Experiments with the interior-point method for solving large scale optimal power flow problems

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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

, Louis Wehenkel

aUniversityofLuxembourg,InterdisciplinaryCentreforSecurity,ReliabilityandTrust(SnT),6,rueRichardCoudenhove-Kalergi,L-1359Luxembourg,Luxembourg

bUniversityofLiège,InstitutMontefiore,Sart-TilmanB28,B-4000Liège,Belgium

a rt 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)fornonlinearpro- 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)isageneralizationoftheOPFproblemthatensuresaddi- tionallythesystemsecuritywithrespecttoa setofpostulated contingencies[2].OPFisnowadaysanessentialtoolinpowersys- 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 foreseethattheneedforusinganautomaticadaptiveoptimalcon- trolscheme,andinparticulara(SC)OPFinreal-time,ismoreand morestringent.Thisraisesthesameoldconcernsregardingmostly thereliabilityandspeedofsuchanapproach[4,5].Furthermore, inEuropethereisatrendtocheckthesecurityandproposecoor- dinatedcontrolactionsforalargeinterconnection,composedby independentpowersystems,inabroaderway[6],whichcallsfor

∗ Correspondingauthor.Tel.:+3524666445345;fax:+3524666445669.

E-mailaddresses:florin.capitanescu@uni.lu(F.Capitanescu), l.wehenkel@ulg.ac.be(L.Wehenkel).

toolsabletodealwithverylargescaleoptimizationproblems,with uptohundredsofmillionsofvariablesandconstraints.

IPM[7,8]hasbeensuccessfullyappliedfornearlytwodecades tovariousOPFproblems[9–13].ThemainadvantagesoftheIPM are:(i)easeofhandlinginequalityconstraintsbylogarithmicbar- rierfunctions,(ii)speedofconvergence,and(iii)astrictlyfeasible initialpointisnotrequired.ThedrawbacksoftheIPMare:(i)the heuristictodecreasethebarrierparameter,(ii)therequiredpos- itivityofslackvariablesandtheircorrespondingdualvariablesat everyiteration(whichmaydrasticallyshortentheNewtonstep length),and(iii)itdoesnotwarmstartwell.

Confidentiality of large scale real-life power systems data deprivesthepowersystemscommunityofrealisticbenchmarks atleastinthefieldofOPFandpreventsresearchersfromrepor- tingOPFresultsobtainedinrealisticconditionsandtherebythe fairassessmentofexistingOPFmethodsonvariousdifferentprob- 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.

http://dx.doi.org/10.1016/j.epsr.2012.10.001

appliedtoa2436-busgrid[23],anon-interiorpointcomplemen- 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,isroughlythreetimeslargerthanpreviouslyreportedIPM- basedNLPOPFs,(ii)thecomplexityofdata(e.g.coexistenceofmany verylonglines,stemmingfromsomenetworkequivalents,andvery shortlines),(iii)thetoughnessoftheoptimizationproblemscon- sideredcomparedtotheliterature(e.g.oftenmorethanthousand constraintsarebindingattheoptimum),and(iv)thenumberand varietyofcontrolvariables(e.g.fewthousandscontrolvariables forthecontrolofbothactiveandreactivepowersareconsidered together).Thepaperalsodiscussesinatransparentwayreliability andspeedissuesoftheIPM.

Thepaperisorganizedasfollows.Section2introducestheOPF problemswhileSection3brieflydescribestheinterior-pointalgo- 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,andstherespectivenum- 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]:

V_i=e_i+jf_i, i=1,...,n,

wheree_iandf_iareitsrealandimaginarypart,respectively,the voltagemagnitudebeing:

V_i=



e²_i +f_i².

G

1 : r

B

G

B

j

i B

G

Fig.1.Modelofagenericbranch.

Wemodelanybranchk asasymmetricalquadruplein in serieswithanidealtransformerwithcomplexratio1/r_k(seeFig.1), where:

r_k=r_k1+jr_k2, k=1,...,b,

r_k²=r_k1² +r_k2² , k=1,...,b.

