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Electric
Power
Systems
Research
j o ur na l ho me p ag e :w w w . e l s e v i e r . c o m / l o c a t e / e p s r
Experiments
with
the
interior-point
method
for
solving
large
scale
Optimal
Power
Flow
problems
Florin
Capitanescu
a,∗,
Louis
Wehenkel
baUniversityofLuxembourg,InterdisciplinaryCentreforSecurity,ReliabilityandTrust(SnT),6,rueRichardCoudenhove-Kalergi,L-1359Luxembourg,Luxembourg bUniversityofLiège,InstitutMontefiore,Sart-TilmanB28,B-4000Liège,Belgium
a
r
t
i
c
l
e
i
n
f
o
Articlehistory: Received16March2012 Receivedinrevisedform 17September2012 Accepted1October2012 Keywords:
OptimalPowerFlow
Security-ConstrainedOptimalPowerFlow Interior-pointmethod
Nonlinearprogramming
a
b
s
t
r
a
c
t
Thispaperreportsextensiveresultsobtainedwiththeinterior-pointmethod(IPM)fornonlinear pro-grammes(NLPs)stemmingfromlarge-scaleandseverelyconstrainedclassicalOptimalPowerFlow(OPF) andSecurity-ConstrainedOptimalPowerFlow(SCOPF)problems.Thepaperdiscussestransparentlythe problemsencounteredsuchasconvergencereliabilityandspeedissuesofthemethod.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
TheOptimalPowerFlow(OPF)[1]isastatic,nonlinear, non-convex,large-scale optimization problem withcontinuous and discretevariables.TheSecurity-ConstrainedOptimalPowerFlow (SCOPF)isageneralizationoftheOPFproblemthatensures addi-tionallythesystemsecuritywithrespecttoa setofpostulated contingencies[2].OPFisnowadaysanessentialtoolinpower sys-temsplanning,operationalplanningandreal-timeoperation.
Due to critical environmentalissues, nowadays mostpower systemshavetoaccommodatea significantlevel ofpenetration ofrenewableintermittentgeneration.Thisrequiressmarterways ofcontrolinreal-time accordingtotheprinciples:just-in-time, just-in-place,andjust-in-context[3].Sinceaclassicalday-ahead preventiveSCOPFapproachwouldbenotanymoresustainable,we foreseethattheneedforusinganautomaticadaptiveoptimal con-trolscheme,andinparticulara(SC)OPFinreal-time,ismoreand morestringent.Thisraisesthesameoldconcernsregardingmostly thereliabilityandspeedofsuchanapproach[4,5].Furthermore, inEuropethereisatrendtocheckthesecurityandpropose coor-dinatedcontrolactionsforalargeinterconnection,composedby independentpowersystems,inabroaderway[6],whichcallsfor
∗Correspondingauthor.Tel.:+3524666445345;fax:+3524666445669. E-mailaddresses:fl[email protected](F.Capitanescu),
[email protected](L.Wehenkel).
toolsabletodealwithverylargescaleoptimizationproblems,with uptohundredsofmillionsofvariablesandconstraints.
IPM[7,8]hasbeensuccessfullyappliedfornearlytwodecades tovariousOPFproblems[9–13].ThemainadvantagesoftheIPM are:(i)easeofhandlinginequalityconstraintsbylogarithmic bar-rierfunctions,(ii)speedofconvergence,and(iii)astrictlyfeasible initialpointisnotrequired.ThedrawbacksoftheIPMare:(i)the heuristictodecreasethebarrierparameter,(ii)therequired pos-itivityofslackvariablesandtheircorrespondingdualvariablesat everyiteration(whichmaydrasticallyshortentheNewtonstep length),and(iii)itdoesnotwarmstartwell.
Confidentiality of large scale real-life power systems data deprivesthepowersystemscommunityofrealisticbenchmarks atleastinthefieldofOPFandpreventsresearchersfrom repor-tingOPFresultsobtainedinrealisticconditionsandtherebythe fairassessmentofexistingOPFmethodsonvariousdifferent prob-lems.Untilrecently,whenalarge2746-busmodelofthePolish powersystembecamefreelyavailable[14],thelargesttestbedfor theOPF/SCOPFprogramswasanIEEEsystemof300buses[15].
Thelarge-scale1NLPOPFproblemstackledbyIPMreportedin theliteratureinvolvepowersystemsof2098buses[16,17],2256 buses[18,19],2423buses[9],2746buses[20],2935buses[21], 3012buses[22],and3467buses[10].Otheralternativealgorithms havebeenalsousedforlarge-scaleOPF,e.g.Newtonmethodwas
1Wearbitrarilyconsiderapowersystemas“large”ifitcontainsmorethan2000
buses. 0378-7796/$–seefrontmatter© 2012 Elsevier B.V. All rights reserved.
appliedtoa2436-busgrid[23],anon-interiorpoint complemen-taritymethodwasappliedtoa2098-bussystem[16,24],amodified barrierLagrangianfunctionwasappliedtoa2111-bussystem[25], a modifiedbarrier approach wasappliedtoa 2256-bus system [26],andatrust-regionbasedaugmentedLagrangianmethodwas appliedtonetworkmodelsof2935buses[21]and3012buses[22]. Furthermore,althoughseveralcommercialNLP SCOPFpackages areavailablefromvariousvendors[27–29],andareroutinelyused bymanysystemoperators,thescientific literaturereportingon experimentsusingSCOPFsolversonlarge-scalesystemsisquite limitedexceptofthefollowingworks:[30]usesanSLPapproach foramodelof12,965buses,[29]reliesonaninterior-pointsolver fora2351-busnetwork,[31]employsaconicprogrammingfora 2383-busPolishsystem,[32]reliesontheIPMfora2746-busPolish system,[22]appliesIPMtoa3012-busgrid,[33]usesaSLPforthe 3551busesUKsystem,and[34]usesamethodcombiningIPMand conjugategradientfora15,000busEuropeansystem.
