _b"> </span>the<span class="_ _9"> </span>expectation<span class="_ _9"> </span>of</div><div class="t m0 x1 h6 y1b6 ff1 fs4 fc0 sc0 lsb ws0">function<span class="_ _9"> </span><span class="ff5 ls3">Q<span class="ff8">(<span class="_ _0"></span><span class="ff5">x</span></span></span></div><div class="t m0 x40 h8 y1b7 ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 x7f h8 y1b6 ff5 fs4 fc0 sc0 ls28 ws0">x,</div><div class="t m0 x80 h8 y1b7 ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x80 h8 y1b6 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x80 h8 y1b7 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x80 h6 y1b6 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8">)<span class="_ _9"> </span><span class="ff1 lsb">under<span class="_"> </span>a<span class="_ _b"> </span>distribution<span class="_ _9"> </span></span><span class="ff7">P<span class="_ _26"> </span><span class="ff1">,<span class="_ _9"> </span>which<span class="_"> </span>is<span class="_ _9"> </span>the<span class="_ _9"> </span>worst-case</span></span></span></div><div class="t m0 x13 h6 y1b8 ff1 fs4 fc0 sc0 ls3 ws0">distribution<span class="_ _6"> </span>over<span class="_ _6"> </span>an<span class="_ _8"> </span>ambiguity<span class="_ _7"> </span>set<span class="_ _6"> </span><span class="ff7">F<span class="_ _18"> </span></span><span class="lsa">.<span class="_ _8"> </span>The<span class="_ _8"> </span>function<span class="_ _7"> </span></span><span class="ff5">Q<span class="ff8">(<span class="_ _1"></span><span class="ff5">x</span></span></span></div><div class="t m0 x81 h8 y1b9 ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 x81 h8 y1b8 ff5 fs4 fc0 sc0 ls28 ws0">x,<span class="_ _18"></span>v</div><div class="t m0 x82 h8 y1b9 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x82 h6 y1b8 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8">)<span class="_ _8"> </span><span class="ff1 lsb">in-</span></span></div><div class="t m0 x13 h6 y1ba ff1 fs4 fc0 sc0 lsb ws0">dicates<span class="_ _c"> </span>the<span class="_ _c"> </span>economic<span class="_"> </span>dispatch<span class="_ _9"> </span>cost<span class="_"> </span>associated<span class="_ _9"> </span>with<span class="_"> </span>UC<span class="_ _9"> </span>decision</div><div class="t m0 x13 h8 y1bb ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 x13 h8 y1bc ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 x13 h6 y1bb ff5 fs4 fc0 sc0 ls3 ws0">x<span class="ff1 lsf">,<span class="_ _7"> </span>under<span class="_ _5"> </span>wind<span class="_ _5"> </span>power<span class="_ _7"> </span>outcome<span class="_ _8"> </span></span>v</div><div class="t m0 x83 h8 y1bc ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x83 h6 y1bb ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff1 lsf">,<span class="_ _7"> </span>which<span class="_ _5"> </span>can<span class="_ _5"> </span>be<span class="_ _7"> </span>calculated<span class="_ _5"> </span>by</span></div><div class="t m0 x13 h6 y1bd ff1 fs4 fc0 sc0 lsa ws0">solving<span class="_"> </span>the<span class="_"> </span>second-stage<span class="_"> </span>problem</div><div class="t m0 x84 h8 y1be ff5 fs4 fc0 sc0 ls3 ws0">Q<span class="ff8">(<span class="_ _1"></span><span class="ff5">x</span></span></div><div class="t m0 x1d h8 y1bf ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 x1d h8 y1be ff5 fs4 fc0 sc0 ls28 ws0">x,<span class="_ _18"></span>v</div><div class="t m0 x85 h8 y1bf ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x85 h8 y1be ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8 ls18">)<span class="_ _6"> </span>=<span class="_ _6"> </span>min</span></div><div class="t m0 x86 h11 y1c0 ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 x86 hf y1c1 ff5 fs0 fc0 sc0 ls3 ws0">t<span class="_ _13"></span><span class="ff6 ls30">∈T</span></div><div class="t m0 x87 h11 y1c2 ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 x88 h11 y1c3 ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 x88 hf y1c4 ff5 fs0 fc0 sc0 ls3 ws0">k<span class="_ _33"></span><span class="ff6 ls30">∈K</span></div><div class="t m0 x89 h8 y1c5 ff8 fs4 fc0 sc0 ls3 ws0">(<span class="ff5">c</span></div><div class="t m0 x8a h7 y1c6 ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 x8b h8 y1c5 ff8 fs4 fc0 sc0 ls3 ws0">+<span class="_ _c"> </span><span class="ff5">z</span></div><div class="t m0 x8c h7 y1c6 ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 x8d h8 y1c5 ff8 fs4 fc0 sc0 ls31 ws0">)+</div><div class="t m0 x8e h11 y1c7 ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 x8f hf y1c4 ff5 fs0 fc0 sc0 ls3 ws0">b<span class="_ _13"></span><span class="ff6 ls30">∈B</span></div><div class="t m0 x90 hb y1c5 ff5 fs4 fc0 sc0 ls3 ws0">V<span class="_ _1f"> </span><span class="ff6">·<span class="_ _c"> </span></span>l</div><div class="t m0 x91 h7 y1c6 ff5 fs0 fc0 sc0 ls17 ws0">bt</div><div class="t m0 x92 h11 y1c2 ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 x93 h6 y1c8 ff1 fs4 fc0 sc0 ls3 ws0">(5)</div><div class="t m0 x84 h6 y1c9 ff1 fs4 fc0 sc0 ls28 ws0">s.t.<span class="_ _b"> </span><span class="ff5 ls3">z</span></div><div class="t m0 x18 h7 y1ca ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 x94 hb y1cb ff6 fs4 fc0 sc0 ls3 ws0">≥<span class="_ _6"> </span><span class="ff5">C</span></div><div class="t m0 x95 h7 y1cc ff5 fs0 fc0 sc0 ls3 ws0">s</div><div class="t m0 x96 h7 y1cd ff5 fs0 fc0 sc0 ls3 ws0">k</div><div class="t m0 x97 h8 y1cb ff8 fs4 fc0 sc0 ls3 ws0">(<span class="ff5">x</span></div><div class="t m0 x98 h7 y1ca ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 x99 hb y1cb ff6 fs4 fc0 sc0 ls3 ws0">−<span class="_ _c"> </span><span class="ff5">x</span></div><div class="t m0 x88 hf y1ce ff5 fs0 fc0 sc0 ls3 ws0">k<span class="_ _33"></span><span class="ff8">(<span class="_ _13"></span></span>t<span class="_ _13"></span><span class="ff6">−<span class="ff8 ls32">1)</span></span></div><div class="t m0 x9a h6 y1cb ff8 fs4 fc0 sc0 ls3 ws0">)<span class="_ _f"> </span><span class="ff6">∀<span class="ff5">k<span class="_ _8"> </span></span><span class="ls1f">∈K<span class="_ _1c"></span><span class="ff5 ls33">,t<span class="_ _34"></span><span class="ff6 ls1f">∈T<span class="_ _35"> </span><span class="ff1 lsb">(6)</span></span></span></span></span></div><div class="t m0 x9b h8 y1cf ff5 fs4 fc0 sc0 ls3 ws0">c</div><div class="t m0 x15 h7 y1d0 ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 x9c hb y1d1 ff6 fs4 fc0 sc0 ls3 ws0">≥<span class="_ _6"> </span><span class="ff5">a</span></div><div class="t m0 x9d h7 y1d2 ff5 fs0 fc0 sc0 ls3 ws0">θ</div><div class="t m0 x9d h7 y1d3 ff5 fs0 fc0 sc0 ls3 ws0">k</div><div class="t m0 x9e h8 y1d1 ff5 fs4 fc0 sc0 ls3 ws0">p</div><div class="t m0 x9f h7 y1d2 ff5 fs0 fc0 sc0 ls3 ws0">t</div><div class="t m0 x9f h7 y1d3 ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 xa0 h8 y1d1 ff8 fs4 fc0 sc0 ls3 ws0">+<span class="_ _c"> </span><span class="ff5">b</span></div><div class="t m0 xa1 h7 y1d2 ff5 fs0 fc0 sc0 ls3 ws0">θ</div><div class="t m0 xa1 h7 y1d3 ff5 fs0 fc0 sc0 ls3 ws0">k</div><div class="t m0 x6b h8 y1d1 ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 xa2 h7 y1d0 ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 xa3 h6 y1d1 ff6 fs4 fc0 sc0 ls3 ws0">∀<span class="ff5">θ<span class="_ _8"> </span></span>∈<span class="_ _6"> </span><span class="ff8">Θ<span class="ff5 ls34">,k<span class="_ _36"></span><span class="ff6 ls1f">∈K<span class="_ _1c"></span><span class="ff5 ls33">,t<span class="_ _34"></span><span class="ff6 ls1f">∈T<span class="_ _37"> </span><span class="ff1 lsb">(7)</span></span></span></span></span></span></div><div class="t m0 x6d h11 y1d4 ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 x6d hf y1d5 ff5 fs0 fc0 sc0 ls3 ws0">k<span class="_ _33"></span><span class="ff6 ls30">∈K</span></div><div class="t m0 xa4 h8 y1d6 ff5 fs4 fc0 sc0 ls3 ws0">p</div><div class="t m0 xa5 h7 y1d7 ff5 fs0 fc0 sc0 ls3 ws0">t</div><div class="t m0 xa5 h7 y1d8 ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 x96 h8 y1d6 ff8 fs4 fc0 sc0 ls3 ws0">+</div><div class="t m0 xa6 h11 y1d9 ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 xa6 hf y1da ff5 fs0 fc0 sc0 ls3 ws0">r<span class="_ _33"></span><span class="ff6 ls30">∈R</span></div><div class="t m0 xa7 h8 y1d6 ff5 fs4 fc0 sc0 ls3 ws0">p</div><div class="t m0 xa8 h7 y1d7 ff5 fs0 fc0 sc0 ls3 ws0">w</div><div class="t m0 xa8 h7 y1d8 ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 x88 h8 y1d6 ff8 fs4 fc0 sc0 ls3 ws0">+</div><div class="t m0 xa9 h11 y1d9 ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 x6c hf y1d5 ff5 fs0 fc0 sc0 ls3 ws0">b<span class="_ _13"></span><span class="ff6 ls30">∈B</span></div><div class="t m0 xaa h8 y1d6 ff5 fs4 fc0 sc0 ls3 ws0">l</div><div class="t m0 xab h7 y1db ff5 fs0 fc0 sc0 ls17 ws0">bt</div><div class="t m0 xac h8 y1d6 ff8 fs4 fc0 sc0 ls3 ws0">=</div><div class="t m0 x8c h11 y1d9 ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 xad hf y1d5 ff5 fs0 fc0 sc0 ls3 ws0">b<span class="_ _13"></span><span class="ff6 ls30">∈B</span></div><div class="t m0 xae h8 y1d6 ff5 fs4 fc0 sc0 ls3 ws0">D</div><div class="t m0 x8e h7 y1db ff5 fs0 fc0 sc0 ls17 ws0">bt</div><div class="t m0 xaf h6 y1d6 ff6 fs4 fc0 sc0 ls3 ws0">∀<span class="ff5">t<span class="_ _6"> </span></span><span class="ls23">∈T<span class="_ _38"> </span></span><span class="ff1">(8)</span></div><div class="t m0 x6d h11 y1dc ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 x6d hf y1dd ff5 fs0 fc0 sc0 ls3 ws0">b<span class="_ _13"></span><span class="ff6 ls30">∈B</span></div><div class="t m0 xa4 h8 y1de ff5 fs4 fc0 sc0 ls3 ws0">K</div><div class="t m0 x25 h7 y1df ff5 fs0 fc0 sc0 ls19 ws0">fb</div><div class="t m0 xb0 h11 y1e0 ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 xb1 h11 y1e1 ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 xb2 hf y1dd ff5 fs0 fc0 sc0 ls3 ws0">k<span class="_ _33"></span><span class="ff6 ls30">∈K</span></div><div class="t m0 x99 h12 y1e2 ff5 fs8 fc0 sc0 ls3 ws0">b</div><div class="t m0 xa8 h8 y1de ff5 fs4 fc0 sc0 ls3 ws0">p</div><div class="t m0 xb3 h7 y1e3 ff5 fs0 fc0 sc0 ls3 ws0">t</div><div class="t m0 xb3 h7 y1e4 ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 xb4 h8 y1de ff8 fs4 fc0 sc0 ls3 ws0">+</div><div class="t m0 xb5 h11 y1e5 ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 xb6 hf y1e6 ff5 fs0 fc0 sc0 ls3 ws0">r<span class="_ _33"></span><span class="ff6 ls30">∈R</span></div><div class="t m0 x83 h12 y1e7 ff5 fs8 fc0 sc0 ls3 ws0">b</div><div class="t m0 xb7 h8 y1de ff5 fs4 fc0 sc0 ls3 ws0">p</div><div class="t m0 xb8 h7 y1e3 ff5 fs0 fc0 sc0 ls3 ws0">w</div><div class="t m0 xb8 h7 y1e4 ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 xad h8 y1de ff8 fs4 fc0 sc0 ls3 ws0">+<span class="_ _c"> </span><span class="ff5">l</span></div><div class="t m0 xb9 h7 y1df ff5 fs0 fc0 sc0 ls17 ws0">bt</div><div class="t m0 xba hb y1de ff6 fs4 fc0 sc0 ls3 ws0">−<span class="_ _c"> </span><span class="ff5">D</span></div><div class="t m0 xaf h7 y1df ff5 fs0 fc0 sc0 ls17 ws0">bt</div><div class="t m0 xbb h11 y1e0 ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 x91 hb y1e8 ff6 fs4 fc0 sc0 ls3 ws0">≤<span class="_ _6"> </span><span class="ff5">F</span></div><div class="t m0 xbc h7 y1df ff5 fs0 fc0 sc0 ls3 ws0">f</div><div class="t m0 xbd h6 y1e9 ff6 fs4 fc0 sc0 ls3 ws0">∀<span class="ff5">f<span class="_ _5"> </span></span><span class="ls1f">∈F<span class="_ _1a"></span><span class="ff5 ls3">,<span class="_ _b"> </span><span class="ff6">∀</span>t<span class="_ _6"> </span><span class="ff6 ls1f">∈T<span class="_ _39"> </span></span><span class="ff1">(9)</span></span></span></div><div class="t m0 x6d hb y1ea ff6 fs4 fc0 sc0 ls3 ws0">−</div><div class="t m0 xbe h11 y1eb ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 xbf hf y1ec ff5 fs0 fc0 sc0 ls3 ws0">b<span class="_ _13"></span><span class="ff6 ls30">∈B</span></div><div class="t m0 xc0 h8 y1ed ff5 fs4 fc0 sc0 ls3 ws0">K</div><div class="t m0 xb0 h7 y1ee ff5 fs0 fc0 sc0 ls19 ws0">fb</div><div class="t m0 xc1 h11 y1ef ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 xc2 h11 y1f0 ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 xc3 hf y1ec ff5 fs0 fc0 sc0 ls3 ws0">k<span class="_ _33"></span><span class="ff6 ls30">∈K</span></div><div class="t m0 xb3 h12 y1f1 ff5 fs8 fc0 sc0 ls3 ws0">b</div><div class="t m0 xc4 h8 y1ed ff5 fs4 fc0 sc0 ls3 ws0">p</div><div class="t m0 xb4 h7 y1f2 ff5 fs0 fc0 sc0 ls3 ws0">t</div><div class="t m0 xb4 h7 y1f3 ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 xb6 h8 y1ed ff8 fs4 fc0 sc0 ls3 ws0">+</div><div class="t m0 xc5 h11 y1f4 ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 xaa hf y1f5 ff5 fs0 fc0 sc0 ls3 ws0">r<span class="_ _33"></span><span class="ff6 ls30">∈R</span></div><div class="t m0 xb8 h12 y1f6 ff5 fs8 fc0 sc0 ls3 ws0">b</div><div class="t m0 xc6 h8 y1ed ff5 fs4 fc0 sc0 ls3 ws0">p</div><div class="t m0 xc7 h7 y1f2 ff5 fs0 fc0 sc0 ls3 ws0">w</div><div class="t m0 xc7 h7 y1f3 ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 xc8 h8 y1ed ff8 fs4 fc0 sc0 ls3 ws0">+<span class="_ _c"> </span><span class="ff5">l</span></div><div class="t m0 x8e h7 y1ee ff5 fs0 fc0 sc0 ls17 ws0">bt</div><div class="t m0 xc9 hb y1ed ff6 fs4 fc0 sc0 ls3 ws0">−<span class="_ _c"> </span><span class="ff5">D</span></div><div class="t m0 xca h7 y1ee ff5 fs0 fc0 sc0 ls17 ws0">bt</div><div class="t m0 x91 h11 y1ef ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 xcb hb y1f7 ff6 fs4 fc0 sc0 ls3 ws0">≤<span class="_ _6"> </span><span class="ff5">F</span></div><div class="t m0 xcc h7 y1ee ff5 fs0 fc0 sc0 ls3 ws0">f</div><div class="t m0 xbd h6 y1f8 ff6 fs4 fc0 sc0 ls3 ws0">∀<span class="ff5">f<span class="_ _5"> </span></span><span class="ls1f">∈F<span class="_ _3a"> </span>∀<span class="_ _1c"></span><span class="ff5 ls3">t<span class="_ _6"> </span><span class="ff6 ls1f">∈T<span class="_ _3b"> </span><span class="ff1 lsa">(10)</span></span></span></span></div><div class="t m0 x9b h8 y1f9 ff5 fs4 fc0 sc0 ls3 ws0">P</div><div class="t m0 xcd h7 y1fa ff5 fs0 fc0 sc0 ls3 ws0">k</div><div class="t m0 xbe h8 y1fb ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 xa4 h7 y1fc ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 xce hb y1fb ff6 fs4 fc0 sc0 ls3 ws0">≤<span class="_ _6"> </span><span class="ff5">p</span></div><div class="t m0 xb2 h7 y1fd ff5 fs0 fc0 sc0 ls3 ws0">t</div><div class="t m0 xb2 h7 y1fe ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 xc3 hb y1fb ff6 fs4 fc0 sc0 ls3 ws0">≤<span class="_ _6"> </span><span class="ff5">P</span></div><div class="t m0 x88 h7 y1fc ff5 fs0 fc0 sc0 ls3 ws0">k</div><div class="t m0 xcf h8 y1fb ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 xa9 h7 y1fc ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 xc5 h6 y1fb ff6 fs4 fc0 sc0 ls3 ws0">∀<span class="ff5">k<span class="_"> </span></span><span class="ls23">∈K<span class="_ _2a"> </span>∀<span class="_ _1c"></span><span class="ff5 ls3">t<span class="_ _6"> </span><span class="ff6 ls23">∈T<span class="_ _3c"> </span><span class="ff1 lsa">(11)</span></span></span></span></div><div class="t m0 x9b hb y1ff ff8 fs4 fc0 sc0 ls3 ws0">0<span class="_ _6"> </span><span class="ff6">≤<span class="_ _6"> </span><span class="ff5">p</span></span></div><div class="t m0 xa5 h7 y200 ff5 fs0 fc0 sc0 ls3 ws0">w</div><div class="t m0 xa5 h7 y201 ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 x96 hb y202 ff6 fs4 fc0 sc0 ls3 ws0">≤<span class="_ _29"> </span><span class="ff8">¯<span class="_ _2d"></span><span class="ff5">w</span></span></div><div class="t m0 xa0 h7 y203 ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 x27 h8 y202 ff8 fs4 fc0 sc0 ls3 ws0">+<span class="_ _c"> </span><span class="ff5">v</span></div><div class="t m0 xa2 h7 y203 ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 xa3 h6 y202 ff6 fs4 fc0 sc0 ls3 ws0">∀<span class="ff5">r<span class="_ _8"> </span></span><span class="ls1f">∈R<span class="_ _2a"> </span>∀<span class="_ _1c"></span><span class="ff5 ls3">t<span class="_ _6"> </span><span class="ff6 ls23">∈T<span class="_ _3d"> </span><span class="ff1 lsa">(12)</span></span></span></span></div><div class="t m0 x9b h8 y204 ff5 fs4 fc0 sc0 ls3 ws0">p</div><div class="t m0 x18 h7 y205 ff5 fs0 fc0 sc0 ls3 ws0">t</div><div class="t m0 x18 hf y206 ff5 fs0 fc0 sc0 ls3 ws0">k<span class="_ _33"></span><span class="ff8">(<span class="_ _13"></span></span>t<span class="_ _13"></span><span class="ff6">−<span class="ff8 ls32">1)</span></span></div><div class="t m0 x9d hb y207 ff6 fs4 fc0 sc0 ls3 ws0">−<span class="_ _c"> </span><span class="ff5">p</span></div><div class="t m0 xb1 h7 y205 ff5 fs0 fc0 sc0 ls3 ws0">t</div><div class="t m0 xb1 h7 y208 ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 xc2 hb y207 ff6 fs4 fc0 sc0 ls3 ws0">≤<span class="_ _6"> </span><span class="ff5 ls1b">RD</span></div><div class="t m0 xd0 h7 y209 ff5 fs0 fc0 sc0 ls3 ws0">k</div><div class="t m0 x89 hb y207 ff6 fs4 fc0 sc0 ls3 ws0">·<span class="_ _c"> </span><span class="ff5">x</span></div><div class="t m0 xd1 h7 y209 ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 xd2 h8 y207 ff8 fs4 fc0 sc0 ls3 ws0">+<span class="_ _c"> </span><span class="ff5">P</span></div><div class="t m0 xd3 h7 y209 ff5 fs0 fc0 sc0 ls3 ws0">k</div><div class="t m0 xd4 hb y207 ff8 fs4 fc0 sc0 ls18 ws0">(1<span class="_ _c"> </span><span class="ff6 ls3">−<span class="_ _c"> </span><span class="ff5">x</span></span></div><div class="t m0 xd5 h7 y209 ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 xd6 h8 y207 ff8 fs4 fc0 sc0 ls3 ws0">)</div><div class="t m0 xbd h6 y20a ff6 fs4 fc0 sc0 ls3 ws0">∀<span class="ff5">k<span class="_ _8"> </span></span><span class="ls1f">∈K<span class="_ _3e"> </span>∀<span class="_ _1c"></span><span class="ff5 ls3">t<span class="_ _6"> </span><span class="ff6 ls1f">∈T<span class="_ _3f"> </span><span class="ff1 lsa">(13)</span></span></span></span></div><div class="t m0 x9b h8 y20b ff5 fs4 fc0 sc0 ls3 ws0">p</div><div class="t m0 x18 h7 y20c ff5 fs0 fc0 sc0 ls3 ws0">t</div><div class="t m0 x18 h7 y20d ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 x9c hb y20b ff6 fs4 fc0 sc0 ls3 ws0">−<span class="_ _c"> </span><span class="ff5">p</span></div><div class="t m0 xd7 h7 y20c ff5 fs0 fc0 sc0 ls3 ws0">t</div><div class="t m0 xd7 hf y20e ff5 fs0 fc0 sc0 ls3 ws0">k<span class="_ _33"></span><span class="ff8">(<span class="_ _13"></span></span>t<span class="_ _13"></span><span class="ff6">−<span class="ff8 ls32">1)</span></span></div><div class="t m0 xc2 hb y20b ff6 fs4 fc0 sc0 ls3 ws0">≤<span class="_ _6"> </span><span class="ff5 ls1b">RU</span></div><div class="t m0 xd8 h7 y20f ff5 fs0 fc0 sc0 ls3 ws0">k</div><div class="t m0 xd9 hb y20b ff6 fs4 fc0 sc0 ls3 ws0">·<span class="_ _c"> </span><span class="ff5">x</span></div><div class="t m0 xaa hf y210 ff5 fs0 fc0 sc0 ls3 ws0">k<span class="_ _33"></span><span class="ff8">(<span class="_ _13"></span></span>t<span class="_ _13"></span><span class="ff6">−<span class="ff8 ls32">1)</span></span></div><div class="t m0 xc7 h8 y20b ff8 fs4 fc0 sc0 ls3 ws0">+<span class="_ _c"> </span><span class="ff5">P</span></div><div class="t m0 xda h7 y20f ff5 fs0 fc0 sc0 ls3 ws0">k</div><div class="t m0 xdb h11 y211 ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 xdc hb y212 ff8 fs4 fc0 sc0 ls3 ws0">1<span class="_ _c"> </span><span class="ff6">−<span class="_ _c"> </span><span class="ff5">x</span></span></div><div class="t m0 xdd hf y210 ff5 fs0 fc0 sc0 ls3 ws0">k<span class="_ _33"></span><span class="ff8">(<span class="_ _13"></span></span>t<span class="_ _13"></span><span class="ff6">−<span class="ff8 ls32">1)</span></span></div><div class="t m0 xde h11 y211 ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 xc2 h6 y213 ff6 fs4 fc0 sc0 ls3 ws0">∀<span class="ff5">k<span class="_"> </span></span><span class="ls23">∈K<span class="_ _2a"> </span>∀<span class="_ _1c"></span><span class="ff5 ls3">t<span class="_ _6"> </span><span class="ff6 ls1f">∈T</span>.<span class="_ _40"> </span><span class="ff1 lsa">(14)</span></span></span></div><div class="t m0 xdf h6 y214 ff1 fs4 fc0 sc0 lsb ws0">In<span class="_ _9"> </span>this<span class="_ _c"> </span>formulation,<span class="_ _9"> </span>the<span class="_ _c"> </span>generation<span class="_ _9"> </span>outputs<span class="_ _c"> </span>from<span class="_ _9"> </span>thermal<span class="_ _9"> </span>units</div><div class="t m0 x13 h6 y215 ff1 fs4 fc0 sc0 lsa ws0">and<span class="_ _1f"> </span>wind<span class="_ _1f"> </span>power<span class="_ _1f"> </span>generation<span class="_ _1f"> </span>are<span class="_ _1f"> </span>denoted<span class="_ _14"> </span>by<span class="_ _1f"> </span><span class="ff5 ls3">p</span></div><div class="t m0 xe0 h7 y216 ff5 fs0 fc0 sc0 ls3 ws0">t</div><div class="t m0 xe0 h7 y217 ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 xe1 h6 y218 ff1 fs4 fc0 sc0 ls18 ws0">and<span class="_ _1f"> </span><span class="ff5 ls3">p</span></div><div class="t m0 xe2 h7 y216 ff5 fs0 fc0 sc0 ls3 ws0">w</div><div class="t m0 xe2 h7 y219 ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 xcc h6 y218 ff1 fs4 fc0 sc0 ls35 ws0">,r<span class="_ _41"></span>e<span class="_ _41"></span>-</div><div class="t m0 x13 h6 y21a ff1 fs4 fc0 sc0 lsa ws0">spectiv<span class="_ _1"></span>ely<span class="_ _1"></span>.<span class="_"> </span>The<span class="_ _6"> </span>objective<span class="_"> </span>function<span class="_"> </span>(5)<span class="_"> </span>includes<span class="_ _6"> </span>the<span class="_ _6"> </span>start-up<span class="_"> </span>cost</div><div class="t m0 x13 h6 y21b ff1 fs4 fc0 sc0 lsa ws0">and<span class="_ _7"> </span>generation<span class="_ _7"> </span>cost<span class="_ _5"> </span>of<span class="_ _7"> </span>thermal<span class="_ _5"> </span>units,<span class="_ _7"> </span>denoted<span class="_ _7"> </span>by<span class="_ _5"> </span><span class="ff5 ls3">z</span></div><div class="t m0 xe3 h7 y21c ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 xe4 h6 y21d ff1 fs4 fc0 sc0 ls18 ws0">and<span class="_ _7"> </span><span class="ff5 ls3">c</span></div><div class="t m0 x21 h7 y21c ff5 fs0 fc0 sc0 ls1c ws0">kt</div><div class="t m0 x22 h6 y21d ff1 fs4 fc0 sc0 ls3 ws0">,</div><div class="t m0 x13 h6 y21e ff1 fs4 fc0 sc0 lsb ws0">as<span class="_ _7"> </span>well<span class="_ _7"> </span>as<span class="_ _7"> </span>the<span class="_ _7"> </span>cost<span class="_ _7"> </span>of<span class="_ _5"> </span>possible<span class="_ _7"> </span>load<span class="_ _7"> </span>loss<span class="_ _7"> </span><span class="ff5 ls3">l</span></div><div class="t m0 xba h7 y21f ff5 fs0 fc0 sc0 ls17 ws0">bt</div><div class="t m0 xe5 h6 y220 ff1 fs4 fc0 sc0 lsb ws0">.<span class="_ _7"> </span>The<span class="_ _7"> </span>start-up<span class="_ _7"> </span>cost</div><div class="t m0 x13 h6 y221 ff1 fs4 fc0 sc0 lsb ws0">and<span class="_ _7"> </span>the<span class="_ _5"> </span>piecewise<span class="_ _7"> </span>linear<span class="_ _7"> </span>generation<span class="_ _5"> </span>cost<span class="_ _7"> </span>are<span class="_ _5"> </span>expressed<span class="_ _7"> </span>by<span class="_ _7"> </span>(6)</div><div class="t m0 x13 h6 y222 ff1 fs4 fc0 sc0 lsa ws0">and<span class="_"> </span>(7),<span class="_ _9"> </span>respectively<span class="_ _3"></span>.