optimal power flow paper 10: using the iv-acopf linearization · 2020. 5. 12. · mixed integer...

November2013

OptimalPowerFlowPaper10

PaulaA.Lipka,RichardP.O’Neill,ShmuelS.Oren,AnyaCastillo,MehrdadPirnia,ClayCampaigne

OptimalTransmissionSwitchingUsing

theIV‐ACOPFLinearizationOptimalPowerFlowPapersPaper10.

PaulaA.Lipka,RichardP.O’Neill,ShmuelS.Oren,AnyaCastillo,MehrdadPirnia,ClayCampaigne

[email protected];[email protected];[email protected]@ferc.gov;[email protected];[email protected]

November2013

AbstractInthispaper,weseektoinvestigatetheperformanceoftransmission

switchingusingtheiterativelinearprogramapproximationtotheCurrentVoltageACOptimalPowerFlow IV‐ACOPF .SeveraldifferentmethodsofusingthisswitchingareinvestigatedtofindamethodthataddressestheMIPchallengesandisbothfastandaccurate.Weconsideropeningonlyoneline,openinguptofivelines,andprogressivelyopeningonelineatatime.ThelinearmethodwithswitchingismuchfasterthanthenonlinearACOPFandgenerallyfindssolutionswithin1%ofthenonlinearACOPF.

Disclaimer:TheviewspresentedarethepersonalviewsoftheauthorsandnottheFederalEnergyRegulatoryCommissionoranyofitsCommissioners.

TableofContents1.Introduction....................................................................................................42.Notation............................................................................................................63.NonlinearIV‐ACOPFTransmissionSwitchingModel...................74.IterativeLinearIV‐ACOPFTransmissionSwitchingModel.......85.ComputationalAnalysis.............................................................................96.Summary..........................................................................................................387.Conclusions.....................................................................................................38References............................................................................................................40Appendix...............................................................................................................42

1. IntroductionInagivenelectricpowernetwork,sometransmissionlinesareoftenoutof

serviceduetooutagesormaintenance.However,ofthelinesthatremainoperable,removingadditionallinesmayreducethetotalpowergenerationcost.Powercompaniesusethispracticeintimesoflowload,butoftenuseexperienceandintuitionwithreliabilitytesting;thatis,withoutexplicitlysearchingfortheoptimaltopology.ThebenefitoftransmissionswitchinghasbeendemonstratedbyFisheret.al. 2008 andHedmanet.al. 2008 bysolvinga‘DC’approximationofthealternatingcurrentoptimalpower ACOPF flowsystem.Hedmanet.al. 2008 achievedsavingsofover20%byswitchinglinesoffversustheoriginalnetworkontheIEEE118bustestcase.Evenwhentheconsideringnetworkcontingencies,transmissionswitching againusingthe‘DC’approximation canstillprovidesubstantialbenefits.Hedmanet.al. 2008 showthatopeningfivelinesreducescostsby8%intheN‐1compliantRTS‐96system.PotluriandHedman 2012 suggestthatswitchinghassignificantbenefitsovermaintainingastaticnetworkwhensolvingthefullACOPF.TheyalsodemonstratethatanimprovementinthesystemcostusingtheDCsolutionproceduredoesnotalwaysgivealowercostintheoriginalACsystem.Infact,theDCsolutionwithlineswitchingisACinfeasibleorincreasesthesystemcostovernoswitchinginseveralinstances.

ThenonlinearACOPFtransmissionswitchingproblemisrarelysolvedduetoitshighcomputationalintensity.Inpractice,theOPFissolvedusingtheDCapproximationanditerationsthattrytoobtainACfeasibilityandN‐1reliability.InordertosolvethenonlinearACOPFwithbinaryvariables,thesolvercouldpossiblybranchoneverycombinationofopened/closedlinesinthenetworkandsolveanonlinearprogram NLP foreachdifferentinstance,whichisprohibitivelycomputationallyexpensiveatthecurrenttime.IfthereareNlines,thiscouldrequiresolving2Nnonlinearprograms.Inaddition,thenonlinearprogramisnonconvex.

Mixedintegerlinearprograms MIPs generallysolvemuchfasterthanmixedintegernonlinearprograms.However,evensolvingaMIPcanalsobequitetime‐intensive,asonemayneedtosolvethelinearprogram LP 2Ntimes;however,techniquesforsolvingMIPshavesignificantlyreducedthisupperbound.