Motivatedbythefactthattheseworkspresentgenerallyonly oneexampleofa successfulapplicationofa givenalgorithmto agivenOPF/SCOPFproblem,themaincontributionofthispaper liesin thetransparent report of extensiveresults with a non-commercialIPM-basedOPFprogram,developedbytheauthorsfor researchpurposes[35],onseverelyconstrainedlarge-scaleOPFand SCOPFproblems.Thechallengesofthesecomputationsarerelated to:(i)thesizeofour8387-bussystem,whichmodelsalargepartof Europe,isroughlythreetimeslargerthanpreviouslyreported IPM-basedNLPOPFs,(ii)thecomplexityofdata(e.g.coexistenceofmany verylonglines,stemmingfromsomenetworkequivalents,andvery shortlines),(iii)thetoughnessoftheoptimizationproblems con-sideredcomparedtotheliterature(e.g.oftenmorethanthousand constraintsarebindingattheoptimum),and(iv)thenumberand varietyofcontrolvariables(e.g.fewthousandscontrolvariables forthecontrolofbothactiveandreactivepowersareconsidered together).Thepaperalsodiscussesinatransparentwayreliability andspeedissuesoftheIPM.
Thepaperisorganizedasfollows.Section2introducestheOPF problemswhileSection3brieflydescribestheinterior-point algo-rithmusedforsolvingthem.Section4providesnumericalresults obtainedwiththeIPMforseveralOPFandSCOPFproblems.Section 5concludes.
2. FormulationoftheOptimalPowerFlowproblem
For the sake of facilitating the reader’ interpretation of our resultsaswellastomakethepaperself-containedwedescribe inthissectiontheformulationoftwoclassicalOPFproblems.
2.1. Notations
Letusdenoteby:n,g,c,b,l,t,o,a,andstherespective num-bersof:buses(n),generators(g),loads(c),branches(b),lines(l), alltransformers(t),transformerswithcontrollabletaps(o),phase shifters(a),andshuntelements(s),respectively.
We formulate the OPF problem with complex voltages expressedinrectangularcoordinates[12,35]:
Vi=ei+jfi, i=1,...,n,
whereeiandfiareitsrealandimaginarypart,respectively,the
voltagemagnitudebeing:
Vi=
e2 i +f 2 i .G
sk1
:
r
kB
skG
kB
skj
i
B
kG
sk Fig.1.Modelofagenericbranch.Wemodelanybranchk asasymmetricalquadruplein in serieswithanidealtransformerwithcomplexratio1/rk(seeFig.1), where:
rk=rk1+jrk2, k=1,...,b, rk2=rk21+rk22, k=1,...,b.
Atransformerwithtap-changerisaparticularcasewhererk2=0 whilealineisaparticularcasewhererk1=1andrk2=0.
2.2. Objectivefunctions
Inthispaperwedealwithtwoclassicalobjectivesnamely min-imumgenerationdeviationwithrespecttothebasecase(MD): MD=min g
i=1 (Pgi−Pgi0) 2, (1)andminimumactivepowerlosses(MPL): MPL=min b
k=1 GskV2 i +Gsk V2 j r2 k +Gk Vi2+V 2 j r2 k −2rk1(eiej+fifj)+rk2(eifj−fiej) r2 k , (2) wherePgiandPgi0 istheactivepowerandbasecaseactivepowerofgeneratori,Gk(respectivelyGsk)istheconductance(respectively
halfshuntconductance)ofthebranchklinkingbusesiandj. 2.3. Controlvariables
Weconsiderthefollowingcontrolvariables:generatoractive power,generatorreactivepower,controllabletransformerratio, shuntreactanceandphaseshifterangle.