<span class="_"> </span>Constraints<span class="_"> </span>(8)<span class="_ _9"> </span>enforce<span class="_"> </span>po<span class="_ _0"></span>wer<span class="_"> </span>balance<span class="_ _c"> </span>be-</div><div class="t m0 x13 h6 y223 ff1 fs4 fc0 sc0 lsf ws0">tween<span class="_ _8"> </span>generation<span class="_ _8"> </span>supply<span class="_ _7"> </span>and<span class="_ _6"> </span>demand,<span class="_ _7"> </span>where<span class="_ _8"> </span><span class="ff5 ls3">D</span></div><div class="t m0 xe6 h7 y224 ff5 fs0 fc0 sc0 ls17 ws0">bt</div><div class="t m0 xe1 h6 y225 ff1 fs4 fc0 sc0 lsb ws0">is<span class="_ _8"> </span>the<span class="_ _8"> </span>power</div><div class="t m0 x13 h6 y226 ff1 fs4 fc0 sc0 ls18 ws0">demand<span class="_ _7"> </span>at<span class="_ _7"> </span>bus<span class="_ _8"> </span><span class="ff5 ls3">b</span><span class="lsb">,<span class="_ _7"> </span>during<span class="_ _7"> </span>time<span class="_ _7"> </span>step<span class="_ _7"> </span><span class="ff5 ls3">t</span><span class="lsa">.<span class="_ _7"> </span>Inequalities<span class="_ _7"> </span>(9)<span class="_ _7"> </span>and<span class="_ _7"> </span>(10)</span></span></div><div class="t m0 x13 h6 y227 ff1 fs4 fc0 sc0 lsa ws0">follo<span class="_ _0"></span>w<span class="_ _7"> </span>the<span class="_ _5"> </span>dc<span class="_ _7"> </span>power<span class="_ _7"> </span>ﬂo<span class="_ _0"></span>w<span class="_ _7"> </span>model<span class="_ _5"> </span>in<span class="_ _7"> </span>[29]<span class="_ _7"> </span>to<span class="_ _5"> </span>conﬁne<span class="_ _7"> </span>the<span class="_ _5"> </span>power</div><div class="t m0 x13 h6 y228 ff1 fs4 fc0 sc0 lsb ws0">transmission<span class="_ _7"> </span>below<span class="_ _7"> </span>the<span class="_ _7"> </span>line<span class="_ _7"> </span>capacity<span class="_ _7"> </span><span class="ff5 ls3">F</span></div><div class="t m0 xc8 h7 y229 ff5 fs0 fc0 sc0 ls3 ws0">f</div><div class="t m0 xe7 h6 y22a ff1 fs4 fc0 sc0 lsa ws0">.<span class="_ _7"> </span>Constraints<span class="_ _7"> </span>(11)<span class="_ _7"> </span>de-</div><div class="t m0 x13 h6 y22b ff1 fs4 fc0 sc0 lsa ws0">ﬁne<span class="_ _8"> </span>the<span class="_ _8"> </span>maximum<span class="_ _8"> </span>capacity</div><div class="t m0 xd9 h8 y22c ff5 fs4 fc0 sc0 ls3 ws0">P</div><div class="t m0 xe8 h7 y22d ff5 fs0 fc0 sc0 ls3 ws0">k</div><div class="t m0 xe9 h6 y22c ff1 fs4 fc0 sc0 lsa ws0">and<span class="_ _8"> </span>the<span class="_ _8"> </span>minimum<span class="_ _8"> </span>capacity<span class="_ _8"> </span><span class="ff5 ls3">P</span></div><div class="t m0 xea h7 y22e ff5 fs0 fc0 sc0 ls3 ws0">k</div><div class="t m0 x13 h6 y22f ff1 fs4 fc0 sc0 lsa ws0">of<span class="_"> </span>thermal<span class="_ _c"> </span>units,<span class="_"> </span>and<span class="_"> </span>constraints<span class="_ _c"> </span>(12)<span class="_"> </span>ensure<span class="_ _c"> </span>that<span class="_"> </span>the<span class="_ _c"> </span>generation</div><div class="t m0 x13 h6 y230 ff1 fs4 fc0 sc0 lsb ws0">from<span class="_"> </span>wind<span class="_ _6"> </span>farms<span class="_"> </span>must<span class="_ _6"> </span>be<span class="_ _6"> </span>lower<span class="_"> </span>than<span class="_ _6"> </span>the<span class="_ _6"> </span>av<span class="_ _0"></span>ailable<span class="_"> </span>wind<span class="_ _6"> </span>power<span class="_ _1"></span>.</div><div class="t m0 x13 h6 y231 ff1 fs4 fc0 sc0 lsa ws0">The<span class="_ _5"> </span>ramp-up<span class="_ _17"> </span>and<span class="_ _5"> </span>ramp-down<span class="_ _5"> </span>limits<span class="_ _17"> </span>of<span class="_ _5"> </span>units<span class="_ _17"> </span>are<span class="_ _5"> </span>enforced<span class="_ _17"> </span>by</div><div class="t m0 x13 h6 y232 ff1 fs4 fc0 sc0 lsa ws0">constraints<span class="_"> </span>(13)<span class="_"> </span>and<span class="_"> </span>(14).</div><div class="t m0 xdf h6 y233 ff1 fs4 fc0 sc0 lsb ws0">In<span class="_"> </span>order<span class="_ _6"> </span>to<span class="_"> </span>facilitate<span class="_ _6"> </span>the<span class="_ _6"> </span>following<span class="_"> </span>discussion,<span class="_"> </span>the<span class="_ _6"> </span>UC<span class="_"> </span>model</div><div class="t m0 x13 h6 y234 ff1 fs4 fc0 sc0 ls3 ws0">abov<span class="_ _1"></span>e<span class="_ _7"> </span>is<span class="_ _7"> </span>written<span class="_ _7"> </span>in<span class="_ _7"> </span>a<span class="_ _7"> </span>general<span class="_ _7"> </span>matrix<span class="_ _8"> </span>form.<span class="_ _7"> </span>W<span class="_ _1"></span>e<span class="_ _7"> </span>will<span class="_ _8"> </span>use<span class="_ _7"> </span><span class="ff6">N</span></div><div class="t m0 xeb h7 y235 ff8 fs0 fc0 sc0 ls3 ws0">1</div><div class="t m0 xec h6 y236 ff1 fs4 fc0 sc0 lse ws0">to</div><div class="t m0 x13 h6 y237 ff1 fs4 fc0 sc0 lsb ws0">denote<span class="_"> </span>the<span class="_"> </span>set<span class="_ _c"> </span>of<span class="_"> </span>all<span class="_"> </span>ﬁrst-stage<span class="_"> </span>decision<span class="_ _c"> </span>v<span class="_ _0"></span>ariables,<span class="_"> </span>and<span class="_"> </span>the<span class="_ _c"> </span>set<span class="_"> </span>of</div><div class="t m0 x13 h6 y238 ff1 fs4 fc0 sc0 lsa ws0">all<span class="_ _6"> </span>second-stage<span class="_ _8"> </span>decision<span class="_ _8"> </span>variables<span class="_"> </span>is<span class="_ _8"> </span>denoted<span class="_ _6"> </span>by<span class="_ _8"> </span><span class="ff6 ls3">N</span></div><div class="t m0 xed h7 y239 ff8 fs0 fc0 sc0 ls3 ws0">2</div><div class="t m0 x92 h6 y23a ff1 fs4 fc0 sc0 ls36 ws0">.T<span class="_ _1c"></span>h<span class="_ _42"></span>es<span class="_ _42"></span>e<span class="_ _1c"></span>t<span class="_ _42"></span>s</div><div class="t m0 x13 h6 y23b ff1 fs4 fc0 sc0 lsa ws0">of<span class="_ _8"> </span>all<span class="_ _7"> </span>constraints<span class="_ _8"> </span>for<span class="_ _7"> </span>the<span class="_ _8"> </span>ﬁrst<span class="_ _8"> </span>and<span class="_ _7"> </span>second<span class="_ _8"> </span>stage<span class="_ _7"> </span>are<span class="_ _8"> </span>denoted<span class="_ _8"> </span>by</div><div class="t m0 x13 hb y23c ff6 fs4 fc0 sc0 ls3 ws0">M</div><div class="t m0 xee h7 y23d ff8 fs0 fc0 sc0 ls3 ws0">1</div><div class="t m0 xef h6 y23e ff1 fs4 fc0 sc0 ls18 ws0">and<span class="_ _9"> </span><span class="ff6 ls3">M</span></div><div class="t m0 xa5 h7 y23d ff8 fs0 fc0 sc0 ls3 ws0">2</div><div class="t m0 xf0 h6 y23e ff1 fs4 fc0 sc0 lsa ws0">,<span class="_ _9"> </span>respectively<span class="_ _3"></span>.<span class="_ _c"> </span>Hence<span class="_ _9"> </span>the<span class="_ _c"> </span>ﬁrst-stage<span class="_ _9"> </span>problem<span class="_ _c"> </span>can<span class="_ _9"> </span>be</div><div class="t m0 xf he ya3 ffa fs6 fc1 sc0 ls3 ws0">Authorized licensed use limited to: SICHUAN UNIVERSITY. Downloaded on January 05,2021 at 09:02:32 UTC from IEEE Xplore. Restrictions apply. </div></div><div class="pi" data-data='{"ctm":[1.616162,0.000000,0.000000,1.616162,0.000000,0.000000]}'></div></div>

<div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://csdnimg.cn/release/download_crawler_static/85190235/bg4.jpg"><div class="t m0 x2e h2 y1 ff1 fs0 fc0 sc0 ls1 ws0">42<span class="_ _2"> </span><span class="ls0">IEEE<span class="_"> </span>TRANSA<span class="_ _1"></span>CTIONS<span class="_"> </span>ON<span class="_"> </span>POWER<span class="_"> </span>SYSTEMS,<span class="_"> </span>V<span class="_ _0"></span>OL.<span class="_"> </span>32,<span class="_"> </span>NO.<span class="_"> </span>1,<span class="_"> </span>J<span class="_ _1"></span>ANUAR<span class="_ _1"></span>Y<span class="_"> </span>2017</span></div><div class="t m0 x2e h6 y13b ff1 fs4 fc0 sc0 lsb ws0">expressed<span class="_"> </span>as</div><div class="t m0 xf1 hb y23f ff8 fs4 fc0 sc0 ls18 ws0">min<span class="_"> </span><span class="ff6 ls3">Q<span class="ff8">(<span class="_ _1"></span><span class="ff5">x</span></span></span></div><div class="t m0 xf2 h8 y240 ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 xf2 h6 y23f ff5 fs4 fc0 sc0 ls3 ws0">x<span class="ff8">)<span class="_ _43"> </span><span class="ff1 lsa">(15)</span></span></div><div class="t m0 xf3 h6 y241 ff1 fs4 fc0 sc0 ls28 ws0">s.t.<span class="_ _6"> </span><span class="ff5 lsa">Ax</span></div><div class="t m0 xf4 h8 y242 ff5 fs4 fc0 sc0 lsa ws0">Ax</div><div class="t m0 xf4 hb y241 ff5 fs4 fc0 sc0 lsa ws0">Ax<span class="_ _6"> </span><span class="ff6 ls3">≤<span class="_ _c"> </span><span class="ff5">b</span></span></div><div class="t m0 x78 h8 y242 ff5 fs4 fc0 sc0 ls3 ws0">b</div><div class="t m0 x78 h6 y241 ff5 fs4 fc0 sc0 ls3 ws0">b<span class="_ _44"> </span><span class="ff1 lsa">(16)</span></div><div class="t m0 x2e h6 y243 ff1 fs4 fc0 sc0 lsd ws0">with<span class="_ _9"> </span><span class="ff5 ls3">A</span></div><div class="t m0 xf5 h8 y244 ff5 fs4 fc0 sc0 ls3 ws0">A</div><div class="t m0 xf5 hb y243 ff5 fs4 fc0 sc0 ls3 ws0">A<span class="_ _6"> </span><span class="ff6">∈<span class="_ _6"> </span><span class="ff7">R</span></span></div><div class="t m0 xf6 hf y245 ff6 fs0 fc0 sc0 ls0 ws0">|M</div><div class="t m0 xf7 h12 y246 ff8 fs8 fc0 sc0 ls3 ws0">1</div><div class="t m0 xf8 hf y245 ff6 fs0 fc0 sc0 ls37 ws0">|×|N</div><div class="t m0 xf9 h12 y246 ff8 fs8 fc0 sc0 ls3 ws0">1</div><div class="t m0 x73 hf y245 ff6 fs0 fc0 sc0 ls3 ws0">|</div><div class="t m0 xfa h6 y247 ff1 fs4 fc0 sc0 ls18 ws0">,<span class="_"> </span>and<span class="_ _c"> </span><span class="ff5 ls3">b</span></div><div class="t m0 xfb h8 y248 ff5 fs4 fc0 sc0 ls3 ws0">b</div><div class="t m0 xfb hb y247 ff5 fs4 fc0 sc0 ls3 ws0">b<span class="_ _6"> </span><span class="ff6">∈<span class="_ _6"> </span><span class="ff7">R</span></span></div><div class="t m0 x5c hf y245 ff6 fs0 fc0 sc0 ls0 ws0">|M</div><div class="t m0 xfc h12 y246 ff8 fs8 fc0 sc0 ls3 ws0">1</div><div class="t m0 xfd hf y245 ff6 fs0 fc0 sc0 ls3 ws0">|</div><div class="t m0 xfe h6 y247 ff1 fs4 fc0 sc0 lsa ws0">.<span class="_"> </span>The<span class="_"> </span>function<span class="_"> </span><span class="ff6 ls3">Q<span class="ff8">(<span class="_ _0"></span><span class="ff5">x</span></span></span></div><div class="t m0 xff h8 y248 ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 xff h6 y247 ff5 fs4 fc0 sc0 ls3 ws0">x<span class="ff8">)<span class="_ _6"> </span><span class="ff1 lsd">is<span class="_"> </span>the</span></span></div><div class="t m0 x2e h6 y249 ff1 fs4 fc0 sc0 lsb ws0">expected<span class="_ _5"> </span>cost<span class="_ _5"> </span>under<span class="_ _17"> </span>the<span class="_ _17"> </span>worst-case<span class="_ _5"> </span>distribution,<span class="_ _5"> </span>which<span class="_ _5"> </span>is<span class="_ _17"> </span>ex-</div><div class="t m0 x2e h6 y24a ff1 fs4 fc0 sc0 lsb ws0">pressed<span class="_"> </span>as<span class="_"> </span>belo<span class="_ _0"></span>w</div><div class="t m0 x100 hb y24b ff6 fs4 fc0 sc0 ls3 ws0">Q<span class="ff8">(<span class="_ _1"></span><span class="ff5">x</span></span></div><div class="t m0 x101 h8 y24c ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 x101 h8 y24b ff5 fs4 fc0 sc0 ls3 ws0">x<span class="ff8 ls38">)=<span class="_ _13"></span>s<span class="_ _1c"></span>u<span class="_ _42"></span>p</span></div><div class="t m0 xf1 hf y24d ff9 fs0 fc0 sc0 ls3 ws0">P<span class="_ _33"> </span><span class="ff6">∈</span>F</div><div class="t m0 x102 hc y24e ff7 fs4 fc0 sc0 ls3 ws0">E</div><div class="t m0 x103 hd y24f ff9 fs0 fc0 sc0 ls3 ws0">P</div><div class="t m0 xf2 hb y24e ff6 fs4 fc0 sc0 ls3 ws0">{<span class="ff5">Q<span class="ff8">(<span class="_ _1"></span><span class="ff5">x</span></span></span></div><div class="t m0 x62 h8 y250 ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 x62 h8 y24e ff5 fs4 fc0 sc0 ls28 ws0">x,</div><div class="t m0 x49 h8 y250 ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x104 h8 y24e ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x104 h8 y250 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x104 h6 y24e ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8">)<span class="ff6">}<span class="_ _b"></span></span></span>.