Onewaytolimitthenumberofbranchesistoconstrainthenumberoflinesallowedtobeopen.Althoughnotnecessarilyoptimal,theapproachfindsabettersolutionthanthestatusquo SeeHedmanetal,Fulleretal,andRuizetal,2012 andisoftenveryclosetotheoptimalsolution.Ourworkinthispaperalsoshowsthatopeningmorelinespastacertainnumberhasnoorlittlebenefit.

IfthesystemisforcedtoopenSlines,thenthenumberofbranchesforLPsintheMIPisNchooseS.However,intheDCmodel,itisneveroptimaltoopenlinesthatislandpartsofthenetwork seeOstrowskietal,2012 .Inaconnected

network,youcantravelfromanybusofthenetworktoanyotherbusviatransmissionlines althoughonemayhavetotravelthroughotherbuses .“Islanding”meansthattherearepairsofbusesthatarenotconnectedbyanypath.Aka,aconnectednetworkwouldbeonecircle,andanetworkwithislandingwouldbetwocircles;youcouldnevertravelfromabusinthefirstcircletothebusinthesecondcircle.Therefore,onecanreducetheMIPbranchandboundtreebyremovinganypathsthatresultinislanding.Toafirstorderapproximation,webelievethisistrueforACnetworks.Inthispaper,wehavenotimplementedthisprocedure.

Anadditionalissueisthatourlinearprogramisnotstatic;rather,itisbasedonthelastiteration seeO’Neilletal,2012andCampaigneetal,2013 .Ourlinearizationdependsonthepreviousoptimalpoint.However,wemaythenswitchthenetworkbasedonarevisedlinearizationinalateriteration.

Thispaperseekstoaddressthefollowingquestions: DoesthelinearizedACOPFselectlinestoswitchinaccordancewiththenonlinear

ACOPF? HowmuchfasteristheiterativelinearizedACOPFcomparedtothenonlinear

ACOPF? Howwelldotheproposedheuristicsperform?

2. Notation.Variablesandparametersareindexedoverbusesusingsubscriptsnandm,

n,mϵN.Herewerefertotransmissionassetsaslines.Transmissionlinesareindexedbyk,kϵK,withoneterminaloflinekbeingnandtheotherterminalbeingm.SetW n isthesetoflineskthatconnecttobusn.Foracomplexvariableorparameter,thesuperscriptrdenotestherealportionandthesuperscriptjdenotestheimaginaryportion.Forexample,ifx a jb,xr a,xj bwherej ‐1 1/2.Theindexofamajoriterationish.DecisionVariablespn realpowerinjectedatbusnqn reactivepowerinjectedatbusnvrn realpartofthevoltageatbusnvjn imaginarypartofthevoltageatbusnirn realpartofthecurrentinjectionatbusnijn imaginarypartofthecurrentinjectionatbusnirk realpartofthecurrentonlinekijkimaginarypartofthecurrenton

linekzk binaryvariablethatis1,iftheswitchatnforlinekisopen,and0,if

closedParameterscpn pn quadraticcostofrealpoweratbusncqn qn quadraticcostofreactivepoweratbusncpln pn stepwiselinearapproximationofcpn pn cqln qn stepwiselinearapproximationofcqn qn bk susceptanceoflinekbn0 selfsusceptanceofnoden toground gk conductanceoflinekgn0 selfconductanceofnoden toground yk gk jbk admittanceoflinekyn0 admittancefrombusntogroundpnd realpowerdemandatbusnqnd reactivepowerdemandatbusnpminn minimumrequiredrealpowergenerationatbusnpmaxn maximumallowedrealpowergenerationatbusnqminn minimumrequiredreactivepowergenerationatbusnqmaxn maximumallowedreactivepowergenerationatbusnvminn minimumrequiredvoltagemagnitudeatbusnvmaxn maximumallowedvoltagemagnitudeatbusnvrn realvoltagevalueatbusnfromthepreviouslinearprogramsolutionvjn imaginaryvoltagevalueatbusnfromthepreviouslinearprogram

solution

irn realcurrentvalueatbusnfromthepreviouslinearprogramsolutionijn imaginarycurrentvalueatbusnfromthepreviouslinearprogram