2.4. Equalityconstraints
Equalityconstraintsmainlyinvolvenodalactiveandreactive powerbalanceequations,which,fortheithbus(i=1,...,n),take ontheform: Pgi−Pci−(e2i+f 2 i)
k∈Bi (Gsk+Gk) + k∈Bi [(eiejk+fifjk)(rk1Gk+rk2Bk)−(fiejk−eifjk)(rk2Gk−rk1Bk)]=0, (3) Qgi−Qci+(ei2+fi2)[Bsi+ k∈Bi (Bsk+Bk)] + k∈Bi [(eiejk+fifjk)(rk2Gk−rk1Bk)+(fiejk−eifjk)(rk1Gk+rk2Bk)=0, (4)wherePgiandQgiaretheactiveandreactivepowersofthegenerator
connectedatbusi,PciandQciaretheactiveandreactivedemandsof
theloadconnectedatbusi,Bsiistheshuntsusceptanceconnectedat
busi,Bidenotesthesetofbranchesconnectedtobusi(and
∀
k∈Bi, jkdenotestheotherbustowhichbranchkisconnected),GkandBk(respectivelyGskandBsk)arethelongitudinal(respectivelyshunt)
conductanceandsusceptanceofthekthbranchconnectedtobusi. Additionalequalityconstraintsmayexist,asfor examplethe settingofaphaseshifterratiotoaspecifiedreferencerk0: r2
k1+rk22−rk20=0, k=1,...,a, (5) orthesettingofgeneratorvoltagetoaspecifiedreference: e2i+fi2−(Viref)2=0, i=1,...,g. (6) 2.5. Inequalityconstraints
Theoperationallimitson(longitudinal)branchescurrentand voltagesmagnitudetakeontheform:
(G2 k+B2k) V2 i + V2 j r2 k −2rk1(eiej+fifj)+rk2(eifj−fiej) r2 k ≤Ikmax, k=1,...,b (7) (Vmin i ) 2≤e2 i +fi2≤(Vimax) 2, i=1,...,n. (8) Physicallimitsofpowersystemdevicescanbeexpressedas: Pmin gi ≤Pgi≤Pgimax, i=1,...,g, (9) Qmin gi ≤Qgi≤Qgimax, i=1,...,g, (10) r1mini ≤r1i≤r1maxi , i=1,...,o, (11) Bmin i ≤Bsi≤Bmaxi , i=1,...,s, (12) tanimin≤r2i r1i ≤ tanmaxi , i=1,...,a, (13)wherefortheithgeneratorPmin gi ,P max gi (respectivelyQ min gi ,Q max gi )are itsactive(respectivelyreactive)outputlimits,fortheith control-labletransformerrmin
1i andr1maxi areboundsonitsratio,fortheith shuntBmin
i andB max
i areboundsonitssusceptance,andfortheith phaseshiftermin
i and
max
i areboundonitsangle.
3. Multiplecentralitycorrectionsinterior-pointalgorithm WesolvetheaboveOPFproblemsusingthemultiple central-itycorrections(MCC)interior-pointalgorithm[17,35,36],thatthe authorsfoundafterextensiveexperimentsovertheyearsasthe mostreliable IPM algorithm. In order to make the paper self-containedwe brieflydescribethisalgorithminthis sectionand refer the interested reader to [35] for further implementation details.
3.1. OptimalityconditionsintheIPM
TheOPFformulationsoftheprevioussectioncanbecompactly writtenasageneralnonlinearprogrammingproblem:
minf(x), (14)
s.t. g(x)=0, (15)
h(x)≥0, (16)
wheref(x),g(x)and h(x)areassumedtobetwicecontinuously differentiable,xisa(dim(C)+2n)-dimensionalvectorthat encom-passes both control variables (vectorof size dim(C)) and state variables(realandimaginarypartofvoltageatallbuses),gisa p-dimensionalvectoroffunctionsandhisaq-dimensionalvector offunctions.
TheLagrangianassociatedwiththisNLPproblemwithintheIP frameworkcanbedefinedas:
L(y)=f(x)− q
i=1
lnsi−Tg(x)−T[h(x)−s],
wheres=[s1,...,sq]Tisthevectorofslackvariables,andare
thevectorsofLagrangemultipliers(alsocalleddualvariables),is apositivescalarcalledbarrierparameter,andy=[sx]Tgroups
allvariables.
TheperturbedKKTfirstordernecessaryoptimalityconditions areobtainedbysettingtozerothederivativesoftheLagrangian withrespecttoallunknowns[8]:
⎡
⎢
⎢
⎢
⎢
⎣
∇
sL(y)∇
L(y)∇
L(y)∇
xL(y)⎤
⎥
⎥
⎥
⎥
⎦
=⎡
⎢
⎢
⎣
−e+S −h(x)+s −g(x)∇
f(x)−Jg(x)T−Jh(x)T⎤
⎥
⎥
⎦
=0, (17)whereSisadiagonalmatrixofslackvariables,e=[1,...,1]T,
∇
f(x)isthegradientof f,Jg(x)istheJacobianof g(x)andJh(x)isthe Jacobianofh(x).Notethatinordertofacilitatethepresentation wehaverewrittenthecomplementarityslacknessconstraintsas e+S=0butaproperimplementationshouldrelyonthe con-straints/s+=0soastopreservethesymmetryoftheHessian.
Thepureprimal-dualIPalgorithmconsistsinsolvingiteratively thelinearizedKKTconditionsfortheNewtondirectionykwhile
decreasingthebarrierparameterkgraduallytozeroasiterations
progress: H(yk)
⎡
⎢
⎢
⎢
⎢
⎣
sk k k xk⎤
⎥
⎥
⎥
⎥
⎦
=⎡
⎢
⎢
⎢
⎢
⎣
ke−Skk h(xk)−sk g(xk) −∇
f(xk)+J g(xk)Tk+Jh(xk)Tk⎤
⎥
⎥
⎥
⎥
⎦
(18)where H(yk) is the Hessian matrix (of second derivatives)
(
∂
2L(yk)/
∂
y2).Wedenotebydim(KKT)thesizeofthisHessian.3.2. TheMCCalgorithm
WebrieflyoutlinetheMCCalgorithmtosolvetheKKT optimal-ityconditions(17):
I.Initialization.
Settheiterationcountk=0.Chose0>0.Initializey0,taking carethatslackvariablesandtheircorrespondingdualvariables arestrictlypositive(s0,0)>0.
II.Thepredictorstep.
