survey of approaches to solving the acopf

OptimalPowerFlowPaper4

AstaffpaperbyAnyaCastilloRichardP.O’NeillMarch2013TheviewspresentedarethepersonalviewsoftheauthorsandnottheFederalEnergyRegulatoryCommissionoranyofitsCommissioners.

SurveyofApproachestoSolvingtheACOPF

OptimalPowerFlowPaper4

AnyaCastilloandRichardPO’Neill

[email protected];[email protected]

March2013

Abstract:WepresentabackgroundonapproacheshistoricallyappliedtosolvetheACOPF,manywhichareusedinourfollowingcompanionstudyontestingsolutiontechniques Castillo,2013 .Inthispaperwepresentanintroductionontheassociatedtheoryinnonlinearoptimization,andthendiscussthesolversandpublishedalgorithmsthathavebeenappliedtotheACOPF,datinginitiallyfromCarpentierin1962tocurrentdayapproaches.WeprovideinsightintothemajorcontributionsinsolutionmethodsappliedtotheACOPFtodate.

Disclaimer:Theviewspresentedarethepersonalviewsoftheauthorsandnotthe

FederalEnergyRegulatoryCommissionoranyofitsCommissioners.

TableofContents

1. Introduction 4

2. Definitions 4

3. UnconstrainedNonlinearOptimization 5

4. ConstrainedNonlinearOptimization 14

5. DecompositionTechniques 23

6. CommercialSolvers 28

7. HistoryofACOPFSolutionTechniques 29

8. SummaryandConclusions 40

References 41

1.Introduction Recently,mathematicaloptimizationhasbecomeapowerfultoolinmanyreal‐lifeapplicationsalongwiththemoretraditionalusesinengineeringandscience.Thehistoryisdriven,andwillcontinuetobedriven,bythedemandtosolveincreasinglylargernonlinearoptimizationproblemsunderthelimitationsofstorageandcomputingtime. Here,wepresentbackgroundonapproachesthatareusedinacompanionstudyandareoftenappliedtosolvetheACOPF.WearefocusedonunderstandingthereliabilityofsuchapproachesandwhetherthereisgoodreasontobelievethatconvergenceoftheACOPFwouldbeguaranteedatareasonablerate.Therefore,weintroduceanddiscusssomeoftheassociatedtheory. Insection2,wepresentdefinitions.Insection3,wereviewunconstrainedoptimizationfornonlinearprogrammingfollowedbyareviewofconstrainedoptimizationfornonlinearprogramminginsection4;therearenumerousnonlinearprogrammingresourcesand,forexample,thereadercanreferto Nocedal,1999 ,Panos,2006 foramoredetaileddiscussion.Insection5,wereviewproblemdecompositiontechniquesandpopularapproximationstotheACOPF.Insection6,wereviewthenonlinearcommercialsolversappliedtotheACOPFinourcompanioncomputationalstudy Castillo,2013 .Theninsection7,wereviewtheliteratureonsolvingtheACOPFfollowedbyabriefdiscussioninsection8.2.Definitions

Definition1.IfxєXRn,xiscalledafeasiblesolution.

Definition2.Thefunctionf x withdomainRnandcodomainRisdenotedby

f x :Rn→R.

Definition3.Iff x* f x forallxєX,x*iscalledanoptimalsolution.

Definition4.Iff x* f x forallxєX’X,x*iscalledalocaloptimalsolution.

Definition5.ThesetTRXiscalledatrustregionwhereanapproximationoff x is‘trusted.’

Definition6.ThesetXisconvex,ifandonlyifαx1 1‐α x2єXforallαє 0,1 andallx1,x2єX.

Definition7.Thefunctionf x isconvexindomainXifandonlyiff αx1 1‐α x2 αf x1 1‐α f x2 foranyαє 0,1 andallx1,x2єX.

Definition8.IfdєRnandx sdєXforsomes 0,discalledafeasibledirection.

Definition9.ThefunctionF xk xk 1istheresultofiterationkandtheapproachisadescentalgorithmiff xk 1 f xk thereductioncondition .

Defintion10.TheL2‐normis||.

Definition11.Iff x isoncedifferentiable,thevectoroffirstderivativesoff x withrespecttoxisf x ;f x isalsocalledtheJacobianorgradientvector.

Definition12.Iff x istwicedifferentiable,thematrixofsecondderivativesoff x withrespecttoxisH x 2f x ;H x iscalledtheHessian.

Definition13.IfQisasymmetricmatrixandd1TQd2 0,d1andd2areQ‐orthogonal,alsoknownasconjugatedirections.

Definition14.If A‐eI v 0,eisaneigenvalueandvisaneigenvectorofA.

Definition15.Forunconstrainedoptimizationwesolveinf f x |xєX wheretheobjectivefunctionf x :Rn→R

Definition16.Forequalityconstrainedoptimizationwesolveinf f x |g x 0,xєX wheretheconstraintsg x :Rn→Rm m n .

Definition17.Forinequalityconstrainedoptimizationwesolveinf f x |g x 0,xєX wheretheconstraintsg x :Rn→Rm mmaybelargerthann .

Definition18.TheindexsetASistheactivesetofconstraintswhereAS x j|gj x 0 .

Definition19.TheLagrangianfunctionisL x,λ f x λTg x .

Definition20.TheLagrangemultiplierλexistssuchthat x*,λ* isastationarypointoftheLagrangianfunctionwhereλL x*,λ* 0,whichimpliesg x* 0.

3.UnconstrainedNonlinearOptimizationWestartwiththe‘unconstrained’nonlinearoptimizationproblem.Wewanttominimizethenonlinearfunctionf x :

f* inf f x |xєX

wherex*isasolutiontotheaboveproblem P .MethodsforsolvingthisproblemdatebacktoNewtonandGauss.Inunconstrainedoptimization,thesetXisRn.IfXisasubsetofRn,XcouldbethesetofintegersZnordenoteaconstraintsetofequations.IfPisinfeasible,wedefinef* ∞;ifPisunbounded,wedefinef* ‐∞.Thefunctionfistypicallycalledtheobjectivefunction.Equivalently,theproblemcanbestatedasamaximizationof‐f x .Wepresentthefollowingtheoremswithoutproof.FormoredetailsseeMangasarian 1969 andLuenberger 1984 .

Theorem1.Iff x isdifferentiable,x*isarelativeorlocalminimumifforanyfeasibledirectiond,f x* d 0.

Theorem2.Iff x istwicedifferentiable,x*isarelativeorlocalminimumifforanydirectionfeasibled,f x* d 0;andiff x* d 0,dTH x* d 0.

Theorem3.IfXisaconvexsetandf x isconvexfunctiononX,arelativeorlocalminimumisaglobalminimum.

Theorem4.IfXisaconvexsetandf x isconvexfunctiononXandtwicedifferentiable,H x* ispositivesemidefiniteonX.

Theorem5.IfAissymmetric,alleigenvaluesarereal.

Theorem6.Ifalleigenvaluesarepositive negative ,Aispositive negative definite.

Theorem7.Ifoneormoreeigenvaluesare0,Aissingular.

Theorem8.IfAissymmetric,VistheeigenvectormatrixandEisdiag e whereAVVEandVcanbechosenorthonormal V‐1 VT ,thusA VEVT.

IterativeMethods.Thenonlinearfunctionf x cannotbesolveddirectlythroughfactorizationmethods.Thealgorithmsforsolvingunconstrainednonlinearoptimizationproblemscanbebroadlydefinedasderivative‐freemethods,methodsusingfirstderivativesandmethodsusingsecond‐orderderivatives.SincethestandardACOPFiscontinuousanddifferentiable,wefocusonderivative‐basedmethods. MostofthemethodsdescribedareiterativemethodswhichgenerateasequenceofXK x1,x2,…xk,…xK forKiterations.Wecallanyspecificprocessasolveroranalgorithm.Generally,alliterativesearchmethodshaveafivestepprocess:Step1.Chooseafunctiontooptimize;setk 0.Chooseaninitialpoint:x0.

Step2.Chooseasearchdirection:dk.

Step3.Choosestepsizeskwhereskisapositivescalarandcalculateanewpoint:xk 1 xk skdk.

Step4.Testforstopping:Ifxk 1satisfiestheconvergencecriteriaorexceedsthetimeallotted,thensetK k 1andstop.Otherwise,ifxk 1doesnotsatisfytheconvergencecriteria,thensetk k 1andgotoStep2.

Therearenumerousapproachesforeachstep.Whereaslinesearchmethodscomputeasearchdirectiondkandthenastepsizesk,trustregionmethodsdefinea

regionTRX,typicallyanelipsearoundthecurrentiteratexk,tochooseastepsizeskєTRanddirectiondksimultaneously.Thechoiceofthestepsizeandthendirectioninlinesearchmethods,orinthecaseoftrustregionmethodsthechoiceofthetrustregionandthendeterminingskdk,isimportantinpromotingconvergencefromremotestartingpoints.

InStep1,theusercansupplythestartingpoint.Iftheinitialpointisinfeasible,findingafeasiblesolutioncanbeanoptimizationprobleminitself.Forparallelalgorithms,theprocessmaystartwithmanyinitialpoints.Thesimplestparallelalgorithmisthehorseracewheretheapproachesarestartedinparallelandthefirsttoconvergeterminatesthealgorithm.Morecomplexparallelalgorithmsinteract. Steps2and3areoftendefinedasasinglestepxk 1 F xk whereFisthedescentalgorithm.Whetheralinearsearchortrustregionmethodisapplied,theresultiswhere

f xk skdk f xk andhopefully

f xk skdk f xk .Typicallydkisadescentdirectionwhere dkTf xk 0.ForinstancewecandeterminethesearchdirectiondkbyNewton‐Raphson’smethod: 2f xk dk ‐f xk .WewilldiscussNetwon‐Raphsoninfurtherdetailbelow.NotethatwemayreplacetheaboveHessianH xk 2f xk withanysymmetricandnonsingularmatrixBk.IfBk I,thedescentalgorithmiscalledsteepestdescent.IfBk H xk ,themethodiscalledquasi‐Newton.Forexample,therestrictedstepvariesfromaNewtonsteptoasmallsteepestdecentstep.TheNewtonstepapproachhasquadraticconvergence,butcanfailtoconverge.Thesteepestdecentapproachhasslowerlinearconvergence,butfailslessfrequently.

