电气自动化专业论文 基于蒙特卡洛法的概率潮流算法研究

[27]3.4本章小结本章建立了蒙特卡洛潮流算法的基本模型，通过Matlab和Matpower进行编程，利用IEEE14节点系统对负荷的有功和无功以及发电机的出力进行了概率潮流计算。在本章开始了解了本论文用到的软件Matlab和Matpower的基本信息和基本的使用方法，参照使用手册进行了程序的编写。

第四章算例分析4.1用IEEE14标准测试系统进行算例分析本文提出的概率潮流计算方法用于IEEE14节点标准测试系统，采用matpower进行确定性的潮流计算。选取和做为代表来显示概率潮流输出随机变量的精度。4.1.1IEEE14节点系统IEEE14节点系统共有24个输入随机变量，负荷为正态分布和离散分布，发电机输出为二项式分布。设采样规模为20000次的结果为准确值，以下图4-1是IEEE14节点系统的结构图，表4-1是IEEE14节点系统正态分布负荷数据，表4-2是IEEE14节点系统中二项式分布的节点数据，表4-3是IEEE14节点系统的线路和变压器的数据，表4-4是IEEE14节点系统中离散分布的节点数据。图4-1IEEE14节点系统结构图表4-SEQ表\*ARABIC1IEEE14节点系统正态分布负荷数据表4-SEQ表\*ARABIC2IEEE14节点系统中二项式分布的节点数据表4-SEQ表\*ARABIC3IEEE14节点系统的线路和变压器数据表4-SEQ表\*ARABIC4IEEE14节点系统中离散分布的节点数据4.1.2算例结果分析通过对IEEE14节点系统的概率潮流计算，分别得到了线路56有功功率和无功功率以及5，6节点的电压，相角和幅值的概率密度曲线和累积分布函数曲线。图4-SEQ图\*ARABIC2线路56有功功率的概率密度曲线图4-SEQ图\*ARABIC3线路56无功功率的概率密度曲线图4-SEQ图\*ARABIC4节点5电压的概率密度曲线图4-SEQ图\*ARABIC5节点6电压的概率密度曲线图4-SEQ图\*ARABIC6节点5的相角概率密度曲线图4-SEQ图\*ARABIC7节点6的相角概率密度曲线图4-SEQ图\*ARABIC8线路56有功的累积分布函数曲线图4-SEQ图\*ARABIC9线路56无功的累积分布函数曲线图4-SEQ图\*ARABIC10节点5电压累积分布函数曲线图4-SEQ图\*ARABIC11节点5相角累积分布函数曲线图4-SEQ图\*ARABIC12节点6相角累积分布函数曲线从图2-12我们知道其分布符合正态分布，线路56的有功功率主要分布在（37，44）之间，峰值在（40，41）之间；其无功功率主要分布在（9，16）之间，峰值在（12，13）间。节点5电压主要分布在（-2，4）之间，集中分布在01之间；而节点6电压相比节点5则靠右集中。节点5的相角在（-11，-5）间，峰值在-8左右；节点6的相角分布在（-15，-10）之间，峰值在-12左右。以下是用蒙特卡罗概率潮流方法得到的节点电压和相角的计算结果。表4-5各节点电压和相角的期望值与标准差节点号电压期望值电压标准差相角期望值相角标准差11.062.00E-150021.0452.77E-15-4.99670.372231.015.49E-15-12.78470.973441.01440.0020.-10.28670.688751.01740.0017-8.76350.583661.075.76E-15-14.23550.990571.05980.0026-13.34341.017481.098.68E-15-13.34341.017491.05450.0048-14.92501.2095101.04940.0041-15.08741.1787111.05610.0025-14.79121.1176121.05500.0029-15.09151.1471131.05010.0012-15.16631.0697141.03440.0035-16.02781.1600IEEE14节点系统蒙特卡罗方法概率潮流计算线路有功和无功计算结果见表4-6表4-6各条线路有功和无功功率的期望值与标准差始端节点号末端节点号有功期望有功标准差无功期望无功标准差12157.325811.2545-20.46922.63041575.50034.83034.82300.50182373.64805.68523.55150.55402456.14493.45400.34160.61022541.55882.54522.36420.454634-23.31764.41806.47162.040945-61.07454.401212.83061.38214728.01913.7768-10.45910.82764916.03632.1541-0.70510.74165644.26724.410611.53630.54996117.39952.05573.96030.98246127.82E1.98112.56140.370361317.77961.50577.42140.661178-3.69E-163.10E-14-18.13761.56637928.01913.77685.90952.18999105.16112.00393.84611.02409149.34321.40733.36360.64431011-3.82761.8417-1.97870.970612131.62041.43160.80660.293313145.66751.21311.98630.66094.1.3计算时间和精度这里把常用的变异系数U（精度）作为采样过程收敛的判据，其定义为： (4-1)式中E(F)为抽样数学期望，V(F)为抽样方差。使用蒙特卡洛模拟法对IEEE14节点系统进行概率潮流计算，分别抽样200，500，1000，2000，3000，4000，5000，6000，7000，8000，9000，10000次，对线路5-6潮流的均值、方差、精度以及计算时间进行了计算，表4-7列出了蒙特卡洛算法的时间和精度。通过计算知道在不同次数下得到的精度不同，抽样次数和精度的平方成反比，抽样次数越高，精度越低，方法就越准确。