Acopf 1 History Formulation Testing

HistoryofOptimalPowerFlowandFormulations•December2012

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HistoryofOptimalPowerFlowandFormulations

OptimalPowerFlowPaper1

MaryB.Cain,RichardP.O’Neill,AnyaCastillo

[email protected];[email protected];[email protected]

December,2012

Abstract:ThepurposeofthispaperistopresentaliteraturereviewoftheACOptimalPowerFlow(ACOPF)problemandproposeareaswheretheACOPFcouldbeimproved.TheACOPFisattheheartofIndependentSystemOperator(ISO)powermarkets,andissolvedinsomeformeveryyearforsystemplanning,everydayforday‐aheadmarkets,everyhour,andevenevery5minutes.Itwasfirstformulatedin1962,andformulationshavechangedlittleovertheyears.Withadvancesincomputingpowerandsolutionalgorithms,wecanmodelmoreoftheconstraintsandremoveunnecessarylimitsandapproximationsthatwerepreviouslyrequiredtofindasolutioninreasonabletime.Oneexampleisnonlinearvoltagemagnitudeconstraintsthataremodeledaslinearthermalproxyconstraints.Inthispaper,werefertothefullACOPFasanACOPFthatsimultaneouslyoptimizesrealandreactivepower.Today,50yearsaftertheproblemwasformulated,westilldonothaveafast,robustsolutiontechniqueforthefullACOPF.FindingagoodsolutiontechniqueforthefullACOPFcouldpotentiallysavetensofbillionsofdollarsannually.Basedonourliteraturereview,wefindthattheACOPFresearchcommunitylacksacommonunderstandingoftheproblem,itsformulation,andobjectivefunctions.However,wedonotclaimthatthisliteraturereviewisacompletereview—ourintentwassimplytocapturethemajorformulationsoftheACOPF.Instead,inthispaper,weseektoclearlypresenttheACOPFproblemthroughclearformulationsoftheproblemanditsparameters.Thispaperdefinesanddiscussesthepolarpower‐voltage,rectangularpower‐voltage,andrectangularcurrent‐voltageformulationsoftheACOPF.Additionally,itdiscussesthedifferenttypesofconstraintsandobjectivefunctions.ThispaperlaysthegroundworkforfurtherresearchontheconvexapproximationoftheACOPFsolutionspace,asurveyofsolutiontechniques,andcomputationalperformanceofdifferentformulations.

Disclaimer:TheviewspresentedarethepersonalviewsoftheauthorsandnottheFederalEnergy

RegulatoryCommissionoranyofitsCommissioners.


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TableofContents

1.Introduction.......................................................................................4

2.HistoryofPowerSystemOptimization....................................7

3.Conventions,Parameters,SetsandVariables........................13

4.AdmittanceMatrixandACPowerFlowEquations...............16

5.ACOPFFormulations.......................................................................22

6.LiteratureReviewofFormulations...........................................28

7.Conclusions........................................................................................32

References


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1.Introduction

TheheartofeconomicallyefficientandreliableIndependentSystem

Operator(ISO)powermarketsisthealternatingcurrentoptimalpowerflow

(ACOPF)problem.Thisproblemiscomplexeconomically,electricallyand

computationally.Economically,anefficientmarketequilibriumrequiresmulti‐part

nonlinearpricing.Electrically,thepowerflowisalternatingcurrent(AC),which

introducesadditionalnonlinearities.Computationally,theoptimizationhas

nonconvexities,includingbothbinaryvariablesandcontinuousfunctions,which

makestheproblemdifficulttosolve.Thepowersystemmustbeabletowithstand

thelossofanygeneratorortransmissionelement,andthesystemoperatormust

makebinarydecisionstostartupandshutdowngenerationandtransmission

assetsinresponsetosystemevents.Forinvestmentplanningpurposes,theproblem

needsbinaryinvestmentvariablesandamultipleyearhorizon.

Even50yearsaftertheproblemwasfirstformulated,westilllackafastand

robustsolutiontechniqueforthefullACOPF.Weuseapproximations,

decompositionsandengineeringjudgmenttoobtainreasonablyacceptable

solutionstothisproblem.Whilesuperiortotheirpredecessors,today’s

approximate‐solutiontechniquesmayunnecessarilycosttensofbillionsofdollars

peryear.Theymayalsoresultinenvironmentalharmfromunnecessaryemissions

andwastedenergy.UsingEIAdataonwholesaleelectricitypricesandU.S.and

Worldenergyproduction,Table1givesarangeofpotentialcostsavingsfroma5%

increaseinmarketefficiencyduetoimprovementstotheACOPF.(EIA2012).Small

increasesinefficiencyofdispatcharemeasuredinbillionsofdollarsperyear.Since

theusualcostofpurchasingandinstallingnewsoftwareforanexistingISOmarket

islessthan$10milliondollars(O’Neillet.al.2011),thepotentialbenefit/costratios

ofbettersoftwareareintherangeof10to1000.


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TABLE1:POTENTIALCOSTSAVINGSOFINCREASEDEFFICIENCYOFDISPATCH(EIA2012)

2009gross

electricity

production

(MWh)

Productioncost

($billion/year)

assuming

$30/MWhenergy

price

Savings

($billion/year)

assuming5%

increasein

efficiency

Productioncost

($billion/year)

assuming

$100/MWh

energyprice

Savings

($billion/year)

assuming5%

increasein

efficiency

U.S. 3,724,000 112 6 372 19

World 17,314,000 519 26 1731 87

AnultimategoalofISOmarketsoftware,andatopicoffutureresearch,isthe

security‐constrained,self‐healing(correctiveswitching)ACoptimalpowerflow

withunitcommitmentovertheoptimalnetwork.Theoptimalnetworkisflexible,

withassetsthathavetime‐varyingdynamicratingsreflectingtheassetcapability

undervaryingoperatingconditions.Theoptimalnetworkisalsooptimally

configured–openingorclosingtransmissionlinesbecomesadecisionvariable,or

controlaction,ratherthananinputtotheproblem,orstate.Whenpossible,the

securityconstraintsarecorrectiveratherthanpreventive.Withpreventivesecurity

constraints,thesystemisoperatedconservativelytosurvivelossofany

transmissionelementorgenerator.Incontrast,correctiveconstraintsreconfigure

thesystemwithfast‐actingequipmentsuchasspecialprotectionsystemsor

remedialactionschemesimmediatelyfollowinglossofageneratorortransmission

element,allowingthesystemtobereliablyusedclosertoitslimits.Thisproblem

mustbesolvedweeklyin8hours,dailyin2hours,hourlyin15minutes,eachfive

minutesin1minuteandforself‐healingpost‐contingencyin30seconds.Currently,

theproblemissolvedthroughvaryinglevelsofapproximation,dependingon

applicationandtimescale,butwithincreasesincomputingpoweritmaybe

possibletoreducethenumberofapproximationsandtakeadvantageofparallel

computing.

Today,thecomputationalchallengeistoconsistentlyfindaglobaloptimal

solutionwithspeedsuptothreetofiveordersofmagnitudefasterthanexisting

solvers.Thereissomepromisingrecentevidencethatthiscouldbearealityinfive

totenyears.Forexample,inthelasttwodecadesmixed‐integerprogramming(MIP)

hasachievedspeedimprovementsof107;thatis,problemsthatwouldhavetaken


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10yearsin1990canbesolvedinoneminutetoday.Asaconsequence,MIPis

replacingotherapproachesinISOmarkets.ImplementationofMIPintotheday‐

aheadandreal‐timemarkets,withtheCommission’sencouragement,hassaved

Americanelectricitymarketparticipantsoverone‐halfbilliondollarsperyear

(FERC2011).MorewillbesavedasallISOsimplementMIPandthenew

formulationsitpermitsinthenextseveralyears.