solutionimaxk maximumcurrentmagnitudeonlinekirktherealcurrentvalueon

linekfromthepreviouslinearprogramsolutionirk imaginarycurrentvalueonlinekfromthepreviouslinearprogram

solutionijk imaginarycurrentvalueonlinekfromthepreviouslinearprogram

solutionMk alargeconstant,usedtocreateaneither‐orconstraint3. NonlinearIV‐ACOPFTransmissionSwitchingModel Thenonlinearcurrent‐voltage IV ‐ACOPFisusedasabenchmarkforlinearapproximationILIV‐ACOPF.TheIV‐ACOPFformulationis

Minimize∑ncpn pn cqn qn 1 Subjecttoirk gk vrn‐vrm ‐bk vjn‐vjm forallk 2 ijk bk vrn‐vrm gk vjn‐vjm forallk 3 irn ∑k∊W n irk–gn0vrn bn0vjn foralln 4 ijn ∑k∊W n ijk–gn0vjn–bn0vrn foralln 5 pn vrnirn vjnijn pnD foralln 6 pminn pn pmaxn foralln 7 qnG vjnirn‐vrnijn qnD foralln 8 qminn qn qmaxn foralln 9 vrn 2 vjn 2 vmaxn 2 foralln 10 vminn 2 vrn 2 vjn 2 foralln 11 irk 2 ijk 2 imaxk 2 forallk 12

Thenetworkflowequations, 2 ‐ 5 arelinear.Theupperboundsonthelinecurrentmagnitudes 12 andvoltagemagnitudesatbuses 10 arecircleswiththeirinteriors convex ,andthuscanbeapproximatedtoanydegreeofaccuracywithcircumscribingpolygons.Thelowerboundonvoltagemagnitude,whilenon‐convex,isseldombindingbecausetheoptimizationpushesvoltageshighertoreducelosses.Inrectangularform,theequationsforreal 6 andreactivepower 8 injectionsandwithdrawalsintermsofcurrentandvoltagearesecond‐ordernon‐convexpolynomials.TheIV‐ACOPFwithtransmissionswitchingisformedbyaddingbinarydecisionvariablezkandreplacingequations 2 , 3 ,and 12 withequations 2.1 , 2.2 , 3.1 , 3.2 ,and 12.1 .zk 1iflinekisdisconnectedandzk 0iflinekisconnected.

irk gk vrn‐vrm ‐bk vjn‐vjm zkM 2.1

irk gk vrn‐vrm ‐bk vjn‐vjm ‐zkM 2.2 ijk bk vrn‐vrm gk vjn‐vjm zkM 3.1 ijk bk vrn‐vrm gk vjn‐vjm –zkM 3.2 irk 2 ijk 2 1‐zk imaxk 2 12.1

4. IterativeLinearIV‐ACOPFTransmissionSwitchingModel

Weapproximatethequadraticconstraintequations whichexpressrealandreactivepowerintermsofcurrentsandvoltages withhyperplanesthataretangenttotheconstrainthypersurfacesusingfirstorderTaylorapproximations.Withtheresultinglinearapproximations,theILIV‐ACOPF h ateachmajoriterationhis:Minimize∑ncpln pn cqln qn 21Subj. irk gk vrn‐vrm ‐bk vjn‐ vjm zkM forallk 2.1to irk gk vrn‐vrm ‐bk vjn‐ vjm ‐ zkM 2.2 ijk bk vrn‐vrm gk vjn‐ vjm zkM 3.1 ijk bk vrn‐vrm gk vjn‐ vjm – zkM 3.2 irn ∑k∊W n irk–gn0vrn bn0 vjn foralln 4 ijn ∑k∊W n ik–gn0vjn– bn0 vrn foralln 5 pnG vrnirn vjnijn vrnirn vjnijn ‐ vrnirn vjnijn pnD for alln 26 pnmin pnG pnmax for alln 27 qnG vjnirn‐vrnijn‐vrnijn vjnirn ‐ vjnirn ‐ vrnijn qnD for alln 28 qnmin qnG qnmin for alln 29 vrfnvrn vjfnvjn vmaxn 2 forf 01,…,fmax

andalln30

vrdnvrn vjdnvjn vmaxn 2 ford 0,…,h‐1andalln

31

irfkirk ijfkijk 1‐zk imaxk forf 1,…,fmax andallk

32

irdkirk ijdkijbk imaxk 2 ford 0,…,h‐1andallk

33

‐2vrn a/hb vrn‐vrn 2vrn a/hb for alln 34 ‐2vjn a/hb vjn‐vjn 2vjn a/hb for alln 35

wherefmaxisthenumberofsidesofthepreprocessedcircumscribingpolygonsanddindexestheiterativetightcuts.

