外文翻译--基于优化的牛顿——拉夫逊法和牛顿法的潮流计算Word下载.doc
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外文翻译--基于优化的牛顿——拉夫逊法和牛顿法的潮流计算Word下载.doc
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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
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