Step4testsforconvergence.NumerousstrategiesassumeLipschitzcontinuityofthegradient Nocedal,1999 .Intheory,toproveconvergenceaninfinitesequenceisshowntoconvergetoastationarypoint.Asubsequencehasalimitingpropertythatsatisfiesfirst‐ordernecessaryorsecond‐ordernecessaryconditions.Inpractice,asolverstopsinafinitenumberofiterationswhenthesolutionis‘closeenough’totheoptimalsolution.Forexample,forsomeuserdefinedconvergencecriteriaδ 0,stopif

| xk–xk 1 |/|xk| δorfk–fk 1 δor|f xk | δ.

Furthermore,thegloballyconvergentalgorithmsarethosewhichconvergefromanystartingpointtoalocaloptima,andthereforehavethepropertythatthegradientnormsconvergetozero: limk∞||fk|| 0RateofConvergence.Theerrortermisdefinedas

εk xk–x*

If0 β limsupkє∞|εk 1|/|εk|p ∞,βiscalledtheconvergenceratioandpistheorderofconvergence.Ifp 1andβ 0,theconvergencerateiscalledlinear.Ifp 1andβ 0,theconvergencerateiscalledsuperlinear.Ifp 2andβ 0,theconvergencerateiscalledquadratic.FeasibleDirectionMethods.FeasibledirectionmethodsrequirethatforagiveniteratexkєX,wecanfindadescentdirectiondkwhichisalsoafeasibledirectionatxk.Thereforeateachiterationtheremustexistanewfeasiblepointoftheformxk skdkthatmeetsthereductioncondition. Feasibledirectionmethodsincludebothderivativefreeandderivativebasedmethods.Linesearchwithoutderivatives,suchastheFibonaccisearch,onlyrequiresfunctionevaluations.TheFibonaccisequenceisthebasisforthisapproachinwhichsequentialpointsarechosensuchthatthediscrepancyf xk 1 f xk isminimized.Howeveriffisdifferentiable,wesolvesf xk sdk 0.Thisisastandardalgorithmintheinitialstageofalinesearch. First‐derivativegradientapproachesusethefirstderivativeasthefeasibledirection,dk ‐f xk .Thisisthedirectionatwhichfdecreasesmostquickly.Belowwesummarizethefirst‐derivativemethodsGauss‐Seidel,steepestdescent,conjugategradient,andQuasi‐Newton,andthenalsothesecond‐derivativebasedNewton’smethod.Gauss‐SeidelMethod.TheGauss‐Seidelapproachchoosestheunitvectorasacoordinatedirectiondk 0,…1k,…0 andalinesearchinonevariableisperformedforsomes 0:

minsf xk sdk

Thisproblemissequentiallysolvedineachcoordinatedirectionandrepeateduntiltheconvergencecriteriaismet.Thisapproachhaslinearconvergencerate.

SteepestDescentMethod.Thesteepestdescentmethod,whichisfundamentally

xk 1 F xk xk–f xk

forunitstepsizeandmorespecifically,

xk 1 F xk xk skdk

forsk,dk 0suffersfromslowconvergenceduetozigzagging,asshowninFigure1.Numerousmethodshavebeenproposedtoavoidzigzagging.

Ideallyinchoosingthestepsizes,wefindthelocalminimizertotheaboveobjectivefunctioninordertoobtainasubstantialreductioninf.Howeverthismayrequiretoomanyevaluationsoff,andinthecaseiffisdifferentiable,thensf.Thereforestrategiesthatperforminexactlinesearchidentifyanadequatereductioninfbytryingcandidatevaluesfors.Asweshallsee,exactlinesearchesarenotneededandmaynotbethebestapproachbecausealgorithmsthatachieverapidconvergencecansometimesconflictwiththeglobalconvergencerequirements,andviceversa.

Figure1.Zigzaggingwithsteepestdescentandlackofzigzagwithconjugatedirections.

ConjugateGradientMethod.TheintroductionoftheconjugategradientmethodbyFletcherandReevesin1964wastheinceptionoflarge‐scalenonlinearoptimization.TheconjugategradientmethoddeterminesconjugatedirectionstotheHessian

(x0, y0)

Steepest Descent Conjugate Gradient

throughevaluatingfandf,butwithoutdirectlyevaluatingtheHessian2f.Theprocedureisasfollows:

Step1.Letg0 f x0 andd0 ‐g0;setk 0.

Step2.Solveforxk 1 xk skdkwheresk argminsf xk sdk .

Step3.Determinedirectiondk 1 gk 1 rkdkwhere rk gk 1Tgk 1/gkTgk,,and gk 1 f xk 1 .

Step4.GotoStep2andrepeatfork 1…,n 1.

Quasi‐NewtonMethod. Inthelate1950’s,Davidonintroducedquasi‐Newtonmethods,basedonNewton’smethodandalsoknownassecantmethods,whichuseasequenceofpositivedefinitematricestoapproximatetheHessian orinverseHessian .Thesemethodsareinexactlinesearchmethodsandhaveasuperlinearconvergencerate.FletcherandPowelldemonstratedthatDavidon’sapproachisequivalenttoconjugategradientmethodwhenappliedwithexactlinesearchestoconvexquadraticfunctions.Quasi‐Newtonmethodsonlyrequirefirstderivatives,approximatetheHessianandadheretothedescentpropertyduetothepositivedefiniteness. Forexample,iff x istwicedifferentiable,thesecond‐orderTaylor’sseriesexpansionoff x atxkis:

f x f xk f xk T x xk x xk TH xk x xk /2 O |x xk |2

DroppingO |x xk |2 ,f x isaquadraticapproximationoff x

f x f xk f xk T x xk x xk TH xk x xk /2

Iff x isquadratic,then

f x bTx xTQx/2, f x bT xTQ, H x Q,and O |x xk |2 0.

Theaboveexpansionisexactandx*isasolutionto

Qx* b.

IfQispositivesemidefiniteatx*,wehavethat

f x 0,x*TQx* 0,and

f x f x* .

Thereforeinquasi‐NewtonmethodstheHessianapproximationHkischosentosatisfy

f xk Δx f xk HkΔx

whichiscalledthesecantequationandistheTaylorseriesofthegradientitself.StepsintheQuasi‐NewtonMethod:

Step1.AnapproximateinitialvalueofH0 Iisfrequentlysufficientforconvergence.Setk 0.

Step2.Hkispositivedefinite.Letdk Hk‐1f xk andcomputesk argminsf xk sdk .Setxk 1 xk skdk,wheresksatisfiesWolfeconditions:

f xk skdk f xk α1skdkTf xk

dkTf xk skdk α2skdkTf xk

where0 α2 α2 1.TheabovefirstinequalityissometimesreferredtoastheArmijorule,whichensuresthatthesteplengthskresultsinasufficientdecreaseinf,andthesecondisthecurvaturecondition,whichensuresthattheslopehasbeenreducedsufficiently Nocedal,1999 .Anotherwayofstatingthecurvatureconditionisasfollows:

vkT f xk skdk f xk vkTHkvk 0

wherevk skdk.TheWolfeconditionscanbeappliedinmostlinesearchmethodsandareimportantinimplementingquasi‐Newtonapproaches.

Step3.ComputeHk 1byarank‐two‐update,wheretheinverseoftheHessianBk Hk‐1isupdatedbythesumoftwosymmetricrank1matrices:

Bk 1 Bk βkvkvkT γkHkykykTHk

wherevk skdk,yk f xk skdk f xk ,βk 1/vkTyk,andγk 1/ykTHkyk.TheHessianapproximationischosentosatisfythequasi‐Newtoncondition: Hk 1 f xk 1 f xk xk 1 xk.Setk k 1;gotoStep2.TheabovealgorithmisknownastheDavidon‐Fletcher‐Powell DFP ;theBroyden‐Fletcher‐Goldfarb‐Shanno BFGS methodisthedualofDFPbecausewhereastheDFPconvergestotheinverseoftheHessian,theBFGSmethodconvergestotheHessianinitselfandthereforeisamoredirectapproach.

In1970,Broyden,Fletcher,Goldfarb,andShannoindependentlyintroducedasecantapproachnowknownastheBroyden‐Fletcher‐Goldfarb‐Shanno BFGS method.Inthisdirectapproach,

Hk 1 Hk βkykykT γkHkvkvkTHk

whereinsteadγk 1/vkTHkvk. TheLevenberg‐Marquardtmethodusesatrust‐regiontosolveasystem.Themethoddeterminesavalueν 0suchthatH xk νIispositivedefiniteandthensolves

sk∆xk H xk νI ‐1f xk .

Aniterationradiusrkmayimposealimitonthestepsizesk,oralternativelyνcontrolsthelength.Howeverbynotcontrollingthelengthrk,degeneracy i.e.non‐invertible H xk νI andill‐conditioningissuescanresultinastepsizeanddirectionthatarebadlydeterminedbyν.Newton’sMethod.Newton’smethodhasbeenappliedwithbothlinesearchandtrustregionapproaches.Non‐monotonemethodsinwhichthefunctionvaluesareallowedtoincreaseatsomeiterationshavebeenusefulinsolvinghighlynonconvexproblems.ThebasicNewton’salgorithmis:

Step1.Guessx0,andsetk 0.