表4-7不同次数下蒙特卡洛算法的时间和精度次数时间精度方差均值2004.6845.42E-0518.523843.968350011.48282.03E-0519.283343.9769100022.30621.05E-0519.022144.1722200041.49754.93E-0620.528444.1941300067.14923.42E-0619.493944.1835400092.92892.55E-0619.13254411411.98E-0619.526144.22426000146.61961.68E-0619.522244.24027000166.98051.42E-0619.685044.10598000190.40901.24E-0619.495844.20919000213.11501.13E-0620.027944.208610000237.27081.01E-0619.803544.13774.2本章小结本章进行了算例分析。首先系统的了解了IEEE14节点系统的相关知识，得到了节点结构图以及IEEE14节点系统的相关数据。通过对IEEE14节点系统的程序运行，得到了大量的概率计算结果，包括各支路的有功功率和无功功率分布，各节点的相角，电压幅值的概率分布曲线和累积分布函数曲线等。然后对算法进行了时间和精度的校验。结果表示：在对负荷的有功无功以及发电机出力进行的基于蒙特卡洛法的概率潮流计算时，取20000次进行抽样，根据发电机及负荷的概率分布，得到各节点电压幅值以及各支路有功无功等计算结果并统计其概率分布情况，验证了蒙特卡洛法在电力系统潮流计算中的可行性。

第五章总结和展望5.1结论潮流计算是电力系统研究的最基础、最重要的课题之一。本文在总结前人研究成果的基础上，主要对概率潮流计算进行了深入的研究。本文研究了蒙特卡洛法的概率潮流计算方法，并将此方法应用于电力系统概率潮流计算当中。研究主要成果成总结如下：1.建立了概率模型：负荷概率模型、发电机概率模型；2.给出了利用蒙特卡洛方法进行概率潮流计算的方法与步骤，利用matlab编制了概率潮流程序；3.本文应用了蒙特卡洛方法对IEEE14节点系统进行概率潮流计算。结果表明在负荷变化以及发电机出力不确定的情况下，蒙特卡洛方法可以使电力系统中的问题简单化，准确地反映系统中存在的问题。5.2展望本次设计中的不足之处：1.本文仅考虑了发电机出力和节点负荷的不确定性，而电力系统中存在着其他大量的不确定性，概率潮流中可以尝试考虑尽可能多的系统不确定因素，如网络结构的随机变化等。2.本论文的研究工作主要集中在理论研究，加快基于概率方法的系统规划及安全预警方法在电力系统的应用，收集电力系统的运行数据以及故障数据，通过与实际情况的比较，不断修正所提出的评估方法可以对完善理论分析作出有益的补充。现如今，蒙特卡洛方法越来越广泛地被应用到解决实际问题，也形成了一些特定的蒙特卡洛方法，如马尔科夫链蒙特卡洛方法等等，蒙特卡洛方法的应用研究会随着科技的发展更迫切，我相信经过更多人更深入的研究，蒙特卡洛方法一定能取得更大的发展，进一步推广概率潮流算法在电力系统中的应用，蒙特卡洛方法也必将被越来越普遍地应用到解决国民生活的各种各样的问题。

致谢本论文是在贾燕冰老师和麻金碧学姐的悉心指导和热情关怀下完成的。导师严谨求实的治学态度，虚怀若谷的高尚情操，诲人不倦的育人精神、平易近人的师长风范和不辞辛劳、精益求精的工作态度给学生留下了深刻的印象，是学生永远学习的楷模。导师工作繁忙，百忙之中仍时刻关心本论文的研究进展，指导学生的工作也一丝不苟。本论文从选题到论文的完成无不凝聚着恩师的心血和辛劳。两个月来，贾老师在学习、工作和生活中给予了诸多关怀与帮助，学姐也在论文的研究中给了我莫大的帮助，我铭记在心，值此论文完稿之际，对我的导师表示崇高的敬意和衷心的感谢。在这里，对所有指导和关心过我的老师和学长学姐们表示真诚的谢意。另外，我还要深深地感谢我的父母和亲人。在我漫长的求学生涯中，他们含辛茹苦地承受着生活的艰辛，在物质和精神上支持我的学业。正是他们的理解与支持，正是他们深沉而无私的爱，激励着我奋发向上，自强不息，使我能胜利地走完这段求学之路。谨以此文献给所有关心我和爱护我的老师、亲人、同学与朋友。

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Flow.ELECTRICPOWERCOMPONENTSANDSYSTEMS，2016.

附录外文文献AModifiedNatafTransformation-basedExtendedQuasi-MonteCarloSimulationMethodforSolvingProbabilisticLoadFlowAbstract—Comparedtoconventionalprobabilisticloadflowanal-ysis，theextendedprobabilisticloadflowcanbestoppedbyitselfwhentheerrorrestrictionsatisfied，whichismoresuitableinpracticaluse.Sincethepowerflowresultsalreadyobtainedcanberetained，thecomputationalburdensofextendedprobabilisticloadflowlargelydependontheextensionofsamples.Muchoftheliteraturesuggeststhatthequasi-MonteCarlosimulationmethodismoreefficientthanLatinhypercubesampling.Inthisarticle，theextendedtechniqueforquasi-MonteCarlosimulationisproposed.Afterthat，splinerecon-structionisutilizedtomodifytheconventionalNataftransformationtoobtaincorrelatedsamplesbythefirstseveralordersofmoments.Basedontheabovetwotechniques，amodifiedNataftransformation-basedextendedquasi-MonteCarlosimulationmethodisproposedtosolveprobabilisticloadflowproblems.ComparedtoextendedLatinhypercubesampling，theproposedmethodnotonlyhashighercompu