外文翻译--基于优化的牛顿——拉夫逊法和牛顿法的潮流计算文档格式.doc

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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.Thisallowstoestimateastaticstabilitymarginofthestateandtofindthemostperilouspathofitsweighting.

III.INVESTIGATIONSONTHETESTSCHEME

ConvergenceoftheNewton'

smethodinoptimizationwithafullHessianmatrixhasbeeninvestigated.CalculationsweremadebasedonprogramMathCADforanetworkcomprisingthreebusestheparametersofwhicharepresentedinFigure1.Dependantvariableswereanglesofvectorsofbusvoltage1and2，independentvariableswerecapacitiesinnodes1and2,andabsolutevaluesofvoltagesofnodes1,2and3werefixed.

Fig.1–TheTestscheme

InFigure2,theboundaryofexistencedomainforasolutionofthesteady-stateispresentedinangularcoordinatesδ1-δ2.ThisboundaryconformstoapositivevalueoftheJacobiandeterminant:

）6（

AsaresultofthepowerflowcalculationbasedontheNewtonmethodinoptimization,theanglevalueshavebeenreceived,thesevaluescorrespondingtothegivencapacitiesinFig.2（generationispositiveandloadingisnegative）.

Forthestatepointswhichareinsidetheexistencedomain,theobjectivefunction

（2）hasbeenreducedtozero.Forthestatepointswhichareontheboundaryoftheexistencedomain,objectivefunction

（2）hasnotbeenreducedtozeroandthecalculatedvaluesofcapacitiesdifferedfromthegivencapacities.

Fig.2–DomainofExistenceforaSolution

Fig.3-Boundaryofexistencedomain

InFig.3,theboundaryoftheexistencedomainispresentedincoordinatesofcapacitiesP1-P2.Statepointsoccurringontheboundaryoftheexistencedomain（6）havebeensetbythecapacitieswhichwereoutsidetheexistencedomain.Asa

resultofpowerflowcalculationbyminimization

（2）basedontheNewton'

smethodinoptimization,theiterativeprocessconvergestothenearestboundarypoint.Itisduetothefactthatsurfacesoftheequallevelofobjectivefunction

（2）incoordinatesofnodalcapacitiesarepropercircles（forthreemachinesystem）havingthecentreonthepointdefinedbygivenvaluesofnodalcapacities.Thegraphicinterpretationofsurfacesoftheequallevelofobjectivefunctionforoperatingpointstatewith13000MWloadingbus1and15000MWgeneratingbus2ispresentedinFig.3.

Hessianmatrixisremarkableinitsbeingnotsingularontheboundaryofexistencedomain.ThedeterminantofaHessianmatrix（5）ispositivearoundzeroandanegativevalueoftheJacobianmatrixdeterminant.Thisfactallowsthepowerflowtobecalculatedevenfortheunstablepointswhichareoutsideexistencedomain.Theiterativeprocessbasedonthesystemofthelinearequations（4）solutionhasconvergedtothecriticalstabilitypointwithin3-5iteration.Naturally,theiterativeprocessbasedonNewton-Rapsonmethodisdivergentforsuchunsolvableoperatingpoints.

Theconvergencedomainofthemethodunderconsiderationhasbeeninvestigated.Whatismeantisthatnotallunsolvableoperatingpointswillbepulledontothe

boundaryofexistencedomain.Acertainthresholdhavingbeenexceededtheiterativeprocesshasbeguntoconvergetotheimaginarysolutionwithanglesexceeding360°

.

Itisnecessarytonotethattoreceiveacriticalstabilityoperatingpointincasewheninitialnodalcapacitiesaresetoutsidetheboundaryoftheexistencedomain,thereisnonecessitytomakeanyadditionaltermsastheiterativeprocessconvergesnaturallytothenearestboundarypoint.

Pullingtheoperationpointontofeasibilityboundaryisnotalwayspossiblebytheshortestandoptimalpath.Thereareanumberofconstraints,suchasimpossibilityofload（consumption）increaseatbuses,constraintsofgenerationshedding/gainingatstations.Loadfollowingcapabilityofgeneratorunitsisvarious,consequentlyforfasterpullingtheoperationpointontothefeasibilityboundaryitisnecessaryto

carryoutthispullingprobablybylonger,butfasterpath.

Thealgorithmprovidespossibilityofpathcorrectionofpulling.Itiscarriedoutbyusingoftheweightingcoefficients,whichdefinedegreeofparticipationofeach

nodeintotalcontrolaction.ForthispurposediagonalmatrixAoftheweightingcoefficientsforeachnodeisincludedintotheobjectivefunction

（2）:

）7（

AlldiagonalelementsoftheweightingcoefficientmatrixAshouldbegreater-thanzero:

Wheninitialapproximationliesintothefeasibilitydomain,coefficientsarenotinfluenceonthecomputationalprocessandontheresult.

Inthefigure4differentpathsofthepullingthesameoperationpointontofeasibilityboundarydependingontheweightingcoefficientsarepresented.Pathsarepresentedfortwodifferentoperatingpoints.

IntablesIandIIeffectofweightingcoefficientsontheoutputcomputationispresented.IntablesIandIIk1andk2areweightingcoefficientforbuses1and2,respectively.

TABLEI

WEIGHTINGCOEFFICIENTEFFECTONOUTPUTCOMPUTATION