<span class="_ _45"> </span><span class="ff1 lsa">(17)</span></div><div class="t m0 x2e h6 y251 ff1 fs4 fc0 sc0 lsa ws0">Inequalities<span class="_ _7"> </span>(16)<span class="_ _5"> </span>are<span class="_ _5"> </span>minimum<span class="_ _7"> </span>up/down<span class="_ _7"> </span>time<span class="_ _5"> </span>constraints<span class="_ _5"> </span>[30]</div><div class="t m0 x2e h6 y252 ff1 fs4 fc0 sc0 lsa ws0">that<span class="_"> </span>deﬁne<span class="_"> </span>the<span class="_"> </span>feasible<span class="_"> </span>region<span class="_"> </span><span class="ff7 ls3">T<span class="_ _7"> </span></span><span class="lse">in<span class="_"> </span>(3).</span></div><div class="t m0 x43 h6 y253 ff1 fs4 fc0 sc0 lsb ws0">Similarly<span class="_ _1"></span>,<span class="_ _b"> </span>the<span class="_ _b"> </span>second-stage<span class="_ _b"> </span>problem<span class="_ _9"> </span>is<span class="_ _b"> </span>expressed<span class="_ _b"> </span>in<span class="_ _b"> </span>the<span class="_ _9"> </span>matrix</div><div class="t m0 x2e h6 y254 ff1 fs4 fc0 sc0 lse ws0">form</div><div class="t m0 x101 h8 y255 ff5 fs4 fc0 sc0 ls3 ws0">Q<span class="ff8">(<span class="_ _1"></span><span class="ff5">x</span></span></div><div class="t m0 x2c h8 y256 ff5 fs4 fc0 sc0 ls3 ws0">x</div><div class="t m0 x2c h8 y255 ff5 fs4 fc0 sc0 ls28 ws0">x,<span class="_ _18"></span>v</div><div class="t m0 xf3 h8 y256 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x105 h8 y255 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8 ls18">)<span class="_ _6"> </span>=<span class="_ _6"> </span>min<span class="_"> </span></span>q</div><div class="t m0 x106 h8 y256 ff5 fs4 fc0 sc0 ls3 ws0">q</div><div class="t m0 x106 h8 y255 ff5 fs4 fc0 sc0 ls3 ws0">q</div><div class="t m0 x107 h7 y257 ff5 fs0 fc0 sc0 ls3 ws0">T</div><div class="t m0 x108 h8 y258 ff5 fs4 fc0 sc0 ls3 ws0">y</div><div class="t m0 x108 h8 y259 ff5 fs4 fc0 sc0 ls3 ws0">y</div><div class="t m0 x108 h6 y258 ff5 fs4 fc0 sc0 ls3 ws0">y<span class="_ _46"> </span><span class="ff1 lsa">(18)</span></div><div class="t m0 x101 h6 y25a ff1 fs4 fc0 sc0 ls28 ws0">s.t.<span class="_ _6"> </span><span class="ff5 ls39">Tx</span></div><div class="t m0 x109 h8 y25b ff5 fs4 fc0 sc0 ls39 ws0">Tx</div><div class="t m0 x109 h8 y25a ff5 fs4 fc0 sc0 ls39 ws0">Tx<span class="_ _33"></span><span class="ff8 ls3">+<span class="_ _b"> </span></span><span class="ls3a">Wy</span></div><div class="t m0 x102 h8 y25b ff5 fs4 fc0 sc0 ls3a ws0">Wy</div><div class="t m0 x102 hb y25a ff5 fs4 fc0 sc0 ls3a ws0">Wy<span class="_ _b"> </span><span class="ff6 ls3">≤<span class="_ _6"> </span><span class="ff5">h</span></span></div><div class="t m0 x107 h8 y25b ff5 fs4 fc0 sc0 ls3 ws0">h</div><div class="t m0 x107 h8 y25a ff5 fs4 fc0 sc0 ls3 ws0">h<span class="ff8">(<span class="_ _1"></span><span class="ff5">v</span></span></div><div class="t m0 x10a h8 y25b ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x10a h6 y25a ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8">)<span class="_ _47"> </span><span class="ff1 lsa">(19)</span></span></div><div class="t m0 x2e h6 y25c ff1 fs4 fc0 sc0 lsd ws0">with<span class="_ _c"> </span><span class="ff5 ls3">q</span></div><div class="t m0 xf5 h8 y25d ff5 fs4 fc0 sc0 ls3 ws0">q</div><div class="t m0 xf5 hb y25c ff5 fs4 fc0 sc0 ls3 ws0">q<span class="_"> </span><span class="ff6">∈<span class="_ _6"> </span><span class="ff7">R</span></span></div><div class="t m0 x10b hf y25e ff6 fs0 fc0 sc0 ls0 ws0">|N</div><div class="t m0 x80 h12 y25f ff8 fs8 fc0 sc0 ls3 ws0">2</div><div class="t m0 xf7 hf y25e ff6 fs0 fc0 sc0 ls3 ws0">|</div><div class="t m0 x4c h6 y260 ff1 fs4 fc0 sc0 ls3 ws0">,<span class="_ _c"> </span><span class="ff5">T</span></div><div class="t m0 xd h8 y261 ff5 fs4 fc0 sc0 ls3 ws0">T</div><div class="t m0 xd hb y260 ff5 fs4 fc0 sc0 ls3 ws0">T<span class="_ _17"> </span><span class="ff6">∈<span class="_ _8"> </span><span class="ff7">R</span></span></div><div class="t m0 x8 hf y25e ff6 fs0 fc0 sc0 ls0 ws0">|M</div><div class="t m0 xf3 h12 y25f ff8 fs8 fc0 sc0 ls3 ws0">2</div><div class="t m0 xfb hf y25e ff6 fs0 fc0 sc0 ls37 ws0">|×|N</div><div class="t m0 x10c h12 y25f ff8 fs8 fc0 sc0 ls3 ws0">1</div><div class="t m0 x10d hf y25e ff6 fs0 fc0 sc0 ls3 ws0">|</div><div class="t m0 x103 h6 y260 ff1 fs4 fc0 sc0 ls3 ws0">,<span class="_ _c"> </span><span class="ff5">W</span></div><div class="t m0 x10e h8 y261 ff5 fs4 fc0 sc0 ls3 ws0">W</div><div class="t m0 x10e hb y260 ff5 fs4 fc0 sc0 ls3 ws0">W<span class="_ _17"> </span><span class="ff6">∈<span class="_ _8"> </span><span class="ff7">R</span></span></div><div class="t m0 x49 hf y25e ff6 fs0 fc0 sc0 ls3b ws0">|M|</div><div class="t m0 x10f h12 y25f ff8 fs8 fc0 sc0 ls3 ws0">2</div><div class="t m0 x68 hf y25e ff6 fs0 fc0 sc0 ls3c ws0">×|N</div><div class="t m0 xa h12 y25f ff8 fs8 fc0 sc0 ls3 ws0">2</div><div class="t m0 x110 hf y25e ff6 fs0 fc0 sc0 ls3 ws0">|</div><div class="t m0 x61 h6 y260 ff1 fs4 fc0 sc0 ls18 ws0">,<span class="_"> </span>and<span class="_"> </span><span class="ff5 ls3">h</span></div><div class="t m0 x51 h8 y261 ff5 fs4 fc0 sc0 ls3 ws0">h</div><div class="t m0 x51 h8 y260 ff5 fs4 fc0 sc0 ls3 ws0">h<span class="ff8">(<span class="_ _1"></span><span class="ff5">v</span></span></div><div class="t m0 x57 h8 y261 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x57 hb y260 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8">)<span class="_ _6"> </span><span class="ff6">∈</span></span></div><div class="t m0 x2e hc y262 ff7 fs4 fc0 sc0 ls3 ws0">R</div><div class="t m0 x3b hf y263 ff6 fs0 fc0 sc0 ls0 ws0">|M</div><div class="t m0 x75 h12 y264 ff8 fs8 fc0 sc0 ls3 ws0">2</div><div class="t m0 x66 hf y263 ff6 fs0 fc0 sc0 ls3 ws0">|</div><div class="t m0 x111 h6 y265 ff1 fs4 fc0 sc0 lsf ws0">.<span class="_ _5"> </span>The<span class="_ _5"> </span>vector<span class="_ _7"> </span><span class="ff5 ls3">y</span></div><div class="t m0 xf9 h8 y266 ff5 fs4 fc0 sc0 ls3 ws0">y</div><div class="t m0 xf9 hb y265 ff5 fs4 fc0 sc0 ls3 ws0">y<span class="_"> </span><span class="ff6">∈<span class="_ _6"> </span><span class="ff7">R</span></span></div><div class="t m0 xf3 hf y263 ff6 fs0 fc0 sc0 ls0 ws0">|N</div><div class="t m0 x77 h12 y264 ff8 fs8 fc0 sc0 ls3 ws0">2</div><div class="t m0 x112 hf y263 ff6 fs0 fc0 sc0 ls3 ws0">|</div><div class="t m0 x102 h6 y265 ff1 fs4 fc0 sc0 lsa ws0">consists<span class="_ _5"> </span>of<span class="_ _5"> </span>all<span class="_ _17"> </span>second-stage<span class="_ _5"> </span>de-</div><div class="t m0 x2e h6 y267 ff1 fs4 fc0 sc0 lsb ws0">cision<span class="_ _1f"> </span>variables<span class="_ _1f"> </span>in<span class="_ _14"> </span>(5)–(14).<span class="_ _1f"> </span>Constraints<span class="_ _14"> </span>of<span class="_ _14"> </span>the<span class="_ _1f"> </span>second-stage</div><div class="t m0 x2e h6 y268 ff1 fs4 fc0 sc0 lsb ws0">problem<span class="_ _7"> </span>above<span class="_ _7"> </span>are<span class="_ _5"> </span>written<span class="_ _5"> </span>as<span class="_ _7"> </span>i<span class="_ _13"></span>nequality<span class="_ _7"> </span>(19).<span class="_ _5"> </span>The<span class="_ _5"> </span>expression</div><div class="t m0 x2e h6 y269 ff1 fs4 fc0 sc0 lsb ws0">(12)<span class="_ _5"> </span>suggests<span class="_ _17"> </span>that<span class="_ _17"> </span>the<span class="_ _5"> </span>right-hand-side<span class="_ _17"> </span>vector<span class="_ _7"> </span><span class="ff5 ls3">h</span></div><div class="t m0 x3e h8 y26a ff5 fs4 fc0 sc0 ls3 ws0">h</div><div class="t m0 x3e h8 y269 ff5 fs4 fc0 sc0 ls3 ws0">h<span class="ff8">(<span class="_ _1"></span><span class="ff5">v</span></span></div><div class="t m0 x58 h8 y26a ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x58 h6 y269 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8">)<span class="_ _5"> </span><span class="ff1 lsb">is<span class="_ _17"> </span>affected</span></span></div><div class="t m0 x2e h6 y26b ff1 fs4 fc0 sc0 lsb ws0">by<span class="_ _7"> </span>the<span class="_ _8"> </span>uncertain<span class="_ _7"> </span>term<span class="_ _8"> </span><span class="ff5 ls3">v</span></div><div class="t m0 x7c h8 y26c ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x7c h6 y26b ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff1 lsa">,<span class="_ _7"> </span>which<span class="_ _7"> </span>is<span class="_ _8"> </span>commonly<span class="_ _7"> </span>expressed<span class="_ _7"> </span>as<span class="_ _8"> </span>the</span></div><div class="t m0 x2e h6 y26d ff1 fs4 fc0 sc0 lsb ws0">follo<span class="_ _0"></span>wing<span class="_"> </span>equation</div><div class="t m0 x113 h8 y26e ff5 fs4 fc0 sc0 ls3 ws0">h</div><div class="t m0 x113 h8 y26f ff5 fs4 fc0 sc0 ls3 ws0">h</div><div class="t m0 x113 h8 y26e ff5 fs4 fc0 sc0 ls3 ws0">h<span class="ff8">(<span class="_ _1"></span><span class="ff5">v</span></span></div><div class="t m0 x7b h8 y26f ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x7c h8 y26e ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8 ls38">)=<span class="_ _1"></span><span class="ff5 ls3">h</span></span></div><div class="t m0 x77 h8 y26f ff5 fs4 fc0 sc0 ls3 ws0">h</div><div class="t m0 x77 h8 y26e ff5 fs4 fc0 sc0 ls3 ws0">h</div><div class="t m0 x114 h7 y270 ff8 fs0 fc0 sc0 ls3 ws0">0</div><div class="t m0 x5e h8 y271 ff8 fs4 fc0 sc0 ls3 ws0">+</div><div class="t m0 x115 h11 y272 ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 xf2 hf y273 ff5 fs0 fc0 sc0 ls3 ws0">s<span class="_ _13"></span><span class="ff6 ls30">∈S</span></div><div class="t m0 x107 h8 y271 ff5 fs4 fc0 sc0 ls3 ws0">h</div><div class="t m0 x107 h8 y274 ff5 fs4 fc0 sc0 ls3 ws0">h</div><div class="t m0 x107 h8 y271 ff5 fs4 fc0 sc0 ls3 ws0">h</div><div class="t m0 x108 h7 y270 ff5 fs0 fc0 sc0 ls3 ws0">v</div><div class="t m0 x108 h7 y275 ff5 fs0 fc0 sc0 ls3 ws0">s</div><div class="t m0 x116 h8 y271 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x117 h7 y276 ff5 fs0 fc0 sc0 ls3 ws0">s</div><div class="t m0 x118 h6 y271 ff1 fs4 fc0 sc0 lsa ws0">(20)</div><div class="t m0 x2e h6 y277 ff1 fs4 fc0 sc0 lsa ws0">where<span class="_ _9"> </span><span class="ff5 ls3">h</span></div><div class="t m0 x2f h8 y278 ff5 fs4 fc0 sc0 ls3 ws0">h</div><div class="t m0 x2f h8 y277 ff5 fs4 fc0 sc0 ls3 ws0">h</div><div class="t m0 x119 h7 y279 ff8 fs0 fc0 sc0 ls3 ws0">0</div><div class="t m0 x65 hb y27a ff6 fs4 fc0 sc0 ls3 ws0">∈<span class="_ _6"> </span><span class="ff7">R</span></div><div class="t m0 xc hf y279 ff6 fs0 fc0 sc0 ls0 ws0">|M</div><div class="t m0 x39 h12 y27b ff8 fs8 fc0 sc0 ls3 ws0">2</div><div class="t m0 x11a hf y279 ff6 fs0 fc0 sc0 ls3 ws0">|</div><div class="t m0 x11b h6 y27a ff1 fs4 fc0 sc0 ls18 ws0">and<span class="_ _9"> </span><span class="ff5 ls3">h</span></div><div class="t m0 x8 h8 y27c ff5 fs4 fc0 sc0 ls3 ws0">h</div><div class="t m0 x2c h8 y27a ff5 fs4 fc0 sc0 ls3 ws0">h</div><div class="t m0 x11c h7 y279 ff5 fs0 fc0 sc0 ls3 ws0">v</div><div class="t m0 x11c h7 y27d ff5 fs0 fc0 sc0 ls3 ws0">s</div><div class="t m0 xfb hb y27a ff6 fs4 fc0 sc0 ls3 ws0">∈<span class="_ _6"> </span><span class="ff7">R</span></div><div class="t m0 x10c hf y279 ff6 fs0 fc0 sc0 ls0 ws0">|M</div><div class="t m0 x10e h12 y27b ff8 fs8 fc0 sc0 ls3 ws0">2</div><div class="t m0 x4 hf y279 ff6 fs0 fc0 sc0 ls3 ws0">|</div><div class="t m0 x11d h6 y27a ff1 fs4 fc0 sc0 lsb ws0">are<span class="_"> </span>constants<span class="_"> </span>implying<span class="_ _c"> </span>how</div><div class="t m0 x2e h6 y27e ff1 fs4 fc0 sc0 lsa ws0">the<span class="_"> </span>vector<span class="_ _9"> </span><span class="ff5 ls3">h</span></div><div class="t m0 x40 h8 y27f ff5 fs4 fc0 sc0 ls3 ws0">h</div><div class="t m0 x40 h8 y27e ff5 fs4 fc0 sc0 ls3 ws0">h<span class="ff8">(<span class="_ _1"></span><span class="ff5">v</span></span></div><div class="t m0 x6a h8 y27f ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x6a h6 y27e ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8">)<span class="_ _6"> </span><span class="ff1 lsa">is<span class="_"> </span>inﬂuenced<span class="_"> </span>by<span class="_"> </span>each<span class="_"> </span>uncertain<span class="_"> </span>term<span class="_"> </span></span></span>v</div><div class="t m0 x11e h7 y280 ff5 fs0 fc0 sc0 ls3 ws0">s</div><div class="t m0 x11f h6 y281 ff1 fs4 fc0 sc0 ls3 ws0">.