Duetoidiosyncrasiesindesign,currentsoftwareoversimplifiestheproblem

indifferentways,andrequiresoperatorinterventiontoaddressreal‐timeproblems

thatdonotshowupinmodels.Thisoperatorinterventionunnecessarilyalters

settlementpricesandproducessuboptimalsolutions.TheJointBoardonEconomic

DispatchfortheNortheastRegionstatedin2006thatimprovedmodelingofsystem

constraintssuchasvoltageandstabilityconstraintswouldresultinmoreprecise

dispatchesandbettermarketsignals,butthattheswitchtoAC‐basedsoftware

wouldincreasethetimetorunasinglescenariofromminutestooveranhour,

makinguseofACOPFimpractical,evenfortheday‐aheadmarket(FERC2006).One

exampleistheMidwestIndependentSystemOperator(MISO),whereoperators

havetocommitresourcesbeforetheunitcommitmentandeconomicdispatch

softwaremodelsareruntoaddresslocalvoltageissuesthatMISOhashaddifficulty

modelinginitsmarketsoftware(FERC2012).PJMInterconnection(PJM)employs

anapproach,calledPerfectDispatch,thatex‐postsolvesthereal‐timemarket

problemwithperfectinformation(PJM2012).ThePerfectDispatchsolutionisused

totrainoperators,wheretheycancomparethe“perfectdispatch,”whichisbasedon

“perfect”after‐the‐factinformationtotheactualdispatch,whichisbasedonthe

informationavailableatthetime.ISOmodelssolveproxiesorestimatesforreactive

powerandvoltageconstraints,wheretheycalculatelinearthermalconstraintsto

approximatequadraticvoltagemagnitudeconstraints.Thedetailsoftransmission

constraintmodelingandtransmissionpricinghavebeenneglected,butneedtobe

consideredtoimprovetheaccuracyofACOPFcalculations.Transmission

constraintscanbemodeledintermsofcurrent,realpower,apparentpower,voltage

magnitudedifferences,orangledifferences.Thechoiceofconstraintdependsonthe

typeofmodel,dataavailability,andphysicallimit(voltage,stability,orthermal


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limit).Surrogateconstraintscanbecalculatedbasedonthelineflowequations,but

thesecalculationshaveinherentassumptions.OneexampleistheArizona‐Southern

Californiaoutagein2011,wheresomelinelimitsweremodeledandmonitoredas

realpowertransferlimitswhileothersweremodeledascurrenttransferlimits

(FERC/NERC2012).ThispaperseekstobetterunderstandtheACOPFproblem

throughclearformulationsoftheproblem,theoreticalpropertiesoftheproblem

anditsparameters,approximationstothenonlinearfunctionsthatarenecessaryto

maketheproblemsolvable,andtoproducecomputationalresultsfromlargeand

smalltestproblemsusingvarioussolversandstartingpoints.Discretevariables

suchasequipmentstates,generatorcommitments,andtransmissionswitching

furthercomplicatetheACOPF,butwedonotdiscusstheseinthispaper.Withthe

increasedmeasurementsandcontrolsinherentinsmartgridupgrades,thepotential

savingsaregreater,althoughtheproblemmaybecomemorecomplexwithmore

discretedevicestomodel.

Intherestofthepaper,weprovideabriefhistoryofpowersystem

optimization,presentnotationandnomenclature,formulatetheadmittancematrix

andpowerflowequations,formulateconstraints,presentdifferentformulationsof

theACOPF,andpresentaliteraturereviewofACOPFformulations.

2.HistoryofPowerSystemOptimization

Powersystemoptimizationhasevolvedwithdevelopmentsincomputing

andoptimizationtheory.Inthefirsthalfofthe20thcentury,theoptimalpowerflow

problemwas“solved”byexperiencedengineersandoperatorsusingjudgment,

rulesofthumb,andprimitivetools,includinganalognetworkanalyzersand

specializedsliderules.Gradually,computationalaidswereintroducedtoassistthe

intuitionofoperatorexperience.Theoptimalpowerflowproblemwasfirst

formulatedinthe1960’s(Carpentier1962),buthasproventobeaverydifficult

problemtosolve.Linearsolversarewidelyavailableforlinearizedversionsofthe

optimalpowerflowproblem,butnonlinearsolverscannotguaranteeaglobal

optimum,arenotrobust,anddonotsolvefastenough.Ineachelectricitycontrol

room,theoptimalpowerflowproblemoranapproximationmustbesolvedmany

timesaday,asoftenasevery5minutes.


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Therearethreetypesofproblemscommonlyreferredtoinpowersystem

literature:powerflow(loadflow),economicdispatch,andoptimalpowerflow.

Threeotherclassesofpowersystemoptimization,specificallyunitcommitment,

optimaltopology,andlong‐termplanning,involvebinaryandintegervariables,and

arenotdiscussedinthispaper;butcombinedwiththeinsightsonformulationsin

thispaper,couldbepromisingareasforfutureresearch.

Table2comparesthemajorcharacteristicsofthepowerflow,economic

dispatch,andoptimalpowerflowproblems.Thepowerfloworloadflowrefersto

thegeneration,load,andtransmissionnetworkequations.Powerflowmethodsfind

amathematicallybutnotnecessarilyphysicallyfeasibleoroptimalsolution.The

powerflowequationsthemselvesdonottakeaccountoflimitationsongenerator

reactivepowerlimitsortransmissionlinelimits,buttheseconstraintscanbe

programmedintomanypowerflowsolvers.

Asecondtypeofproblem,economicdispatch,describesavarietyof

formulationstodeterminetheleast‐costgenerationdispatchtoserveagivenload

withareservemargin,buttheseformulationssimplifyorsometimesaltogether

ignorepowerflowconstraints.

Athirdtypeofproblem,theoptimalpowerflow,findstheoptimalsolutionto

anobjectivefunctionsubjecttothepowerflowconstraintsandotheroperational

constraints,suchasgeneratorminimumoutputconstraints,transmissionstability

andvoltageconstraints,andlimitsonswitchingmechanicalequipment.Optimal

powerflowissometimesreferredtoassecurity‐constrainedeconomicdispatch

(SCED);mostimplementationsofSCEDincludeonlythermallimits,andproxiesfor

voltagelimits.Thereareavarietyofformulationswithdifferentconstraints,

differentobjectivefunctions,anddifferentsolutionmethodsthathavebeenlabeled

optimalpowerflow;thesearediscussedintheformulationssectionlaterinthis

paper.FormulationsthatusetheexactACpowerflowequationsareknownas

“ACOPF.”Simplerversions,knownasDCOPF,assumeallvoltagemagnitudesare

fixedandallvoltageanglesareclosetozero.DCstandsfordirectcurrent,butisabit

ofamisnomer;aDCOPFisalinearizedformofafullalternatingcurrentnetwork

(ACOPF)andnotapowerflowsolutionforadirectcurrentnetwork.Weusethe


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generaltermOPFtoincludebothACOPFandDCOPF.TheACOPFisoftensolved

throughdecoupling,whichtakesadvantageofthestructureoftheproblem,where

realpower(P)andvoltageangle(θ)aretightlycoupledandvoltagemagnitude(V)

andreactivepower(Q)aretightlycoupled,buttheP‐θandV‐Qproblemsare

weaklycoupledduetotheassumptionsthatthephaseangledifferencesbetween

adjacentbusesarerathersmall,andhigh‐voltagetransmissionnetworkshavemuch

higherreactancecomparedtoresistance.ThedecoupledOPFdividestheACOPF

intotwolinearsubproblems,onewithpowerandvoltageangleandanotherwith

voltagemagnitudeandreactivepower.Inthispaper,weusethetermACOPFto

refertothefullACOPFthatsimultaneouslyoptimizesrealandreactivepower,and

decoupledOPFtorefertothedecoupledproblemsthatseparatelyoptimizerealand

reactivepoweranditeratebetweenthetwotoreachanoptimalsolution.


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TABLE2:MAJORTYPESOFPOWERSYSTEMPROBLEMS

Generalproblemtype

Problemname Includesvoltageangleconstraints?

Includesbusvoltagemagnitudeconstraints?

Includestransmissionconstraints?

Includeslosses?

Assumptions Includesgeneratorcosts?

Includescontingencyconstraints?