Step2.Atxk,evaluatef xk .

Step3.EvaluateBk H xk oranapproximationBk H xk .Forexample,whenH xk isnotpositivedefinite,thenmodificationscouldbeappliedtodetermineH xk andguaranteeadecreaseinf xk .SuchmethodsmaymodifyH xk sothat f xk TBkf xk 0,orsuchmethodsmaycomputeanegativesearchdirectionwhichsatisfies

∆xkTBk∆xk 0and

xf xk T∆xk 0.

Step4.Solve:Bkdk f xk .

Step5.Iftheconvergenceerrorislessthanthetolerance, e.g.,||f xk || εand||dk|| ε thenstop;elsecontinue.

Step6.Findsksothat0 sk 1andf xk skdk f xk .

Step7.xk 1 xk skdk.Setk k 1andgotoStep2.

Notethatrecalculatingf xk ,Bk,andskdk xk 1 xkateachiterationmaybeverycostlyintermsofcomputationaltime.Letdk f xk ,xk 1 xk sdk.IfH xk ispositivesemidefinite,

u dkTH xk dk 0andv dkTdk 0

f xk 1 f xk dkTsdk sdkTH xk sdk vs us2

forsomes 0,wherewehavef xk 1 f xk .ForgloballyconvergentNewtonandquasi‐Newtonmethods,Bkmustbepositivedefiniteandalsohaveaboundedconditionnumberbecauseahighconditionnumbercanresultinill‐conditionedproblems Nocedal,1999 .Intuitively,thesearchdirectiondkisnotnearorthogonaltothegradientinordertopromoteconvergence.Howeverforproblemsthatareill‐conditioned,itmaybenecessarytosearchalongdirectionsthatarenearlyorthogonaltothegradient.NumericalEfficiencyandStability.Applyingtheappropriatetechniquesresultsincomputationallyefficientandnumericallystableprocedures.Forexamplethecoreofsolvingasystemoflinearalgebraicequations,Ax b,isdecomposingorfactoringthecoefficientmatrix.Throughthisprocess,thesolutioncanbeobtainedwithmuchlesscomputationaleffort,andbettermethodsaresolvedfasterwithlessnumericalerror.Matrixinversionisusuallyunstableandresultsindensematricesandunnecessaryfloating‐pointcalculations.Sometimesforsimplicityandsometimesoutofignorance,analgorithmispresentedasinvertingamatrixateachiteration:xA‐1b.Thisapproachisnotonlycomputationallyexpensive,butalsonumerically

unstable. Forlargeproblems,Aisusuallysparse,buttheinverseisusuallydense.Sparsematricesarestoredefficientlybystoringonlynonzeroesandtheirrowandcolumnindex.Whenoneoftheoperandsinafloating‐pointoperationiszero,itcanbeskipped,savingtime.Problemsarereferredtoasill‐conditionedwhensmallperturbationstothecoefficientsresultinanunpredictablylargechangeofthesolution.Certainconditionssuchaspositivedefinitenesscanbechecked,andcorrectionstoanindefinitematrixcanbeapplied. Accumulatingfloatingpointround‐offandalsocancellationerrorscausesearchvectorstoloseorthogonalityinconjugatesearchmethods.Similarly,steepestdescentmethodscanaccumulatefloatingpointround‐off,whichmaycausexktoconvergetosomepointnearx.Thiseffectcanbedeterredbyoccasionallyrecomputingthecorrectresidual.

Scaling.Numericalstabilitystartswithscalingtheproblem.Scalingisparttheoryandpartartform.Properconsistentscalingmeansprimal,dualsolutionvalues,constantsandderivativesofnonlinearterms Jacobianelements inabsolutevaluebearound1,e.g.from0.01to100.Mostsolvershaveautomaticscalingoptions.FactoringLinearEquations.Inpractice,ateachiterationdk Bkf xk issolvedbysparsematrixfactorizationtechniquesoftenamatrixisrepresentedastheproductorsumofmatrices;sucharepresentationistermedasadecompositionorfactorizationoftheoriginalmatrix.TheGaussianeliminationproductformoftheinverseofAstorestheinverseasaproductofelementarycolumnmatrices,where

A 1 EkEk 1 E2E1.

andEisanelementarycolumnoperationwhereallbutonecolumnistheidentitymatrix.Thereforewehavethat

x A 1b EkEk 1 E2E1b.

Foriterativeprocedures,askbecomelarge,numericalerrorsbuild‐upandexcessivefloating‐pointoperationsoccur.Thereforethematrixneedstoberefactoredintok nfactors.

AnothertechniqueisLUfactorization,whereLisalowertriangularmatrixandUisanuppertriangularmatrix.ForanymatrixA,

Ax LUx L Ux b

WecansolveLy bbysubstitutionandthenUx ybysubstitution.Foradditionaldetails,seeMarkowitzLUdecomposition Reid,1976;1982 andBartels‐Golubupdate BartelsandGolub,1969;1971 .FactoringLinearEquationswithSymmetricMatrices.Symmetricmatriceshavemoreefficientfactorizations.TheCholeskydecompositionofaHermitian,positive‐definitematrix,A,istheproductofalowertriangularmatrixanditsconjugatetranspose.A LL*whereL*istheconjugatetranspose.CholeskydecompositionisroughlytwiceasefficientastheLUdecompositionforsolvingsystemsoflinearequations.WithQRfactorization,AresultsinanorthogonalmatrixQthatisabasisforthecolumnspaceofAandanuppertriangularmatrixRwhereA QR.TheseapproacheslosecertaininvariancepropertiesassociatedwithNewton’smethodandNewton’smethodwithlinesearch,andthereforemaynottransformcorrectly,whentheHessianisill‐conditioned.4.ConstrainedNonlinearOptimizationTheconstrainednonlinearproblemorprogramincanonicalprimal P formis:

f* inf f x |g x 0,h x 0,xX P

Thisformulationexplicitlyintroducesanadditionalsetofinequality,g x 0,andequalityconstraints,h x 0,thatarelinearornonlinear.IfxX,h x 0andg x 0,xissaidtobeafeasiblesolution.Nonlinearequalityconstraintscanbe

representedasg x 0and‐g x 0.Wedefinec x 0as g x s 0,h x 0 .LagrangianFunction.ThealgorithmsforsolvingconstrainednonlinearoptimizationproblemstypicallystartwithatransformationoftheLagrangian.TheLagrangiandual D formulationis

L* supλ 0 infL x,λ |xX,h x 0

whereL x,λ f x λTg x istheLagrangianfunctionandλistheLagrangianmultipliervector.Thevariableλissometimescalledthedualorslackvariable.Letx*,λ* bethesolutionto D .IfL x,λ isunboundedforxX,thenL* ∞.Furthermore,ifthefeasibleregionto D isanemptyset,thenL* ‐∞. IterativemethodssuchaspenaltyandaugmentedLagrangianmethods,barrierorinteriorpointmethods,variablemetricmethods,sequentiallinearprogrammingmethodsandsequentialquadraticprogrammingmethodsoftensolveasequenceofgeneralizedLagrangianfunctions.ThegeneralizedLagrangianfunctionis:

L x,λ f x Λ g x ,λ,μ

thatincludesapositivepenaltyparameterμ 0withapenaltyfunctionΛ.Karush‐Kuhn‐Tucker KKT Conditions.Iff x andg x aredifferentiabletheKKTnecessary first‐order conditionsare

f x λTg x 0, λ 0, λTg x 0,and h x 0.

Undercertainconstraintqualifications,forexample,ifthereisg x 0andh x 0,or g x ,h x hasfullrankwheretheactivesetASdenotesanactiveconstraint,theKKTarenecessaryforalocaloptimum.IfxisalocaloptimalsolutiontoPandalsosatisfiestheKKTsufficiencyconditionswithstrictcomplementaritytoλ,then x,λ X Rm isasaddlepointoftheLagrangian:

L x,λ L x,λ L x,λ

forall x,λ X Rm .Therefore,thexandλareoptimalsolutionstotheprimalanddualproblems,respectivelyand

inf f x |g x 0,h x 0,xX supλ 0inf L x,λ |h x 0,xX .

ThispropertyisknownasweakdualitywhenxisfeasibleinPandλisfeasibleinD.f*‐L* 0iscalledtheduality complementarity gap.Strongdualityresultsinazerodualitygap:

f*‐L* 0.

Furthermore,ifxisfeasibleandthereexistsλsuchthatstrongdualityholdsundercertainconstraintqualificationswherexisalocaloptima.CertainstabilitypropertiesleadtonecessaryandsufficientconditionswhichholdwithequalityfortheexistenceofaglobalsaddlepointoftheLagrangianfunctionwithrespecttoX Rm .Ifg x andf x areconvexfunctions,thentheKKTaresufficientforglobaloptimality.Iff x andg x aretwicedifferentiable,andtheHessian

H x,λ 2L x,λ 2f x λT2g x

ispositivedefinite,thenwehavealocalminima,orifH x,λ isnegativedefinite,thenwehavealocalmaxima.Theresultisanalogoustostrongduality. OftentheKKTmaynotholdattheoptimalsolutioniftheproblemisnonconvex,andthereforesuchKKTnecessaryconditionsarenotsufficienttoproveglobaloptimality.Whentheprimalproblemisnonconvex,theremaybelocaloptimathatarenotgloballyoptimal. Thefollowingsubsectionsdetailthedominantapproaches,whicharebaseddirectlyonsolvingtheKKTconditionsforconstrainedoptimization.AugmentedLagrangianandPenaltyMethods.In1969inseparatepapers,HestenesandPowellintroducedtheaugmentedLagrangian.Inthe1970s,itgainedastrongfollowingandisstillappliedtoday.TheaugmentedLagrangianistheLagrangianfunctionwithaquadraticpenaltyterm:

Λ g x ,λ,μ λTg x μTg x 2

ThebasicalgorithmfortheaugmentedLagrangianandgeneralpenaltyfunctionmethods:

Step1.Chooseλ0,μ0initerationk 0.