Atransformerwithtap-changerisaparticularcasewherer_k2=0 whilealineisaparticularcasewherer_k1=1andr_k2=0.

2.2. Objectivefunctions

Inthispaperwedealwithtwoclassicalobjectivesnamelymin- imumgenerationdeviationwithrespecttothebasecase(MD):

MD=min



(P_gi−P_gi⁰)², (1)

andminimumactivepowerlosses(MPL):

MPL=min



G_skV_i²+G_skV_j² r_k²

+G_k



V_i²+V_j²

r²_k −2r_k1(e_ie_j+f_if_j)+r_k2(e_if_j−f_ie_j) r_k²



, (2)

whereP_giandP_gi⁰ istheactivepowerandbasecaseactivepowerof generatori,G_k(respectivelyG_sk)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:

P_gi−P_ci−(e²_i+f_i²)



k∈Bi

(G_sk+G_k)

+



k∈Bi

[(e_ie_jk+f_if_jk)(r_k1G_k+r_k2B_k)−(f_ie_jk−e_if_jk)(r_k2G_k−r_k1B_k)]=0, (3) Q_gi−Q_ci+(e_i²+f_i²)[B_si+



k∈Bi

(B_sk+B_k)]

+



k∈Bi

[(e_ie_jk+f_if_jk)(r_k2G_k−r_k1B_k)+(f_ie_jk−e_if_jk)(r_k1G_k+r_k2B_k)=0, (4)

whereP_giandQ_giaretheactiveandreactivepowersofthegenerator connectedatbusi,PciandQciaretheactiveandreactivedemandsof theloadconnectedatbusi,B_siistheshuntsusceptanceconnectedat busi,Bidenotesthesetofbranchesconnectedtobusi(and∀^k∈Bi, j_kdenotestheotherbustowhichbranchkisconnected),G_kandB_k (respectivelyG_skandB_sk)arethelongitudinal(respectivelyshunt) conductanceandsusceptanceofthekthbranchconnectedtobusi.

Additionalequalityconstraintsmayexist,asfor examplethe settingofaphaseshifterratiotoaspecifiedreferencer_k0: r_k1² +r_k2² −r_k0² =0, k=1,...,a, (5) orthesettingofgeneratorvoltagetoaspecifiedreference:

e²_i+f_i²−(V_i^ref)²=0, i=1,...,g. (6) 2.5. Inequalityconstraints

Theoperationallimitson(longitudinal)branchescurrentand voltagesmagnitudetakeontheform:

 

 

k+B²_k)



V_i²+V_j²

r_k² −2r_k1(e_ie_j+f_if_j)+r_k2(e_if_j−f_ie_j) r²_k



≤I_k^max,

k=1,...,b (7)

(V_i^min)²≤e²_i +f_i²≤(V_i^max)², i=1,...,n. (8) Physicallimitsofpowersystemdevicescanbeexpressedas:

P_gi^min≤P_gi≤P_gi^max, i=1,...,g, (9) Q_gi^min≤Q_gi≤Q_gi^max, i=1,...,g, (10) r_1i^min≤r_1i≤r_1i^max, i=1,...,o, (11) B^min_i ≤B_si≤B^max_i , i=1,...,s, (12) tan_i^min≤r_2i

r_1i ≤tan^max_i , i=1,...,a, (13) wherefortheithgeneratorP_gi^min,P_gi^max(respectivelyQ_gi^min,Q_gi^max)are itsactive(respectivelyreactive)outputlimits,fortheithcontrol- labletransformerr_1i^minandr_1i^maxareboundsonitsratio,fortheith shuntB^min_i andB^max_i areboundsonitssusceptance,andfortheith phaseshifter^min_i and^max_i areboundonitsangle.

3. Multiplecentralitycorrectionsinterior-pointalgorithm WesolvetheaboveOPFproblemsusingthemultiplecentral- 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:

min f(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)-dimensionalvectorthatencom- passes both control variables (vectorof size dim(C))and state variables(realandimaginarypartofvoltageatallbuses),gisa p-dimensionalvectoroffunctionsandhisaq-dimensionalvector offunctions.