OPF ACOPF,orFullACOPF

Yes Yes Yes Yes Yes No

OPF DCOPF No No;allvoltagemagnitudesfixed

Yes Maybe Voltagemagnitudesareconstant

Yes No

OPF DecoupledOPF Yes Yes Yes Yes Power‐voltageangleareindependentofvoltagemagnitude‐reactivepower

Yes No

OPF Security‐ConstrainedEconomicDispatch(SCED)

Yes No Yes Yes Voltagemagnitudesareconstant

Yes Yes

Powerflow

PowerFlow,orLoadFlow

No,butcanbeadded

Yes No,butcanbeadded

Yes No No

Economicdispatch

EconomicDispatch

No No No Depends Notransmissionconstraints

Yes No

OPF SecurityConstrainedOPF(SCOPF)

Yes Depends Yes Yes Depends Yes Yes


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Wenowdiscussearlyresearchofthethreetypesofproblemsinpower

systemoptimization:economicdispatch,powerflow,andoptimalpowerflow.

Asearlyasthe1930’s,theeconomicdispatchproblemwassolvedbyhandor

specially‐developedslideruleusingtheprincipleofequalincrementalloading,

takingaslongas8hourstocomplete(Happ1977).Earlycomputationsofeconomic

dispatchwereslow.Kirchmayerestimatedthatitwouldtake10minutesof

computationaltimetoproducetheschedulesfora10generatorsystematagiven

systemprice(Kirchmayer1958).Incontrast,RTOstodaysolvesystemsofhundreds

ofgeneratorsinamatterofseconds.Inthesurveyofeconomicdispatchmethodsup

throughthe1970’s,Happprovidesanoverviewoftheevolutionofeconomic

dispatchformulationsanddifferentwaystoaccountforlosses.

Priortodigitalcomputers,asearlyas1929,thepowerflowproblemwas

solvedwithanalognetworkanalyzersthatsimulatedpowersystems(Sasson1967).

WardandHalepublishedthefirstautomateddigitalsolutiontothepowerflow

problemin1956(Ward1956).SassonandJaimesprovideasurveyandcomparison

ofearlyloadflowsolutionmethods,whicharevariousiterativemethodsbasedon

thenodaladmittancematrix(Ymatrix)oritsinverse,thenodalimpedancematrix

(Zmatrix)(Sasson1967).Earlyresearchers,includingCarpentier,usedtheGauss‐

Seidelmethod.TheNewton‐Raphsonmethodbecamethecommonlyusedsolution

methodduringthe1960’s(Peschonet.al.1968),afterTinneyandothersdeveloped

sparsitytechniquestotakeadvantageofthestructureoftheadmittancematrixin

theOPFproblem.Theadmittancematrixissparse,meaningithasmanyzero

elements;thisisbecausepowersystemnetworksarenotdenselyconnected.

Sparsitytechniquescanbeusedtoreducedatastorageandincreasecomputation

speed(Stott1974).

EarlyresearchonOPFusedclassicalLagrangiantechniquesforthe

optimalityconditions,butneglectedboundsonvariables(Squires1961).In1962,

CarpentierpublishedtheoptimalityconditionsforanOPF,includingvariable

bounds,basedontheKuhn‐Tuckerconditions;thisisgenerallyconsideredthefirst

publicationofafullyformulatedOPF(Carpentier1962).Carpentierassumesthat

theapplicablefunctionsdisplay“suitableconvexity”fortheKuhn‐Tucker(now


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referredtoastheKarush‐Kuh‐TuckerorKKT)conditionstoapply(Carpentier

1962).Giventhestructureofthepowerflowequations,thismaybeabig

assumption(Hiskens2001andSchecter2012).CarpentierincludesthefullAC

powerflowequations,generatorrealandreactivepowerconstraints,busvoltage

magnitudeconstraints,andbusvoltageangledifferenceconstraintsforbuses

connectedbytransmissionelements.

HuneaultandGallianaprovideanextensivesurveyofoptimalpowerflow

literatureupto1991,surveyingover300articlesandciting163(Huneault1991).

Theyconclude,“Thehistoryofoptimalpowerflow(OPF)researchcanbe

characterizedastheapplicationofincreasinglypowerfuloptimizationtoolstoa

problemwhichbasicallyhasbeenwell‐definedsincetheearly1960’s.”Thepaper

outlinestheevolutionofOPFliterature,groupedbysolutionmethod.Thesolution

methodsincludevariousformsofgradientmethods,linearprogramming,quadratic

programming,andpenaltymethods.Theauthorsconcludethat“commercially

availableOPFalgorithmsallsatisfythefullnonlinearloadflowmodelandafullset

ofboundsonvariables.”TheauthorsfurtherconcludethattheOPFremainsa

difficultmathematicalproblem.Thepresentalgorithmscannotcomputequickly

enough,andarepronetoseriousill‐conditioningandconvergenceproblems.

Anotherareaofresearch,security‐constrainedOPF,accountsfor

transmissioncontingencyconstraintsandposesadditionalcomputational

challenges(Carpentier1979,Stott1987).Ourdiscussioninthispaperfocuseson

ACOPF.Futureresearchcouldextendtheformulationstoincludecontingency

constraintsthatarerequiredtomaintainthesystemafteranoutage.This

formulationincreasesthesizeoftheproblemformulationbyafactorequaltothe

numberofcontingenciesstudied.

ResearchershaveidentifiedchallengestosolvingtheOPF,including

modelingdiscretevariables,localminima,lackofuniformproblemdefinition,

solutionreliabilityandcomputingtime.Someofthesehavebeensolved:both

Tinneyetal.andMomohetal.discussedthechallengesinmodelingdiscrete

variablesinOPFsolutions(Tinney1988),(Momoh1997).Today,withadvancesin

mixedintegerprogramming(MIP),discretevariablescanbemodeledandincluded


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inACOPFsolutions.Otherchallengespersisttoday:Koesslerstatesthatthe“lackof

uniformityinusageanddefinition”hasbeenachallengetousersanddevelopersin

OPF,andspecificallydiscusseslocalminima,whichindicatethattheproblemis

nonconvex(Momoh1997).HuneaultandGallianafoundthatalgorithmsavailablein

1991couldnotcomputeOPFsolutionsquicklyandreliablyenough,andthatthe

OPF,likemanynonlinearproblems,ispronetoill‐conditioninganddifficult

convergence(Huneault1991).

3.Conventions,Parameters,SetsandVariables

NotationandNomenclature

Whennandmaresubscripts,theyindexbuses;kindexesthetransmission

elements.Whenjisnotasuperscript,j=(‐1)1/2;iisthecomplexcurrent.Whenjisa

superscript,itisthe‘imaginary’partofacomplexnumber.Matricesandvectorsare

representedwithuppercaseletters.Scalarsandcomplexnumbersareinlowercase

letters.ForcolumnvectorsAandBoflengthn,whereakandbkarethekth

componentsofAandBrespectively,theHadamardproduct‘·’isdefinedsothatA·B

=C,whereCisacolumnvectoralsooflengthn,withkthcomponentck=akbk.

Thecomplexconjugateoperatoris*(superscript)and*(nosuperscript)isan

optimalsolution.

Weassumebalanced,three‐phase,steady‐stateconditions.Allvariablesare

associatedwithasingle‐linemodelofabalanced,three‐phasesystem.Acommon

practiceinpowersystemmodelingistheper‐unit(p.u.)representation,wherebase

quantitiesforvoltagemagnitude,current,power,andimpedance(oradmittance)

areusedtonormalizequantitiesinanetworkwithmultiplevoltagelevels.Such

normalizationisaconvenience.Weusetheconventionthataninjectionoccurs

whentherealpartofthecomplexnumberispositiveandawithdrawaloccurswhen

therealpartofthecomplexnumberisnegative.

Thetopologyofthenetworkconsistsoflocationsknownasbusesornodes

andtransmissionelementsconnectingpairedbuses.Thenetworkisanundirected

planargraph.


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IndicesandSets

n,marebus(node)indices;n,mϵ{1,…,N}whereNisthenumberofbuses.(misan

aliasforn)

kisathree‐phasetransmissionelementwithterminalbusesnandm.

kϵ{1,…,K}whereKisthenumberoftransmissionelements;kcountsfrom1tothe

totalnumberoftransmissionelements,anddoesnotstartoverforeachbuspairnm.

K’isthesetofconnectedbuspairsnm(|K’|≤|K|).

Unlessotherwisestated,summations(∑)areoverthefullsetofindices.