-tationalefficiencybutcanalsoextendthesamplesbyarbitrarysteps.Theresultsalsoshowthatbothtemporalandspatialcorrelationscanbemanagedbytheproposedmethod.INTRODUCTIONTheincreasingintegrationofrenewablesourceshasenhancedtherandomnatureofpowersystems[1].Probabilisticloadflow(PLF)，whichwasfirstproposedbyBorkowskain1974[2]，isavitalapproachtoaccesstheperformancesofthepowersystemconsideringtheuncertaintiesofloadsorrenewablegenerations.Fortheabilitytoexposepotentialcrisis，PLFhasbecomeoneofthemostusefultoolsinpowersystemanalysis[2–6].MonteCarlosimulation(MCS)，whichsimulatesvariousuncertaintiesthroughaseriesofdeterministiccalculations，isthemostaccurate，flexible，androbustmethodadoptedinPLFcomparedtootherwell-studiedmethods，includingthecumulantsmethod[7]andpointestimationmethod[8].Asaresult，itisoftenadoptedasthetouchstoneofothermethods.AlthoughMCShastheaboveadvantages，itusuallysuffersfromgreatcomputationalburden.TheimprovementoftheElectricPowerComponentsandSystems，Vol.44(2016)，computationalefficiencyofMCSisthemainfocusofthisarticle.Accordingtowhetherthesamplesizeisfixedornot，MCScanbeclassifiedintotwomaincategories:conventionalMCS[2–5]andextendedMCS[6].InconventionalMCS，thesamplesizeisgivenandfixed;themainfocusrestsonthegenerationsofcorrelatedsamples[9，10].InextendedMCS，thesampleex-tensionwillpersistuntiltheconvergencecriteriaaresatisfied，whichexpandstheapplicationsofPLFandismoresuitableinpracticaluse.Inthegenerationofsamples，pseudo-samplingmethodsrepresentedbyLatinhypercubesampling(LHS)[11]per-formedastratifiedsamplingtechniquethatprovidesanef-ficientapproachtosamplerandomvariablesfromtheirentiredistributions，whichenhancestheuniformityofsampledse-quences.Consequently，theefficiencyofPLFcanbeenhancedbytheLHS.TheextensionofLHSwasfirstproposedin[12]，whichwasutilizedinPLFcalculation[6].Unfortunately，thesamplingstructuredeterminesthattheextensionofLHScanonlyperformfromsampleswithsizeNtosize2N，whichwillprobablyreducetheefficiencyofextendedPLF.Forinstance，iftheoverallalgorithmnearlyconvergeswhenthesamplesizeisN，thentheextensionto2Nwillleadtoplentyofunnec-essarycalculations.Therefore，extendedPLFwitharbitraryextensionstepscanreducethecomputationalburden.Manystudieshaveshownthatthelowdiscrepancysequence(LDS)basedMCSismorecomputationalefficientthanLHS[13，14].Furthermore，LDSsaregeneratedbitwise;thustheextensionwitharbitrarystepsispossible，buttheextensionisstillnotwithinthecurrentresearchfield.Afterthegeneration，thoseobtainedsamplesshouldbetransformedintotheexactdistributionsofrenewablesourcesorloads.Severalmethodsexistinthisfield，includingpolyno-mialnormaltransformation(PNT)[15]，Nataftransformation(awidelyutilizeddistributionreconstructionmethod)[16，17]，andtheRosenblattmethod[15].PNTcanreconstructthedis-tributionsonlybythefirstseveralordersofmoments，uttheaccuracyislowerthantheothertwomethods.Also，theappli-cationsofNataftransformationandtheRosenblattmethodarestilllimitedevenfortheirhigheraccuracybythedemandsofvariables’marginaldistributions.Theproblemthenbecomeswhethersuchamethodexiststhatcancombinebothofthoseadvantages，includingthetransformationaccuracyandabilitytoutilizethefirstseveralmoments.Tosettletheabove-mentionedproblems，amodifiedNataftransformation-basedextendedquasi-MCS(NEQMCS)methodisproposedinthisarticle，whichisabletoenlargesamplesizewitharbitrarystep.Singularvaluedecomposi-tion(SVD)isadoptedinplaceofCholeskydecompositiontohandlethenon-positivedefinitecorrelationcoefficientmatrix，includingbothspatialandtemporalcorrelations.SimulationsonamodifiedIEEE30-bussystemandamodifiedIEEE118-bussystemdemonstratethevalidityoftheproposedmethod.Themainhighlightsofproposedmethodcanbeconcludedas:AnextensiontechniqueofLDSswitharbitraryextensionstepsisproposedinthisarticle，andthistechniqueisfirstutilizedinthecalculationsofPLF;thesplinereconstructionmethodisutilizedtomodifyconventionalNataftransformation.Thismodificationmakethesampleswithcorrelationsobtainableonlybythefirstseveralmoments，whichexpandstheapplica-tionsofNataftransformation.Therestofthisarticleisorganizedasfollows.DetailsofthemodifiedNataftransformationareintroducedinSection2.Theextendedquasi-MonteCarlomethodbasedonSobolsequencesisdepictedinSection3.InSection4，theperformancesoftheproposedmethodarestudiedwiththemodifiedIEEE30-and118-bussystems.Finally，Section5bringstheconclusion.MODIFIEDNATAFTRANSFORMATIONIngeneral，themarginaldistributionsofrenewableresourcesareextremelydifferentininstallationsiteandtimescale;thedistributionsarehardtoobtaininpractice，whilethemomentsaremucheasiertoattain.Thus，reconstructionsofdistributionsbythemomentsarethefirstchoiceofpracticalPLF.PNTisabletoobtaindistributionsbythefirstseveralordersofmoments，buttheaccuracyislowerthanNataftransforma-tionandtheRosenblattmethod.In[18]，anon-normaldistribu-tionreconstructionmethodwasproposed，whichwasutilizedin[19]toreconstructthedistributionofvoltagemagnitudes.Uponthisbase，modifiedNataftransformationisproposedinthisarticle.Thismethodisabletoutilizethemoments，whichexpandstheapplicationsofconventionalNataftransformation，depictedinwhatfollows.DistributionReconstructionMethodBasedonSplineReconstructionIn[18]，splinereconstructionwasutilizedtoobtainthemarginaldistributionsofvariables.Theobtainedprobabilitydensityfunctionsaredisplayedaspiecewisepolynomials.Theprocessisshownnext.Assumexisarandomvariablebelongingtoarbitrarydis-tributionininterval[a，b]withthefirstνcordersofmomentsgiven，saywindpowerorsolarintensity.[a，b]canbedi-videdintoνc+3sub-intervals，saya=x1≤x2···≤xνc+4=b.Ineachsub-interval，thethird-orderpiecewisepolyno-mials(x)={si(x)|iνc+3}isutilizedtoapproximatethedistributionofx，notedasf(x):FromEq.