</div><div class="t m0 x43 h6 y282 ff1 fs4 fc0 sc0 lsb ws0">It<span class="_ _17"> </span>can<span class="_ _17"> </span>be<span class="_ _1f"> </span>seen<span class="_ _17"> </span>from<span class="_ _17"> </span>(15)<span class="_ _1f"> </span>that<span class="_ _17"> </span>this<span class="_ _17"> </span>formulation<span class="_ _1f"> </span>attempts<span class="_ _17"> </span>to</div><div class="t m0 x2e h6 y283 ff1 fs4 fc0 sc0 lsb ws0">minimize<span class="_ _c"> </span>the<span class="_ _c"> </span>worst-case<span class="_ _9"> </span>expected<span class="_"> </span>cost<span class="_ _9"> </span>over<span class="_ _9"> </span>an<span class="_"> </span>ambiguity<span class="_ _9"> </span>set<span class="_"> </span><span class="ff7 ls3">F<span class="_ _33"></span><span class="ff1">.</span></span></div><div class="t m0 x2e h6 y284 ff1 fs4 fc0 sc0 lsa ws0">This<span class="_ _8"> </span>feature<span class="_ _8"> </span>differentiates<span class="_ _8"> </span>our<span class="_ _8"> </span>approach<span class="_ _7"> </span>from<span class="_ _6"> </span>the<span class="_ _8"> </span>conv<span class="_ _1"></span>entional</div><div class="t m0 x2e h6 y285 ff1 fs4 fc0 sc0 lsb ws0">stochastic<span class="_"> </span>programming<span class="_"> </span>and<span class="_"> </span>rob<span class="_ _1"></span>ust<span class="_"> </span>optimization.<span class="_"> </span>Details<span class="_"> </span>of<span class="_"> </span>the</div><div class="t m0 x2e h6 y286 ff1 fs4 fc0 sc0 lsb ws0">ambiguity<span class="_"> </span>set<span class="_"> </span><span class="ff7 ls3">F<span class="_ _7"> </span></span>are<span class="_"> </span>provided<span class="_"> </span>in<span class="_"> </span>the<span class="_"> </span>next<span class="_"> </span>section.</div><div class="t m0 x2e h9 y287 ff2 fs4 fc0 sc0 lsa ws0">B.<span class="_ _25"> </span>An<span class="_"> </span>Ambiguity<span class="_"> </span>Set<span class="_"> </span>for<span class="_"> </span>Uncertain<span class="_"> </span>W<span class="_ _1"></span>ind<span class="_"> </span>P<span class="_ _3"></span>ower<span class="_"> </span>Generation</div><div class="t m0 x43 h6 y288 ff1 fs4 fc0 sc0 lsa ws0">This<span class="_"> </span>section<span class="_ _c"> </span>focuses<span class="_"> </span>on<span class="_ _c"> </span>modeling<span class="_"> </span>the<span class="_"> </span>random<span class="_ _c"> </span>term</div><div class="t m0 x38 h8 y289 ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x38 h8 y288 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x38 h8 y289 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x38 h6 y288 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _6"> </span><span class="ff1 lsf">of<span class="_"> </span>wind</span></div><div class="t m0 x2e h6 y28a ff1 fs4 fc0 sc0 lsa ws0">po<span class="_ _0"></span>wer<span class="_ _6"> </span>based<span class="_ _6"> </span>on<span class="_ _6"> </span>an<span class="_ _8"> </span>ambiguity<span class="_ _6"> </span>set<span class="_ _6"> </span>that<span class="_ _8"> </span>characterizes<span class="_ _6"> </span>a<span class="_ _6"> </span>family<span class="_ _6"> </span>of</div><div class="t m0 x2e h6 y28b ff1 fs4 fc0 sc0 ls3 ws0">distributions.<span class="_"> </span>Let<span class="_"> </span><span class="ff6">S<span class="_ _5"> </span></span><span class="lsf">be<span class="_"> </span>the<span class="_ _6"> </span>set<span class="_ _8"> </span>of<span class="_"> </span>all<span class="_ _8"> </span>random<span class="_"> </span>variables</span></div><div class="t m0 x6f h8 y28c ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x6f h8 y28b ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x6f h8 y28c ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x6f h6 y28b ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff1 ls18">,<span class="_"> </span>and<span class="_ _8"> </span></span><span class="ff7">P</span></div><div class="t m0 x2e h6 y28d ff1 fs4 fc0 sc0 ls3 ws0">be<span class="_"> </span>the<span class="_ _9"> </span>distribution<span class="_"> </span>of</div><div class="t m0 x120 h8 y28e ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 xfa h8 y28d ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 xfa h8 y28e ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x120 h6 y28d ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff1 lsb">,<span class="_"> </span>then<span class="_ _9"> </span>the<span class="_"> </span>ambiguity<span class="_"> </span>set<span class="_ _9"> </span></span><span class="ff7">F<span class="_ _7"> </span><span class="ff1 ls18">can<span class="_"> </span>be<span class="_"> </span>deﬁned</span></span></div><div class="t m0 x2e h6 y28f ff1 fs4 fc0 sc0 lsb ws0">in<span class="_"> </span>the<span class="_"> </span>follo<span class="_ _0"></span>wing<span class="_"> </span>standard<span class="_"> </span>form</div><div class="t m0 x64 h8 y290 ff7 fs4 fc0 sc0 ls3 ws0">F<span class="_ _5"> </span><span class="ff8">=</span></div><div class="t m0 x121 h11 y291 ffb fs4 fc0 sc0 ls3 ws0">⎧</div><div class="t m0 x121 h11 y292 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x121 h11 y293 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x121 h11 y294 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x121 h11 y295 ffb fs4 fc0 sc0 ls3 ws0">⎨</div><div class="t m0 x121 h11 y296 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x121 h11 y297 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x121 h11 y298 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x121 h11 y299 ffb fs4 fc0 sc0 ls3 ws0">⎩</div><div class="t m0 x122 hb y29a ff7 fs4 fc0 sc0 ls3 ws0">P<span class="_ _17"> </span><span class="ff6 ls23">∈P</span></div><div class="t m0 x39 h7 y29b ff8 fs0 fc0 sc0 ls3 ws0">0</div><div class="t m0 x123 h8 y29c ff8 fs4 fc0 sc0 ls3 ws0">(<span class="ff7">R</span></div><div class="t m0 x41 hf y29d ff6 fs0 fc0 sc0 ls0 ws0">|S<span class="_ _13"></span>|</div><div class="t m0 x8 h8 y29c ff8 fs4 fc0 sc0 ls38 ws0">):</div><div class="t m0 xfb h8 y29e ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 xfb h8 y29f ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 xfb h8 y29e ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 xfb hb y29f ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_"> </span><span class="ff6">∈<span class="_ _6"> </span><span class="ff7">R</span></span></div><div class="t m0 x124 hf y2a0 ff6 fs0 fc0 sc0 ls0 ws0">|S<span class="_ _13"></span>|</div><div class="t m0 xfb hc y2a1 ff7 fs4 fc0 sc0 ls3 ws0">E</div><div class="t m0 x125 hd y2a2 ff9 fs0 fc0 sc0 ls3 ws0">P</div><div class="t m0 x102 hb y2a1 ff6 fs4 fc0 sc0 ls3 ws0">{</div><div class="t m0 x74 h8 y2a3 ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x74 h8 y2a1 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x74 h8 y2a3 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x74 hb y2a1 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff6">}<span class="_ _6"> </span><span class="ff8 ls3d">=0</span></span></div><div class="t m0 x126 h8 y2a3 ff8 fs4 fc0 sc0 ls3 ws0">0</div><div class="t m0 x126 h8 y2a1 ff8 fs4 fc0 sc0 ls3 ws0">0</div><div class="t m0 xfb hb y2a4 ff7 fs4 fc0 sc0 ls3 ws0">P<span class="_ _6"> </span><span class="ff6">{</span></div><div class="t m0 x30 h8 y2a5 ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x5d h8 y2a4 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x30 h8 y2a5 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x30 hb y2a4 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_"> </span><span class="ff6 ls23">∈V<span class="_ _a"></span>}<span class="ff8">=1</span></span></div><div class="t m0 xfb hc y2a6 ff7 fs4 fc0 sc0 ls3 ws0">E</div><div class="t m0 x125 hd y2a7 ff9 fs0 fc0 sc0 ls3 ws0">P</div><div class="t m0 x102 hb y2a8 ff6 fs4 fc0 sc0 ls3 ws0">{<span class="ff5">g</span></div><div class="t m0 x124 h7 y2a9 ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x10e h8 y2a8 ff8 fs4 fc0 sc0 ls3 ws0">(</div><div class="t m0 x4 h8 y2aa ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x4 h8 y2a8 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x4 h8 y2aa ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x4 hb y2a8 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8">)<span class="ff6 ls38">}≤</span></span>γ</div><div class="t m0 x127 h7 y2a9 ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x3e hb y2a8 ff6 fs4 fc0 sc0 ls3 ws0">∀<span class="ff5">i<span class="_ _6"> </span></span><span class="ls23">∈I</span></div><div class="t m0 x2d h11 y2ab ffb fs4 fc0 sc0 ls3 ws0">⎫</div><div class="t m0 x2d h11 y2ac ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x2d h11 y2ad ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x2d h11 y2ae ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x2d h11 y2af ffb fs4 fc0 sc0 ls3 ws0">⎬</div><div class="t m0 x2d h11 y2b0 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x2d h11 y2b1 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x2d h11 y2b2 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x2d h11 y2b3 ffb fs4 fc0 sc0 ls3 ws0">⎭</div><div class="t m0 x118 h6 y2b4 ff1 fs4 fc0 sc0 lsa ws0">(21)</div><div class="t m0 x2e h6 y2b5 ff1 fs4 fc0 sc0 lsa ws0">where<span class="_ _7"> </span><span class="ff6 ls3">P</span></div><div class="t m0 x128 h7 y2b6 ff8 fs0 fc0 sc0 ls3 ws0">0</div><div class="t m0 x129 h8 y2b7 ff8 fs4 fc0 sc0 ls3 ws0">(<span class="ff7">R</span></div><div class="t m0 x12a hf y2b8 ff6 fs0 fc0 sc0 ls0 ws0">|S<span class="_ _13"></span>|</div><div class="t m0 x4c h6 y2b7 ff8 fs4 fc0 sc0 ls3 ws0">)<span class="_"> </span><span class="ff1 lsb">denotes<span class="_ _8"> </span>the<span class="_ _7"> </span>set<span class="_ _8"> </span>of<span class="_ _7"> </span>all<span class="_ _7"> </span>probability<span class="_ _8"> </span>distributions</span></div><div class="t m0 x2e h6 y2b9 ff1 fs4 fc0 sc0 ls18 ws0">on<span class="_ _6"> </span><span class="ff7 ls3">R</span></div><div class="t m0 xf5 hf y2ba ff6 fs0 fc0 sc0 ls0 ws0">|S<span class="_ _13"></span>|</div><div class="t m0 x12b h6 y2bb ff1 fs4 fc0 sc0 lsa ws0">.