Variables

pnistherealpowerinjection(positive)orwithdrawal(negative)atbusn

qnisthereactivepowerinjectionorwithdrawalatbusn

sn=pn+jqnisthenetcomplexpowerinjectionorwithdrawalatbusn

Wedistinguishbetweenthereal,reactive,orcomplexpowerinjectedataspecific

bus(pn,qn,andsn)andthereal,reactive,orcomplexpowerflowinginatransmission

elementbetweentwobuses:

pnmkistherealpowerflowfrombusntobusmontransmissionelementk

qnmkisthereactivepowerflowfrombusntobusmontransmissionelementk

snmkistheapparentcomplexpowerflowfrombusnontransmissionelementk.snmk

=srnmk+jsjnmk=pnmk+jqnmk

θnisthevoltageangleatbusn

θnm=θn‐θmisthevoltageangledifferencefrombusntobusm

θ–δisthepowerangle.

iisthecurrent(complexphasor);wedistinguishbetweencurrentinjectedata

specificbusandcurrentflowinginatransmissionelementbetweentwobuses:

inisthecurrent(complexphasor)injection(positive)orwithdrawal(negative)at

busnwherein=irn+jijn

inmkisthecurrent(complexphasor)flowintransmissionelementkatbusn(tobus

m).inmk=irnmk+jijnmk

vnisthecomplexvoltageatbusn.vn=vrn+jvjn


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ynmkisthecomplexadmittanceontransmissionelementkconnectingbusnandbus

m(Ifbusesnandmarenotconnecteddirectly,ynmk=0.);yn0istheself‐admittance

(toground)atbusn.

V=(v1,…,vN)Tisthecomplexvectorofbusvoltages;V=Vr+jVj

I=(i1,…,iN)Tisthecomplexvectorofbuscurrentinjections;I=Ir+jIj

P=(p1,…,pN)Tisthevectorofrealpowerinjections

Q=(q1,…,qN)Tisthevectorofreactivepowerinjections

GistheN‐by‐Nconductancematrix

BistheN‐by‐Nsusceptancematrix

NotethatelementsofGandBwillbeconstantforpassivetransmissionelements

suchastransmissionlines,butcanbevariablewhenactivetransmissionelements

suchasphaseshiftingtransformers,switchedcapacitors/reactors,orpower

electronicflexibleACtransmissionsystem(FACTS)devicesareincluded.

Y=G+jBistheN‐by‐Ncomplexadmittancematrix

gnm,bnm,andynmrepresentelementsoftheG,B,andYmatricesrespectively.

FunctionsandTransformations

Re()istherealpartofacomplexnumber,forexample,Re(irn+jijn)=irn

Im()istherealpartofacomplexnumber,forexample,Im(irn+jijn)=ijn

||isthemagnitudeofacomplexnumber,forexample,|vn|=[(vrn)2+(vjn)2]1/2

abs()istheabsolutevaluefunction.

Thetransformationfromrectangulartopolarcoordinatesforcomplexvoltageis:

vrn=|vn|cos(θn)

vjn=|vn|sin(θn)

(vrn)2+(vjn)2=[|vn|sin(θn)]2+[|vn|cos(θn)]2=|vn|2[sin(θn)2+cos(θn)2]=|vn|2

Wedropthebusindexnandletθbethevoltageangleandδbethecurrentangle.

Forrealpower,

p=vrir+vjij=|v|cosθ|i|cosδ+|v|sinθ|i|sinδ=|v||i|[cosθcosδ+sinθsinδ]

=|v||i|(0.5[cos(θ‐δ)+cos(θ+δ)]+0.5[cos(θ‐δ)‐cos(θ+δ)])

=|v||i|cos(θ‐δ)

Forreactivepower,

q=vjir‐vrij=|v|sinθ|i|cosδ‐|v|cosθ|i|sinδ=|v||i|[sinθcosδ‐cosθsinδ]


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=|v||i|.5[sin(θ+δ)+sin(θ‐δ)]‐|v||i|.5[sin(θ+δ)‐sin(θ‐δ)]

q=|v||i|sin(θ‐δ)

θ–δisthepowerangle.

Parameters

rnmkorrkistheresistanceoftransmissionelementk.

xnmkorxkisthereactanceoftransmissionelementk.

smaxkisthethermallimitonapparentpowerovertransmissionelementkatboth

terminalbuses.

θminnm,θmaxnmarethemaximumandminimumvoltageangledifferencesbetweenn

andm

pminn,pmaxnarethemaximumandminimumrealpowerforgeneratorn

qminn,qmaxnarethemaximumandminimumreactivepowerforgeneratorn

C1=(c11,…,c1N)TandC2=(c21,…,c2N)Tarevectorsoflinearandquadraticobjective

functioncostcoefficientsrespectively.

4.AdmittanceMatrixandACPowerFlowEquations

Inthissection,wedeveloptheadmittancematrixandthecurrent‐voltage

flowequations(IVequations),whichareadifferentformulationofthecommonly

usedpowerflowequations.Inthefollowingsections,wedeveloptheadditional

constraintsthatboundthesolutions.

Wedefinetheconductance(G),susceptance(B)andadmittance(Y)matrices,

withelementsgnm,bnm,andynmrespectively,andY=G+jB.Westartwithasimple

admittancematrixdefinedbyresistance,r,andreactance,x.Weassumeshunt

susceptanceisnegligible.TheelementsofG,BandYmatricesarederivedasfollows:

gnmk=rnmk/(rnmk2+xnmk2)forn≠m

bnmk=‐xnmk/(rnmk2+xnmk2)forn≠m

ynmk=gnmk+jbnmkforn≠m

ynmk=0forn=m

ynm=∑kynmkforn≠m

ynn=yn0‐∑n≠mynm

Transformers.Theadmittancematrixabovedoesnotincludetransformer

parameters.Foranidealin‐phasetransformer(assumingzeroresistancein


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transformerwindings,noleakageflux,andnohysteresisloss),theidealvoltage

magnitude(turnsratio)isanmk=|vm|/|vn|andθn=θm,wherenistheprimaryside

andmisthesecondarysideofthetransformer.Sinceθn=θm,

anmk=|vm|/|vn|=vm/vn=‐inm/imn

Thecurrent‐voltage(IV)equationsforidealtransformerkbetweenbusesnandm

are:

inmk=anmk2ynmkvn‐anmkynmkvm

imnk=‐anmkynmkvn+ynmkvm

Forthephaseshiftingtransformer(PAR)withaphaseangleshiftofφ,

vm/vn=tnmk=anmkejφ

inm/imn=tnmk*=‐anmke‐jφ

Thecurrent‐voltage(IV)equationsforthephaseshiftingtransformerkbetween

busesnandmare:

inmk=anmk2ynmkvn‐tnmk*ynmkvm

imnk=‐tnmkynmkvn+ynmkvm

AdmittanceMatrix.IftherearenotransformersorFACTSdevices,Gispositive

semidefiniteandBisnegativesemidefinite.Amatrixwhereynn≥abs(∑mynm)is

calleddiagonallydominantandstrictlydiagonallydominantifynn>abs(∑mynm).

Iftherearenotransformersandyn0=0,GandBareweightedLaplacian

matricesoftheundirectedweightedgraphthatdescribesthetransmissionnetwork.

MuchisknownabouttheweightedLaplacianmatrices.Yisacomplexweighted

Laplacianmatrix.TheadmittancematrixisY=G+jB,whereGandBarereal

symmetricdiagonallydominantmatrices.Asymmetricdiagonallydominantmatrix

hasasymmetricfactorization,forexample,B=UUTwhereeachcolumnofUhasat

mosttwonon‐zerosandthenon‐zeroeshavethesameabsolutevalue.

Forlargeproblems,theadmittancematrix,Y=G+jB,isusuallysparse.The

densityofbothGandBis(N+2K’)/N2whereK’isthenumberofoff‐diagonalnon‐

zeroentries(theaggregateofmultipletransmissionelementsbetweenadjacent

buses)andNisthenumberofbuses.Forexample,inatopologywith1000buses

and1500transmissionelements,GandBwouldhaveadensityof

(1000+3000)/10002=.004.Thelowestdensityforaconnectednetworkisthe


Page18

spanningtree.IthasN‐1transmissionelementsandthedensityis(N+2(N‐1))/N2.