(1)，thefirst-orderandsecond-orderderivativesofsi(x)，notedassi(x)andsi(x)，respectively，canbecalculateddirectly.Insub-interval[xi，xi+1]，thereexistsfourunknowns;thus，4νc+12equationsarerequiredtoobtainEq.(1).Theassumptionsareshownasfollows:f(x)onlyexistsin[a，b];2.f(x)anditsderivativesareallequalto0inorbeyondtheterminalsofinterval[a，b].Atfirst，sixequationscanbeformulatedintheterminalsof[a，b]bytwoassumptions:Inaddition，3νc+6equationscanbeobtainedbythecontinu-ityofEq.(1)anditsderivatives:Therestofthevcequationsaregivenbythefirstvcordersofmoments:wherethedefinitionsofI1，...，I4areshownasfollows:2.2.NatafTransformationInSection2.1，themarginaldistributionscouldbeobtainedbysplinereconstructionwiththefirstseveralordersofmoments.Afterthat，Nataftransformationisutilizedtoobtainthejointdistributionofthosevariables.Theprocedureisdepictedinwhatfollows.whereσiandσjarethestandarddeviations，respectively.andrepresentthecovarianceandcorrelationcoefficientsbetweenAtfirst，Xcanbetransformedintoanormaldistributedspace，notedaswhichisshownasEq.(7):whereFiisthecumulativedensityfunctionofXi，isthecumulativedensityfunctionofnormaldistribution.AssumethecorrelationmatrixofZ.AccordingtothecharacteristicsofNataftransformation，therelationisshownasEq.(8):whererepresentsthejointdistributionofaretheexpectationandstandarddeviationofisthetwo-dimensionalnormaldistributionwiththecorrelationcoefficientofρZij.Uptonow，severalmethodshavebeenadoptedintothetransformationofto[17，20].Anovelmethodwasproposedin[20]thattranslatedEq.(8)intoanotherform，whichisemployedinthisarticle.Whenisobtained，theNataftransformationisperformedtotransformZtoXwiththecorrelationmatrixof.TheprocessisshowninFigure1.2.3.CorrelationControllingMethodBasedonSVDIngeneral，maynotbepositivedefiniteorhavefullrank;hence，theCholeskydecompositionmethodadoptedin[11]maybenotavailable，butissuretobesymmetric.SVDofthematrixcanbecalculated[21]，andthisprocessisdepictedbyTheorem1.FIGURE1.FlowchartofNataftransformationwhereisarealunitarymatrix，andisarealdiagonalmatrixwiththesingularvaluesofsortedontheprincipaldiagonalwithdescendingorders.ThenthecorrelationmatrixofdefinedinEq.(10)equalsρY:andthecorrelationmatrixequalsAccordingtothecharacteristicofSVD，independentsequencecanbeobtainedthroughEq.(13).Inthesamemanner，acorrelatedsequencecanalsobeobtainedbytheinverseprocess:In[21]，boththespatialandtemporalcorrelationscanbemanagedbytheaboveprocess.3.THEEXTENDEDQUASI-MONTECARLOMETHODBASEDONSOBOLSEQUENCES3.1.ASketchofExtendedLHSTheillustrationofLHSisshowninFigure2.LHSdividesthesamplingspaceintoNparts，thuscoveringalargersamplingspace.Ineachpart，LHSselectsapoint(theblackpointinFigure1)asasample.Ascanbeimagined，asamplewithsizeofNcanbegeneratedbyLHS.Asfortheextension，eachintervalamongNpartsmustbedividedintotwopartstogeneratednewpoint[12]，whichiswhytheextendedLHScanonlyperformNto2N.3.2.ExtendedQuasi-MonteCarloMethodFromtheabovediscussions，theextensionfromNto2Nwillprobablyincreasethecomputationalburden.Howtoperformtheextensionwitharbitrarystepsisthemainfocusinthissection.Especially，SobolsequencesareatypeofLDSthatcanbeattainedbybi-levelBooleanoperation[9]，thusextensionwitharbitrarystepsispossible.Inthisarticle，anextendedquasi-MonteCarlomethodisproposedtoobtainsamplesUnewwithasizeofN+NstepthroughaddingNstepnewsamplesUaddonthegivenoldN-sizedsampleUold.Theprocessisdepictedinwhatfollows.AssumethatUold，primitivepolynomialP(x)，anddirectionnumberaregiven.StartthealgorithmwithprimitiveqthorderpolynomialP(x):andNstepnewdirectionnumbervj(N<j≤N+Nstep)withrecurrence:GenerateNstepnewsamplesofUadd:ExampleofExtendedTechniqueBythegraphwayexplanation，theextensionfrom100to150ofatwo-dimensionalSobolsequenceisshowninFigures3(a)and3(b).InFigure3(a)，eachcirclerepresentsasample.Fromthedashedrectangleatthelowerleftofbothfigures，theextensionaddstwonewsamplesandthethreeoldsamplesstillremain.ThatmeanswhenPLFdoesnotconverge，theobtainedpowerflowresultscanberetained.ItcanalsobeconcludedfromthisfigurethattheuniformityoftheLDSiswellpreservedafterextension.ConvergenceCriteriaBeforeperformingextendedPLF，thedefinitionofconvergencemustbegiven.Ingeneral，coefficientofvariationβisusuallyadoptedastheconvergencecriterioninpowersystemreliabilityanalysis[22].Whenthesignificancelevelisα，theconfidenceintervalandcoefficientofvariationaredefinedasfollows:whereμ∗Yandσ∗YaretheestimatedexpectationandstandarddeviationofYwiththesamplesizeofN;z1−α/2isthe1−α/2quantileofnormaldistribution.Accordingthecentrallimittheorem，theprobabilityoftheexpectationofrandomvariableY，notedasμY，locatedintheintervalofEq.(17)equals1−α.AccordingtoEqs.(17)and(18)，thecoefficientofvariationoftheoutputvariableisnotedasβγ，whereγrepresentsthevariabletype，includingnodalvoltageV，voltageangleθ，branchactivepowerP，andbranchreactivepowerQ.Inpractice，whentheMCSisnearlyconvergent，thevariablesusuallyoscillatearoundthereference;itisnoteasytoindicatetheconvergenceofoverallprocess.Themainadvantageoftheaboveindexisthatthecoefficientofvariationisactuallyanintervallengthparameter，andoscillationsintheintervalcannotaffecttheconvergenceindicationofoverallprocessatall.Theprocessconvergeswhentheintervallengthdecreasestoacertainlevel.ErrorDefinitionofMCSInMCS，theresultsobtainedbysimplerandomsamplingwithaverylargesamplesize(500，000inthisarticle)aresetasin[11].Theerrorcomparedtothereferenceareemployedasindicesofthealgorithmiccharacteristic，whichdefinedasEq.