外文翻译基于优化的牛顿拉夫逊法和牛顿法的潮流计算.docx

外文翻译基于优化的牛顿拉夫逊法和牛顿法的潮流计算.docx

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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

外文翻译--基于优化的牛顿—拉夫逊法和牛顿法的潮流计算

英文文献

PowerFlowCalculationbyCombinationofNewton-RaphsonMethodandNewton’sMethodinOptimization.

AndreyPazderin,SergeyYuferev

URALSTATETECHNICALUNIVERSITY?

UPI

E-mail:

pav@//0>.,usv@//.

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

IndexTerms?

Newtonmethod,Hessianmatrix,convergenceofnumericalmethods,steadystatestability

Ⅰ.INTRODUCTION

ThesolutionofthepowerflowproblemisthebasisonwhichotherproblemsofmanagingtheoperationanddevelopmentofelectricalpowersystemsEPSaresolved.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:

whereisthevectorofparametersgivenforpowerflowcalculation.Inpowerflowcalculation,realandreactivepowersaresetineachbusexceptfortheslackbus.Ingenerationbuses,themodulusofvoltagecanbefixed.WX,Yisthenonlinearvectorfunctionofsteady-stateequations.VariablesYdefinethequasi-constantparametersassociatedwithanequivalentcircuitofanelectricalnetwork.Xisarequiredstatevector,itdefinessteadystateofEPS.Thedimensionofthestatevectorcoincideswiththenumberofnonlinearequationsofthesystem1.Therearevariousknownformsofnotationofthesteady-stateequations.Normally,theyarenodal-voltageequationsintheformofpowerbalanceorintheformofcurrentbalance.Complexquantitiesintheseequationscanbepresentedinpolarorrectangularcoordinates,whichleadstoasufficientlylargevarietyformsofthesteady-stateequationsnotation.Therearevariablemethodsofanonlinearsystemofsteady-stateequationssolution.TheyareunitedbytheincrementalvectorofindependentvariablesΔXbeingsearchedandtheconditionofconvergencebeingassessedateachiteration.

B.TheNewton'smethodinoptimization

Anotherwayofsolvingtheproblemofpowerflowcalculationisrelatedtodefiningazerominimumofobjectivefunctionofsquaressumofdiscrepanciesofsteady-state

equations:

2?

Thefunctionminimum2isreachedatthepointwherederivativesonallrequiredvariablesareequaltozero:

3

Itisnecessarytosolveanonlinearsetofequations3tofindthesolutionfortheproblem.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:

AsaresultofthepowerflowcalculationbasedontheNewtonmethodinoptimization,theanglevalueshavebeenreceived,thesevaluescorrespondingtothegivencapacitiesinFig.2generationispositiveandloadingisnegative.

Forthestatepointswhichareinsidetheexistencedomain,theobjectivefunction2hasbeenreducedtozero.Forthestatepointswhichareontheboundaryoftheexistencedomain,objectivefunction2hasnotbeenreducedtozeroandthecalculatedvaluesofcapacitiesdifferedfromthegivencapacities.

Fig.2?

DomainofExistenceforaSolution

Fig.3-BoundaryofexistencedomainInFig.3,theboundaryoftheexistencedomainispresentedincoordinatesofcapacitiesP1-P2.Statepointsoccurringontheboundaryoftheexistencedomain6havebeensetbythecapacitieswhichwereoutsidetheexistencedomain.Asa

resultofpowerflowcalculationbyminimization2basedontheNewton'smethodinoptimization,theiterativeprocessconvergestothenearestboundarypoint.Itisduetothefactthatsurfacesoftheequallevelofobjectivefunction2incoordinatesofnodalcapacitiesarepropercirclesforthreemachinesystemhavingthecentreonthepointdefinedbygivenvaluesofnodalcapacitiesThegraphicinterpretationofsurfacesoftheequallevelofobjectivefunctionforoperatingpointstatewith13000MWloadingbus1and15000MWgeneratingbus2ispresentedinFig.3.

Hessianmatrixisremarkableinitsbeingnotsingularontheboundaryofexistencedomain.ThedeterminantofaHessianmatrix5ispositivearoundzeroandanegativevalueoftheJacobianmatrixdeterminant.Thisfactallowsthepowerflowtobecalculatedevenfortheunstablepointswhichareoutsideexistencedomain.Theiterativeprocessbasedonthesystemofthelinearequations4solutionhasconvergedtothecriticalstabilitypointwithin3-5iteration.Naturally,theiterativeprocessbasedonNewton-Rapsonmethodisdivergentforsuchunsolvableoperatingpoints.

Theconvergencedomainofthemethodunderconsiderationhasbeeninvestigated.Whatismeantisthatnotallunsolvableoperatingpointswillbepulledontothe

boundaryofexistencedomain.Acertainthresholdhavingbeenexceededtheiterativeprocesshasbeguntoconvergetotheimaginarysolutionwithanglesexceeding360Itisnecessarytonotethattoreceiveacriticalstabilityoperatingpointincasewheninitialnodalcapacitiesaresetoutsidetheboundaryoftheexistencedomain,thereisnonecessitytomakeanyadditionaltermsastheiterativeprocessconvergesnaturallytothenearestboundarypoint.

Pullingtheoperationpointontofeasibilityboundaryisnotalwayspossiblebytheshortestandoptimalpath.Thereareanumberofconstraints,suchasimpossibilityofloadconsumptionincreaseatbuses,constraintsofgenerationshedding/gainingatstations.Loadfollowingcapabilityofgeneratorunitsisvarious,consequentlyforfasterpullingtheoperationpointontothefeasibilityboundaryitisnecessaryto

carryoutthispullingprobablybylonger,butfasterpath.

Thealgorithmprovidespossibilityofpathcorrectionofpulling.Itiscarriedoutbyusingoftheweightingcoefficients,whichdefinedegreeofparticipationofeach

nodeintotalcontrolaction.ForthispurposediagonalmatrixAoftheweightingcoefficientsforeachnodeisincludedintotheobjectivefunction2:

AlldiagonalelementsoftheweightingcoefficientmatrixAshouldbegreater-thanzero:

Wheninitialapproximationliesintothefeasibilitydomain,coefficientsarenotinfluenceonthecomputationalprocessandontheresult.

Inthefigure4differentpathsofthepullingthesameoperationpointontofeasibilityboundarydependingontheweightingcoefficientsarepresented.Pathsarepresentedfortwodifferentoperatingpoints.

IntablesIandIIeffectofweightingcoefficientsontheoutputcomputationispresented.IntablesIandIIk1andk2areweightingcoefficientforbuses1and2,respectively.

TABLEI

WEIGHTINGCOEFFICIENTEFFECTONOUTPUTCOMPUTATIONFORINITIALSETCAPACITIESP1-13000MWANDP21