(a) Solve the system (18) for the affine-scaling direction, obtainedbyneglectinginitsright-handside:
H(yk)
⎡
⎢
⎢
⎢
⎢
⎣
sk af k af k af xk af⎤
⎥
⎥
⎥
⎥
⎦
=⎡
⎢
⎢
⎢
⎢
⎣
−Skk h(xk)−sk g(xk) −∇
f(xk)+J g(xk)Tk+Jh(xk)Tk⎤
⎥
⎥
⎥
⎥
⎦
(b) Computetheaffinecomplementaritygapk af: kaf=(sk+˛kafskaf)T(k+˛k af k af) where˛k
af∈(0,1]isthesteplengthwhichwouldbetaken alongtheaffinescalingdirectionifthelatterwasused(19). (c)Estimatethebarrierparameterforthenextiteration:
k af=min
⎧
⎨
⎩
k af k2
,0.2⎫
⎬
⎭
k af qwherek=(sk)Tkdenotesthecomplementaritygapatthe
currentiterate. IIIThecorrectorstep.
(a) Compute a trial2 point ˜yk=yk+˛˜kyk
af, where ˛˜ k= min(˛k
af+ı˛,1)withthedesiredimprovementonthestep lengthı˛=0.2.
(b) Computethecomplementarityproductsatthetrialpoint ˜
vk=S˜k˜k.
(c)Identifycomponentsof ˜vkthatdonotbelongtotheinterval [ˇminkaf,ˇmaxkaf], calledoutliercomplementarity prod-ucts,whereˇmin=0.1andˇmax=10.
(d)Because the corrector step effort focuses on correcting theoutliersonlyinordertoimprovethecentralityofthe nextiterate,sometargetcomplementaryproducts(˜vk)tare defined: (
v
k i) t=⎧
⎪
⎪
⎨
⎪
⎪
⎩
ˇminkaf, if ˜v
ik<ˇminkaf ˇmaxkaf, ifv
˜ik>ˇmaxkaf ˜v
ik, otherwise (e)Thecorrectordirectionykcoisobtainedasthesolutionof thefollowinglinearsystem:
H(yk)
⎡
⎢
⎢
⎢
⎢
⎣
sk co k co k co xk co⎤
⎥
⎥
⎥
⎥
⎦
=⎡
⎢
⎢
⎢
⎣
(vk)t −v˜k 0 0 0⎤
⎥
⎥
⎥
⎦
wherethenonzerocomponentsoftheright-hand-side cor-respondtotheoutliercomplementarityproductsonly. (f)Update3thenewsearchdirection:
yk=ykaf+ykco (g)Choiceofthesteplength.
Determinethemaximumsteplength˛k
p∈(0,1] (respec-tively˛k
d∈(0,1])intheprimal(respectivelydual)variables spacealongtheNewtondirectionyk suchthat sk+1>0 (respectivelyk+1>0): ˛k p=min
1,min sk i<0 −sk i sk i ˛k d=min 1, min k i<0 −k i k i (19) where∈(0,1)isasafetyfactoraimingtoensurestrict positivenessofslackvariablesandtheircorrespondingdual variables.Atypicalvalueofthesafetyfactoris=0.99995. (h)Updatesolution: sk+1=sk+˛k psk k+1=k+˛kdk xk+1=xk+˛k pxk k+1=k+˛kd k2 Notethatatthispointsomeslackvariablesand/ortheircorrespondingdual
variablesmayviolatethestrictpositivityconditions(sk,k)>0.
3 Thecorrectorstepcanbeappliedseveraltimes.Insuchacase,thecurrent
directionykbecomesthepredictorforanewcorrector.
Table1
Testsystemsummary.
n g c b l t o a s
8387 1865 4669 14,561 12,474 2087 589 84 178
(i)Repeatthecorrectorstepapre-definednumberoftimesas longastheimprovementinthesteplengthissatisfactory butinferiorto1.
(IV) Checkconvergence.
A(locally)optimalsolutionisfoundandtheoptimization processterminateswhen:primalfeasibility,scaleddual fea-sibility,scaledcomplementarity gapandobjective function variationfromaniteration tothenextfallbelowsome tol-erances[9,10,12]: max{max i {−hi(x k)},||g(xk)|| ∞}≤ 1 (20) ||
∇
f(xk)−J g(xk)T−Jh(xk)Tk||∞ 1+||xk||2+||k||2+||k||2 ≤ 1 (21) k 1+||xk||2 ≤ 2 (22) |f(xk)−f(xk−1)| 1+|f(xk)| ≤ 2 (23)whereusually 1=10−4and 2=10−6.
Ifconvergencewasnotachievedsetk←k+1andgotostep II.
4. Numericalresults
4.1. Descriptionofthepowersystemmodel
Weusea8387-busmodifiedmodel4oftheinterconnectedEHV EuropeanpowersystemwhichspansfromPortugalandSpainto Ukraine, Russia and Greece. Notice that in this model the real parametersoftheindividualpowersystemcomponents(e.g.lines, transformers,etc.),thenetworktopology,aswellasthelimitson generators active/reactivepowers,transformers ratioandangle, voltages,andbranchcurrentshavebeenbiased.Nevertheless,this modelisrepresentativefortheEuropeaninterconnectioninterms ofsystemsizeandcomplexity.
