潮流不同排序方案的比较文献翻译中英文对照学士学位论文.docx
- 文档编号:29948918
- 上传时间:2023-08-03
- 格式:DOCX
- 页数:19
- 大小:131.12KB
潮流不同排序方案的比较文献翻译中英文对照学士学位论文.docx
《潮流不同排序方案的比较文献翻译中英文对照学士学位论文.docx》由会员分享,可在线阅读,更多相关《潮流不同排序方案的比较文献翻译中英文对照学士学位论文.docx(19页珍藏版)》请在冰豆网上搜索。
潮流不同排序方案的比较文献翻译中英文对照学士学位论文
AComparisonofPowerFlowbyDifferentOrderingSchemes
Abstract—Nodeorderingalgorithms,aimingatkeepingsparsityasfaraspossible,arewidelyusedtoday.Insuchalgorithms,theirinfluenceontheaccuracyofthesolutionisneglectedbecauseitwon’tmakesignificantdifferenceinnormalsystems.While,alongwiththedevelopmentofmodernpowersystems,theproblemwillbecomemoreill-conditionedanditisnecessarytotaketheaccuracyintocountduringnodeordering.Inthispaperweintendtolaygroundworkforthemorerationalityorderingalgorithmwhichcouldmakereasonablecompromisingbetweenmemoryandaccuracy.Threeschemesofnodeorderingfordifferentpurposeareproposedtocomparetheperformanceofthepowerflowcalculationandanexampleofsimplesix-nodenetworkisdiscusseddetailed.
Keywords—powerflowcalculation;nodeordering;sparsity;accuracy;Newton-Raphsonmethod;linearequations
I.INTRODUCTION
Powerflowisthemostbasicandimportantconceptinpowersystemanalysisandpowerflowcalculationisthebasisofpowersystemplanning,operation,schedulingandcontrol[1].Mathematicallyspeaking,powerflowproblemistofindanumericalsolutionofnonlinearequations.Newtonmethodisthemostcommonlyusedtosolvetheproblemanditinvolvesrepeateddirectsolutionsofasystemoflinearequations.ThesolvingefficiencyandprecisionofthelinearequationsdirectlyinfluencestheperformanceofNewton-Raphsonpowerflowalgorithm.Basedonnumericalmathematicsandphysicalcharacteristicsofpowersysteminpowerflowcalculation,scholarsdedicatedtotheresearchtoimprovethecomputationalefficiencyoflinearequationsbyreorderingnodes’numberandreceivedalotofsuccesswhichlaidasolidfoundationforpowersystemanalysis.
Jacobianmatrixinpowerflowcalculation,similarwiththeadmittancematrix,hassymmetricalstructureandahighdegreeofsparsity.Duringthefactorizationprocedure,nonzeroentriescanbegeneratedinmemorypositionsthatcorrespondtozeroentriesinthestartingJacobianmatrix.Thisactionisreferredtoasfill-in.Iftheprogrammingtermsisusedwhichprocessedandstoresonlynonzeroterms,thereductionoffill-inreflectsagreatreductionofmemoryrequirementandthenumberofoperationsneededtoperformthefactorization.Somanyextensivestudieshavebeenconcernedwiththeminimizationofthefill-ins.Whileitishardtofindefficientalgorithmfordeterminingtheabsoluteoptimalorder,severaleffectivestrategiesfordeterminingnear-optimalordershavebeendevisedforactualapplications[2,3].Eachofthestrategiesisatrade-offbetweenresultsandspeedofexecutionandtheyhavebeenadoptedbymuchofindustry.Thesparsity-programmedorderedeliminationmentionedabove,whichisabreakthroughinpowersystemnetworkcomputation,dramaticallyimprovingthecomputingspeedandstoragerequirementsofNewton’smethod[4].
Aftersparsematrixmethods,sparsevectormethods[5],whichextendsparsityexploitationtovectors,areusefulforsolvinglinearequationswhentheright-hand-sidevectorissparseorasmallnumberofelementsintheunknownvectorarewanted.Tomakefulluseofsparsevectormethodsadvantage,itisnecessarytoenhancethesparsityofL-1byorderingnodes.Thisisequivalenttodecreasingthelengthofthepaths,butitmightcausemorefill-ins,greatercomplexityandexpense.Counteringthisproblem,severalnodeorderingalgorithms[6,7]wereproposedtoenhancesparsevectormethodsbyminimizingthelengthofthepathswhilepreservingthesparsityofthematrix.
Uptonow,onthebasisoftheassumptionthatanarbitraryorderofnodesdoesnotadverselyaffectnumericalaccuracy,mostnodeorderingalgorithmstakesolvinglinearequationsinasingleiterationasresearchsubject,aimingatthereductionofmemoryrequirementsandcomputingoperations.Manymatriceswithastrongdiagonalinnetworkproblemsfulfilltheaboveassumption,andorderingtoconservesparsityincreasedtheaccuracyofthesolution.Nevertheless,iftherearejunctionsofveryhighandlowseriesimpedances,longEHVlines,seriesandshuntcompensationinthemodelofpowerflowproblem,diagonaldominancewillbeweaken[8]andtheassumptionmaynotbetenableinvariably.Furthermore,alongwiththedevelopmentofmodernpowersystems,variousnewmodelswithparametersundervariousordersofmagnitudeappearinthemodelofpowerflow.Thepromotionofdistributedgenerationalsoencourageustoregardthedistributionnetworksandtransmissionsystemsasawholeinpowerflowcalculation,anditwillcausemoreseriousnumericalproblem.Allthosethingsmentionedabovewillturntheproblemintoill-condition.Soitisnecessarytodiscusstheeffectofthenodenumberingtotheaccuracyofthesolution.