Step2.Findxk 1 argminx f xk λkTg xk μkTg xk .

Step3.Updateμk 1 μkandλk 1 λk μkTg xk .

Step4.Ifconvergencecriteriaismet,thenstop;elsegotoStep2.

Barrier/Interior‐PointMethods.Barriermethodsrequireaninteriorpointx0,whereg x0 0,andconstructasequenceofunconstrainedproblemswithμ 0

controllingtheweightofthebarriers.Barriertermscannotbedefinedforequalityconstraintsbecauseμincreasestendstozeroasthealgorithmapproachestheboundarytothefeasibleregion.Forexample,themostpopularbarrierfunctionsareoftheform

Λ g x ,λ,μ μTlog g x andΛ g x ,λ,μ 1/μTg x .

OriginallyproposedbyFrisch 1955 andlaterdevelopedbyFiaccoandMcCormick 1968 ,thelogarithmicbarriermethodwasappliedextensivelyinthe1960sand1970sandhasbecomefoundationalfornonlinearinterior‐pointmethods.In1984,interiorpointmethodsgainedmomentumwithKarmarkar’spolynomial‐timeLP.inthelog‐barrierapproach,stepsaregeneratedbyiterativelysolvinganunconstrainedoptimizationproblemsimilartotheform:

min f x μkTlog g x

where

μk 0,and

∂ μkTlog gi xk /∂xk μkTgi xk /gi xk .

Ask→∞,wehaveμk→0andμk/gi xk →λiforalli. Sincethelogarithmofasmallnumberisnegative,the μklog gi xk termispositiveforanyvariablenearthebound;thiseffectivelycreatesbarrierspushingthevariablesintotheinteriorofthefeasibleregion.Barriermethodssufferfromill‐conditioningintheHessianastheoptimumisapproached.Moregenerally,interiorpointmethodsareoftensensitivetotheinitializationandreductionofthebarrierorcentering parameterμ.Suchbarriermethodsarealsosensitivetothestepsizewhenaglobalconvergencestrategyisemployed.Howeverapplyingfactorizationandglobalconvergencestrategiesspecificallydesignedforthegivenbarrierfunctioncanaddresssomeoftheabovecomplications.

In1992,Polyakintroducedamodifiedbarriermethodthatscalesthebarrierfunctionsothatinitialinfeasibilityisnotaproblem.Alsoin1992,Mehrortaintroducedapredictor‐correctormechanismtopromoteconvergencebyreducingthenumberofmatrixfactorizationswhendeterminingsearchdirections.ThemethodusestheCholeskydecompositiontofindtwodifferentdirections:apredictorandacorrector.Thepredictorcomputesanoptimizingsearchdirectionbasedonafirstorderterm.ThecorrectorstepusesthesameCholeskydecompositionfoundduringthepredictorstep.Thesearchdirectionisthesumofthepredictordirectionandthecorrectordirection.Bydeterminingthecenteringparameteradaptively ratherthanpriortotheaffine‐scalingdirection andenablingcorrectionsinthepredicteddirection,Mehrotra’sapproachintroducesextracomputationperiterationbutsignificantlyreducesthenumberofiterations

required.In1996,GondziointroducedmultiplecentralitycorrectionsstrategythatmaximizesthepossiblestepsizeincreaseinMehrotra’sapproachbycomputinguptoseveralcorrections,whichleadstoanevenlargerdecreaseintheresiduals.Mostinteriorpointapproacheshaveleveragedthepredictor‐correctormethod.Althoughthereisnotheoreticalcomplexitybound,themethodiswidelyusedinpractice.Itappearstobecomputationallyefficientandtoconvergeveryfastwhenclosetotheoptimum.

Primal‐DualMethods.Primal‐dualapproachesapply quasi‐ Newton’smethod,andsimilartootheriterativemethodswe’vecoveredthusfar,includesasearchstepanddirection,andaconvergencecriteria.Considertheprimalprogram

minx,yf x subjectto g x y 0 y 0

withtheLagrangian

L x,s,λ,μ f x λT g x y μTy.

Thefirstorderconditionsare

f x λTg x 0,

μ ν 0,

g x y 0,

y 0,

μ 0,

νiyi–1/t 0,and

νy 0

whereiselement‐by‐elementmultiplicationandt→0.Theoptimizationproblemis

minνy subjectto f x λTg x 0 μ ν 0 g x y 0 y 0 μ 0.

Therefore,x,s,λ,μandtareupdatedaftereachNewtonstep.Primal‐dualinteriorpointmethodsallowforprimalanddualiteratestobeinfeasible,e.g.g x s 0.SequentialLinearProgramming SLP Methods.TheSLPmethoddeterminesthesolutiontoanonlinearproblemthroughsolvingasequenceoflinearapproximations,mostlyusingfirst‐orderTaylorseriesexpansions.Foreachiterationk,wesolvethelinearprogram

xk 1 argmin f xk Td|g xk Td g xk 0,dl d du

wheref xk canbereplacedwithlinearizedgeneralizedLagrangians.Sincethelinearizationsarenotnecessarilybounded,trustregionswherexk 1єTRkcanbeusedtoensureconvergence.Furthermore,thisapproachmaybeusedinconjunctionwithapenaltyormeritfunctionandsteprestrictions.Iftheproblemisconvex,thelinearconstraintsarealwaysoutsidethefeasibleregiontothenonlinearproblem,butareguaranteedtoachieveconvergenceask→0. TheSLPcanhandleverylargeproblems,benefitsfromusingLPsolvers,butmayconvergeslowlyandmayviolatenonlinearconstraints.SLPmethodsaresuitableforsolvinglarge‐scalenonlinearprogrammingproblems,sincelargelinearprogramscanbesolvedefficiently.Becauseofthelinearityoftheapproximation,specialscarcitypatternsoftheJacobianmatrixg xk arepassedtotheconstraintmatrixofthelinearprogramdirectly.Aparticularadvantageofsequentiallinearprogrammingmethodsisthatalthoughsecondorderinformationisnotused,convergenceislineareveninthecaseofhighlynonconvexproblems.SequentialQuadraticProgramming SQP Methods.ThismethodisalsoreferredtoasaprojectedLagrangianorLagrange‐NewtonapproachbecausetheaccompanyingsubproblemminimizesaquadraticapproximationtotheLagrangefunctionsubjecttoalinearizedconstraintset. SQPmethodssolveasequenceofquadraticsubproblemstodeterminetheactive‐setandasearchdirectiondasthesolutionforeachiteration.Thekeyideaistoapproximatesecondorderinformationtogetafastfinalconvergencespeed.Thus,wedefineaquadraticapproximationofthegeneralizedLagrangianfunctionL x,λ,μ andanapproximationoftheHessianmatrixbyaquasi‐NewtonNewtonwhereBk2xL xk,λk,μk .Thenweobtainthesubproblem:

mindkTBkdk/2 f xk Tdk subjectto g xk Tdk g xk 0 dkTRk

TheKKTconditionsareappliedtotheobjectiveofthesubproblemtosolvefordk.Ameritfunctionistypicallythesumoftheobjectivefunctionandtheamountofinfeasibilityoftheconstraints. ASQPusuallyrequiresfewerfunctionsandgradientevaluationsthanaSLP,butmayviolatenonlinearconstraintsandishardertosolvethanaSLP.AnotherSQPbasedapproach,thesequentiallinear‐quadraticprogram SLQP ,decouplestheactive‐setidentificationandthestepcomputationintheQPsubproblem.SLQPomitsthequadratictermintheobjectivefunctionandsolvesthesubproblemforatrustregion.Withonlytheactive‐setofconstraints,asubsequentlinearprogramissolvedtodeterminedk.GeneralizedReducedGradientMethods.In1961,Rosenintroducedagradientprojectionmethod.In1969,AbadieandCarpentierintroducedthegeneralizedreducedgradient GRG fornonlinearconstraintscombiningquasi‐NewtonmethodsandWolfe’sreducedgradientmethod.TheactualimplementationhasmanymodificationstomakeitefficientforlargemodelsseeinDrud 1985and1992 .ThereareseveralGRGvariationsincludingchoicesofLagrangiansandlinesearches.Westartwiththeproblem:

minf x subjecttog x 00 x xu

Atmajoriterationk,wesolvethefollowingnonlinearprogram:

GRGk:minF x,λk,μk subjecttogk g xk g xk T x‐xk 0 dualvariableλk 1

where

F x,λk,μk f x –λkT g x –gk μk g x –gk T g x –gk ,and

μkisascalarpenaltyparameter.

Forthesetofbasic dependent variablesB,thesetofsuperbasic independent variablesS,andthesetofnonbasic dependent,fixedatabound variablesNwecanpartitiong xk as

g xk B,S,N

whereBandSarenonsingular,andwecanpartitionxkas

xB xk B,

xS xk S,and

xN xk N

where

BTxB STxS NTxN 0.

Atasolution,thebasicandsuperbasicvariablesarebetweentheirbounds,0 xB xuBand0 xS xuSwhilenonbasicvariableswillnormallybeequaltotheirlowerorupperbound.Atasolution,Swillbenomorethanthenumberofnonlinearvariablesandxsisregardedasasetofindependentvariablesthatareallowedtomoveinadirectionthatwilldecreasetheobjectivefunctionvalue.Thebasicvariablesareeasilyadjustedtosatisfythelinearconstraints.