TheLagrangianassociatedwiththisNLPproblemwithintheIP frameworkcanbedefinedas:

L_(y)=f(x)−



lns_i−␭^Tg(x)−␲^T[h(x)−s],

wheres=[s1,...,sq]^Tisthevectorofslackvariables,␭and␲are thevectorsofLagrangemultipliers(alsocalleddualvariables),is apositivescalarcalledbarrierparameter,andy=[s␲␭x]^Tgroups allvariables.

TheperturbedKKTfirstordernecessaryoptimalityconditions areobtainedbysettingtozerothederivativesoftheLagrangian withrespecttoallunknowns[8]:

⎡

⎢ ⎢

⎢ ⎢

⎣

∇sL_(y)

∇␲L(y)

∇_␭L(y)

∇xL_(y)

⎤

⎥ ⎥

⎥ ⎥

⎦

⎡

⎢ ⎢

⎣

−e+S␲

−h(x)+s

−g(x)

∇^f(x)−J_g(x)^T␭−J_h(x)^T␲

⎤

⎥ ⎥

⎦

whereSisadiagonalmatrixofslackvariables,e=[1,...,1]^T,∇^f(x) isthegradientof f,Jg(x)istheJacobianof g(x)andJ_h(x) isthe Jacobianofh(x).Notethatinordertofacilitatethepresentation wehaverewrittenthecomplementarityslacknessconstraintsas

e+S=0butaproperimplementationshouldrelyonthecon- straints/s+=0soastopreservethesymmetryoftheHessian.

Thepureprimal-dualIPalgorithmconsistsinsolvingiteratively thelinearizedKKTconditionsfortheNewtondirectiony^kwhile decreasingthebarrierparameter^kgraduallytozeroasiterations progress:

H(y^k)

⎡

⎢ ⎢

⎢ ⎢

⎣

s^k

␲^k

␭^k

x^k

⎤

⎥ ⎥

⎥ ⎥

⎦

⎡

⎢ ⎢

⎢ ⎢

⎣

^ke−S^k^k h(x^k)−s^k

g(x^k)

−∇^f(xk)+J_g(x^k)^T␭^k+J_h(x^k)^T␲^k

⎤

⎥ ⎥

⎥ ⎥

⎦

where H(y^k) is the Hessian matrix (of second derivatives) (∂²L(y^k)/∂y²).Wedenotebydim(KKT)thesizeofthisHessian.

3.2. TheMCCalgorithm

WebrieflyoutlinetheMCCalgorithmtosolvetheKKToptimal- 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(y^k)

⎡

⎢ ⎢

⎢ ⎢

⎣

s^k_af

␲^k_af

␭^kaf

x^k_af

⎤

⎥ ⎥

⎥ ⎥

⎦

⎡

⎢ ⎢

⎢ ⎢

⎣

−S^k^k h(x^k)−s^k

g(x^k)

−∇^f(xk)+J_g(x^k)^T␭^k+J_h(x^k)^T␲^k

⎤

⎥ ⎥

⎥ ⎥

⎦

(b) Computetheaffinecomplementaritygap^k_af:

^k_af=(s^k+˛^k_afs^k_af)^T(␲^k+˛^k_af␲^k_af)

where˛^k_af∈(0,1]isthesteplengthwhichwouldbetaken alongtheaffinescalingdirectionifthelatterwasused(19).

(c)Estimatethebarrierparameterforthenextiteration:

^k_af=min

⎧ ⎨

⎩



^k



,0.2

⎫ ⎬

⎭

_af^k q

where^k=(s^k)^T␲^kdenotesthecomplementaritygapatthe currentiterate.

III Thecorrectorstep.

(a) Compute a trial² point ˜y^k=y^k+˛˜^ky^k_af, where ˛˜^k= min(˛^k_af+ı˛,1)withthedesiredimprovementonthestep lengthı_˛=0.2.