Tolimitthenumberoflinestobeopened,weaddtheconstraint:∑kzk k’ 36

Thenetworkflowequations, 22 ‐ 25 arelinearandunchanged.Forthevoltagemagnitudes,thepreprocessedupperboundsarein 30 andtheiterativetightcutsarein 31 .Forthelinecurrentmagnitudes,thepreprocessedupperboundsarein32 andtheiterativetightcutsarein 33 .

Inrectangularform,real 26 andreactivepower 28 injectionsand

withdrawalsintermsofcurrentandvoltageareapproximatedbythefirstorderTaylorseries.

Hereweusetheformulationthatremovestheentiretransmissionelementfromthenetwork;nocurrentcanflowontheline.Ifweopenonlyoneoftheswitches,thelinebecomesacapacitor.

Theiterativelinearizationmethodusedtosolvetheproblemisasfollows:

1 Seth 0.Chooseastartingpointvr0nandvi0n.Addacircumscribingpolygonforeachmaximumvoltagemagnitudeandmaximumcurrentmagnitudeconstraint.ApproximatePandQwithhyperplanesthataretangenttotheconstrainthypersurfaces.

2 Seth h 1.SolvetheresultingLIV‐ACOPF h toobtainoptimalvaluesforthecurrentandvoltage,vrhn,vjhn,irhn,ijhn

3 ChecktheresultforconvergenceoftheoptimalvaluesusingtheactualnonlinearPandQequations 6 through 12 .Ifwithintolerance,stop;otherwisecontinue.

4 Addanothersetoftightvoltageandcurrentconstraintsattheoptimalvoltagesolutiontofurthercutoffinfeasiblevoltageandcurrentsolutions.Relinearizethepandqapproximationusingthecurrentanswerandadjustthestepsizerangeofthebusvoltage.Gotostep2.

Theconvergencecriteriaisasfollows:ifthepercentviolationsofrealpower7 ,reactivepower 9 ,thevoltage 10 ,andthecurrent 12 constraintsisunderacertainthreshold,andthesumoftheaveragepercentviolationsofthevoltage,realpower,andreactivepowerisalsounderathreshold,thesolutionisdeterminedtobeACfeasible.Iftheseviolationcriteriaarenotmet,thesolutionisdeterminednottobeACfeasible.

5. ComputationalTestingProblems.Thetestproblemsconsistentofthe14,30,57,and118busIEEEtestcases seeTable1 athttp://www.ee.washington.edu/research/pstca/index.html.Single‐linediagramsofthe14and30buscasesareshowninFigures1and2.ThequadraticgeneratorcostscomefromMATPOWER Zimmermanetal,2011 .Weformulatethe20‐steplinearapproximationtothequadraticfunction.Wheretherearemultipletransmissionlinesbetweentwonodes,thelinesareaggregatedintoanequivalentsinglelinebetweenthetwonodes.Eachtestproblemhastwolevelstightandloose oflinecurrentconstraints seeLipkaetal,2013 .

Table1:IEEETestBusSystemData

Buses Lines Generators TotalDemandBestKnownValueTightCurrentLimit

BestKnownValueLooseCurrentLimit

No. Capacity quadratic linear quadratic linear14 20 5 7.724 2.590 105.4 107.4 85.3 86.530 41 6 326.80 42.42 5.89 6.10 5.79 5.9857 80 7 326.78 235.26 421.5 432.2 419.2 425.5118 186 54 99.66 42.42 1364.9 1388.4 1300.1 1315.5