Inordertoassesstherobustnessofourtoolwehavechosenvery tightoperationallimitsandphysicalboundsovercontrols.Asa con-sequencethebasecaseisquiteconstrained,e.g.manygenerators havenarrowphysicalactive/reactivepowerlimits,manyvoltage limitsareverytight,theanglerangeofseveralphaseshiftersisvery small,6linesareloadedatmorethan90%,etc.Thesedatacontain manyveryshortlines(e.g.98lineshavetheirreactancelowerthan 0.0002pu)andmanyartificiallines,stemmingfromsomenetwork equivalents,withverylargeimpedancevalues,whichleadto rel-ativelyill-conditionedproblems.Forinstanceforthisdatasetthe ratiobetweenthemaximumimpedanceandminimumimpedance overallbranchesisaround408,854,whilefortheIEEE118 sys-tem[15](respectivelythePoland2746-busmodel[14])thisratio isaround51(respectively15,516).Inaddition,thisdataset con-tains743brancheswithalargerratiothanthemaximumratioof thePoland2746-busmodel.
Asummaryofthecharacteristicsofthistestsystemsisgivenin Table1.
4Morecomprehensivedatasetsofthissystemareavailableuponrequestfrom
Table2
Problemsdefinition.
Problem Basecase Obj. Controlvariables Inequalityconstraints
Pg Pgs Vg r1 Bs a Pg Pgs Qg r1 Bs a V I P1 A MPL × × × × P2 A MPL × × × × × × × × × P3 A MPL × × × × × × × × × × P4 A MD × × × × P5 A MD × × × × × P6 A MD × × × × × × × × × P7 A MD × × × × × × × × × × × P8 A MD × × × × × × × × × × P9 A MD × × × × × × × × × × × × P10 B MD × × × × × × P11 B MD × × × × × × × × × × P12 B MD × × × × × × × × × × × × 4.2. Simulationassumptions
Weconsiderin oursimulationstwobasecasesdenotedasA
andB.InthebasecaseAsomevoltagelimitsarenotmet.Thebase caseBisthesameasAexceptthattheflowlimitofalinehasbeen decreasedtocreateathermalcongestion.
Unlessotherwisespecified,weusethedefaultsettingsofthe MCCalgorithm[35]inallcomputationstoenableafairassessment oftheperformancesofthealgorithm.
AlltestshavebeenperformedonaPCof2.8-GHzand4-GBRAM, usingtheIPM-basedNLPOPFprogram,developedforresearch pur-posesbytheauthorsinC++withintheCygwinenvironment[35]. 4.3. OPFproblemsdefinition
InordertotesttherobustnessandefficiencyoftheIPM,several OPFproblemshavebeensolvedinvolvingdifferentcombinations ofobjectives,controlsandconstraints,asshowninTable2.Inthis tablePg,Psg,Vg,Qg,r1,Bs,a,I,andVrefertogeneratoractivepower,
slackgeneratoractivepower,generatorterminalvoltage,generator reactivepower,controllabletransformerratio,shuntsusceptance, phaseshifterangle,branchcurrent,andbusvoltagemagnitude, respectively,whilesymbol“×”meansthatthevariable/constraint isconsideredintheOPF.
In allproblems the equality constraintsare the power-flow equationsandsometimesphaseshifterangles(e.g.inP7,P9,and P12)andgeneratorsvoltages(e.g.inP4toP7).
4.4. ObservationsabouttheOPFresults
Table 3 provides for each problem the size dim(C) of the setofcontrol variables,thesize dim(KKT)ofthe fullsystemof KKToptimalityconstraints(17),andthenumberofiterationsto
Table3
Problemssizeandconvergencedetails.
Problem dim(C) dim(KKT) Iter. Time(s)
P1 1825 76,221 43 16.0 P2 2592 80,056 45 16.8 P3 2592 105,004 65 47.4 P4 1250 72,204 9 5.0 P5 1250 105,752 10 8.1 P6 2017 109,587 14 10.7 P7 2185 110,175 13 10.0 P8 4413 114,109 9 8.3 P9 4581 114,697 9 7.2 P10 3646 110,274 17 12.1 P11 4413 114,109 14 10.6 P12 4581 114,697 16 50.8 Table4
NumberandtypeofactiveconstraintsforproblemP3.
Activeconstraints Total
Qg I V r1 Bs
330 18 1064 83 63 1558
convergenceaswellasthecorrespondingCPUtimes5(inseconds). Furthermore,ingeneralthelargerthenumberofactiveconstraints thelargerthenumberofiterationstoconvergenceandhencethe computationaltime.Noticethattheconvergenceisachievedwithin areasonableCPUtimeforallproblemsinspiteofthelargesizeof thesystemandallthecomputationalchallengesmentioned pre-viously.Theaveragecomputationaleffortperiterationgenerally increaseswiththeproblemsizewhiletheoverallcomputingtime seemstoberathermoreinfluencedbytheproblemsizethanbythe numberofcontrolvariables.
Table4showsthenumberandthetypeofactiveconstraintsat theoptimumofproblemP3.
Theverylargenumberof1558activeconstraintsconfirmsthe extremedifficultyoftheproblem.Thealgorithm’sabilityto effec-tivelycoordinate 18activelinecurrentconstraints(and3other linesloadedabove96%)togetherwith1064Vlimitsisremarkable. NoticethatmostNLPOPFtestscenariosoftheliterature,andin particularforIPM-basedOPFs,rarelyhavemorethanafewactive line-currentconstraintsanda fewhundredofotheractive con-straints(seehowever[30],aworkonSLP-basedSCOPF,where23 linecurrentconstraintsareactive).