(b) Computethecomplementarityproductsatthetrialpoint

˜v^k= ˜S^k˜^k.

(c)Identifycomponentsof ˜v^kthatdonotbelongtotheinterval [ˇ_min^k_af,ˇmax^k_af], calledoutliercomplementarity prod- ucts,whereˇ_min=0.1andˇmax=10.

(d)Because the corrector step effort focuses on correcting theoutliersonlyinordertoimprovethecentralityofthe nextiterate,sometargetcomplementaryproducts(˜v^k)^tare defined:

(v^k_i)^t=

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

ˇ_min^k_af, if ˜v_i^k<ˇ_min^k_af ˇmax^k_af, ifv˜ik>ˇmax^k_af v˜_i^k, otherwise

(e)Thecorrectordirectiony^k_coisobtainedasthesolutionof thefollowinglinearsystem:

H(y^k)

⎡

⎢ ⎢

⎢ ⎢

⎣

s^k_co

␲^kco

␭^kco

x^k_co

⎤

⎥ ⎥

⎥ ⎥

⎦

⎡

⎢ ⎢

⎢ ⎣

(v^k)^t− ˜v^k 0 0 0

⎤

⎥ ⎥

⎥ ⎦

wherethenonzerocomponentsoftheright-hand-sidecor- respondtotheoutliercomplementarityproductsonly.

(f)Update³thenewsearchdirection:

y^k=y^k_af+y^k_co (g)Choiceofthesteplength.

Determinethemaximumsteplength˛^k_p∈(0,1](respec- tively˛^k_d∈(0,1])intheprimal(respectivelydual)variables spacealongtheNewtondirectiony^k suchthat s^k+1>0 (respectively␲^k+1>0):

˛^k_p=min



1,min

s^k i<0

−s^k_i

s^k_i



˛^k_d=min



1, min

^k i<0

−_i^k

_i^k



(19) where∈(0,1)isasafetyfactoraimingtoensurestrict positivenessofslackvariablesandtheircorrespondingdual variables.Atypicalvalueofthesafetyfactoris=0.99995.

(h)Updatesolution:

s^k+1=s^k+˛^k_ps^k ␲^k+1=␲^k+˛^k_d␲^k x^k+1=x^k+˛^k_px^k ␭^k+1=␭^k+˛^k_d␭^k

2 Notethatatthispointsomeslackvariablesand/ortheircorrespondingdual variablesmayviolatethestrictpositivityconditions(s^k,␲^k)>0.

3 Thecorrectorstepcanbeappliedseveraltimes.Insuchacase,thecurrent directiony^kbecomesthepredictorforanewcorrector.

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,scaleddualfea- sibility,scaledcomplementarity gapandobjective function variationfromaniteration tothenextfallbelowsometol- erances[9,10,12]:

max{max

i {−hi(x^k)},||g(x^k)||∞}≤ 1 (20)

||∇^f(xk)−J_g(x^k)^T␭−J_h(x^k)^T␲^k||∞

1+||x^k||2+||␭^k||2+||␲^k||2

≤ ₁ (21)

^k

1+||x^k||2 ≤ ₂ (22)

|f(x^k)−f(x^k−1)|

1+|f(x^k)| ≤ 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.Asacon- sequencethebasecaseisquiteconstrained,e.g.manygenerators havenarrowphysicalactive/reactivepowerlimits,manyvoltage limitsareverytight,theanglerangeofseveralphaseshiftersisvery small,6linesareloadedatmorethan90%,etc.Thesedatacontain manyveryshortlines(e.g.98lineshavetheirreactancelowerthan 0.0002pu)andmanyartificiallines,stemmingfromsomenetwork equivalents,withverylargeimpedancevalues,whichleadtorel- ativelyill-conditionedproblems.Forinstanceforthisdatasetthe ratiobetweenthemaximumimpedanceandminimumimpedance overallbranchesisaround408,854,whilefortheIEEE118sys- tem[15](respectivelythePoland2746-busmodel[14])thisratio isaround51(respectively15,516).Inaddition,thisdatasetcon- tains743brancheswithalargerratiothanthemaximumratioof thePoland2746-busmodel.