Forlargesparsesystems,(N+2(N‐1))/N2≈3/N.

TransformersandFACTSdeviceschangethestructureoftheYmatrix.If

therearetransformersandFACTSdevices,let

ynmk ifnotransformerynmk= { anmk2ynmk ifanidealtransformer tnmk*ynmk,or‐tnmkynmkvn ifaphaseshiftingtransformer

asappropriateoff‐diagonalelement,thenynn=yn0+∑k,mynmk,ynm=∑kynmk,andYis

thematrix[ynm].Ifthereareonlyidealin‐phasetransformers,theYmatrixis

symmetric.Iftherearephaseshiftingtransformers,thesymmetryoftheYmatrixis

lost.

ACPowerFlowEquations

Kirchhoff’sCurrentLaw.Kirchhoff’scurrentlawrequiresthatthesumofthe

currentsinjectedandwithdrawnatbusnequalzero:

in=∑kinmk (1)

Ifwedefinecurrenttogroundtobeyn0(vn–v0)andv0=0,wehave:

in=∑kynmk(vn‐vm)+yn0vn (2)

inmk=ynmk(vn‐vm)=gnmk(vrn‐vrm)‐bnmk(vjn‐vjm)+j(bnmk(vrn‐vrm)+gnmk(vjn‐vjm))

irnmk=gnmk(vrn‐vrm)‐bnmk(vjn‐vjm)

ijnmk=bnmk(vrn‐vrm)+gnmk(vjn‐vjm)

Currentisalinearfunctionofvoltage.Rearranging,

in=vn(yn0+∑kynmk)‐∑kynmkvm (3)

Inmatrixnotation,theIVflowequationsintermsofcurrent(I)andvoltage(V)in

(3)are

I=YV=(G+jB)(Vr+jVj)=GVr‐BVj+j(BVr+GVj) (4)

whereIr=GVr‐BVjandIj=BVr+GVj

Inanothermatrixformat,(4)is

I=(Ir,Ij)=Y(Vr,Vj)Tor

I=(Ir,Ij)= G ‐B Vr whereY= G ‐B B G Vj B G


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Ifaandφareconstant,theI=YVequationsarelinear.Ifnot,thelinearityislost

sincesomeelementsoftheYmatrixwillbefunctionsofV.

PowerFlowEquations.Thetraditionalpower‐voltagepowerflowequationsdefined

intermsofrealpower(P),reactivepower(Q)andvoltage(V)are

S=P+jQ=diag(V)I*=diag(V)[YV]*=diag(V)Y*V* (5)

Thepowerinjectionsare

S=V•I*=(Vr+jVj)•(Ir‐jIj)=(Vr•Ir+Vj•Ij)+j(Vj•Ir‐Vr•Ij) (6)

where

P=Vr•Ir+Vj•Ij (7)

Q=Vj•Ir‐Vr•Ij (8)

Thepower‐voltagepowerflowequations(5)and(6)arequadratic.TheIVflow

equations(4)arelinear.

Constraints.First,weintroducethephysicalconstraintsofgenerators,load,and

transmission.

GeneratorandLoadConstraints.Thelowerandupperboundconstraintsfor

generation(injection)andload(withdrawal)are:

Pmin≤P≤Pmax (9)

Qmin≤Q≤Qmax (10)

IntermsofVandI,theinjectionconstraintsare:

Vr•Ir+Vj•Ij≤Pmax (11)

Pmin≤Vr•Ir+Vj•Ij (12)

Vj•Ir‐Vr•Ij≤Qmax (13)

Qmin≤Vj•Ir‐Vr•Ij (14)

Inequalities(11)‐(14)alongwithotherthermalconstraintsonequipment

enforcedateachgeneratorbusconstituteafour‐dimensionalreactivecapability

curve,alsoknownasa“D‐curve’sinceitisshapedlikethecapitalletterD,inthePQ

space.AdditionalD‐curvesdefiningthetradeoffbetweenrealandreactivepower

constituteaconvexsetandcanbeeasilylinearized(FERC2005).Equations(11)‐

(14)arenonconvexquadraticconstraints.Sinceherewemodelasingleperiod,

rampratesareunnecessary.


Page20

VoltageMagnitudeConstraints.Thetwoconstraintsthatlimitthevoltage

magnitudeinrectangularcoordinatesateachbusmare

(vrm)2+(vjm)2≤(vmaxm)2 (15)

(vminm)2≤(vrm)2+(vjm)2 (16)

Again,eachnonlinearinequalityinvolvesonlythevoltagemagnitudesatbusm.In

matrixterms,thevoltagemagnitudeconstraintsare:

Vr•Vr+Vj•Vj≤(Vmax)2 (17)

(Vmin)2≤Vr•Vr+Vj•Vj (18)

VminandVmaxaredeterminedbysystemstudies.Thevoltagemagnitudeboundsare

generallyintherange,[.95,1.05]perunit.Highvoltagesareoftenconstrainedby

thecapabilitiesofthecircuitbreakers.Lowvoltagemagnitudeconstraintscanbe

duetooperatingrequirementsofmotorsorgenerators.

LineFlowThermalConstraints.Smaxkisathermaltransmissionlimitonkbasedon

thetemperaturesensitivityoftheconductorandsupportingmaterialinthe

transmissionlineandtransmissionelements.Transmissionassetsgenerallyhave

threethermalratings:steady‐state,4‐hourand30‐minute.Theseratingsvarywith

ambientweather.Theapparentpoweratbusnontransmissionelementktobusm

is:

snmk=vninmk*=vny*nmk(vn‐vm)*.=vny*nmkv*n‐vny*nmkv*m)

Thethermallimitonsnmkis

(srnmk)2+(sjnmk)2=|snmk|2≤(smaxk)2 (19)

Theseconstraintsarequadraticinsrnmkandsjnmkandquarticinvrn,vjn,vrm,vjm.Since

vn=vrn+jvjnandynmk=gnmk+jbnmk,

vny*nmkv*n=(vrn+jvjn)(gnmk+jbnmk)(vrn+jvjn)

Expanding,weobtain

vny*nmkv*n=[gnmkvrn‐bnmkvjn+j(gnmkvjn+bnmkvrn)](vrn+jvjn)

Expandingagain,weobtain

vny*nmkv*n=gnmk(vrnvrn‐vjnvjn)‐bnmk(vrnvjn+vrnvjn)

+j[gnmk(vjnvrn+vjnvrn)+bnmk(vrnvrn‐vjnvjn)]

vny*nmkv*n=gnmk(vrnvrn‐vjnvjn)‐2bnmk(vrnvjn)

+j[2gnmk(vjnvrn)+bnmk(vrnvrn‐vjnvjn)] (20)


Page21

Inmatrixnotation,

Re(vny*nmkv*n)=[vrn,vjn] gnmk ‐bnmk vrn ‐bnmk ‐gnmk vjn

Im(vny*nmkv*n)=[vrn,vjn] bnmk gnmk vrn gnmk ‐bnmk vjn

Similarly,vny*nmkv*m=(vrn+jvjn)(gnmk+jbnmk)(vrm+jvjm)

Expanding,weobtain

=[gnmkvrn‐bnmkvjn+j(gnmkvjn+bnmkvrn)](vrm+jvjm)

Expandingandcollectingterms,

=gnmk(vrnvrm+vjnvjm)+bnmk(vjnvrm‐vrnvrm)+j[gnmk(vjnvrm‐vjnvjm)+bnmk(vrnvrm‐vjnvjm)]

(21)

Inmatrixnotation,

Re(vny*nmkv*m)=[vrm,vjm] gnmk ‐bnmk vrnbnmk gnmk vjn

Im(vny*nmkv*m)=[vrm,vjm] bnmk ‐gnmk vrngnmk ‐bnmk vjn

Inequality(19)becomesaquadraticconstraint.

LineFlowConstraintsasCurrentLimitations.Ascurrentincreases,linessagand

equipmentmaybedamagedbyoverheating.Theconstraintsthatlimitthecurrent

magnitudeinrectangularcoordinatesateachbusnonkare

(irnmk)2+(ijnmk)2≤(imaxnmk)2 (23)

Again,thenonlinearitiesareconvexquadraticandisolatedtothecomplexcurrentat

thebus.Generally,themaximumcurrents,imaxnmk,aredeterminedbymaterial

sciencepropertiesofconductorsandtransmissionequipment,orasaresultof

systemstabilitystudies.