<span class="_ _6"> </span>The<span class="_ _8"> </span>second<span class="_ _6"> </span>line<span class="_ _8"> </span>of<span class="_ _6"> </span>set<span class="_ _8"> </span><span class="ff7 ls3">F<span class="_ _5"> </span></span><span class="lsb">suggests<span class="_ _8"> </span>that<span class="_ _6"> </span>the<span class="_ _8"> </span>expectation</span></div><div class="t m0 x2e h6 y2bc ff1 fs4 fc0 sc0 lsb ws0">of<span class="_ _5"> </span>forecast<span class="_ _17"> </span>error</div><div class="t m0 x11a h8 y2bd ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x12c h8 y2bc ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x12c h8 y2bd ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x12c h6 y2bc ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _17"> </span><span class="ff1 lsb">is<span class="_ _17"> </span>zero,<span class="_ _17"> </span>and<span class="_ _5"> </span>the<span class="_ _17"> </span>third<span class="_ _5"> </span>line<span class="_ _17"> </span>implies<span class="_ _17"> </span>that<span class="_ _5"> </span>all</span></div><div class="t m0 x2e h6 y2be ff1 fs4 fc0 sc0 lsf ws0">outcomes<span class="_"> </span>of</div><div class="t m0 x6a h8 y2bf ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x12a h8 y2be ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x12a h8 y2bf ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x6a h6 y2be ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _6"> </span><span class="ff1 lsb">are<span class="_"> </span>within<span class="_ _9"> </span>an<span class="_"> </span>uncertainty<span class="_"> </span>set<span class="_ _9"> </span></span><span class="ff6">V<span class="_ _33"></span><span class="ff1 lsf">,<span class="_"> </span>deﬁned<span class="_"> </span>the<span class="_ _9"> </span>same</span></span></div><div class="t m0 x2e h6 y2c0 ff1 fs4 fc0 sc0 lsb ws0">as<span class="_ _9"> </span>the<span class="_"> </span>con<span class="_ _1"></span>ventional<span class="_ _9"> </span>robust<span class="_ _9"> </span>optimization.<span class="_ _c"> </span>In<span class="_ _9"> </span>this<span class="_"> </span>paper<span class="_ _1"></span>,<span class="_ _9"> </span>we<span class="_"> </span>use<span class="_ _9"> </span>an</div><div class="t m0 x2e h6 y2c1 ff1 fs4 fc0 sc0 lsb ws0">uncertainty<span class="_ _c"> </span>set<span class="_ _c"> </span>in<span class="_ _c"> </span>(22)<span class="_ _9"> </span>to<span class="_"> </span>specify<span class="_ _9"> </span>the<span class="_"> </span>lo<span class="_ _1"></span>wer<span class="_"> </span>bound<span class="_ _9"> </span><span class="ff5 ls3">V</span></div><div class="t m0 x44 hf y2c2 ff6 fs0 fc0 sc0 ls3 ws0">−</div><div class="t m0 x59 h7 y2c3 ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 x12d h6 y2c4 ff1 fs4 fc0 sc0 ls18 ws0">and<span class="_ _c"> </span>upper</div><div class="t m0 x2e h6 y2c5 ff1 fs4 fc0 sc0 ls18 ws0">bound<span class="_"> </span><span class="ff5 ls3">V</span></div><div class="t m0 x12e h7 y2c6 ff8 fs0 fc0 sc0 ls3 ws0">+</div><div class="t m0 x12f h7 y2c7 ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 x10b h6 y2c8 ff1 fs4 fc0 sc0 lsa ws0">of<span class="_"> </span>each<span class="_"> </span>random<span class="_"> </span>variable<span class="_"> </span><span class="ff8 ls3">˜<span class="_ _23"></span><span class="ff5">v</span></span></div><div class="t m0 x130 h7 y2c9 ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 x131 hb y2ca ff6 fs4 fc0 sc0 ls3 ws0">V<span class="_ _7"> </span><span class="ff8">=</span></div><div class="t m0 x2f h11 y2cb ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 x121 h8 y2ca ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x121 h8 y2cc ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x119 hb y2ca ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_"> </span><span class="ff6">∈<span class="_ _6"> </span><span class="ff7">R</span></span></div><div class="t m0 xc h7 y2cd ff5 fs0 fc0 sc0 ls3 ws0">S</div><div class="t m0 x34 h8 y2ca ff8 fs4 fc0 sc0 ls3 ws0">:<span class="_ _1f"> </span><span class="ff5">V</span></div><div class="t m0 x101 hf y2cd ff6 fs0 fc0 sc0 ls3 ws0">−</div><div class="t m0 x132 h7 y2ce ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 x7b hb y2ca ff6 fs4 fc0 sc0 ls3 ws0">≤<span class="_ _6"> </span><span class="ff5">v</span></div><div class="t m0 x133 h7 y2cf ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 x112 hb y2ca ff6 fs4 fc0 sc0 ls3 ws0">≤<span class="_ _6"> </span><span class="ff5">V</span></div><div class="t m0 x10e h7 y2d0 ff8 fs0 fc0 sc0 ls3 ws0">+</div><div class="t m0 x71 h7 y2d1 ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 x134 hb y2ca ff6 fs4 fc0 sc0 ls3 ws0">∀<span class="ff5">r<span class="_ _8"> </span></span><span class="ls1f">∈R<span class="_ _2a"> </span>∀<span class="_ _1c"></span><span class="ff5 ls3">t<span class="_ _6"> </span><span class="ff6 ls23">∈T</span></span></span></div><div class="t m0 x46 h11 y2cb ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 x118 h6 y2ca ff1 fs4 fc0 sc0 lsa ws0">(22)</div><div class="t m0 x19 h6 y2d2 ff1 fs4 fc0 sc0 lsa ws0">where<span class="_ _6"> </span>the<span class="_ _8"> </span>index<span class="_ _6"> </span><span class="ff5 ls3">r<span class="_"> </span></span>is<span class="_ _6"> </span>the<span class="_ _8"> </span>index<span class="_ _6"> </span>of<span class="_ _8"> </span>wind<span class="_ _8"> </span>power<span class="_"> </span>sources,<span class="_ _8"> </span>and<span class="_ _8"> </span><span class="ff5 ls3">t<span class="_ _6"> </span></span><span class="ls28">is</span></div><div class="t m0 x19 h6 y2d3 ff1 fs4 fc0 sc0 lsb ws0">the<span class="_"> </span>index<span class="_"> </span>of<span class="_"> </span>time<span class="_"> </span>steps.</div><div class="t m0 x1c h6 y2d4 ff1 fs4 fc0 sc0 lsa ws0">Function<span class="_ _8"> </span><span class="ff5 ls3">g</span></div><div class="t m0 x96 h7 y2d5 ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x135 h6 y2d6 ff1 fs4 fc0 sc0 lsb ws0">in<span class="_ _6"> </span>(21)<span class="_ _8"> </span>is<span class="_ _8"> </span>used<span class="_ _8"> </span>for<span class="_ _8"> </span>incorporating<span class="_ _8"> </span>distribution<span class="_ _6"> </span>i<span class="_ _13"></span>n-</div><div class="t m0 x19 h6 y2d7 ff1 fs4 fc0 sc0 lsb ws0">formation<span class="_"> </span>into<span class="_"> </span>the<span class="_"> </span>ambiguity<span class="_"> </span>set.<span class="_"> </span>The<span class="_"> </span>index<span class="_"> </span><span class="ff5 ls3">i<span class="_ _6"> </span><span class="ff8 ls23">=(<span class="_ _1c"></span><span class="ff5 ls20">j,<span class="_ _b"> </span>r<span class="_ _0"></span>,<span class="_ _b"> </span>t<span class="ff8 ls3">)<span class="ff1 lse">,<span class="_"> </span>with<span class="_"> </span></span><span class="ff5">j</span></span></span></span></span></div><div class="t m0 x19 h6 y2d8 ff1 fs4 fc0 sc0 lsb ws0">the<span class="_ _7"> </span>index<span class="_ _8"> </span>of<span class="_ _7"> </span>segments<span class="_ _7"> </span>describing<span class="_ _8"> </span>the<span class="_ _7"> </span>distribution<span class="_ _7"> </span>of<span class="_ _8"> </span>each<span class="_ _7"> </span>ran-</div><div class="t m0 x19 h6 y2d9 ff1 fs4 fc0 sc0 lsa ws0">dom<span class="_ _17"> </span>variable.<span class="_ _5"> </span>These<span class="_ _1f"> </span>expected<span class="_ _17"> </span>expressions<span class="_ _5"> </span>are<span class="_ _1f"> </span>deﬁned<span class="_ _17"> </span>in<span class="_ _17"> </span>the</div><div class="t m0 x19 h6 y2da ff1 fs4 fc0 sc0 ls3 ws0">follo<span class="_ _0"></span>wing<span class="_"> </span>piecewise<span class="_"> </span>linear<span class="_"> </span>forms:</div><div class="t m0 x136 hc y2db ff7 fs4 fc0 sc0 ls3 ws0">E</div><div class="t m0 x28 hd y2dc ff9 fs0 fc0 sc0 ls3 ws0">P</div><div class="t m0 x23 hb y2dd ff6 fs4 fc0 sc0 ls3 ws0">{<span class="ff5">g</span></div><div class="t m0 x137 h7 y2de ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x25 h8 y2dd ff8 fs4 fc0 sc0 ls3 ws0">(</div><div class="t m0 xd7 h8 y2df ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 xce h8 y2dd ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 xce h8 y2df ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 xd7 hb y2dd ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8">)<span class="ff6">}<span class="_ _6"> </span></span>=<span class="_ _6"> </span><span class="ff7">E</span></span></div><div class="t m0 xa8 hd y2dc ff9 fs0 fc0 sc0 ls3 ws0">P</div><div class="t m0 x138 hb y2dd ff6 fs4 fc0 sc0 ls3 ws0">{<span class="ff5">g</span></div><div class="t m0 xd8 h7 y2de ff5 fs0 fc0 sc0 ls3e ws0">jrt</div><div class="t m0 x139 h8 y2dd ff8 fs4 fc0 sc0 ls3 ws0">(</div><div class="t m0 x8a h8 y2df ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x8a h8 y2dd ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x8a h8 y2df ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x8a hb y2dd ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8">)<span class="ff6">}</span></span></div><div class="t m0 xbf h8 y2e0 ff8 fs4 fc0 sc0 ls3 ws0">=<span class="_ _6"> </span><span class="ff7">E</span></div><div class="t m0 x9d hd y2e1 ff9 fs0 fc0 sc0 ls3 ws0">P</div><div class="t m0 x97 hb y2e2 ff8 fs4 fc0 sc0 ls18 ws0">(max<span class="_ _b"> </span><span class="ff6 ls3">{<span class="_ _13"></span><span class="ff8">˜<span class="_ _23"></span><span class="ff5">v</span></span></span></div><div class="t m0 x138 h7 y2e3 ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 xd0 hb y2e2 ff6 fs4 fc0 sc0 ls3 ws0">−<span class="_ _c"> </span><span class="ff5">C</span></div><div class="t m0 xaa h7 y2e3 ff5 fs0 fc0 sc0 ls3f ws0">jrt</div><div class="t m0 xd2 hb y2e2 ff5 fs4 fc0 sc0 ls3 ws0">,<span class="_ _b"> </span><span class="ff8">0<span class="ff6">}</span>)<span class="_ _6"> </span><span class="ff6">≤<span class="_ _8"> </span></span></span>γ</div><div class="t m0 x8e h7 y2e3 ff5 fs0 fc0 sc0 ls3e ws0">jrt</div><div class="t m0 x23 h6 y2e4 ff6 fs4 fc0 sc0 ls3 ws0">∀<span class="ff5">j<span class="_"> </span></span><span class="ls23">∈J<span class="_ _1d"> </span>∀<span class="_ _1c"></span><span class="ff5 ls3">r<span class="_"> </span><span class="ff6 ls1f">∈R<span class="_ _27"> </span>∀<span class="_ _42"></span><span class="ff5 ls3">t<span class="_ _6"> </span><span class="ff6 ls1f">∈T</span>,<span class="_"> </span><span class="ff6 ls21">J×<span class="_ _1a"></span>R<span class="_ _a"></span>×<span class="_ _1a"></span><span class="ff5 ls3">T<span class="_ _17"> </span><span class="ff8">=<span class="_ _8"> </span><span class="ff6">I<span class="_ _48"> </span><span class="ff1 lsa">(23)</span></span></span></span></span></span></span></span></span></div><div class="t m0 x19 h6 y2e5 ff1 fs4 fc0 sc0 lsa ws0">where<span class="_"> </span><span class="ff6 ls3">J<span class="_ _5"> </span></span><span class="lsb">is<span class="_"> </span>a<span class="_ _c"> </span>set<span class="_"> </span>of<span class="_ _9"> </span>segments<span class="_"> </span>describing<span class="_ _9"> </span>the<span class="_"> </span>distribution<span class="_ _9"> </span>of<span class="_"> </span>each</span></div><div class="t m0 x19 h6 y2e6 ff1 fs4 fc0 sc0 lsa ws0">random<span class="_ _14"> </span>v<span class="_ _0"></span>ariable,<span class="_ _14"> </span>and<span class="_ _14"> </span><span class="ff5 ls3">C</span></div><div class="t m0 x13a h7 y2e7 ff5 fs0 fc0 sc0 ls3f ws0">jrt</div><div class="t m0 x13b h6 y2e8 ff1 fs4 fc0 sc0 lsb ws0">is<span class="_ _1f"> </span>a<span class="_ _14"> </span>cut-off<span class="_ _1f"> </span>constant.<span class="_ _29"> </span>Expression</div><div class="t m0 x19 h6 y2e9 ff1 fs4 fc0 sc0 lsa ws0">(23)<span class="_ _17"> </span>suggests<span class="_ _1f"> </span>that<span class="_ _1f"> </span>the<span class="_ _1f"> </span>expected<span class="_ _17"> </span>value<span class="_ _17"> </span>of<span class="_ _1f"> </span>the<span class="_ _1f"> </span>positiv<span class="_ _1"></span>e<span class="_ _1f"> </span>part<span class="_ _1f"> </span>of</div><div class="t m0 x19 h8 y2ea ff8 fs4 fc0 sc0 ls3 ws0">˜<span class="_ _23"></span><span class="ff5">v</span></div><div class="t m0 x16 h7 y2eb ff5 fs0 fc0 sc0 ls1d ws0">rt</div><div class="t m0 xef hb y2ec ff6 fs4 fc0 sc0 ls3 ws0">−<span class="_ _c"> </span><span class="ff5">C</span></div><div class="t m0 xbe h7 y2eb ff5 fs0 fc0 sc0 ls3e ws0">jrt</div><div class="t m0 x25 h6 y2ec ff1 fs4 fc0 sc0 lsb ws0">is<span class="_"> </span>limited<span class="_ _c"> </span>below<span class="_ _c"> </span>a<span class="_"> </span>constant<span class="_"> </span><span class="ff5 ls3">γ</span></div><div class="t m0 xae h7 y2eb ff5 fs0 fc0 sc0 ls3e ws0">jrt</div><div class="t m0 x13c h6 y2ec ff1 fs4 fc0 sc0 ls3 ws0">.