4.5. OPFfailureonaverytoughproblem
WenowfocusonproblemP8butminimizetheoverallactive powergenerationandhencereplacethequadraticobjective(1)by thelinearone:
L=min g
i=1
Pgi. (24)
ForthisproblemtheMCCalgorithmgetsstuckonaninfeasible non-optimalpoint.Furthermore,theconvergenceisnotrestored even after trying few classical heuristic techniques (e.g. differ-entinitialvaluesfor thebarrierparameter,smallervalues of thesafetyparameter,anddifferentinitializationsofvariables) [16,40].However, by using anotherIPM algorithm, namelythe predictor–corrector(thatwegenerallyfoundslightlylessreliable thantheMCCalgorithm)alocallyoptimalsolutionisfound.Table5
Table5
NumberandtypeofactiveconstraintsforthemodifiedproblemP8.
Activeconstraints Total
Pg Qg I V r1 Bs
828 379 76 510 58 80 1931
Table6
SCOPFstatisticsforproblemP2.
Numbercontingencies dim(C) dim(KKT) Iter. Time(s)
0 2592 80,056 45 16.8
1 4417 158,101 53 78.5
2 6242 236,146 55 155.0
3 8067 314,191 75 389.0
Table7
SCOPFstatisticsforproblemP8.
Numbercontingencies dim(C) dim(KKT) Iter. Time(s)
0 4413 114,109 9 8.3
1 6238 217,100 10 30.2
2 8063 320,091 15 93.0
3 9888 423,082 11 140.9
4 11,713 526,673 10 204.3
providesthenumberandthetypeofactiveconstraintsatthis opti-mum.Thechoiceofthislinearobjectiveleads,asexpected,toa highnumberofactivepowergeneratorconstraintsactiveatthe optimum.Theverylargetotalnumberof1931activeconstraints, aswellasthelargenumberof76activebranchcurrentconstraints (plus11linesloadedabove95%attheoptimum),indicateagainthe toughnessofthistestscenarioascomparedwiththosefromthe literature.
4.6. SCOPFsolutions
InordertofurtherassesshowtheIPMscaleswithproblemsize
weconsidertheproblems P2andP8solvedbya SCOPFin
pre-ventiveonlymode[2].Thelatterconsistsinduplicatingbasecase constraints(3)–(13)foreachcontingencyincludedintheSCOPF andassumesthatcontrolvariableshavethesamevalueinboth basecaseandcontingencystates.ThisSCOPFisaugmentedwith onecontingencyatthetime.
Tables 6and 7provide for eachproblemthe sizeof theset ofcontrolvariables,thesizeofthefullsystemofKKToptimality constraints,thenumberofiterationstoconvergenceandtheCPU times.Onecanobservethatforbothproblemsthecomputational effortincreasesmuchmorethanlinearlywiththesizebut still remainsreasonablegiventheproblemsize.Furthermore,asin pre-viousexamples,weobservethatthesizeoftheproblemhasamore significantimpactonthecomputationaleffortthanthenumberof controlvariables.
4.7. Comparisonwithavailablesolvers
Inthis sectionweperforma comparison withthetwo most efficientalternative availablesolvers for largescalesystems on Matpower Version4.1runningunder Matlab7.13[22],namely theMatlabInteriorPointSolver(MIPS)andthePrimalDual Inte-riorPointMethod(PDIPM).AlthoughMatpowerpossessesseveral interestinglessconventionalmodelfeatures,thecurrentversion doesnotallowmodellinganyamongourtwelvemoreclassicalOPF problemsintermsofobjectivefunctionandcontrolvariables.In ordertoenableanasfairaspossiblecomparison,weconsideronly theOPFobjectivefunctionofminimumgenerationcost/deviation, generators active/reactive powers as control variables, and
Table8
OPFcomparisonwithMatpower.
Basecase Obj. Oursolver
it/time MIPS it/time PDIPM it/time A L 97/71.9 66/31.8 76/37.8
A-PST L 346/252 Failed Failed
A Q 13/10.7 47/22.1 Failed
A-PST Q 20/15.7 32/17.8 42/20.6
constraintsonbranchcurrents,voltagemagnitudeand physical limitsongenerators’active/reactivepowers.Stillthereremaintwo slightmodellingdifferencesconcerningthetransformer transver-sal susceptanceand thehandlingof current constraints,asour solverimplementsonlyconstraintsonlongitudinalbranchcurrents
(7),whereasMatpowersolversimplementsuchconstraintsatboth endsofeachbranch.Furthermore,PDIPMimplementsonlyMVA flowconstraints.However,asthemaincomputationalburdenin IPM isthefactorizationofa linearsystemofequations[12],we noticethatasMatpowersolversrelyonthe“reducedKKTsystem” [10,21],thenumberofinequalityconstraintsdoesnotaffectthe sizeofthissystem,contrarytoourimplementationwhichusesthe fullKKTsystem(17).
Table8presentstheresultsofourexperimentsforfour feasi-bleOPFproblems,where“L”denotesthelinearobjective6function (24),“Q”denotesthequadraticobjectivefunctionMD(1),andthe basecaseA-PSThasbeenobtainedfromthebasecaseAbysetting tozerotheangleofall84phaseshiftersandusingasstartingpoint theloadflowsolutionofcaseA.
