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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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-RaphsonMethodand

Newton'sMethodinOptimization.

AndreyPazderin,SergeyYuferev

URALSTATETECHNICALUNIVERSITY?

UPI

E-mail:

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

Abstract--Inthispaper,theapplicationoftheNewton'smethodin

optimizationforpowerflowcalculationisconsidered.Convergence

conditionsofthesuggestedmethodusinganexampleofathree-machine

systemareinvestigated.Itisshown,thatthemethodallowstocalculate

non-existentstatepointsandautomaticallypullsthemontotheboundary

ofpowerflowexistencedomain.Acombinedmethodwhichiscomposedof

Newton-RaphsonmethodandNewton'smethodinoptimizationispresented

inthepaper.

IndexTerms?

Newtonmethod,Hessianmatrix,convergenceofnumerical

methods,steadystatestability

Ⅰ.INTRODUCTION

problemisthebasisonwhichother

flowthepowerofThesolution

problemsofmanagingtheoperationanddevelopmentofelectricalpower

systemsEPSaresolved.Thecomplexityoftheproblemofpowerflow

calculationisattributedtononlinearityofsteady-stateequations

systemanditshighdimensionality,whichinvolvesiterativemethods.The

basicproblemofthepowerflowcalculationisthatofthesolution

feasibilityanditerativeprocessconvergence[1].

Thedesiretofindasolutionwhichwouldbeontheboundaryofthe

existencedomainwhenthegivennodalcapacitiesareoutsidetheexistence

domainofthesolution,anditisrequiredtopullthestatepointback

ontothefeasibilityboundary,motivatestodevelopmethodsand

algorithmsforpowerflowcalculation,providingreliableconvergenceto

thesolution.

ThealgorithmforthepowerflowcalculationbasedontheNewton's

methodinoptimizationallowstofindasolutionforthesituationwhen

initialdataareoutsidetheexistencedomainandtopulltheoperation

pointontothefeasibilityboundarybyanoptimalpath.Alsoitispossible

toestimateastaticstabilitymarginbyutilizingNewton'smethodin

optimization.

AsthealgorithmbasedontheNewton'smethodinoptimizationhas

considerablecomputationalcostandpowercontrolcannotberealizedin

allnodes,thealgorithmbasedonthecombinationoftheNewton-Raphson

utilized

betoofferedisoptimizationinmethodNewton'stheandhodsmet

forcalculatingspeed,enhancingthepowerflowcalculation.

II.THEORETICALBACKGROUND

A.Steady-stateequations

Thesystemofsteady-stateequations,ingeneral,canbeexpressed

asfollows:

whereisthevectorofparametersgivenforpowerflow

calculation.Inpowerflowcalculation,realandreactivepowersareset

ineachbusexceptfortheslackbus.Ingenerationbuses,themodulusof

voltagecanbefixed.WX,Yisthenonlinearvectorfunctionof

steady-stateequations.VariablesYdefinethequasi-constantparameters

associatedwithanequivalentcircuitofanelectricalnetwork.Xisa

requiredstatevector,itdefinessteadystateofEPS.Thedimensionof

thestatevectorcoincideswiththenumberofnonlinearequationsofthe

system1.Therearevariousknownformsofnotationofthesteady-state

equations.Normally,theyarenodal-voltageequationsintheformofpower

balanceorintheformofcurrentbalance.Complexquantitiesinthese

equationscanbepresentedinpolarorrectangularcoordinates,which

leadstoasufficientlylargevarietyformsofthesteady-stateequations

notation.Therearevariablemethodsofanonlinearsystemofsteady-state

equationssolution.Theyareunitedbytheincrementalvectorof

independentvariablesΔXbeingsearchedandtheconditionof

convergencebeingassessedateachiteration.

B.TheNewton'smethodinoptimization

Anotherwayofsolvingtheproblemofpowerflowcalculationis

relatedtodefiningazerominimumofobjectivefunctionofsquaressum

ofdiscrepanciesofsteady-state

equations:

2?

Thefunctionminimum2isreachedatthepointwherederivatives

onallrequiredvariablesareequaltozero:

3

Itisnecessarytosolveanonlinearsetofequations3tofindthe

solutionfortheproblem.Calculatingthepowerflow,whichismadeby

thesystemofthelinearequationswithaHessianmatrixateachiteration,

isreferredtoastheNewton's

methodinoptimization[4]:

4

TheHessianmatrixcontainstwoitems:

5

Duringthepowerflowcalculation,thedeterminantofHessian

matrixispositiveroundzeroandnegativevalueofadeterminantof

Jacobian.Thisallowstofindthestatepointduringthepowerflow

calculation,wheninitialpointhasbeenoutsideoftheexistencedomain.

TheconvergencedomainofthesolutionoftheNewton'soptimization

methodislimitedbyapositivevalueoftheHessianmatrixdeterminant.

Theiterativepr