IfnoimprovementcanbemadewiththecurrentdefinitionofB,SandN,someofthenonbasicvariablesareselectedtobeaddedtoS,andtheprocessisrepeatedwithanincreasedvalueofS.Ifabasicorsuperbasicvariableencountersoneofitsbounds,thevariableismadenonbasicandthevalueofSisreducedbyone.Astepofthereduced‐gradientmethodiscalledaminoriteration.Forlinearproblems,simplexmethodisthereduced‐gradientmethodwiththenumberofsuperbasicvariableoscillatingbetween0and1.ThestepsinaGRGalgorithmare:

Step1.Setk 0andstartwithafeasiblesolution,x0.

Step2.Computeg xk J B,S,N ,f xk ,andthereducedgradient,gr.Selectthesetofsuperbasicvariables,xs,asasubsetofthenonbasicvariablesthatcanbegainfullychanged.Forthesuperbasicvariables

gr xS f xk T JTλk

andforthebasicvariables

gr xB 0 BTλ ∂f/∂xb .

Step3.Findasearchdirection,ds,forthesuperbasicvariablesthatisbasedongrandpossiblysomesecondorderinformation.ThereducedHessianoff x maybeapproximatedbysolvingasystemoftheform

RTRq Zf x 0

where

Z ‐B‐1S,I,0 ,

Zf x isthereducedgradient,and

RTR ZTHZ.

ThedenseuppertriangularmatrixRisupdatedinvariousapproachesinorderto

approximate2f x .Therefore,ds Zqwheregrisappliedtosolvethelinesearchmin0 α βF x αds,λk,μk .

Step4.SolvethesubproblemGRGk;let xk 1,λk 1μk 1 argmin GRGk .

Step5.Ifgrprojectedontheboundsissmallerthantheconvergencecriteria,thenstop;ifnot,incrementk k 1gotoStep2.

Generalizedreducedgradientmethodscanbeextendedeasilytoverylargeproblemsandproblemswithspecialstructure.GRGisprobablymorerobustthanSLPandSQPmethods,andonceafeasiblesolutionisfoundinGRGmethods,itremainsfeasible.Moreover,GRGapproachesarerelatedtoSQPmethodsandthereforethereexistscombinationsofbothapproaches.ConicandSemidefiniteProgrammingMethods.Thesemethodsareformsofquadraticprogramming,andaresimilartolinearprogrammingbecausetheyareconvexprograms.Forlinearprogram

minxcTxsubjectto

Ax=b x≥0

theequivalentconicprogramis

minxcTxsubjectto

||Aix+bi||2≤ciTx+di,i 1,…,m

andtheequivalentsemidefiniteprogramisoftheform

minxcTxsubjectto

Geometrically,linear,conic,andsemidefiniteprogramsareoftheform

minxcTxsubjectto

Ax+bK

ciTx di I Aix bi≻ 0,i 1,…, m

Aix bi T ciTx di I

whereKisapointed1convexconewithanonemptyinterior.Typicallythesemethodsareappliedtoavoidlocalminimabyrelaxingtheobjectivefunctionanddomaintobeconvex.5.DecompositionTechniques Decompositionmethodsareappliedfortwopurposes,tofitthedecomposedproblemintohigh‐speedmemoryandtodecomposetheproblemintoeasiertosolvesubproblems.Decompositionmethodsusuallyemployeithercolumngenerationorcuttingplanes,whereoneapproachisthedualoftheother. Bender’sdecompositionisacuttingplanemethodwheretheprimalnonlinearproblemis

f* minx,yf x cTysubjectto dualvariableg x Ay b λ

anditsdual

L* maxλminx,yf x cTy λ g x Ay b .

Theproblemcanbepartitionedwhere,givenx,wecansolveforvariableyusingthefollowinginnerminimizationoftheprimal

minycTysubjecttoAy b g x

andmaximizationofthedual

maxλ b‐g x TλsubjecttoλA cT.

Theaboveinnerminimizationisalinearprogramsuchthat

λk argmaxλ b‐g xk Tλ|λA b,λ 0 ,f* minimizexf x y0,andy0 b‐g x Tλkk 0,1,…

fork 0,1,…K.Bender’sAlgorithm:

Step1.Setk 0;chooseafeasiblesolutionx0.

1 K is pointed if it does not contain any subspace except the origin.

Step2.Solveλk 1 argminλ b‐g xk Tλ|λA cT .

Step3.Solvexk 1 argminx f xk y0|y0 b‐g xk Tλk wheretheinnerminimizationsolvesfory0.

Step4.Ifxk 1satisfiestheconvergencecriteria,stop;otherwise,setk k 1andgotoStep2.

Dantzig‐Wolfesolvesasimplerorsmallersubproblemthatgeneratesacolumnforthemasterproblem:

minx,yf x cTy

subjectto dualvariable

g x Ay b. λ

yєY

ThereforetheDantzig‐Wolfealgorithmisasfollows:

Step1.Setk 0;chooseafeasiblesolution x0,y0 whereg x0 Ay0 b.

Step2.Generateacolumnandsolvethemodifiedmasterproblemforyanduk:

minuk,y∑kf xk uk cTy

subjectto dualvariable

∑kg xk uk Ay λk

∑kuk 1 βk

uk 0

yєY

Step3.Solvethesubproblemxk 1 argminx f x λkg x .

Step4.Ifxk ∑kxkuksatisfiestheconvergencecriteria,stop;elsek k 1andgotoStep2.

Equivalenttobothsteps2and3,thedualcanbesolvedbygeneratingacuttingplane:

maxλk,yλkb βk subjectto λkA cT λkg xk βk f xk λk 0.

TheninStep4,Iff x andg x areconvex,∑kf xk uk ∑kf xk and∑kg xk uk ∑kg xk .Furthermore,ifu0*,…uk*andyk*isafeasiblesolutionthenxkisafeasiblesolutionandask→∞,wehavethatxkandyk*isanoptimaltotheNLP.ApproximationsoftheACOPF.TheACOPFcanbeformulatedinseveralways seeCainetal.2012 .ThecanonicalACOPFinpolarcoordinatesis:

min∑ncn pn cn qn subjecttopn pdn ∑mvnvm gnmcosθnm bnmsinθnm qn qdn ∑mvnvm gnmsinθnm bnmcosθnm pnmin pn pnmax qnmin qn qnmax vnmin vn vnmaxθminnm θnm θn θm θmaxnm

PleaserefertoCainetal. 2012 fornotationdefinitionsandtheequivalentACOPFformulationinrectangularcoordinates.Fortherealandreactivepowerflowequations,thefirstpartialderivativesonthepowerinjectionspnandqnare:

∂pn/∂vm vn gnmcosθnm bnmsinθnm n m

∂pn/∂θm vnvm gnmsinθnm bnmcosθnm n m

∂pn/∂vn ∑m n vm gnmcosθnm bnmsinθnm 2vngnn

∂pn/∂θn ∑m n vnvm gnmsinθnm bnmcosθnm vn2bnn

∂qn/∂vm vn gnmsinθnm bnmcosθnm n m

∂qn/∂θm vnvm gnmcosθnm bnmsinθnm n m

∂qn/∂vn ∑m n vm gnmsinθnm bnmcosθnm 2vnbnn

∂qn/∂θn ∑m n vnvm gnmcosθnm bnmsinθnm vn2gnn

Thefirstpartialderivativesareusedinapproximationmethods.Notethatinthesederivativesthemagnitudeofthecosineandsinetermsaregenerallyscaledbythemultipleoftheconductanceorsusceptance.Sincereactanceofatransmissionlineismuchlargerthanresistance,theconductance realpartofadmittancematrix,B ismuchsmallerthanthesusceptance imaginarypartofadmittancematrix,G .Inmostcasesthesinetermstendtobesmallwhereasthecosinetermisnearunity.Uponinspection,theresultofthesephysicalpropertiesresultsinrealpowerthatishighlysensitivetochangesinvoltageangleandreactivepowerthatishighlysensitivetochangesinvoltagemagnitude.

DecoupledPowerFlowModel.InitiallyproposedbyStottandAlsacin1974,theDecoupledPowerFlow DPF isbasedontheprinciplethatwhentheJacobianofthepowerflowequations

∂p/∂θ ∂p/∂v

∂q/∂θ ∂q/∂v

isevaluatednumerically,theoff‐diagonalsubmatricesaremuchsmallerinmagnitudethanthediagonalsubmatrices:

∂p/∂θ ∂q/∂θand ∂q/∂v ∂p/∂v.

Wethereforesettheoff‐diagonalentriesoftheJacobiantozero

∂q/∂θ 0and ∂p/∂v 0,

andthendecomposetheproblemintoapairofsubproblemswherethep‐θrealpowermodelminimizesthesystemcostsandtheq‐vreactivepowermodelminimizestherealpowertransmissionlosses.Inthisapproachweassumethatθnm0,sinθnm θnm,cosθnm 1,andgnm bnm.Let

b’nm ∂pn/∂θm vnvmbnmn m, b’nn 0, b”nm ∂qn/∂vm vnbnmn m,and b”nn 2vnbnn

whereb’isanapproximationofthematrixofpartialderivativesoftherealpowerflowequationswithrespecttothebusvoltagephaseanglesandb’’isanapproximationofthematrixofpartialderivativesofthereactivepowerflowequationswithrespecttothebusvoltagemagnitudes.TheDecoupledPowerFlowmodelis:

min∑ncn pn cn qn subjectto pn‐pdn ∑mpnm spn qn‐qdn ∑mqnm sqn pminnm pnm b’nmθmn pmaxnm qminnm qnm b”nmvm qmaxnm pminn pn pmaxn qminn qn qmaxn vminn vn vmaxn θminnm θnm θn‐θm θmaxnm

where

pn ∑mvnvmbnmθnm qn ∑m‐vnvmbnm.

