外文翻译--基于优化的牛顿——拉夫逊法和牛顿法的潮流计算.doc

外文翻译--基于优化的牛顿——拉夫逊法和牛顿法的潮流计算.doc

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英文文献

PowerFlowCalculationbyCombinationofNewton-RaphsonMethodandNewton’sMethodinOptimization.

AndreyPazderin,SergeyYuferev

URALSTATETECHNICALUNIVERSITY–UPI

E-mail:

pav@daes.ustu.ru,usv@daes.ustu.ru

Abstract--Inthispaper,theapplicationoftheNewton’smethodinoptimizationforpowerflowcalculationisconsidered.Convergenceconditionsofthesuggestedmethodusinganexampleofathree-machinesystemareinvestigated.Itisshown,thatthemethodallowstocalculatenon-existentstatepointsandautomaticallypullsthemontotheboundaryofpowerflowexistencedomain.AcombinedmethodwhichiscomposedofNewton-RaphsonmethodandNewton’smethodinoptimizationispresentedinthepaper.

IndexTerms—Newtonmethod,Hessianmatrix,convergenceofnumericalmethods,steadystatestability

Ⅰ.INTRODUCTION

Thesolutionofthepowerflowproblemisthebasisonwhichotherproblemsofmanagingtheoperationanddevelopmentofelectricalpowersystems（EPS）aresolved.Thecomplexityoftheproblemofpowerflowcalculationisattributedtononlinearityofsteady-stateequationssystemanditshighdimensionality,whichinvolvesiterativemethods.Thebasicproblemofthepowerflowcalculationisthatofthesolutionfeasibilityanditerativeprocessconvergence[1].

Thedesiretofindasolutionwhichwouldbeontheboundaryoftheexistencedomainwhenthegivennodalcapacitiesareoutsidetheexistencedomainofthesolution,anditisrequiredtopullthestatepointbackontothefeasibilityboundary,motivatestodevelopmethodsandalgorithmsforpowerflowcalculation,providingreliableconvergencetothesolution.

ThealgorithmforthepowerflowcalculationbasedontheNewton'smethodinoptimizationallowstofindasolutionforthesituationwheninitialdataareoutsidetheexistencedomainandtopulltheoperationpointontothefeasibilityboundarybyanoptimalpath.AlsoitispossibletoestimateastaticstabilitymarginbyutilizingNewton'smethodinoptimization.

AsthealgorithmbasedontheNewton’smethodinoptimizationhasconsiderablecomputationalcostandpowercontrolcannotberealizedinallnodes,thealgorithmbasedonthecombinationoftheNewton-RaphsonmethodsandtheNewton’smethodinoptimizationisofferedtobeutilizedforcalculatingspeed,enhancingthepowerflowcalculation.

II.THEORETICALBACKGROUND

A．Steady-stateequations

Thesystemofsteady-stateequations,ingeneral,canbeexpressedasfollows:

（1）

whereisthevectorofparametersgivenforpowerflowcalculation.Inpowerflowcalculation,realandreactivepowersaresetineachbusexceptfortheslackbus.Ingenerationbuses,themodulusofvoltagecanbefixed.W（X,Y）isthenonlinearvectorfunctionofsteady-stateequations.VariablesYdefinethequasi-constantparametersassociatedwithanequivalentcircuitofanelectricalnetwork.Xisarequiredstatevector,itdefinessteadystateofEPS.Thedimensionofthestatevectorcoincideswiththenumberofnonlinearequationsofthesystem

（1）.Therearevariousknownformsofnotationofthesteady-stateequations.Normally,theyarenodal-voltageequationsintheformofpowerbalanceorintheformofcurrentbalance.Complexquantitiesintheseequationscanbepresentedinpolarorrectangularcoordinates,whichleadstoasufficientlylargevarietyformsofthesteady-stateequationsnotation.Therearevariablemethodsofanonlinearsystemofsteady-stateequationssolution.TheyareunitedbytheincrementalvectorofindependentvariablesΔXbeingsearchedandtheconditionofconvergencebeingassessedateachiteration.

B.TheNewton'smethodinoptimization

Anotherwayofsolvingtheproblemofpowerflowcalculationisrelatedtodefiningazerominimumofobjectivefunctionofsquaressumofdiscrepanciesofsteady-state

equations:

（2）

Thefunctionminimum

（2）isreachedatthepointwherederivativesonallrequiredvariablesareequaltozero:

（（3）

Itisnecessarytosolveanonlinearsetofequations（3）tofindthesolutionfortheproblem.Calculatingthepowerflow,whichismadebythesystemofthelinearequationswithaHessianmatrixateachiteration,isreferredtoastheNewton's

methodinoptimization[4]:

（（4）

TheHessianmatrixcontainstwoitems:

（5）

Duringthepowerflowcalculation,thedeterminantofHessianmatrixispositiveroundzeroandnegativevalueofadeterminantofJacobian.Thisallowstofindthestatepointduringthepowerflowcalculation,wheninitialpointhasbeenoutsideoftheexistencedomain.

TheconvergencedomainofthesolutionoftheNewton'soptimizationmethodislimitedbyapositivevalueoftheHessianmatrixdeterminant.Theiterativeprocessevenforasolvableoperatingpointcanconvergetoanincorrect

solutionifinitialapproximationhasbeenoutsideconvergencedomain.Thisallows