<span class="_"> </span>Intuiti<span class="_ _1"></span>vely<span class="_ _1"></span>,<span class="_"> </span>as<span class="_ _c"> </span>more</div><div class="t m0 x19 h6 y2ed ff1 fs4 fc0 sc0 ls3 ws0">segments<span class="_ _6"> </span>are<span class="_ _8"> </span>incorporated<span class="_ _8"> </span>into<span class="_ _8"> </span>the<span class="_ _8"> </span>ambiguity<span class="_ _8"> </span>set,<span class="_ _6"> </span>distributions</div><div class="t m0 x19 h6 y2ee ff1 fs4 fc0 sc0 lsa ws0">of<span class="_"> </span>wind<span class="_ _8"> </span>can<span class="_ _6"> </span>be<span class="_ _6"> </span>characterized<span class="_ _8"> </span>with<span class="_"> </span>more<span class="_ _8"> </span>details,<span class="_ _6"> </span>which<span class="_ _6"> </span>leads<span class="_ _6"> </span>to</div><div class="t m0 x19 h6 y2ef ff1 fs4 fc0 sc0 ls3 ws0">less<span class="_"> </span>conserv<span class="_ _0"></span>ativ<span class="_ _1"></span>e<span class="_"> </span>solutions.</div><div class="t m0 x1c h6 y2f0 ff1 fs4 fc0 sc0 lsb ws0">W<span class="_ _3"></span>e<span class="_"> </span>are<span class="_"> </span>aware<span class="_ _c"> </span>that<span class="_"> </span>pre<span class="_ _0"></span>vious<span class="_"> </span>literature<span class="_"> </span>[24]–[26]<span class="_"> </span>has<span class="_"> </span>proposed</div><div class="t m0 x19 h6 y2f1 ff1 fs4 fc0 sc0 lsa ws0">v<span class="_ _0"></span>arious<span class="_ _5"> </span>nonlinear<span class="_ _5"> </span><span class="ff5 ls3">g</span></div><div class="t m0 xa7 h7 y2f2 ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x87 h6 y2f3 ff1 fs4 fc0 sc0 lsa ws0">functions<span class="_ _5"> </span>for<span class="_ _5"> </span>characterizing<span class="_ _7"> </span>variances<span class="_ _5"> </span>or</div><div class="t m0 x19 h6 y2f4 ff1 fs4 fc0 sc0 lsa ws0">e<span class="_ _0"></span>ven<span class="_"> </span>higher<span class="_"> </span>order<span class="_"> </span>moments<span class="_"> </span>of<span class="_"> </span>distrib<span class="_ _1"></span>utions.<span class="_"> </span>In<span class="_"> </span>this<span class="_"> </span>paper<span class="_ _1"></span>,<span class="_"> </span>how-</div><div class="t m0 x19 h6 y2f5 ff1 fs4 fc0 sc0 lsa ws0">e<span class="_ _0"></span>ver<span class="_ _1"></span>,<span class="_ _9"> </span>we<span class="_ _9"> </span>will<span class="_ _c"> </span>focus<span class="_ _9"> </span>on<span class="_ _9"> </span>applying<span class="_ _c"> </span>the<span class="_ _9"> </span>linear<span class="_ _9"> </span>piecewise<span class="_ _9"> </span><span class="ff5 ls3">g</span></div><div class="t m0 x13d h7 y2f6 ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x81 h6 y2f7 ff1 fs4 fc0 sc0 lsa ws0">functions</div><div class="t m0 x19 h6 y2f8 ff1 fs4 fc0 sc0 lsf ws0">because<span class="_"> </span>the<span class="_ _6"> </span>computational<span class="_ _6"> </span>cost<span class="_ _6"> </span>of<span class="_ _6"> </span>nonlinear<span class="_ _6"> </span><span class="ff5 ls3">g</span></div><div class="t m0 xaf h7 y2f9 ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 xd6 h6 y2fa ff1 fs4 fc0 sc0 lsb ws0">functions<span class="_"> </span>is<span class="_ _6"> </span>too</div><div class="t m0 x19 h6 y2fb ff1 fs4 fc0 sc0 lsb ws0">high<span class="_"> </span>to<span class="_ _9"> </span>be<span class="_"> </span>implemented<span class="_"> </span>in<span class="_ _c"> </span>large<span class="_ _9"> </span>power<span class="_ _c"> </span>systems.<span class="_"> </span>Linear<span class="_ _c"> </span><span class="ff5 ls3">g</span></div><div class="t m0 x13e h7 y2fc ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x29 h6 y2fd ff1 fs4 fc0 sc0 lsa ws0">func-</div><div class="t m0 x19 h6 y2fe ff1 fs4 fc0 sc0 lsa ws0">tions<span class="_ _5"> </span>cause<span class="_ _17"> </span>much<span class="_ _5"> </span>lower<span class="_ _5"> </span>computational<span class="_ _17"> </span>burden,<span class="_ _5"> </span>thus<span class="_ _5"> </span>allowing</div><div class="t m0 x19 h6 y2ff ff1 fs4 fc0 sc0 lsd ws0">more<span class="_"> </span><span class="ff5 ls3">g</span></div><div class="t m0 xcd h7 y300 ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x13f h6 y301 ff1 fs4 fc0 sc0 lsb ws0">functions<span class="_ _6"> </span>to<span class="_ _6"> </span>be<span class="_ _6"> </span>incorporated<span class="_ _8"> </span>to<span class="_ _6"> </span>better<span class="_ _6"> </span>capture<span class="_ _8"> </span>distribu-</div><div class="t m0 x19 h6 y302 ff1 fs4 fc0 sc0 lsb ws0">tional<span class="_"> </span>characteristics.</div><div class="t m0 x1c h6 y303 ff1 fs4 fc0 sc0 lsa ws0">Because<span class="_ _7"> </span>the<span class="_ _7"> </span>e<span class="_ _1"></span>xpectation<span class="_ _7"> </span>of<span class="_ _7"> </span>each<span class="_ _7"> </span>function<span class="_ _7"> </span><span class="ff5 ls3">g</span></div><div class="t m0 xe0 h7 y304 ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x140 h6 y305 ff1 fs4 fc0 sc0 lsf ws0">under<span class="_ _7"> </span>ambigu-</div><div class="t m0 x19 h6 y306 ff1 fs4 fc0 sc0 ls3 ws0">ous<span class="_ _7"> </span>distributions<span class="_ _5"> </span>is<span class="_ _7"> </span>difﬁcult<span class="_ _7"> </span>to<span class="_ _5"> </span>assess<span class="_ _5"> </span>directly<span class="_ _1"></span>,<span class="_ _7"> </span>some<span class="_ _5"> </span>auxiliary</div><div class="t m0 x19 h6 y307 ff1 fs4 fc0 sc0 lsa ws0">v<span class="_ _0"></span>ariables,<span class="_ _7"> </span>denoted<span class="_ _7"> </span>by<span class="_ _7"> </span>the<span class="_ _5"> </span>vector</div><div class="t m0 xb8 h8 y308 ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 xb8 h8 y307 ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 xb8 h8 y308 ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 xb8 hb y307 ff5 fs4 fc0 sc0 ls3 ws0">u<span class="_ _6"> </span><span class="ff6">∈<span class="_ _6"> </span><span class="ff7">R</span></span></div><div class="t m0 xae hf y309 ff6 fs0 fc0 sc0 ls0 ws0">|I<span class="_ _13"></span>|</div><div class="t m0 x141 h6 y30a ff1 fs4 fc0 sc0 lsb ws0">,<span class="_ _7"> </span>are<span class="_ _7"> </span>introduced<span class="_ _7"> </span>into</div><div class="t m0 x19 h6 y30b ff1 fs4 fc0 sc0 lsa ws0">the<span class="_ _6"> </span>model<span class="_ _8"> </span>to<span class="_ _8"> </span>express<span class="_ _8"> </span>the<span class="_ _8"> </span>upper<span class="_ _8"> </span>bounds<span class="_ _8"> </span>of<span class="_ _6"> </span>these<span class="_ _8"> </span>functions.<span class="_ _8"> </span>The</div><div class="t m0 x19 h6 y30c ff1 fs4 fc0 sc0 lsb ws0">original<span class="_"> </span>ambiguity<span class="_"> </span>set<span class="_"> </span>is<span class="_"> </span>extended<span class="_"> </span>to<span class="_"> </span>set<span class="_"> </span><span class="ff7 ls3">G<span class="_ _13"></span></span>,<span class="_"> </span>expressed<span class="_"> </span>as</div><div class="t m0 x142 h8 y30d ff7 fs4 fc0 sc0 ls3 ws0">G<span class="_ _8"> </span><span class="ff8">=</span></div><div class="t m0 x143 h11 y30e ffb fs4 fc0 sc0 ls3 ws0">⎧</div><div class="t m0 x143 h11 y30f ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x143 h11 y310 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x143 h11 y311 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x143 h11 y312 ffb fs4 fc0 sc0 ls3 ws0">⎨</div><div class="t m0 x143 h11 y313 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x143 h11 y314 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x143 h11 y315 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x143 h11 y316 ffb fs4 fc0 sc0 ls3 ws0">⎩</div><div class="t m0 x94 hb y317 ff7 fs4 fc0 sc0 ls3 ws0">Q<span class="_ _6"> </span><span class="ff6 ls23">∈P</span></div><div class="t m0 xa6 h7 y318 ff8 fs0 fc0 sc0 ls3 ws0">0</div><div class="t m0 x98 h8 y319 ff8 fs4 fc0 sc0 ls3 ws0">(<span class="ff7">R</span></div><div class="t m0 xa7 hf y31a ff6 fs0 fc0 sc0 ls0 ws0">|S<span class="_ _13"></span>|</div><div class="t m0 x144 hb y319 ff6 fs4 fc0 sc0 ls3 ws0">×<span class="_ _c"> </span><span class="ff7">R</span></div><div class="t m0 xd9 hf y31a ff6 fs0 fc0 sc0 ls0 ws0">|I<span class="_ _13"></span>|</div><div class="t m0 xe8 h8 y319 ff8 fs4 fc0 sc0 ls38 ws0">):</div><div class="t m0 xd2 h8 y31b ff8 fs4 fc0 sc0 ls3 ws0">(</div><div class="t m0 xb8 h8 y31c ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 xb8 h8 y31b ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 xb8 h8 y31c ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 xb8 h8 y31b ff5 fs4 fc0 sc0 ls40 ws0">v,</div><div class="t m0 xc7 h8 y31c ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 xad h8 y31b ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 xad h8 y31c ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 xad hb y31b ff5 fs4 fc0 sc0 ls3 ws0">u<span class="ff8">)<span class="_ _6"> </span><span class="ff6">∈<span class="_ _6"> </span><span class="ff7">R</span></span></span></div><div class="t m0 xdc hf y31d ff6 fs0 fc0 sc0 ls0 ws0">|S<span class="_ _13"></span>|</div><div class="t m0 xaf hb y31e ff6 fs4 fc0 sc0 ls3 ws0">×<span class="_ _c"> </span><span class="ff7">R</span></div><div class="t m0 x145 hf y31d ff6 fs0 fc0 sc0 ls0 ws0">|I<span class="_ _13"></span>|</div><div class="t m0 xd2 hc y31f ff7 fs4 fc0 sc0 ls3 ws0">E</div><div class="t m0 x146 hd y320 ff9 fs0 fc0 sc0 ls3 ws0">Q</div><div class="t m0 x147 hb y31f ff6 fs4 fc0 sc0 ls3 ws0">{</div><div class="t m0 x148 h8 y321 ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x8d h8 y31f ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x8d h8 y321 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x148 hb y31f ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff6">}<span class="_ _6"> </span><span class="ff8 ls1f">=0</span></span></div><div class="t m0 xd2 hc y322 ff7 fs4 fc0 sc0 ls3 ws0">Q</div><div class="t m0 x149 h11 y323 ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 x147 h8 y322 ff8 fs4 fc0 sc0 ls3 ws0">(</div><div class="t m0 x8d h8 y324 ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 x14a h8 y322 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x14a h8 y324 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x14a h8 y322 ff5 fs4 fc0 sc0 ls40 ws0">v,</div><div class="t m0 x14b h8 y324 ff8 fs4 fc0 sc0 ls3 ws0">˜</div><div class="t m0 xae h8 y322 ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 xe7 h8 y324 ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 xe7 hb y322 ff5 fs4 fc0 sc0 ls3 ws0">u<span class="ff8">)<span class="_ _6"> </span><span class="ff6">∈</span></span></div><div class="t m0 x90 h8 y325 ff8 fs4 fc0 sc0 ls3 ws0">¯</div><div class="t m0 xd5 hb y322 ff6 fs4 fc0 sc0 ls3 ws0">V</div><div class="t m0 x14c h11 y323 ffb fs4 fc0 sc0 ls3 ws0"></div><div class="t m0 xdd h8 y322 ff8 fs4 fc0 sc0 ls23 ws0">=1</div><div class="t m0 xd2 hc y326 ff7 fs4 fc0 sc0 ls3 ws0">E</div><div class="t m0 x146 hd y327 ff9 fs0 fc0 sc0 ls3 ws0">Q</div><div class="t m0 x147 hb y328 ff6 fs4 fc0 sc0 ls3 ws0">{<span class="_ _33"></span><span class="ff8">˜<span class="_ _21"></span><span class="ff5">u</span></span></div><div class="t m0 xb9 h7 y329 ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x14b hb y328 ff6 fs4 fc0 sc0 ls38 ws0">}≤<span class="ff5 ls3">γ</span></div><div class="t m0 xaf h7 y329 ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 xdd hb y328 ff6 fs4 fc0 sc0 ls3 ws0">∀<span class="ff5">i<span class="_ _6"> </span></span><span class="ls1f">∈I</span></div><div class="t m0 x20 h11 y32a ffb fs4 fc0 sc0 ls3 ws0">⎫</div><div class="t m0 x20 h11 y32b ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x20 h11 y32c ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x20 h11 y32d ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x20 h11 y32e ffb fs4 fc0 sc0 ls3 ws0">⎬</div><div class="t m0 x20 h11 y32f ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x20 h11 y330 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x20 h11 y331 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x20 h11 y332 ffb fs4 fc0 sc0 ls3 ws0">⎭</div><div class="t m0 x21 h8 y333 ff5 fs4 fc0 sc0 ls3 ws0">.