TheDecoupledPowerFlowalgorithm:

Step1.Setk 0,andinitializevnk 1.

Step2.SolveDPFforpnk,θnk,qnk,andvnk.

Step3.Ifpnk,θnk,qnk,andvnksatisfytheconvergencecriteria,e.g.,acfeasibility,thenstop.Ifnot,setb’nm vnkvmkbnmandb”nm vnkbnmandk k 1,andgotoStep2.

Sincetheunderlyingproblemisnonconvex,thereisnoguaranteeofconvergenceintheoriginalACOPFproblem.BθModel.Afurthersimplificationdropsreactivepowercompletely.Ifgij 0,sinθ θ,cosθ 1,andvm 1,theBthetamodelssolves:

min∑ncn pn subjectto pn pdn ∑mpnm pminnm pnm bnmθmn pmaxnmpminn pn pmaxn θminnm θnm θn θm θmaxnm

DistributionFactorModel.Afurthersimplificationisthedistributionfactor DF modelwherealltransactionsaredecomposedintoa‘sale’toareferencenodeand‘purchase’fromareferencenode. min∑ncnpn subjectto ∑npn ∑npdn 0 pminn pn pmaxn pnmk ∑ndfkn pn pdn pmaxnmk

Sincewehaveassumedlossesarezero,superpositioningmakesDFroughlyequivalenttotheBθformulationifθmaxnmisequivalenttopmaxnmk.Thedifferenceisthesizeoftheproblem;whereasBθhasN 2Kconstraints,thedistributionfactormodelhas1 KMconstraintswhereMisthesubsetoflinesmonitoredforbindingthermallimits.

6.CommercialSolvers

Commercialoptimizationsolversfornonlinearproblemsvaryinalgorithmictechniquesandimplementation.Wesurveyfivewell‐knownnonlinearsolvers:MINOS,IPOPT,SNOPT,KNITRO,andCONOPT.Pleasereferto Castillo,2013 foracomputationalstudyontheperformanceofthesesolversappliedtotheACOPF.MINOS.MINOSisaGRGmethodandisdesignedtosolvelarge‐scaleoptimizationproblems MurtaghandSaunders,1982and2003 .MINOSpartitionsvariablesintolinearandnonlinearelementsandtheniterativelysolvesthesubproblemswithlinearizedconstraintsandanaugmentedLagrangianobjectivefunction Robinson,1972 .Insteadofanonlinearconjugate‐gradientapproach,aquasi‐Newtonapproachisappliedtocertainsubspaces.SparseLUbasisfactorsaremaintainedbyLUSOL Gilletat,1987 .Thenonlinearconstraintsmaybesatisfiedonlyinthelimit,sothatfeasibilityandoptimalitymayoccursimultaneously.Animportantfeatureisastableimplementationofaquasi‐Newtonalgorithmforoptimizingthesuperbasicvariables.IPOPT InteriorPointOPTimizer .IPOPTconvertstheproblemtoabarrierproblem.Ideallythefunctionsf x andg x aretwicecontinuouslydifferentiable.IPOPTuseslinefilteredsearchesandincludesafeasibilityrestorationphase.Filtermethodspromoteglobalconvergencethroughmeasuringdecreasesintheobjectivefunctionandinfeasibilityastwoseparatecriteriathatarecontrolledsimultaneously.IPOPThasoptionsinlinesearchstrategiesforglobalization,includinganexactpenaltymeritfunction,augmentedLagrangianmeritfunction,filtermethod R.Fletcher,S.Leyffer,andP.Toint ,HessianandseveralHessianapproximationmethods.SNOPT SparseNonlinearOPTimizer .SNOPTimplementsasparseactive‐setsequentialquadraticprogramming SQP methodthatemploysquasi‐NewtonapproximationstodeterminetheHessianinthequadraticprogrammingsubproblem;thenanaugmentedLagrangianmeritfunctionguidesthelinesearchdirection.SparsebasisfactorsaremaintainedbyLUSOL.Ifonlytheobjectiveisnonlinear,theproblemislinearlyconstrained LC andtendstosolvemoreeasilythanthegeneralcasewithnonlinearconstraints NC .SNOPTuseslimited‐memoryquasi‐NewtonapproximationstotheHessianoftheLagrangian.Themeritfunctionforstep‐lengthcontrolisanaugmentedLagrangian,asinthedenseSQPsolverNPSOL Gilletat,1992 .Ingeneral,SNOPTrequireslessmatrixcomputationthanNPSOLandfewerevaluationsofthefunctionsthanthenonlinearalgorithmsinMINOS.Itismostefficientifonlysomeofthevariablesenternonlinearly,ortherearerelativelyfewdegreesoffreedomatasolution i.e.,manyconstraintsareactive .

KNITRO.KNITROimplementsbothinterior‐pointandactive‐setmethodsforsolvingnonlinearoptimizationproblems.Thevariablescanbecontinuous,binary,orinteger.Inthebarrier/interiormethod,KNITROsolvesaseriesofbarriersub‐problemscontrolledbyabarrierparameter.Thealgorithmusestrustregionsandameritfunctiontopromoteconvergence.Thealgorithmperformsoneormoreminimizationstepsoneachbarrierproblem,thendecreasesthebarrierparameter,andrepeatstheprocessuntiltheoriginalproblemhasbeensolvedtothedesiredaccuracy.KNITROprovidestwoproceduresforcomputingthesteps.Oneversionwhereeachstepiscomputedusingaprojectedconjugategradientiteration.Thisapproachfactorsaprojectionmatrixtoapproximatelyminimizeaquadraticmodelofthebarrierproblem.Theotherprocedurealwaysattemptstocomputeanewiteratebysolvingtheprimal‐dualKKTmatrixusingdirectlinearalgebra,butifthestepqualitycannotbeguaranteedorifnegativecurvatureisdetected,thenthenewiterateiscomputedbythefirstprocedure.KNITROalsoimplementsanactive‐setsequentiallinear‐quadraticprogramming SLQP algorithmthatuseslinearprogrammingsub‐problemstoestimatetheactive‐setateachiteration.KNITROincludesanactive‐setSLQPwithtrustregiontopromoteconvergence.CONOPT CONstrainedOPTimization .CONOPTisageneralizedreducedgradientGRG methodthatsearchesalongthesteepestdescentdirectioninthesuperbasicvariables.Thegeneralizedprojectionmethodprojectsthesearchdirectionintothesubspacetangenttotheactive‐setofconstraints.Theactive‐setisthesubsetofequalityandinequalityconstraintsatapointthatsatisfywithequality;forexample,equalityconstraintsareactiveatallfeasiblepoints.Therefore,thesearchdirectionisprojectedintothenullspaceofthegradientsfortheequalityandbindinginequalityconstraints.Theprojectedgradientcouldbeinfeasible,whichthenrequiresacorrectionstep.7.HistoryofACOPFSolutionTechniques In1962,CarpentierintroducedthealternatingcurrentoptimalpowerflowACOPF foreconomicdispatchbaseduponKarush‐Kuhn‐Tucker KKT conditions.CarpentieremployedtheGauss‐Seidelmethodrepresentingtheloadflowsaspowerinjectionsinthevoltagepolarform.Carpentierincludedoperationalconstraintsonrealpowercontrol,generatorbusvoltagemagnitudelimits,reactivepowercontrolofswitchableVARsources,andtransformertapsetting.Thisformulationhasnonconvexequalityconstraintswithquadraticandtrigonmetricfunctions seeCainetal,2012 .Thecomplexvariablescanbeexpressedinpolarorrectangularcoordinatesandresultsindifferenttypesofnonconvexconstraints.SincetheACOPFproblemisanonconvexmathematicalprogram,theKKTconditionsyield

onlyalocaloptimalsolution;evenfeasibilitycannotbeguaranteedbymostnonlinearprogramming NLP algorithms. In1969,CarpentierandAbadiepublishedthegeneralizedreducedgradientmethod,whichisageneralizationofCarpentier’s‘differentialinjections’methodthatwasoriginallyconceivedin1964tosolvetheoptimalpowerflowproblemCarpentier,1979 .SinceCarpentier’scontributions,therehasbeenawealthofresearchdoneonalgorithmicmethodstosolvetheACOPF.In1968,DommelandTinneypresentedanapproachbasedonpowerflowsolutionbyNewton’smethodwithagradientadjustmentfactorforthepenaltyfunctiontoaccountfordependentconstraints.In1969,SassonusedtheFletcher‐Powellmethod;thenin1973,SassondirectlycomputedtheHessianandutilizeditssparsity.In1973,AlsacandStottincorporatedexactoutage‐contingencyconstraintsandusedanaugmentedLagrangian.In1977,BiggsandLaughtonusedarecursiveequalityquadraticprogram.Intheabovestudies,testproblemswithlessthan30‐buseswereused.DuringthistimetheadvancementsofgradienttechniqueswithNewton’smethodwerewidelyadoptedintheearlyACOPFalgorithms,seeHapp 1977 .In1979,Wuetal.usedamodifiedreducedgradientwithpenaltyfunctionmethodsandincludedthelargesttestcasepublishedto‐date,a1410‐bussystem. Inthefollowingtablewesummarizethepublishedoptimalpowerflowstudiesbythetestproblems,convergencecriteria,initialization,andperformancemetricsreported.ThestudiesvaryinallthemetricslistedinTable1,andveryfewofthestudiesincludedacomparisonagainstcommercialsolversorpreviouslypublisheddata.Forexampleoutofthestudieslisted,Burchettetal. 1982 ,AokiandKanezashi 1985 ,HuneaultandGaliana 1990 ,PonrajahandGaliana 1990 ,MomohandZhu 1999 ,andWangetal.employedMINOSintheiralgorithmicimplementationorcomparedcomputationalresultstothatofMINOS. TheabbreviationsinTable1areasfollows:N/A NotApplicable ,NR NotReported ,FS FlatStart ,LFS LoadFlowStart ,RS RandomStart ,HS HotStart ,WS WarmStart ,US UserSpecifiedStart ,C.G. ComplementarityGap ,Opt.OptimalityTolerance ,andFeas. FeasibilityTolerance .NotefortheSDPmethodsinBai 2008 andLavaei 2012 ,initializationisnotrequired. ThestudieswithUS UserSpecified listedastheinitializationmethodemployedanon‐random,selectiveapproach.NumerousstudiesreportedadditionalmetricstothoselistedinTable1.However,weidentifythishandfulofmetricsasbasicinformationneededforfairtestingandcomparisonofproposedalgorithmsbyotherresearchers. ThefollowingTable2summarizesthetestproblemspublishedoninACOPFstudiesto‐date.Numerousofthetestproblemshaveanetworkoriginthatisnotdescribedwell.Althoughmanyofthetestproblemsareusedacrossmultiplestudies,asdenotedinthe“SourceData”columnasmallpercentageofthetest