LineFlowConstraintsasVoltageAngleConstraints.Thepowerflowingoveran

AClineisapproximatelyproportionaltothesineofthevoltagephaseangle

differenceatthereceivingandtransmittingends.Forstabilityreasons,thevoltage

angledifferenceforterminalbusesnandmconnectedbytransmissionelementk

canbeconstrainedasfollows:


Page22

θminnm≤θn‐θm≤θmaxnm (24)

Intherectangularformulation,thearctanfunctionappearsinsomeconstraints.

θminnm≤arctan(vjn/vrn)‐arctan(vjm/vrm)≤θmaxnm (25)

Thetheoreticalsteady‐statestabilitylimitforpowertransferbetweentwo

busesacrossalosslesslineis90degrees.Ifthislimitwereexceeded,synchronous

machinesatoneendofthelinewouldlosesynchronismwiththeotherendofthe

line.Inaddition,transientstabilityandrelaylimitsonreclosinglinesconstrain

voltageangledifferences.TheangleconstraintsusedintheACOPFshouldbethe

smallestoftheseangleconstraints,whichdependontheequipmentinstalledand

configuration.However,manytestcasesdonotincludeanyvoltageangleorline

flowconstraints.Ingeneral,systemengineersdesignandoperatethesystem

comfortablybelowthevoltageanglelimittoallowtimetorespondifthevoltage

angledifferenceacrossanylineapproachesitslimit.

5.ACOPFFormulations

Webeginwithadiscussionofobjectivefunctions,thenanoteonbustypes,

andfinallydiscussdifferentformulationsoftheACOPF.Theformulationsofthe

ACOPFpresentedhereincludealltheconstraints,butmaytakedifferentapproaches

tomodelingtheconstraints.Asdiscussedabove,current,voltagemagnitude,and

voltageangleconstraintscanbecalculatedthataresurrogatesforeachother.We

discussconstraintsfurtherin(O’Neill2012).

ObjectiveFunction.VariousauthorsformulatetheACOPFwithdifferentobjective

functions.Theyincludeminimizinggenerationcosts,maximizingmarketsurplus,

minimizinglosses,minimizinggeneration(equivalenttominimizinglosses),and

maximizingtransfers.Withoutdemandfunctions,minimizinggenerationcostsand

maximizingmarketsurplusareequivalent.

AfullACOPFthataccuratelymodelsallconstraintsandcontrolswithan

objectivefunctionofminimizingcostwouldinherentlymeettheobjectivesof

minimizinggeneratorfuelcosts,minimizinggenerationoutput,minimizinglosses,

minimizingloadshedding,andminimizingcontrolactions.

WhenitisnotfeasibletorunafullACOPFduetotimeconstraints,computing

power,orlackofarobustsolutionalgorithm,acommonsubstituteistodecouple


Page23

theproblemanditeratebetweenaDCOPFthatminimizescostsbyvaryingreal

power,thenfixthegeneratoroutputsfromtheDCOPFandrunanACOPFthat

minimizeslossesbyvaryingreactivepowerofgenerators,capacitors,etc.For

economicallydispatchingresourcesinanACOPFthatfullymodelsvoltageand

stabilityconstraints,minimizingcostisthecorrectobjectivefunction;objective

functionsofminimizinglosses,minimizinggeneration,andmaximizingtransfersfor

anACOPFareinconsistentwitheconomicprinciples,andresultinsub‐optimal

dispatch.WedonotdiscussthedetailsofdecoupledOPFhere,butsaveitfora

futurereviewofsolutionalgorithms.

Stottetal.discussbadly‐posedproblemswhenanOPFformulationdoesnot

adheretothenormalengineeringprinciplesofpowersystemoperation(Stott

1987).Theymentionafewexamplesindecoupledformulations:minimizinglosses

withgeneratorrealpoweroutputasvariableswouldmoveawayfromaminimum‐

costsolution;imposinglimitsonMWreserveswithonlygeneratorvoltagecontrols

andtransformervoltagetapcontrols,butnorealpowercontroltomeetthereserve

limit.Theystatethatitishelpfultoassociateeachcontrol,constraint,andobjective

inadecoupledOPFwitheitherorboththeactiveandreactivepowersubproblems.

Theyfurthernotethatsomeobjectivefunctionsandconstraintsarenotalgebraicor

differentiable,andthatmultiplesolutionsarelikelytoexist,inparticularwhen

therearemanyreactivepowercontrols(suchasswitchedcapacitors,FACTS

devices,orgenerators)innetworkloops.

Itispossibletoformulateanobjectivefunctionthatincludesthecostof

reactivepower.Forageneratorthecostofgenerationisafunctionoftheapparent

powergenerated,c(S)=cP(P)+cQ(Q),whereS=(P2+Q2)1/2.Ifweassumethatthe

costofreactivepowerissmallcomparedtothecostofrealpowerandifthecost

function,c(S),islinearinS,anapproximationofc(S)is

c(S)≈cP(P)+cQ(|Q|).

Bus‐type.InP,Q,|V|,θspace,therearefourquantitiesateachbus:voltage

magnitude(V),voltageangle(θ),realpower(P),andreactivepower(Q).Inapower

flowsolutionwithoutoptimization,busesareclassifiedintothreebustypes:PQ,PV

andslack.PQbusesgenerallycorrespondtoloadsandPVbusestogenerators.


Page24

GeneratorbusesarecalledPVbusesbecausepowerandvoltagemagnitudeare

fixed;loadbusesareknownasPQbusesbecauserealandreactivepowerarefixed,

thatis,Pmin=PmaxandQmin=Qmax;slackorreferencebuseshaveafixedvoltage

magnitudeandvoltageangle.Forapowerflowtosolve,theslackbusneedstohave

sufficientrealandreactivepowertomakeupforsystemlossesandholdtheslack

busvoltagemagnitudeat1;forthisreason,abuswithalargegeneratoris

commonlychosenasaslackbus.Table3comparesthedifferenttypesofbuses.

Table3:Busclassificationusedinpowerflowproblems

BusType Fixedquantities Variablequantities Physical

interpretation

PV realpower,voltage

magnitude

reactivepower,voltage

angle

generator

PQ realpower,reactive

power

voltagemagnitude,

voltageangle

load,orgeneratorwith

fixedoutput

Slack voltagemagnitude,

voltageangle

realpower,reactive

power

anarbitrarilychosen

generator

Inapowerflow,theslackbusservespartlytoensureanequalnumberof

variablesandconstraints;withoutadesignatedslackbus,thesystemwouldbe

over‐determined,withmoreequationsthanunknowns.Stottstatesthattheneedfor

aslackbusalsoarisesbecausethesystemI2Rlossesarenotpreciselyknownin

advanceoftheload‐flowcalculationforlinearDCmodelsandthereforecannotbe

assignedtoaparticulargeneratordispatch(Stott1974).Somemodelsusea

distributedslackbuswheregeneratorsatseveraldifferentbusesprovidesystem

slack.

WenotethatanACOPFthatiteratesbetweenasimplifiedOPFandanAC

powerflowmayneedaslackbusforthepowerflowiterations,buteventhenthe

voltagemagnitudeattheslackbusdoesnothavetobefixed.

WhenusinganiterativemethodsuchasNewtonorGauss‐Seideltosolvethe

powerflowequations,theconvergencetoleranceisgenerallysetbasedonthe

“mismatch”terms.Mismatchreferstothedifferencebetweenknownvaluesateach


Page25

bus,suchasPandQatloadbuses,andthevaluesP(x)andQ(x)computedwiththe

powerflowequationsateachiteration.

SincetheACOPFisanoptimizationproblem,wherethenumberofvariables

doesnothavetoequalthenumberofconstraints,specifyingaslackorreferencebus

isunnecessary.Infact,Carpentiernotedthisasearlyas1962(Carpentier1962).1In

alloptimizationformulationsherein,weforgothebustypedesignation.Inan

optimizationcontext,thesecategorizationsseemoverlyprescriptive,andcould

unnecessarilyover‐constraintheproblem.Forexample,fixingthereferencevoltage

magnitudeat1.0perunitwheninnormaloperationsgeneratorsvaryvoltage

magnitudebetween0.95and1.05perunitcouldresultinasub‐optimalsolution.

