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

《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》由会员分享，可在线阅读，更多相关《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（14页珍藏版）》请在冰点文库上搜索。

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.

Theiterativeprocessevenforasolvableoperatingpointcanconverge

toanincorrect

solutionifinitialapproximationhasbeenoutsideconvergence

domain.Thisallowstoestimateastaticstabilitymarginofthestate

andtofindthemostperilouspathofitsweighting.

III.INVESTIGATIONSONTHETESTSCHEME

ConvergenceoftheNewton'smethodinoptimizationwithafull

Hessianmatrixhasbeeninvestigated.Calculationsweremadebasedon

programMathCADforanetworkcomprisingthreebusestheparametersof

whicharepresentedinFigure1.Dependantvariableswereanglesofvectors

ofbusvoltage1and2,independentvariableswerecapacitiesinnodes

1and2,andabsolutevaluesofvoltagesofnodes1,2and3werefixed.

Fig.1?

TheTestscheme

InFigure2,theboundaryofexistencedomainforasolutionofthe

steady-stateispresentedinangularcoordinatesδ1-δ2.Thisboundary

conformstoapositivevalueoftheJacobiandeterminant:

AsaresultofthepowerflowcalculationbasedontheNewtonmethod

inoptimization,theanglevalueshavebeenreceived,thesevalues

correspondingtothegivencapacitiesinFig.2generationispositiveand

loadingisnegative.

Forthestatepointswhichareinsidetheexistencedomain,the

objectivefunction2hasbeenreducedtozero.Forthestatepointswhich

areontheboundaryoftheexistencedomain,objectivefunction2hasnot

from

differedcapacitiesofvaluescalculatedtheandzerotoreducedbeen

thegivencapacities.

Fig.2?

DomainofExistenceforaSolution

Fig.3-BoundaryofexistencedomainInFig.3,theboundaryofthe

existencedomainispresentedincoordinatesofcapacitiesP1-P2.State

pointsoccurringontheboundaryoftheexistencedomain6havebeenset

bythecapacitieswhichwereoutsidetheexistencedomain.Asa

resultofpowerflowcalculationbyminimization2basedonthe

Newton'smethodinoptimization,theiterativeprocessconvergestothe

nearestboundarypoint.Itisduetothefactthatsurfacesoftheequal

levelofobjectivefunction2incoordinatesofnodalcapacitiesare

propercirclesforthreemachinesystemhavingthecentreonthepoint

definedbygivenvaluesofnodalcapacitiesThegraphicinterpretationof

surfacesoftheequallevelofobjectivefunctionforoperatingpoint

statewith13000MWloadingbus1and15000MWgeneratingbus2ispresented

inFig.3.

Hessianmatrixisremarkableinitsbeingnotsingularonthe

boundaryofexistencedomain.ThedeterminantofaHessianmatrix5is

positivearoundzeroandanegativevalueoftheJacobianmatrix

determinant.Thisfactallowsthepowerflowtobecalculatedevenfor

theunstablepointswhichareoutsideexistencedomain.Theiterative

processbasedonthesystemofthelinearequations4solutionhas

Naturally,

iteration.3-5withinpointstabilitycriticalthetoconverged

theiterativeprocessbasedonNewton-Rapsonmethodisdivergentforsuch

unsolvableoperatingpoints.

Theconvergencedomainofthemethodunderconsiderationhasbeen

investigated.Whatismeantisthatnotallunsolvableoperatingpoints

willbepulledontothe

boundaryofexistencedomain.Acertainthresholdhavingbeen

exceededtheiterativeprocesshasbeguntoconvergetotheimaginary

solutionwithanglesexceeding360Itisnecessarytonotethattoreceive

acriticalstabilityoperatingpointincasewheninitialnodalcapacities

aresetoutsidetheboundaryoftheexistencedomain,thereisnonecessity

tomakeanyadditionaltermsastheiterativeprocessconvergesnaturally

tothenearestboundarypoint.

Pullingtheoperationpointontofeasibilityboundaryisnotalways

possiblebytheshortestandoptimalpath.Thereareanumberof

constraints,suchasimpossibilityofloadconsumptionincreaseatbuses,

constraintsofgenerationshedding/gainingatstations.Loadfollowing

capabilityofgeneratorunitsisvarious,consequentlyforfasterpulling

theoperationpointontothefeasibilityboundaryitisnecessaryto

carryoutthispullingprobablybylonger,butfasterpath.

Thealgorithmprovidespossibilityofpathcorrectionofpulling.

Itiscarriedoutbyusingoftheweightingcoefficients,whichdefine

degreeofparticipationofeach

nodeintotalcontrolaction.ForthispurposediagonalmatrixAof

theweightingcoefficientsforeachnodeisincludedintotheobjective

function2:

AlldiagonalelementsoftheweightingcoefficientmatrixAshould

begreater-thanzero:

Wheninitialapproximationliesintothefeasibilitydomain,

coefficientsarenotinfluenceonthecomputationalprocessandonthe

result.

Inthefigure4differentpathsofthepullingthesameoperation

pointontofeasibilityboundarydependingontheweightingcoefficients

arepresented.Pathsarepresentedfortw