problemsarereadilyavailableinthepublicdomaintouse;thesetestproblemsarepostedonlinebytheUniversityofWashingtonElectricalEngineeringorbyMATPOWER.Therawdatafortestproblems23and39areavailableinBiggs 1977 andPai 1989 ,respectively.Welistthereferencingstudyifthereaderisinterestedinfurtherinformationonthetestproblemortocontacttheauthorforthesourcedata.

Table1.ACOPFStudiesthatincludeNumericalAnalysisYear Author TestProblems Convergence Initial‐ Work‐ Code CPU Iter‐

No. ofNodes Criteria ization station Lang. Time ations1969 Sasson 30 10^‐5Opt. NR x x x x1973 Alsac 30 10^‐3Opt. US x1974 Mukherjee 25 NR US x x x1977 Biggs 23 NR US x1979 Wu 11,136,333,1410 NR LFS x x x1982 Burchett 118,597 NR NR x x x1982 Divi 9,10,11 10^‐5Feas. FS x x1982 Shoults 5,30,962 NR US x x x x1984 Burchett 350,1100,1600,1900 NR RS x x x1984 Sun 912 NR US x x1985 Aoki 14,135 NR NR x x x x1988 Santos 118,129 NR NR x x x1989 Nanda 14,30,89 10^‐4Feas. LFS x x1989 Ponrajah 6,10,30,118 NR FS x x x x1990 Alsac 1330,1200,700 NR HS x x x1990 Huneault 30,118 NR US x x1990 Salgado 14,30,39,57,89,118 10^‐3Feas. LFS x x x1993 Almeida 14,30,34 NR FS,LFS x x1994 Momoh 9,14,30,118 NR US x x x x

1994 Wu 9,30,39,118,244 10^‐6C.G. FS x x x x1995 Chebbo 706 50^‐5 Opt.,50^‐4Feas. US x x x1997 Lai 30 NR US x x1998 Torres 30,57,118,300 10^‐4Opt. FS,LFS x x1998 Wei 14,30,57,118,344,703,1047 10^‐6Opt. US x x x x

Year Author TestProblems Convergence Initial‐ Work‐ Code CPU Iter‐ No.ofNodes Criteria ization station Lang. Time ations

1999 Momoh 14 10^‐6Opt. US x x1999 Yan 118,1062 10^‐8Opt.,Feas. US x x x2000 Nejdawi 30,57,118,300 NR FS x2000 Torres 30,57,118,300 10^‐4Opt. US x x x x2001 Castronuovo 118,352,750,1704 10^‐6Opt.,Feas. NR x x x2001 Lima 810,2256 10^‐3Feas. US x2001 Torres 118,256,300,555,2098 10^‐4Opt. US x x x2001 Xie 30,57,118 10^‐5C.G.,Feas. WS x2002 Jabr 14,24,30,57,118,175,300 10^‐4Opt. US x x x x2002 Lin 118,244 10^‐3Opt. US x2002 Torres 30,57,118,256,30,555,2098 10^‐4Opt. US x x x x2003 Oliveira 30,118,1564,1732,1993 Sqrtofmachineeps. US x x x x2004 Lin 118,244 10^‐3Opt. US x2005 Min 14,30,57,118,254,300, 662 10^‐3Opt. LFS x x x2005 Tate 2,118,10274 10^‐3Feas. FS x2007 Capitanescu 60,118,300 10^‐6C.G. RS x x x2007 Lin 9,14,30,57,118 10^‐4C.G.,Feas. FS x x x2007 Sousa 30,57,118,300 10^‐4Opt. US x x2007 Wang 30,57,118,300,2383,2935 NR FS x x x x2008 Bai 4,14,57,118,300 10^‐5Opt. N/A x x x2008 Jabr 9,14,30,39,57,118,300,2383 10^‐8Opt. FS,LFS x x x x2008 Lin 9,14,30,57,118,300 10^‐4C.G.,Feas. FS x x x x2009 Bedrinana 2,14,57 NR NR x2009 Chiang 678,2052,2383 10^‐6C.G.,Feas. FS x x x

Year Author TestProblems Convergence Initial‐ Work‐ Code CPU Iter‐ No.ofNodes Criteria ization station Lang. Time ations

2009 Jiang 14,30,39,57,118,300,701,2052,2383,2736,2746

10^‐5C.G.,Feas. FS x x x x

2009 Sousa 30,118,300,2256 10^‐3Feas. NR x x x x2009 Yang 30,118,300 NR NR x x2010 Jiang 5,6,9,14,30,39,57,118,2383 NR FS x x x x2010 Jiangetal. 118,300,678,2052,2383,2746 NR NR x x x x2010 Xie 57,118,300,2052,2790 10^‐6C.G. US x x x x2011 Chung 14,118 10^‐4Feas. FS x x x x2011 Phan 6,9,14,30,39,57,118,300,2746 10^‐5Opt. US x x x2011 Sousa 30,57,118,300,1211 10^‐4Opt. FS,LFS,RS x x2011 Zimmerman 9,30,36,118,300,2383,2736,3120,

2935,21000,42000NR NR x x x

2012 Lavaei 14,30,57,118,300 NR N/A x

Table2.TestProblemsusedinACOPFStudiesTestProblem NetworkOrigin SourceData ReferencingStudy

14 MidwesternUSSystem UWElectricalEng Multiplestudies23 N/A Biggs,1977 Biggs,197724 IEEEReliabilityTest System UWElectricalEng Multiplestudies25 N/A N/A Mukherjee,197430 MidwesternUSSystem UWElectricalEng Multiplestudies34 N/A N/A Almeida,199339 NewEngland US System Pai,1989 Phan,201157 MidwesternUS System UWElectricalEng Multiplestudies60 Nordic32System N/A Capitanescu,200789 N/A N/A Salgado,1990118 MidwesternUS System UWElectricalEng Multiplestudies129 CompanhiaEnergeticade SaoPaulo N/A Santos,1988135 ChugokuElectricPowerCo.,Japan N/A Aoki,1985136 N/A N/A Wu,1979175 N/A N/A Jabr,2002244 N/A N/A Wu,1994 Initialstudy254 N/A N/A Min,2005256 N/A N/A Torres,2001and2002300 MidwesternUS System UWElectricalEng Multiplestudies333 N/A N/A Wu,1979344 JapanSystem N/A Wei,1998350 NortheasternUS Utility N/A Burchett,1984352 South‐SoutheasternBrazilSystem N/A Castronuovo,2001555 N/A N/A Torres,2001and2002597 InterconnectionofSeveralUtilities N/A Burchett,1982662 N/A N/A Min,2005678 N/A N/A Chiang,2009

TestProblem NetworkOrigin SourceData ReferencingStudy

700 InterconnectionofSeveralUtilities N/A Alsac,1990701 UndisclosedRealPowerSystem N/A Jiang,2009703 ChinaSystem N/A Wei,1998706 N/A N/A Chebbo,1995750 South‐SoutheasternBrazilSystem N/A Castronuovo,2001810 South‐SoutheasternBrazilSystem N/A Lima,2001912 NortheasternUS System N/A Sun,1984962 16InterconnectedAreas N/A Shoults,19821047 SimulationPowerSystem N/A Wei,19981062 N/A N/A Yan,19991100 EasternUnitedStatesPool N/A Burchett,19841200 UtilityCompany N/A Alsac,19901211 UndisclosedRealPowerSystem N/A Sousa,20111330 InterconnectionofSeveral Utilities N/A Alsac,19901410 N/A N/A Wu,19791564 South‐SoutheasternBrazilSystem N/A Oliveira,20031600 WesternUnitedStatesUtility N/A Burchett,19841704 South‐SoutheasternBrazilSystem N/A Castronuovo,20011732 South‐SoutheasternBrazilSystem N/A Oliveira,20031900 NortheasternUS Utility N/A Burchett,19841993 South‐SoutheasternBrazilSystem N/A Oliveira,20032052 N/A N/A Multiplestudies2098 ModifiedBrazilSystem N/A Torres,2001and20022256 South‐SoutheasternBrazilSystem N/A Lima,20012383 PolishSystem Zimmerman,2011 Zimmerman,20112736 PolishSystem Zimmerman,2011 Zimmerman,20112746 PolishSystem Zimmerman,2011 Zimmerman,2011