Mostmodernsolverspre‐processtheproblem,removingvariablesthathaveequal

lowerandupperboundsandreplacingthemwithaconstant.

ACOPFPower‐Voltage(PQV)Formulation.MostoftheACOPFliteratureusesthe

polarpower‐voltageformulationsbasedontheearlyworkofCarpentierduringthe

1960’s(Carpentier1962).

PolarPower‐VoltageFormulation.Thepolarpower‐voltage(polarPQV)ACOPF

(polarACOPF‐PQV)replacesquadraticequalityconstraintsin(32)withthepolar

formulationof(27)‐(28):

Network‐wideobjectivefunction:Minc(S) (26)

Network‐wideconstraints:

Pn=∑mkVnVm(Gnmkcosθnm+Bnmksinθnm) (27)

Qn=∑mkVnVm(Gnmksinθnm‐Bnmkcosθnm) (28)

Vmin≤V≤Vmax (29)

1Roughtranslationof(Carpentier1962):IfvoltageandanglearetakenasvariablesinplaceofPandQ,therestrictionof

fixingthereferencevoltagecanbelifted;voltageandangleareineffectindependentvariablesthatfixthestateofthenetwork,

anditsufficestowriteanobjectivefunctionthatisminimizedwithrespecttothesevariables.Thearbitrarilychosenreference

busdisappearsandtheproblemisthemostgeneralthatonecanpose.


Page26

θminnm≤θn‐θm≤θmaxnm. (30)

Inthisformulation,(27)and(28)represent2Nnonlinearequalityconstraintswith

quadratictermsandsineandcosinefunctionsthatapplythroughoutthenetwork.

Inthisformulation,voltageangledifferenceconstraintsarelinear.Inthe

rectangularformulationdiscussedbelow,arctanfunctionsappearintheangle

differenceconstraints.

RectangularPowerVoltageFormulation.Therectangularpower‐voltage

formulation,shownbelow,islesscommonintheliterature.Therectangularpower‐

voltage(rectangularPQV)ACOPF(rectangularACOPF‐PQV)formulationisshown

below.


Network‐wideconstraint:P+jQ=S=V•I*=V•Y*V* (32)

Bus‐specificconstraints

Pmin≤P≤Pmax (33)

Qmin≤Q≤Qmax (34)

(|snmk|)2≤(smaxk)2 forallk (35)

(29)isreplacedby:



(30)isreplacedby:


Inthisformulation,(32)represents2Nquadraticequalitiesthatapplythroughout

thenetwork;(33)‐(34)aresimplevariableboundsateachbus;(35)and(37)

representsconvexquadraticinequalitiesateachbus;(37)representsanonconvex

quadraticinequalitiyateachbus;and(37)and(38)representsnonconvex

inequalitiesbetweeneachsetofconnectedbuses.

ACOPFCurrentInjection(IV)Formulation.Currentinjectionformulationsuse

powerflowequationsbasedoncurrentandvoltageratherthanpowerflow

equationsbasedonpowerandvoltagediscussedabove.Weonlyconsiderthe

rectangularcurrent‐voltage(rectangularIV)ACOPF(rectangularACOPF‐IV)

formulationduetotheadvantagesinexpressingthecurrentinjectionsaslinear


Page27

equalityconstraints;however,thepolarcurrent‐voltageformulationcouldbeeasily

derived.

TheIVformulationhas6Nvariables(P,Q,Vr,Vj,Ir,Ij)andtheVΘhas4Nvariables

(P,Q,|V|,Θ).

RectangularACOPF‐IVformulation.TherectangularACOPF‐IVformulationis

shownbelow.


Network‐wideconstraint:I=YV (41)

Bus‐specificconstraints:

P=Vr•Ir+Vj•Ij≤Pmax (42)

Pmin≤P=Vr•Ir+Vj•Ij (43)

Q=Vj•Ir‐Vr•Ij≤Qmax (44)

Qmin≤Q=Vj•Ir‐Vr•Ij (45)



(inmk)2≤(imaxk)2 forallk (48)


Inthisformulation,(41)represents2Nlinearequalityconstraintsthatapply

throughoutthenetwork.ThisisincontrasttothePQVformulationswhere

quadraticandtrigonometricconstraintsapplythroughoutthenetworkandlinear

constraintsareisolatedateachbus.Equations(42)to(45)arelocalquadratic

nonconvexconstraints.Equations(46)and(48)arelocalconvexquadratic

inequalityconstraints,but(47)arenon‐convexlocalquadraticinequality

constraints.Overall,theconstraintsetisstillnonconvex,butwehypothesizethat

thisformulationmaybeeasiertosolvethanthepower‐voltageformulations,since

thenonlinearitiesareisolatedtoeachbusandeachtransmissionelement,whilethe

constraintsthatapplythroughoutthenetworkarelinear.Ingeneral,linearsolvers

solveproblemsfasterthannonlinearsolvers.Asdiscussedpreviously,thevoltage

anglelimit(49)couldbereplacedwithananalogouscurrentlimitandtheproblem

becomeslocallyquadraticwithlinearnetworkequations,and(48)and(49)are

essentiallyredundantconstraints.


Page28

PolarPQV RectangularPQV RectangularIV

Networkconstraints

2Nnonlinearequalityconstraintswithquadratictermsandsineandcosinefunctions

2Nquadraticequalities 2Nlinearequalityconstraints

Voltageangledifferenceconstraints

Linear Nonconvex(arctan) Linearifreplacedwithcurrentorapparentpowerconstraint

Busconstraints Linear Noncovexquadraticinequalities

Locallyquadratic,somenonconvex,someconvex

6.LiteratureReviewofFormulations

Mostliteratureusesthepolarpower‐voltageformulation,whileasmaller

groupofpapersusetherectangularpower‐voltageformulation.Somehavealso

proposedhybridandalternativeformulations.So,ratherthanattempttoreviewthe

vastliteratureonthetraditionalformulationbasedonpowerandreactivepower

equations,wefocusonalternativeformulationsinthissection.

Stottetal.criticizethatmuchOPFresearchsincetheclassicalformulationsof

Carpentier,DommelandTinneyhaveaddressedsimilarformulationswithout

consideringtheadditionalrequirementsneededforpracticalreal‐timeapplications,

partlybecauseOPFproblemsarestillstretchingthelimitsofappliedoptimization

technology,andalsothatutilitieshavebeenslowtoadoptsoftwaretocalculateOPF

“on‐line,”orinnear‐real‐time(Stott1987).Theyfurthernotethatitisamistaketo

analyticallyformulateOPFproblemsbyregardingthemassimpleextensionsof

conventionalpowerflow;oncethepowerflowproblemisformulatedasan

optimizationproblemwithdegreesoffreedom,problemsthatappeareasytosolve

canturnouttobebadlyposed,forexamplewithconflictingobjectivefunction,

controls,andconstraints.ForOPF,theynotethatresearchershavenotagreedon

“rulesofsolvability,”whicharetheengineeringcriterianeededforanOPFsolution


Page29

tobephysicallyvalid,especiallyforvoltageandreactivepower,andthatthese

“rulesofsolvability”havehardlyifeverbeenmentionedinthevastliteratureon

OPF.TheyalsoidentifyseveralcommonproblemswiththeOPFformulation.Mostof

theserelatetomodelingvoltagecharacteristicsofgeneration,load,and

transformers,butalsoincludeproblemswithincompatibilityofobjective,controls,

andconstraints.Forexample,oneincompatibilityproblemusesanobjectiveof

minimizinglosseswithgeneratorrealpoweroutputsasvariables,ratherthanfixing

generatorrealpoweroutputsattheminimumcostdispatchandadjustingreactive

powersettingstominimizelosses(Stott1987).