TheresultsshowthatbothMIPSandPDIPMsolvershavealower computationaleffortperiterationthanoursolver,whichis cer-tainlyduetotheuseofthereducedKKTsystemandpossiblythe libraryusedtosolvethissystem.However,oursolverconverges inreasonabletimeandisbothfasterandneedsalowernumber ofiterationsforthequadraticobjective.Theresultsalsoindicate that,similarlytothefailureofoursolverfortheproblemdescribed inSection4.5,otherexcellentsolversmayalsooccasionallyfail, especiallyonverytoughoptimizationproblemsasincaseA-PST. Inthelattercasetheverylargenumberofiterationsofoursolverto convergenceisnotsatisfactoryandcouldbepracticallyconsidered asacaseoffailure,e.g.ifamaximumrunningtimewasrequired. Wepresumethatbyrunningagainthesolverwithdifferentsetsof parameters[16]theconvergenceoffailedcasescouldberestored. Acomprehensivecomparisonconcerningtherelativereliabilityof thesesolversisoutofthescopeofthepaper,asitshouldconsider manydifferentOPFproblemsundervariedoperatingconditions. WeonlyunderlinetheneedtoimprovethereliabilityofIPMcodes forverytoughlargescaleOPFproblems.
Takingintoaccounttheslightmodelingdifferencesandthelarge numberof84branchcurrentbindingconstraints,weassessedthat inallcasesMIPSandoursolverconvergedpracticallytothesame solution.
4.8. SCOPFsolutionsona3012-bussystem
WeconsideramodelofthePolandpowersystemwhichOPF dataareavailableonMatpowerweb-site[22].Thesystem com-prises3012buses,3371lines,201transformers,and298generators (obtained after aggregating coexisting generators at the same bus).
Weconsidertheproblemofminimizinggenerationcostsolved byaSCOPFincorrectivemode[37].Weuse292generatorsactive
6NoticethatforthisobjectivefunctionoursolverfailedforamorecomplexOPF
Table9
SCOPFresultsforthePolandsystem.
Numbercontingencies dim(C) dim(KKT) Iter. Time(s)
0 480 33,640 35 6.7 1 960 68,446 38 24.4 2 1440 103,252 37 60.1 3 1920 138,058 42 122.3 4 2400 172,864 41 177.2 5 2880 207,670 41 68.7 6 3360 242,476 39 76.0 7 3840 277,282 40 88.2 8 4320 312,088 46 113.8 9 4800 346,894 47 129.0 10 5280 381,700 47 144.9 11 5760 416,506 47 158.1 12 6540 451,312 48 176.3
powerand188generatorsreactivepowerascontrolvariables,and constraintsonbranchcurrents,voltages,physicallimitson genera-tors’active/reactivepowers,andactivepowerre-dispatchof292 generatorsascorrectiveactions.
Table9showstheresultsobtainedwithoursolverforincreasing numbersofcontingenciesincludedintheSCOPF.Exceptmaybefor thecaseswherethenumberofcontingenciesincludedintheSCOPF rangesfrom1to4,thealgorithmscalesverywell.
TheseresultsshowthattheSCOPFsolutionforthisalreadyquite largesystemisreasonablyfastandcanbeenvisagedevencloseto real-time.Furthermore,comparingSCOPFresultsofTables7and9 for problems of close sizes one can observe that, the average computationaleffortperiterationforthe3012-bussystemis con-siderablyfasterthanforthe8387-bussystem,whichweexplain bythemuchworsenumericalconditioningofthelattersystem,as alreadydiscussedinSection4.1.
5. Discussion,conclusionsandperspectives
ThispaperhaspresentedextensivenumericalresultswithIPM forlargescaleOPF/SCOPFproblems,withthegoalofadvancingthe stateoftheknowledgeaboutthismethodintermsofrobustness andscalability.
Ourresultsshowthatnowadays,evenwithoutusingthemost powerfulcomputersavailableonthemarket,itiscertainly feasi-bletorunanNLPOPFforalargesysteminreal-timeprovidedthat areliablesolveris used.Furthermore,weprovethatourSCOPF codeis practicalona 3012-bus system.Ontheother hand,for thepoorlyconditioned8387-busgrid,computermemory limita-tionspreventedusfromincludingmorethanfewcontingencies (usingthefullnetwork model)intotheSCOPF,andthereforeto assesshowthecomputationaleffortfurtherscaleswiththe prob-lemsize.However,even assumingsufficientcomputermemory, asonecouldforesee thatforsuchasystemfewtensof contin-genciesmight bebindingattheoptimum,oursolvercouldnot complywithclosetoreal-timerequirements,whileitcould pos-siblystill beused in day-ahead operational planning.For such large-scalesystems,inordertorendertheSCOPFtractablein real-timeapplicationsonewillhavetoadoptapproximatemodelsfor post-contingencystates[27,34]orresorttoBendersdecomposition [37].
AsthefactorizationoftheHessianmatrix isbyfarthemost expensivecomputationaltaskofaninterior-pointalgorithm itera-tionthechoiceofasuitablelibraryforthesolutionofasymmetric, verysparse,and veryill-conditionedsystemoflinearequations isparamount.Inourimplementationweusedthesparse object-oriented library SPOOLES [38] which has several options, and wecouldexperimentthestrongdependenceofperformancevs. robustnesstradeoffsontheparticularoptionschosen.