Asummaryofthecharacteristicsofthistestsystemsisgivenin Table1.

4Morecomprehensivedatasetsofthissystemareavailableuponrequestfrom theprojectPEGASEweb-page:www.fp7-pegase.eu.

Table2

Problemsdefinition.

Problem Basecase Obj. Controlvariables Inequalityconstraints

Pg P_g^s Vg r1 Bs a Pg P_g^s 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,developedforresearchpur- posesbytheauthorsinC++withintheCygwinenvironment[35].

4.3. OPFproblemsdefinition

InordertotesttherobustnessandefficiencyoftheIPM,several OPFproblemshavebeensolvedinvolvingdifferentcombinations ofobjectives,controlsandconstraints,asshowninTable2.Inthis tablePg,P^s_g,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 thesystemandallthecomputationalchallengesmentionedpre- viously.Theaveragecomputationaleffortperiterationgenerally increaseswiththeproblemsizewhiletheoverallcomputingtime seemstoberathermoreinfluencedbytheproblemsizethanbythe numberofcontrolvariables.

Table4showsthenumberandthetypeofactiveconstraintsat theoptimumofproblemP3.

Theverylargenumberof1558activeconstraintsconfirmsthe extremedifficultyoftheproblem.Thealgorithm’sabilitytoeffec- tivelycoordinate 18activelinecurrentconstraints(and3other linesloadedabove96%)togetherwith1064Vlimitsisremarkable.

NoticethatmostNLPOPFtestscenariosoftheliterature,andin particularforIPM-basedOPFs,rarelyhavemorethanafewactive line-currentconstraintsanda fewhundredofotheractivecon- straints(seehowever[30],aworkonSLP-basedSCOPF,where23 linecurrentconstraintsareactive).

4.5. OPFfailureonaverytoughproblem

WenowfocusonproblemP8butminimizetheoverallactive powergenerationandhencereplacethequadraticobjective(1)by thelinearone:

L=min



P_gi. (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

5CPUtimeconcernstheoptimizationprocessonly.

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

providesthenumberandthetypeofactiveconstraintsatthisopti- mum.Thechoiceofthislinearobjectiveleads,asexpected,toa highnumberofactivepowergeneratorconstraintsactiveatthe optimum.Theverylargetotalnumberof1931activeconstraints, aswellasthelargenumberof76activebranchcurrentconstraints (plus11linesloadedabove95%attheoptimum),indicateagainthe toughnessofthistestscenarioascomparedwiththosefromthe literature.

4.6. SCOPFsolutions

InordertofurtherassesshowtheIPMscaleswithproblemsize weconsidertheproblems P2andP8solvedbya SCOPFinpre- 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,asinpre- 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)andthePrimalDualInte- 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 slightmodellingdifferencesconcerningthetransformertransver- 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).

Table8presentstheresultsofourexperimentsforfourfeasi- bleOPFproblems,where“L”denotesthelinearobjective⁶function (24),“Q”denotesthequadraticobjectivefunctionMD(1),andthe basecaseA-PSThasbeenobtainedfromthebasecaseAbysetting tozerotheangleofall84phaseshiftersandusingasstartingpoint theloadflowsolutionofcaseA.

TheresultsshowthatbothMIPSandPDIPMsolvershavealower computationaleffortperiterationthanoursolver,whichiscer- 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].Thesystemcom- prises3012buses,3371lines,201transformers,and298generators (obtained after aggregating coexisting generators at the same bus).

Weconsidertheproblemofminimizinggenerationcostsolved byaSCOPFincorrectivemode[37].Weuse292generatorsactive

6NoticethatforthisobjectivefunctionoursolverfailedforamorecomplexOPF problemintermsofcontrolsandconstraints(seeSection4.5).

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,physicallimitsongenera- 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-bussystemiscon- siderablyfasterthanforthe8387-bussystem,whichweexplain bythemuchworsenumericalconditioningofthelattersystem,as alreadydiscussedinSection4.1.