Afewresearchershavedevelopedacurrentinjectionformulationforthe

powerfloworoptimalpowerflowequations.Currentinjectionandreactivecurrent

aretermsusedintheliteratureforaformulationsimilartotheIVformulation

discussedearlierinthispaper.Additionally,someliteratureusestheterm“in

phase”fortherealcomponentofcurrent(Ir)and“quadrature”fortheimaginary

componentofcurrent(Ij);inthiscontext,quadraturereferstobeing90degreesout

ofphase.Mostofthesepapersidentifychallengesmodelinggenerator,orPVbuses,

wheretherealpowerinjectionandvoltagemagnitudeareknownbutthereactive

powerinjectionisnot.SeveralauthorshaveidentifiedwaystomodelPVbuses.We

discusstheseformulationshere.

Dommeletal.presentapowerflowformulationusingcurrentinjectionsand

amixofpolarandrectangularcoordinates,whereeachPQbusisrepresentedby

twoequationsfortherealandimaginarycomponentsofcurrentmismatchesin

termsofcomplexvoltageinrectangularcoordinates,whilePVbusesare

representedbyasingleactivepowermismatchequationandassociatedvoltage

angledeviation(Dommel1970).Tinneylatermentionsthatacurrentinjection

algorithmwithaconstantnodaladmittancematrixcouldnotbeusedforgeneral

powerflowapplicationsbecauseasatisfactorymethodofmodelingPVbuseshad

notbeendeveloped(Tinney1991).OtherauthorsalludetodifficultiesmodelingPV

busesusingcurrentinjections,andmuchofliteratureusingcurrentinjection

formulationsappliestoradialdistributionnetworkswherePVbusesareless

common.Forsomesolutiontechniques,modelingPVbuseswithcurrentinjection


Page30

equationsintroducessingularitiesintosomematricesinthesolutiontechnique.

SubstitutionsintroducedependenciesintheJacobian,meaningthattheentire

Jacobianwouldhavetoberecalculatedateachstep(GómezRomero2002).Various

authorshaveproposedsubstitutionsandapproximationstomodelPVbusesina

currentinjectionformulation.

StadlinandFletcherdiscussa“voltageversusreactivecurrent”modelfor

voltageandreactivecontrolthatiswellsuitedforusewithalinearprogramming

algorithm(Stadlin1982).ThispaperdoesnotdirectlydiscussanOPF,butprovides

amodelthatcouldbeusedinalinearprogrammingoptimizationforreactive

dispatchandvoltagecontrol.Themodelwouldbeusedafterarealpowerdispatch

model,suchasadecoupledpowerflow,wasrun,andwouldassumefixedreal

powergeneration,exceptattheswingbus.Thismodelusesrealandreactive

current(computedasP/VandQ/V,respectively).Theauthorsuseanincremental

currentmodelratherthananincrementalpowermodelbecausetheJacobian

matricesofacurrentmodelarelesssensitivetobusvoltagevariations.Inaddition,

thesensitivitycoefficientofvoltagetoreactivecurrentismuchlesssensitivethan

thesensitivitycoefficientofvoltagetoreactivepower.Theauthorsfixtheswingbus

voltageangleatzero,butallowthevoltagemagnitudetofloat.Theauthorsnote

theirassumptionsresultinamoreaccurate“decoupled”relationshipbetween

incrementalreactivecurrentandvoltagethanisgivenbytheBmatrixusedinB‐θ

decoupledOPF,andthatthismoreaccurateandmorelinearmodelreducesthe

iterationsinanoptimizationalgorithm.ThesensitivitycoefficientsintheBmatrix

areaccurateonlyinasmallrangeofvoltage,requiringrecalculationoftheBmatrix

forlargechangesinvoltage;StadlinandFletcher’smodelisaccurateandlinearover

alargervoltageoperatingrangethanaB‐θmodel.StadlinandFletcherwantedto

defineamodelwhichremainsnearlylinearforchangesinvoltageandreactive

variablessothatefficientlinearprogrammingtechniquescouldbeapplied.

DaCostaandRosanotethatfortherectangularformulation,generationor

PVbuseshavedifferentequationsthanloadorPQbuses.Atloadbuses,activeand

reactivepowermismatchesareknown.Atgenerationbuses,reactivepower

mismatchesarenotknownbutvoltagemagnitudeconstraintsareknown,because


Page31

inatraditionallyformulatedpowerflow,generatorreactivepoweroutputis

variable(DaCosta2008).Therefore,avoltagemagnitudeconstraintisaddedto

eachloadbus,resultinginadifferentJacobianmatrix.

DaCostaetal.presentarectangularformulationofaNewton‐Raphson

powerflowbasedoncurrentinjections,forbothPQandPVbuses(DaCosta1999,

Lin2008).Inthisformulation,theJacobianmatrixhasthesamestructureasthe

nodaladmittancematrix,exceptforPVbuses.ForPVbuses,theauthorsintroducea

newdependentvariable,ΔQ,andanadditionalconstraintonvoltagemagnitude

deviation.Thevoltagemagnitudeconstraintislinearized:

ΔVn=0≈(Vrn/Vn)ΔVrn+(Vjn/Vn)ΔVjn,whereVnisthevoltagemagnitudeatbus

n,Vrnistherealcomponentofvoltageatbusn,andVjnistheimaginarycomponent

ofvoltageatbusn.

DaCostaandRosanotethatthecurrentinjectionequationsarelinearfor

electricalnetworkswithonlyPQbusesandaconstantimpedanceloadmodel(Da

Costa2008).

Jiangetal.publishedapower‐currenthybridrectangularOPFformulation.

Theydividebusesintotwotypes,thosewithnon‐zeroinjections,andthosewith

zeroinjections(Jiang2009).Forbuseswithnon‐zeroinjections,thepower

mismatchformulationisused,whilethecurrentmismatchformulationisusedfor

buseswithzeroinjections.Theauthorsnotethatinthecurrentmismatch

formulation,whichissimilartotheIVformulationpresentedabove,thefirst‐order

derivativesoftheequationsareconstantsandthesecond‐orderderivativesare

zeros.Bydividingthebusesintotwogroups,thehybridmethodsavescomputation

timefortheJacobianandHessianmatrices.

MeliopoulosandTaouseaformulationreferredtoas“QuadraticPower

Flow,”withcurrentconservationequationsfromKirchhoff’scurrentlawin

rectangularcoordinatesinsteadofpowerflowequations,andaddoperational

constraintstothemodelonlywhentheyareviolatedinthepreviousiteration

(Meliopoulos2011).Theequationsmodelinggenerators,constantpowerloads,and

transformersarequadraticequationsseparatedintorealandimaginaryparts.The

objectivefunctionistominimizethesumofapenaltyfactortimesthesumof


Page32

currentmismatchesandthetotalgeneratorcosts.Themodelincludesaslackbusas

the“mismatchcurrentsource”wherethevoltagemagnitudeisastatevariableand

therealandimaginarycomponentsofcomplexvoltagearecontrolvariables,whilea

PVbushasthevoltagemagnitudeasacontrolvariableandrealandimaginary

componentsofcomplexvoltageasstatevariables.Theauthorslinearizetoeliminate

integerstatevariables.Thequadraticconstraintsarelinearizedwhentheyare

addedtothemodel.

7.Conclusions

Thispaperhaspresentedaliteraturereviewofdifferentformulationsofthe

ACOPFanddiscussedareasforfutureresearchwheretheACOPFcouldbe

improved.TheACOPFproblemisinherentlydifficultduetononconvexities,

multipartnonlinearpricing,andalternatingcurrent.Wedonotyethavepractical

approachestosolvingnonconvexproblems.TheACOPFisawell‐structured

problem,andhasdevelopedduring50yearsofresearch.Academiaandindustry

havedevelopedvariousapproachestosolvingtheACOPF,withdifferent

formulations,algorithms,andassumptions.Thetraditionalapproachhasbeento

linearizethefullACOPFproblemanddecomposeitintosubproblems.TheACOPFis

notahypotheticalproblem–itissolvedevery5minutesthroughapproximations

andjudgment.After50years,thereisnotyetacommerciallyviablefullACOPF.

ManypossibilitiesandwaystoexaminetheACOPFremain.Today’ssolversdonot

returnthegapbetweenthegivenandgloballyoptimalsolution;ifwemakearough

estimatethattoday’ssolversareonaverageoffby10%,andworldenergycostsare

$400billion,closingthegapby10%isahugefinancialimpact.


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