(19):BecauseoftherandomnessofMCS，eachsamplesizeiscalculated100times.Theaverageerrorεγsissetasperformanceindices.Fordemonstratingtheoverallaccuracy，εallisdefinedastheaverageerrorofPLF，asfollows:ProcessoftheProposedExtendedQuasi-MonteCarloMethodTheextensioncanbeperformedinarbitrarystep，assumetheinitialsamplesizeisN0，themaximumextensiontimesisNmax，andtheextensionstepisNstep.InSections4.3and4.5，forthecomparisonbetweenextendedLatinhypercubesampling(ELHS)andNEQMCS，theextensionstepoftheNEQMCSaresettobethesameastheELHS;inSections4.4，4.6，and4.7，theextensionstepissetasconstant.TheprocessofNEQMCSisshowninFigure4.CASESTUDY4.1.SettlementofTestCaseInthisarticle，amodifiedIEEE30-bussystemandamodifiedIEEE118-bussystemareadoptedtotesttheperformance.ThesimulationplatformisbasedonanIntelCoreduali7-3820with3.6GHzand8GBRAM(LenovoCorporation，Beijing，China).ThewindspeeddataforprobabilitydensityfunctionreconstructionaremeasuredinwindfarmsinnorthwesternChinawiththemarginalprobabilitydensityfunctionsunknown.Therelationbetweenactivepoweroutputandwindspeedisasfollows:wherePistheratedactivepowerofthewindgenerator;vci，vr，andvcoarecut-inwindspeed，ratedwindspeed，andcut-outwindspeed，respectively，whicharesetas2.5，13，and25m/s.InthemodifiedIEEE30-bussystem，sixwindgeneratorswiththecapacityof10MWareinstalledatnodes3，4，6，16，17，and28，respectively;inthemodifiedIEEE118-bussystem，fourwindfarmswithacapacityof120MWareinstalledatnodes67，68，93，94，95，and96.Thepowerfactorofwindfarmiskeptconstantas0.9.Fortheconsiderationofbothtemporalandspatialcorrelations，fourloadscenariosrepresentingdifferentoperatingtimesareproposed.ThetemporalandspatialcorrelationmatrixesareshownasCtandCs，respectively.Allvariablesareassumedtobepositivecorrelated.AsforthetemporalcorrelationmatrixCt，correlationsonlyexistinadjacentoperatingtimes;thus，Ctisatridiagonalmatrix;Csissetasnon-negativedefinite:Theloadsareassumedtobenormallydistributed，withtheexpectationequalingtheirinitialvalueandstandarddeviationequalin5%oftheirexpectation.Thecorrelationbetweendifferentnodalloadsequals.ResultsofDistributionReconstructionThewindspeeddataaremeasuredinarealwindfarminnorthwesternChina.Byutilizationofthosedata，themarginaldistributionsofwindspeedcanbeobtainedbysplinereconstruction.ThenwindspeedsamplewithcorrelationsareobtainedbytheCombinationwithNataftransformation.Theresultsoftheproposedmethodandthird-andfifth-orderPNTsarecomparedinFigure5.FromFigure5，thecombinationofsplinereconstructionandNataftransformationcanutilizethemomentstoreconstructprobabilitydensityfunctiondirectly.Theresultsalsoshowthattheproposedmethodhashigheraccuracythanboththird-orderandfifth-orderPNTs;thatis，themodifiedNataftransformationmethodproposedhereinexpandstheapplicationsofNataftransformation.Theproposedmethodalsohasahiddenadvantageofcomputationefficiency.FromEqs.