Figure1:14BusDiagram

Figure2:30BusDiagram

HardwareandSoftware.TheproblemsweresolvedonanIntelXeonE7458serverwith864‐bit2.4GHzprocessorsand64GBmemory.However,allproblemsonlyusedoneprocessoratatime.Minordifferencesinsolutiontimeswererecordedwhentheproblemswererunatdifferenttimesofday,butthedifferencesweresmallenoughtobeconsideredbackgroundnoise.TheproblemswereformulatedinGAMS23.6.2.ThenonlinearmixedintegerprogramusedwasKNITRO.ThenonlinearprogramswithoutintegervariablesusedsolverIPOPTversion3.8.LinearprogramsusedGUROBIversion4.0.0withtheaggressivepresolveoption.Theimplementationwassimplisticinthattheproblemwassolvedfromscratchateachmajoriteration.StartingfromthepreviouslinearprogramwasnotanoptionintheGAMSsolver.Theremaybeeasilygainedspeedupsbynotstartingeachmajoriterationfromscratch.TheallowableMIPgapwassetto0.1%.OptimizationParametersSettings.ForIPOPT,weusethedefaultparameters.Forstep‐sizeconstraints,weexamineb 1 linearstepsize andb 2 quadraticstepsize anda 0.5and1in 34 and 35 .Wechoose16and32sidedcircumscribingpolygonsbasedonthetestinginPirniaetal 2012 .

Themaximumnumberofmajoriterations linearprogram wassetto20.Earliertestingrevealedthatthesolutionusuallyconvergedtoafeasiblesolution

withintolerance in20iterationsorless,andproblemsthatranformorethan20iterationswouldruntothemaximumnumberofallowediterations.Thelinearproblemisconsideredfeasiblewithintolerancewhentheaveragedeviationofvoltages,realpower,andreactivepowerislessthan0.1%andthemaximumdeviationofallindividualnodesislessthan0.5%.Initialization.Astheinitialstartingpoint,weuseaflatstart,vrn 1andvin 0,forallbusesn.Otherstartingpointscanbeused:theseincludehotstarts,randomstarts,andDCstarts seeCastilloetal2013 .Incommercialpractice,ahotstartmaybeavailablefromprevioussolutionssinceeachdispatchissolvedinsequentialtimesteps.Termination.Themaximumnumberofmajoriterationswassetto100.Therelativeconvergencetolerancesweresetat0.001or0.005.ThemaximumCPUtimewasneverreached. Weexaminefivedifferentapproaches:bruteforce totalenumeration one‐lineoptimization,MIPone‐lineoptimization,iterativelinearizedACOPFwithaMIPateveryiteration openingupto5lines ,linearizedACOPFwiththeMIPatthefirstiteration openingupto5lines ,andaprogressiveapproachwhereoneadditionallineisallowedtobeopenateachstage upto5lines .Brute‐ForceOne‐LineApproach.First,weexaminetheresultsofthelinearizedACOPFversusthenonlinearACOPFwhenthenetworkisfixedtoonelineremovedusingthefollowingprocedure,whereNsubproblemsaresolved.Theprocedureinsteps1‐5isrepeatedfor4differentapproaches:16and32preprocessedvoltagecutswithlinearandquadraticvoltagestepsizereduction.

1 ForeachlinekinthetransmissionsystemwithKlines:2 Inthemodeldescribedinsection4,zisaparameterinsteadofavariable.Letalllinesbeinservice aka,z 0 exceptforlinek setz 1 .Notethatsincezisaparameterinsteadofavariable,anLPinsteadofaMIPissolved.3 Usetheiterativelinearizationmethodtosolvetheproblem.4 Setlinektobelinek 1.Gotostep2ifkislessthanorequaltoK.5 Whenalloftheresultshavebeentabulatedforeachlinetakenout k K 1 ,thesubproblemwiththelowestcostisconsideredtobetheoptimalconfiguration.6 Toobtainanonlinearcomparison,steps1‐5arerepeatedexceptinstep2,thenonlinearmodeldescribedinsection3isusedandstep3isreplacedwiththesolver’snonlinearsolutionmethods.

MIPOne‐LineApproach.WealsocomparetheresultsofthelinearizedACOPFwithaMIPateveryiterationtotheresultsofbruteforcelinearizedACOPFinordertodemonstratewhethertheMIPateachiterationworksappropriately.

Theprocedureinsteps1‐2isrepeatedfor4differentapproaches:16and32preprocessedvoltagecutswithlinear b 1 andquadratic b 2 voltagestepsizereduction.