5. Discussion,conclusionsandperspectives

ThispaperhaspresentedextensivenumericalresultswithIPM forlargescaleOPF/SCOPFproblems,withthegoalofadvancingthe stateoftheknowledgeaboutthismethodintermsofrobustness andscalability.

Ourresultsshowthatnowadays,evenwithoutusingthemost powerfulcomputersavailableonthemarket,itiscertainlyfeasi- bletorunanNLPOPFforalargesysteminreal-timeprovidedthat areliablesolveris used.Furthermore,weprovethatourSCOPF codeis practicalona 3012-bus system.Ontheother hand,for thepoorlyconditioned8387-busgrid,computermemorylimita- tionspreventedusfromincludingmorethanfewcontingencies (usingthefullnetwork model)intotheSCOPF,andthereforeto assesshowthecomputationaleffortfurtherscaleswiththeprob- lemsize.However,even assumingsufficientcomputermemory, asonecouldforesee thatforsuchasystemfewtensof contin- genciesmight bebindingattheoptimum,oursolvercouldnot complywithclosetoreal-timerequirements,whileitcouldpos- siblystill beused in day-ahead operational planning.For such large-scalesystems,inordertorendertheSCOPFtractableinreal- timeapplicationsonewillhavetoadoptapproximatemodelsfor post-contingencystates[27,34]orresorttoBendersdecomposition [37].

AsthefactorizationoftheHessianmatrix isbyfarthemost expensivecomputationaltaskofaninterior-pointalgorithmitera- tionthechoiceofasuitablelibraryforthesolutionofasymmetric, verysparse,and veryill-conditionedsystemoflinearequations isparamount.Inourimplementationweusedthesparseobject- oriented library SPOOLES [38] which has several options, and wecouldexperimentthestrongdependenceofperformancevs.

robustnesstradeoffsontheparticularoptionschosen.

Furthermore,inourimplementation,forthesakeoffacilitat- ing theprogramming effort,wesolve ateach iteration the full systemofKKToptimalityconditionswhereas mostauthorsrec- ommendtheuseofthe“reducedsystem”only[10],whichmay break-downthesizeofthesystemuptoaround30%byappropriate eliminationofthecomplementarityslacknessandinequalitycon- straints.Althoughthecomplementarityslacknessconstraintsare trivialequationsandthereforeshouldnotposeadditionalprob- lemstoasmartlibrary,a furtherdecreaseofthecomputational timeshouldbeexpectedifonereliesonthe“reducedsystem”only.

Our study also shows that the IPM algorithms proposedin thepowersystemsliteraturemayleadtoconvergenceproblems onveryhard optimizationproblems.Themaincauseofconver- 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],buttheytakeadditionalcomputationaleffortwithoutguar- anteeingsolutionrestoration.

WeenvisagetoimprovethereliabilityofourIPMimplementa- 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-timeinvolvesamongothersthesuc- cessivesolutionsofcloselyrelatedNLPproblems.TheIPMdoesnot naturallywarmstartwellandhencecannotdirectlytakeadvan- tageofthesolutionsgottenattheprevioustime-steps.Although thecurrentspeedofIPMissatisfactorysince,aswithanyNewton- basedmethodthenumberofiterationsgenerallyscalesacceptably withtheproblemsize,anyimprovementofitswarmstart abil- ity(or,moregenerally,ofitslearningability)iswelcome.Welook forwardtoassessonSCOPFproblemstheimprovementsreported forgenericNLPs(seee.g.[41]).Furthermore,recentresults[45]

showthatmethodssupposedtowarmstartwell(e.g.SQP)mayfail onlargescaleproblemswithaverylargenumberofbindingcon- 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.handlingofdiscretevari- ables,useoflimitednumbersofcontrolactions,etc.)remainalso tobecarefullyassessed[4,5,46,47].Inaddition,improvedwaysto analyzeandvisualizetheresultsoflarge-scaleOPF,e.g.incaseof gridcongestion,shouldbedevised[30].