Furthermore,inourimplementation,forthesakeof facilitat-ing theprogramming effort,wesolve ateach iteration the full systemofKKToptimalityconditionswhereas mostauthors rec-ommendtheuseofthe“reducedsystem”only[10],whichmay break-downthesizeofthesystemuptoaround30%byappropriate eliminationofthecomplementarityslacknessandinequality con-straints.Althoughthecomplementarityslacknessconstraintsare trivialequationsandthereforeshouldnotposeadditional prob-lemstoasmartlibrary,a furtherdecreaseofthecomputational timeshouldbeexpectedifonereliesonthe“reducedsystem”only. Our study also shows that the IPM algorithms proposedin thepowersystemsliteraturemayleadtoconvergenceproblems onveryhard optimizationproblems.Themaincauseof conver-genceproblems oftheIPMis thattheiterationsbecomesstuck atanon-optimalpointifoneapproachestooearlythefeasibility boundaryandthereforetheslackvariablesprematurelygoto0, aphenomenoncalledjamming[39].Fortunately,therearesome well-knownremediestorestoretheconvergenceofanIPM(e.g. changingtheinitialvalueofsomeparametersandespeciallythe barrier parameter, using a less aggressive decrease of the bar-rierparameter,usinganalternativeinterior-pointalgorithm,etc.) [16,40],buttheytakeadditionalcomputationaleffortwithout guar-anteeingsolutionrestoration.
WeenvisagetoimprovethereliabilityofourIPM implementa-tionbyfocusingonthreeaspects:theuseofanti-jammingremedies [39],regularization(ortoensurethesmoothnessofthesolution byaugmentingtheproblemwithpenaltyfunctionstobetterdeal withanill-posedHessianmatrix)[41],anduseofmeritfunctionsto ensurethatjointprogressismadebothtowardsalocalminimizer andtowardsfeasibility(thisprogressisachievedbyshorteningthe step-lengthalongtheNewtonsearchdirection)[42].
Asensiblesolutiontomitigatereliabilityissues,beyondusing appropriate librariesfor thesolutionof linearsystemsof equa-tions,wouldbetoembedarobustandfastgenericNLPsolverin theOPF[34].Nowadaysthereexistindeedseveralmaturegeneric NLPsolvers(e.g.KNITRO,IPOPT,LOQO,CONOPT,SNOPT,MINOS, etc.)thatprovedexcellentperformancesongenericproblemsas wellasforsomepowersystemproblemsmodeledinAMPL[43]or GAMS[44].
UsingOPF/SCOPFinreal-timeinvolvesamongothersthe suc-cessivesolutionsofcloselyrelatedNLPproblems.TheIPMdoesnot naturallywarmstartwellandhencecannotdirectlytake advan-tageofthesolutionsgottenattheprevioustime-steps.Although thecurrentspeedofIPMissatisfactorysince,aswithany Newton-basedmethodthenumberofiterationsgenerallyscalesacceptably withtheproblemsize,anyimprovementofitswarmstart abil-ity(or,moregenerally,ofitslearningability)iswelcome.Welook forwardtoassessonSCOPFproblemstheimprovementsreported forgenericNLPs(seee.g.[41]).Furthermore,recentresults[45] showthatmethodssupposedtowarmstartwell(e.g.SQP)mayfail onlargescaleproblemswithaverylargenumberofbinding con-straintsattheoptimum:inparticular,itwasfoundthatforNLPOPFs withequilibriumconstraints(inwhichequilibriumconstraintsare modeledbyRNSfunctions)thewarmstartofKNITRO(withSQP option)failedwhiletheSQPsolverSNOPTdidnotperformfaster thanthecoldstartofKNITRO(withIPMoption);also,theSQPsolver SNOPTfailedoncoldstartonmostproblems,whileKNITRO(with IPMoption)wasfoundthemostreliableamongthesolverstested [45].
Finally,thepresentpaperfocusedonNLPaspectsoftheOPF problem,buttheimpactontheoverallreliabilityandspeedofthe OPFprocessofotherkeyfeatures(e.g.handlingofdiscrete vari-ables,useoflimitednumbersofcontrolactions,etc.)remainalso tobecarefullyassessed[4,5,46,47].Inaddition,improvedwaysto analyzeandvisualizetheresultsoflarge-scaleOPF,e.g.incaseof gridcongestion,shouldbedevised[30].
Acknowledgments
TheauthorswouldliketothankRTEFrance(theFrench trans-missionsystem operator) for theirsupport and advices on the problemformulation.
FlorinCapitanescuissupportedbytheNationalResearchFund LuxembourgintheframeoftheCOREproject“Reliableandefficient distributedelectricitygenerationinsmartgrids”,C11/SR/1278568. Louis Wehenkel acknowledges the support of the Belgian NetworkDYSCO, fundedby theInteruniversityAttractionPoles Programme,initiatedbytheBelgian State,SciencePolicyOffice. Thescientificresponsibilityrestswiththeauthors.
WethankCleveAshcraft,themaindeveloperoftheSPOOLES library,for usefulsuggestionsconcerningtheproperuseof the libraryforoptimizationproblems.
WethankRayZimmerman,themaindeveloperofMatpower, for useful suggestions concerning the comparisons made with Matpower.
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