</div><div class="t m0 x14d h6 y334 ff1 fs4 fc0 sc0 lsa ws0">(24)</div><div class="t m0 x19 h6 y335 ff1 fs4 fc0 sc0 lsb ws0">This<span class="_ _9"> </span>extended<span class="_ _9"> </span>set<span class="_ _9"> </span>deﬁnes<span class="_ _c"> </span>a<span class="_ _9"> </span>family<span class="_ _9"> </span>of<span class="_ _9"> </span>joint<span class="_ _9"> </span>distributions,<span class="_ _9"> </span>denoted</div><div class="t m0 x19 h6 y336 ff1 fs4 fc0 sc0 ls18 ws0">by<span class="_ _b"> </span><span class="ff7 ls3">Q</span><span class="lsa">,<span class="_ _9"> </span>for<span class="_ _9"> </span>both<span class="_ _b"> </span>the<span class="_ _9"> </span>random<span class="_ _9"> </span>variables<span class="_ _18"></span><span class="ff5 ls3">v</span></span></div><div class="t m0 x149 h8 y337 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x14e h6 y336 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _c"> </span><span class="ff1 lsb">and<span class="_ _b"> </span>the<span class="_ _9"> </span>auxiliary<span class="_ _b"> </span>variables</span></div><div class="t m0 x19 h8 y338 ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 x19 h8 y339 ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 x19 h6 y338 ff5 fs4 fc0 sc0 ls3 ws0">u<span class="ff1 lsb">.<span class="_"> </span>The<span class="_"> </span>uncertainty<span class="_"> </span>set</span></div><div class="t m0 x144 h8 y33a ff8 fs4 fc0 sc0 ls3 ws0">¯</div><div class="t m0 xb3 h6 y338 ff6 fs4 fc0 sc0 ls3 ws0">V<span class="_ _7"> </span><span class="ff1 lsb">in<span class="_"> </span>the<span class="_ _c"> </span>third<span class="_"> </span>line<span class="_"> </span>of<span class="_"> </span>(24)<span class="_ _c"> </span>is<span class="_"> </span>the<span class="_"> </span>e<span class="_ _0"></span>xtended</span></div><div class="t m0 x19 h6 y33b ff1 fs4 fc0 sc0 lsb ws0">form<span class="_"> </span>of<span class="_"> </span><span class="ff6 ls3">V<span class="_ _33"></span></span>,<span class="_"> </span>expressed<span class="_"> </span>by</div><div class="t m0 x19 h8 y33c ff8 fs4 fc0 sc0 ls3 ws0">¯</div><div class="t m0 x1e hb y33d ff6 fs4 fc0 sc0 ls3 ws0">V<span class="_ _7"> </span><span class="ff8">=</span></div><div class="t m0 x14f h11 y33e ffb fs4 fc0 sc0 ls3 ws0">⎧</div><div class="t m0 x14f h11 y33f ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x14f h11 y340 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x14f h11 y341 ffb fs4 fc0 sc0 ls3 ws0">⎨</div><div class="t m0 x14f h11 y342 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x14f h11 y343 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x14f h11 y344 ffb fs4 fc0 sc0 ls3 ws0">⎩</div><div class="t m0 xcd h8 y345 ff8 fs4 fc0 sc0 ls3 ws0">(<span class="_ _1"></span><span class="ff5">v</span></div><div class="t m0 xbe h8 y346 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 xbe h8 y345 ff5 fs4 fc0 sc0 ls26 ws0">v,<span class="_ _33"></span>u</div><div class="t m0 x137 h8 y346 ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 x150 hb y345 ff5 fs4 fc0 sc0 ls3 ws0">u<span class="ff8">)<span class="_ _6"> </span><span class="ff6">∈<span class="_ _6"> </span><span class="ff7">R</span></span></span></div><div class="t m0 xa0 hf y347 ff6 fs0 fc0 sc0 ls0 ws0">|S<span class="_ _13"></span>|</div><div class="t m0 x151 hb y348 ff6 fs4 fc0 sc0 ls3 ws0">×<span class="_ _c"> </span><span class="ff7">R</span></div><div class="t m0 x13a hf y347 ff6 fs0 fc0 sc0 ls0 ws0">|I<span class="_ _13"></span>|</div><div class="t m0 xb5 h8 y348 ff8 fs4 fc0 sc0 ls3 ws0">:</div><div class="t m0 x13b h8 y349 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x13b h8 y34a ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x13b hb y349 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_"> </span><span class="ff6 ls1f">∈V</span></div><div class="t m0 x13b h8 y34b ff5 fs4 fc0 sc0 ls3 ws0">g</div><div class="t m0 xd1 h7 y34c ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x83 h8 y34d ff8 fs4 fc0 sc0 ls3 ws0">(<span class="_ _1"></span><span class="ff5">v</span></div><div class="t m0 x152 h8 y34e ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x152 hb y34d ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8">)<span class="_ _6"> </span><span class="ff6">≤<span class="_ _6"> </span></span></span>u</div><div class="t m0 x153 h7 y34c ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 xdb hb y34d ff6 fs4 fc0 sc0 ls3 ws0">∀<span class="ff5">i<span class="_ _6"> </span></span><span class="ls1f">∈I</span></div><div class="t m0 x13b h8 y34f ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 xc5 h7 y350 ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x152 hb y351 ff6 fs4 fc0 sc0 ls3 ws0">≤<span class="_ _6"> </span><span class="ff8 ls18">max</span></div><div class="t m0 x149 h7 y352 ff5 fs0 fc0 sc0 ls3 ws0">v</div><div class="t m0 x149 h7 y353 ff5 fs0 fc0 sc0 ls3 ws0">v</div><div class="t m0 x149 hf y352 ff5 fs0 fc0 sc0 ls3 ws0">v<span class="_ _33"></span><span class="ff6 ls30">∈V</span></div><div class="t m0 x154 h8 y351 ff5 fs4 fc0 sc0 ls3 ws0">g</div><div class="t m0 xda h7 y350 ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x8e h8 y351 ff8 fs4 fc0 sc0 ls3 ws0">(<span class="_ _1"></span><span class="ff5">v</span></div><div class="t m0 x13c h8 y354 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x13c hb y351 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8">)<span class="_ _f"> </span><span class="ff6">∀</span></span>i<span class="_ _6"> </span><span class="ff6 ls1f">∈I</span></div><div class="t m0 x155 h11 y355 ffb fs4 fc0 sc0 ls3 ws0">⎫</div><div class="t m0 x155 h11 y356 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x155 h11 y357 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x155 h11 y358 ffb fs4 fc0 sc0 ls3 ws0">⎬</div><div class="t m0 x155 h11 y359 ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x155 h11 y35a ffb fs4 fc0 sc0 ls3 ws0">⎪</div><div class="t m0 x155 h11 y35b ffb fs4 fc0 sc0 ls3 ws0">⎭</div><div class="t m0 x20 h6 y35c ff5 fs4 fc0 sc0 ls3 ws0">.<span class="_ _49"> </span><span class="ff1 lsa">(25)</span></div><div class="t m0 x19 h6 y35d ff1 fs4 fc0 sc0 lsa ws0">The<span class="_ _17"> </span>second<span class="_ _17"> </span>line<span class="_ _17"> </span>of<span class="_ _17"> </span>expression<span class="_ _17"> </span>(25)<span class="_ _17"> </span>suggests<span class="_ _17"> </span>that<span class="_ _17"> </span>function<span class="_ _17"> </span><span class="ff5 ls3">g</span></div><div class="t m0 x156 h7 y35e ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x19 h6 y35f ff1 fs4 fc0 sc0 lsb ws0">is<span class="_ _8"> </span>bounded<span class="_ _7"> </span>by<span class="_ _8"> </span>the<span class="_ _8"> </span>auxiliary<span class="_ _7"> </span>v<span class="_ _1"></span>ariable<span class="_ _7"> </span><span class="ff5 ls3">u</span></div><div class="t m0 xd4 h7 y360 ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x154 h6 y35f ff1 fs4 fc0 sc0 lsb ws0">,<span class="_ _8"> </span>so<span class="_ _7"> </span>that<span class="_ _8"> </span>the<span class="_ _8"> </span>fourth<span class="_ _7"> </span>line</div><div class="t m0 x19 h6 y361 ff1 fs4 fc0 sc0 lsb ws0">of<span class="_ _8"> </span>ambiguity<span class="_ _7"> </span>set<span class="_ _8"> </span><span class="ff7 ls3">F<span class="_ _17"> </span></span>in<span class="_ _7"> </span>(21)<span class="_ _8"> </span>holds<span class="_ _7"> </span>if<span class="_ _8"> </span>the<span class="_ _8"> </span>fourth<span class="_ _7"> </span>constraint<span class="_ _8"> </span>of<span class="_ _8"> </span>set</div><div class="t m0 x19 h6 y362 ff7 fs4 fc0 sc0 ls3 ws0">G<span class="_ _5"> </span><span class="ff1 lsb">in<span class="_ _5"> </span>(24)<span class="_ _7"> </span>is<span class="_ _5"> </span>satisﬁed.<span class="_ _7"> </span>The<span class="_ _5"> </span>last<span class="_ _5"> </span>inequality<span class="_ _7"> </span>in<span class="_ _5"> </span>(25)<span class="_ _7"> </span>implies<span class="_ _5"> </span>that</span></div><div class="t m0 x19 h8 y363 ff5 fs4 fc0 sc0 ls3 ws0">u</div><div class="t m0 xdf h7 y364 ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x1b h6 y365 ff1 fs4 fc0 sc0 lsa ws0">is<span class="_ _8"> </span>conﬁned<span class="_ _7"> </span>belo<span class="_ _1"></span>w<span class="_ _7"> </span>the<span class="_ _8"> </span>worst-case<span class="_ _8"> </span>value<span class="_ _6"> </span>of<span class="_ _7"> </span><span class="ff5 ls3">g</span></div><div class="t m0 xe6 h7 y364 ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 xbb h8 y365 ff8 fs4 fc0 sc0 ls3 ws0">(<span class="_ _1"></span><span class="ff5">v</span></div><div class="t m0 xca h8 y366 ff5 fs4 fc0 sc0 ls3 ws0">v</div><div class="t m0 x157 h6 y365 ff5 fs4 fc0 sc0 ls3 ws0">v<span class="_ _13"></span><span class="ff8">)<span class="ff1 lsb">,<span class="_ _8"> </span>which<span class="_ _7"> </span>is<span class="_ _8"> </span>a</span></span></div><div class="t m0 x19 h6 y367 ff1 fs4 fc0 sc0 lsa ws0">constant<span class="_ _b"> </span>and<span class="_ _9"> </span>can<span class="_ _9"> </span>be<span class="_ _9"> </span>separately<span class="_ _9"> </span>calculated<span class="_ _9"> </span>based<span class="_ _9"> </span>on<span class="_ _b"> </span>the<span class="_ _9"> </span>deﬁnition</div><div class="t m0 x19 h6 y368 ff1 fs4 fc0 sc0 ls18 ws0">of<span class="_"> </span><span class="ff5 ls3">g</span></div><div class="t m0 x136 h7 y369 ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x6d h6 y36a ff1 fs4 fc0 sc0 lsb ws0">in<span class="_"> </span>(23).<span class="_"> </span>This<span class="_"> </span>constraint<span class="_"> </span>is<span class="_"> </span>added<span class="_"> </span>to<span class="_"> </span>enforce<span class="_"> </span>a<span class="_"> </span>tighter<span class="_"> </span>upper</div><div class="t m0 x19 h6 y36b ff1 fs4 fc0 sc0 lsf ws0">bound<span class="_ _9"> </span>of<span class="_ _c"> </span>function<span class="_ _9"> </span><span class="ff5 ls3">g</span></div><div class="t m0 x99 h7 y36c ff5 fs0 fc0 sc0 ls3 ws0">i</div><div class="t m0 x158 h6 y23e ff1 fs4 fc0 sc0 lsa ws0">,<span class="_ _9"> </span>thus<span class="_ _c"> </span>giving<span class="_ _b"> </span>a<span class="_"> </span>less<span class="_ _b"> </span>conservati<span class="_ _0"></span>ve<span class="_ _9"> </span>uncertainty</div><div class="t m0 xf he ya3 ffa fs6 fc1 sc0 ls3 ws0">Authorized licensed use limited to: SICHUAN UNIVERSITY. Downloaded on January 05,2021 at 09:02:32 UTC from IEEE Xplore. Restrictions apply. </div></div><div class="pi" data-data='{"ctm":[1.616162,0.000000,0.000000,1.616162,0.000000,0.000000]}'></div></div>