Basedontheexistingnodeorderingalgorithmmentionedabove,thispaperfocusattentiononthecontradictionbetweenmemoryandaccuracyduringnodeordering,researchhowcouldnodeorderingalgorithmaffecttheperformanceofpowerflowcalculation,expectingtolaygroundworkforthemorerationalityorderingalgorithm.Thispaperisarrangedasfollows.ThecontradictionbetweenmemoryandaccuracyinnodeorderingalgorithmisintroducedinsectionII.NextasimpleDCpowerflowisshowedtoillustratethatnodeorderingcouldaffecttheaccuracyofthesolutioninsectionIII.Then,takinga6-nodenetworkasanexample,theeffectofnodeorderingontheperformanceofpowerflowisanalyzeddetailedinsectionIV.ConclusionisgiveninsectionVI.
II.CONTRADICTIONBETWEENMEMORYANDACCURACY
INNODEORDERINGALGORITHM
Accordingtonumericalmathematics,completepivotingisnumericallypreferabletopartialpivotingforsystemsoflineralgebraicequationsbyGaussianEliminationMethod(GEM).Manymathematicalpapers[9-11]focustheirattentiononthediscriminationbetweencompletepivotingandpartialpivotingin(GEM).Reference[9]showshowpartialpivotingandcompletepivotingaffectthesensitivityoftheLUfactorization.Reference[10]proposesaneffectiveandinexpensivetesttorecognizenumericaldifficultiesduringpartialpivotingrequires.Oncetheassessmentcriterioncannotbemet,completepivotingwillbeadoptedtogetbetternumericalstability.Inpowerflowcalculations,partialpivotingisrealizedautomaticallywithoutanyrow-interchangesandcolumn-interchangesbecauseofthediagonallydominantfeaturesoftheJacobinmatrix,whichcouldguaranteenumericalstabilityinfloatingpointcomputationinmostcases.Whileduetoroundingerrors,thepartialpivotingdoesnotprovidethesolutionaccurateenoughinsomeill-conditionings.Ifcompletepivotingisperformed,ateachstepoftheprocess,theelementoflargestmoduleischosenasthepivotalelement.Itisequivalenttoadjustthenodeorderinginpowerflowcalculation.Sothenoderelatetotheelementoflargestmoduleistendtoarrangeinfrontforthepurposeofimprovingaccuracy.
Thenodereorderingalgorithmsguidedbysparsematrixtechnologyhavewildlyusedinpowersystemcalculation,aimingatminimizingmemoryrequirement.Inthesealgorithms,thenodeswithfeweradjacentnodestendtobenumberedfirst.Theresultisthatdiagonalentriesinnodeadmittancematrixtendtobearrangedfromleasttolargestaccordingtotheirmodule.Analogously,everydiagonalsubmatricesrelatetoanodetendtobearrangedfromleasttolargestaccordingtotheirdeterminants.Sotheresultsobtainedformsuchalgorithmswilljustdeviateformtheprinciplefollowwhichtheaccuracyofthesolutionwillbeenhance.Thatiswhatwesaythereiscontradictionbetweennodeorderingguidedbymemoryandaccuracy.
III.DIFFERENCEPRECISIONOFTHESOLUTIONUSINGPARTICALPIVOTINGANDCOMPLETEPIVOTING
Itissaidthatcompletepivotingisnumericallypreferabletopartialpivotingforsolvingsystemsoflinearalgebraicequations.Whenthesystemcoefficientsarevaryingwidely,theaccuracyofthesolutionwouldbeaffectbyroundingerrorshardlyanditisnecessarytotaketheinfluenceoftheorderingontheaccuracyofthesolutionintoconsideration.
Fig.1DCmodelofSample4-nodenetwork
Asanexample,considertheDCmodelofsample4-nodesystemshowninFigure1.Node1istheswingnodehavingknownvoltageangle;nodes2-4areloadnodes.Followingtheoriginalnodenumber,theDCpowerflowequationis:
Tosimulatecomputernumericalcalculationoperations,foursignificantfigureswillbeusedtosolvetheproblem.ExecutingGEMwithoutpivotingon
(1)yieldsthesolution[θ2,θ3,θ4]T=[-0.3036,-0.3239,-0.3249]T,whosecomponentsdifferfromthatoftheexactsolution[θ2,θ3,θ4]T=[-0.3,-0.32,-0.321]T.Amoreexactsolutioncouldbeobtainedbycompletepivoting:
[θ2,θ3,θ4]T=[-0.3007,-0.3207,-0.3217]T,andtheorderofthenodeafterrowandcolumninterchangesis3,2,4.Sothisisamorereasonableorderingschemeforthepurposeofgettingmorehighaccuracy.
IV.THEINFLUENCEOFNODEREODERINGONTHEPERFORMANCEOFNEWTON-RAPHSONPOWERFLOWMETHOD
Fig.2Sample6-nodenetwork
Onthebasisoftheabove-mentionedanalysis,theschemefornodereorderingwillnotonlyaffectmemoryrequirementbutalsotheaccuracyofthesolutioninsolvinglinearsimultaneousequations.SoperformanceofNewton-Raphsonpowerflowmethodwillbedifferentwithvariousnodeordering.Inthissectionthreeschemesoforderingfordifferentpurposewillbeappliedtoasample6-nodenetworkshowninFig2tocomparetheinfluenceofthemontheaccuracyofthesolution,theconvergencerate,thecalculatedamountandthememoryneededinpowerflowcomputation.ThedetailoftheperformanceisshownintableIV.
A.Pur
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- 潮流 不同 排序 方案 比较 文献 翻译 中英文 对照 学士 学位 论文