(7)and(8)，Nataftransformationhasmanyintegralanddifferentialcalculations.AndfromEq.(1)，itisfoundthattheprobabilitydensityfunction(PDFs)obtainedbytheproposedmethodareallinpolynomialform;itisquiteconvenientforthemtoperformintegrallyordifferentially.Inaddition，from[13，15]，itisnotworthwhileforPNTtousemomentshigherthantheninthorderforthegreatcomputationalburdens;however，thecomputationalstressoftheproposedmethodinhigherordermomentscanstillbemaintainedforthefastcalculationofpolynomialintegralordifferential.ErrorAnalysisoftheProposedMethodAlthoughtheSobolsequencecanextendinarbitrarystep，forthecomparisonwithextendedLHS，theextensionstepofNEQMCSaresetasthesamewithELHSinthispart.TheerrorcurvesofthemodifiedIEEE30-and118-bussystemsareshowninFigure6.FromFigure6，itcanbeconcludedthatNEQMCShasfasterconvergencecomparedtoELHS，whichverifiestheacceleratingeffectoftheSobolsequenceonMonteCarloconvergence[9];furthermore，inbothsystems，NEQMCSconvergeswithNof800，whileELHSare1600and3200.Thus，theconvergenceofNEQMCSisnotsensitivetothesystemscalecomparedtoELHS.ResultsaftertheConsiderationofTemporalCorrelationsSections4.2and4.3demonstratethatproposedmethodisabletomanagethespatialcorrelation.Inthispart，thecaseconsideringtemporalcorrelationsisstudied.Probabilitydensityfunctionsofactivepowerinbranches104-105and100-103withtemporalcorrelationsandwithoutcorrelationsinfouroperatingtimesareshowninTable1.t1，t2，t3，andt4representthosefouroperatingtimesmentionedinSection4.1.ItiseasilyseenfromTable1thatthetemporalcorrelationsareabletoreducethedifferencesofbothexpectationsandStandarddeviationsindifferentoperatingtimes.Thatismainlybecausethosecorrelationswillreducethedifferencesofwindspeedsamplesint1，t2，t3，andt4.TheConvergenceCharacteristicoftheProposedMethodToobtainregulationbetweenthenumericalcharacteristicofoutputvariablesandsamplesize，thevoltageofthe11thnodeandtheactivepowerofbranch6-8inthemodifiedIEEE30systemaresetasexamples.TheconvergencetrendsofthevariablesareshowninFigures7(a)–7(d).FromFigure7，itcanbeseenthattheproposedmethodhasbetterconvergencecharacteristicscomparedtoELHS.Theoverallprocesscanconvergeafterafewextensions.FIGURE7.Comparisonsofthreemethodswithdifferentsamplesizes.AdoptionofConvergenceCriteriaInthissection，theactivepowerofline6-8intheIEEE30-busSystemistakenasanexample.Aftertheadoptionofsimplerandomsampling(SRS)，ELHS，andNEQMCS，theconvergencecriteria，extensiontimes，deterministicpowerflowcalculationtimes，consumingtimes，confidencelevel，andcomputationalaccuracyareshowninTable2andTable3，respectively.TheobtainedprobabilitydensityfunctionsareshowninFigure8.Equations(17)and(18)aretakenasconvergencecriteriaandβsetissetto0.1%.TheextensionstepsofSRSandNEQMCSaresetasfive;theextensionstepofELHSisNto2N.Inthispart，NEQMCSisextendedinaconstantstep，whereasLHSisext