TestProblem NetworkOrigin SourceData ReferencingStudy

2790 UndisclosedRealPowerSystem N/A Xie,20102935 PolishSystem Zimmerman,2011 Zimmerman,20113120 PolishSystem Zimmerman,2011 Zimmerman,201110274 N/A N/A Tate,200521000 N/A N/A Zimmerman,201142000 N/A N/A Zimmerman,2011

Sequentialquadraticprogramming SQP methodsappliedtotheOPFwereeitherlinearprogramming LP ‐basedQPmethodsorQPmethodsbasedontheconceptofanactive‐setoflinearlyindependentconstraints.TheLPapproaches,basedonWolfe’sorBeale’salgorithm,solvealinearprogrambyarevisedsimplextechnique.Exploitingmanyofthestrengthsoflinearprogramming,theseapproachespermittheuseofartificialvariablestoguidefeasibility,parametricprogramming,andeaseofequalityconstraintsintotheformulation.Furthermore,someofthesuccessivequadraticprogramsapplyNewton’smethodtothesubproblem.In1984bycomparison,Burchettetal.presentamethodwheretheQPsubproblemsareequivalenttoasequenceofNewtonstepstotheoptimalsolution. In1982ShoultsandSundecomposedtheproblemintoactiveandreactivesubproblems.In1984,SunappliedNewton’smethodtotheSQPandadvancedsparsitytechniquestothedecomposition,butthemethodexhibitedproblemsininitializationandill‐conditioning.In1986,Contaxisetal.andin1989,Nandaetal.appliedFletcher’smethodtothedecoupledsubproblems. In1981,Girasetal.appliedPowell’squasi‐NewtonmethodtotheACOPF,whichperformsaBroyden‐Fletcher‐Goldfarb‐Shanno BFGS updateandshowsconvergencefrominfeasiblestartingpoints.In1982,Divietal.appliedFletcher’squasi‐Newtonmethodwithashiftedpenaltyfunction,andtheBroyden‐Fletcher‐Shanno BFS updatingformulatonumericallystabilizetheHessianasapositivedefinitematrixandpromoteglobalconvergencebymeansofapenaltyfunctionthatensuressufficientprogressalongtheline‐searchtowardstheactive‐setofconstraints.In1982,Burchettetal.andin1985Aokietal.solveasequenceofsparse,linearlyconstrainedsubproblemsbasedonMINOS.In1988,SantosproposedthedualaugmentedLagrangianmethodthatattemptstoaddressill‐conditioningoftheHessianbyapplyingaquasi‐Newtonmethodtothedualfunction.In1992,Monticellietal.,presentanimprovementuponMariaetal. 1987 inwhichanadaptivepenaltystrategyisusedtoensurepositivedefinitenessoftheHessian,withoutnegativelyaffectingthequadraticconvergencecharacteristicofNewton’smethod. ThesemethodsbuilduponNewton’smethodforunconstrainedoptimization,Lagrange’smethodforoptimizationwithequalities,andFiaccoandMcCormick’sbarriermethod.MeliopoulosandXia 1993 ,Vargasetal. 1992 ,Momohetal.1994 ,MomohandZhu 1999 ,andNejdawietal. 2000 appliedaninteriorpointalgorithm IPM toaLPorQP,wheretheconstraintsarelinearized.Moreover,primal‐dualIPMshavebeensuccessfulinsolvingtheACOPFbyintroducingalogarithmicbarrierfunctioninplaceoftheinequalityconstraints.Wuetal. 1994 ,Weietal. 1998 ,TorresandQuintana 1998 ,YanandQuintana 1999 ,Castronuovoetal. 2001 ,XieandSong 2001 ,Wangetal. 2007 ,XieandChiang

2010and2011 ,Sousaetal. 2011 ,andChungetal. 2011 presentaprimal‐dualinteriorpointmethodtosolvingtheACOPF. In1994,Granvillesolvestheprimal‐dualwithoutMehrotra’spredictor‐correctormechanism,andnotesthatthecontributionofbarriertermsintothediagonaloftheHessianmatrixareveryeffectiveinbringingpositivedefinitenesstotheproblem,thereforemakingauxiliarypenaltyfunctionsunnecessary.In1998,Torresetal.exploittherectangularformulationofthepowerflowconstraints,whichisquadratic,usinganinteriorpointmethod.ThisresultsinaconstantHessianandaTaylorexpansionterminatingatthesecond‐ordertermthatreducesthecomputationalburdenanditerationstoconvergence.Furthermore,Torresetal.1998 perturbtheboundarytodealwithnumericalill‐conditioningthatmayoccurwithbindingconstraints.In1998,Weietal.useaninterior‐pointmethodbasedonapplyingNewton’smethodtothenonlinearsystemofperturbedKKTconditions,whichissimilartotheapproachbyTorresandQuintana 2001 ;thismethodpromotesglobalconvergence.In1998,Weietal.presentadatastructurethatfurtherreducesthenodalblockfill‐inelements.Incomparisontostoringtheaugmentedsystemincompactblocks,asnotedbySunetal. 1984 andWeietal.1998 ,Castronuovoetal. 2001 proposeavectorizationtechniquethatonlyconsidersnonzerotermsinordertodecreasecomputationalcostperoperation.In1999,YanandQuintana 1999 presentadynamicadjustmentofthestepsizeandtolerancetoimproveconvergencespeed. Wuetal. 1994 ,TorresandQuintana 1998 ,YanandQuintana 1999 ,Castronuovoetal. 2001 ,Linetal. 2007 ,Wangetal. 2007 ,Linetal. 2008 ,XieandChiang 2010and2011 ,Sousaetal. 2011 andChungetal. 2011 useMehrotra’spredictor‐corrector.In2009,Sousaetal.presentapredictor‐correctormodifiedbarrierapproach,basedonPolyak’smodifiedbarriermethod,toaddressill‐conditioning.However,thecorrectorstepcanleadtoveryslowconvergenceorfailure.In2001,TorresandQuintanaapplyGondzio’smultiplecentralitycorrectionsstrategy.Furthermore,XieandSong 2001 andChungetal. 2011 leverageanodalblockdatastructurewithintheinteriorpointmethoditeratesinordertoreducethecorrectionequation. Duetothenonlinearitiesandill‐conditioningoftheACOPF,recentresearchhasfocusedonapplyingmethodswithparticularlyrobustglobalconvergenceproperties.Jabretal. 2002 presentaprimal‐dualinteriorpointmethodthatreplacestheHessianwitha“2‐normpositiveapproximant”witha“watch‐dog”strategy.Chiangetal. 2009 presentatwo‐stagesolutionalgorithmwhereanactive‐setquotientgradientmethodisappliedinstageonetoinduceglobalconvergence,andaninterior‐pointmethodisappliedinstagetwotoobtainalocaloptimalsolution.MinandShengsong 2005 ,PajicandClements 2005 ,Zhouetal.2005 ,SousaandTorres 2007 ,Wangetal. 2007 ,Chiangetal. 2009 ,and

Sousaetal. 2011 applyatrustregionmethod.SimilartoTorresandQuintana2001 ,MinandShengsong 2005 alsoapplyGondzio’smultiplecentralitycorrectionsstrategytotheprimal‐dualIPM,butinsteadincorporatethetrust‐regionmethodinsolvingthesubproblems.Theinfinitynorm,whichformsaclosed,compact,convexhypercubeinn‐dimensionalspace,isusedtodefinethetrustregion,andameritfunctionisusedtodeterminewhethertoacceptorrejectatrialsteptothetrustregionsubproblem.Theauthorsalsointroduceafeasibilityrestorationvariable,whichcontrolsthefeasibilityrequirementsofthetrustregionsubproblem. Alsoknownasparametriccontinuationmethods,homotopymethodsarepath‐followingapproacheswhichexplicitlyprogresstowardsasolutiontotheoriginalnonlinearproblem.Indoingsothesemethodsconstructandsolveanewsimplerproblemcomparedtotheoriginaloneandthengraduallyreconstructtheoriginalsystemofequationsinordertosolvefortheunknownsolution.PonrajahandGaliana 1989 ,HuneaultandGaliana 1990 ,Almeidaetal. 1993 ,Limaetal.2001 ,andJiangetal. 2010 applyhomotopymethodstosolvetheACOPF. Mostrecently,convexoptimizationtechniqueshavebeenappliedtotheACOPF.In2007,JabrreformulatedtheACOPFasasecond‐orderconicprogramandappliedaninteriorpointmethodinMOSEKthatisspecifictoconicquadraticoptimizationthathaspolynomialconvergence.In2012,LavaeiandLowapplysemidefiniteprogrammingoptimizationtosolvetheACOPF,andproveglobaloptimalityunderasufficientzero‐dualitygapcondition. Anotherclassofoptimization,derivativefreeoptimization,istypicallyappliedwhenfirst‐andsecond‐derivativesarenotavailableorareexpensivetocompute.Laietal. 1997 ,Iba 1994 ,Abido 2002 andmanyothersapplyderivativefreeapproachestosolvetheOPF.8.SummaryandConclusions Forthelastfiftyyears,thelatestdevelopmentsinnonlinearoptimizationhavebeenappliedtotheACOPFinhopesofbettersolutiontechniquesforlarge‐scale,practicalnetworkoperationsandplanning.Untilrecently,thedominantformulationhasbeeninpolarcoordinates.Althoughtheresearchto‐datepresentsarelativelypositivepicture,clearlythepublishedexperimentalresultshavebeenlimited.Thelackofreportedmetricsandavailabletestproblemsmakesitdifficulttoperformacomparativeassessmentofproposedsolutiontechniquesto‐date.Furthermore,thereisasignificantlackofindependentstudiesthatcomparethenumerousapproachesinasystematicmanner;forexample,refertothebenchmarkingstudycompletedbyMittelman 2012 wherehereportsonsolverperformanceforoptimizinggeneralnonlinearproblems.Inacompanionstudy

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