毕业设计外文翻译.docx

毕业设计外文翻译.docx

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毕业设计外文翻译

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.Puropse1SavingMemo