1 Usethemodeldescribedinsection4withtheaddedconstraint∑kzk 1.Thismeansthatinanyanswer,onetransmissionlinewillbeopen.2 Usetheiterativelinearizationmethodtosolvetheproblem.Ateveryiteration,aMIPwillbesolved.3 Then,toobtainanonlinearcomparison,steps1and2arerepeatedexceptinstep1,thenonlinearmodel KNITRO describedinsection3isusedandstep2isreplacedwiththesolver’snonlinearsolutionmethods.

IterativeLinearizedACOPFwithaMIPateveryiterationopeningupto5linesMoreextensively,theACOPFwasrunforopeningupto5lines.Thislimitonthenumberoflinesopenedwasplacedtoachieveareasonableruntime.Additionally,allowingtheprogramopenuptotenlinesdidnotresultinsignificantreductionofthetotalpowerproductioncostversusallowingittoopen5lines.Theprocedureisrepeatedfor4differentapproaches:16and32preprocessedvoltagecutswithlinearandquadraticvoltagestepsizereduction.

1 Usethemodeldescribedinsection4withtheaddedconstraint∑kzk k’wherek’ 5.Thismeansthatinanyanswer,upto5transmissionlinescanbeopen.2 Usetheiterativelinearizationmethodtosolvetheproblem.Ateveryiteration,aMIPwillbesolved.3 Whentheiterativelinearizationmethodhaseitherconvergedorexceededtheiterationlimit,solvethenonlinearACOPFwithtransmissionswitchingasinsection3butagainthelineconfiguration zk variablesaresetasparametersaccordingtothesolutioninstep2 ,againtoevaluatewhethertheLIV‐ACOPFsolutionwastrulyACfeasible.

LinearizedACOPFwiththeMIPatthefirstiterationopeningupto5lines.ToreducethecomputationaltimeofsolvingtheACOPF,asecondapproachistorunaMIPonlyonthefirstiterationoftheiterativelinearizationmethod,thenruntheotherLPiterationsonthefixednetworkthatwastheoptimalanswerfromtheMIPrepeatedly.Thestepsusedareasfollows:Theprocedureisrepeatedfor4differentapproaches:16and32preprocessedvoltagecutswithlinearandquadraticvoltagestepsizereduction.

1 Usethemodeldescribedinsection4withtheaddedconstraint∑kzk k’wherek’ 5.Thismeansthatinanyanswer,upto5transmissionlinescanbeopen.

2 Dooneiterationoftheiterativelinearizationmethodtosolvetheproblem.ThisiterationwillsolveaMIP.3 Inthemodeldescribedinsection4,replacethevariableszwithparameters.Thelineconfigurationwillbesettotheoptimalsolutionfoundinstep2.Forexample,ifstep2foundthatalllinesareinthenetworkexceptthelinesfrombus1to4andfrombus2to5,thenz141 1,z251 1,andallotherz 0.4 Solvethemodelinstep3usingtheiterativelinearizationmethod.5 Whentheiterativelinearizationmethodhaseitherconvergedorexceededtheiterationlimit,solvethenonlinearACOPFwithtransmissionswitchingasinsection3,butagainthelineconfiguration zk variablesaresetasparametersaccordingtothesolutioninstep2 ,againtoevaluatewhethertheLIV‐ACOPFsolutionwastrulyACfeasible.

Itisexpectedthatthismodelwilltakelesstimethanthepreviousmodel,sincelessMIPswillberunduringtheprocess.ProgressiveMIP.ThepurposeofthismethodistoreducethenumberofbranchestheMIPhastoconsiderinthecaseofopeningupto5transmissionlines.Thepreviousapproachhas Nchoose5 Nchoose4 Nchoose3 Nchoose2 Nchoose1 potentialbranches.Inthisapproach,ateachiterationk,withthefirstiterationask 0,thealgorithmdecideswhichofN‐klinestoopenorwhethertonotopenanyadditionallines,withatotalofN‐k 1possibilities.Thisapproachhasupto N 1 N N‐1 N‐2 N‐3 5N‐5potentialbranches,whichismanyfewerthanthepreviousapproach.Table2showsthissignificantreduction.Table2.NumberofPotentialMIPbranches:Normalvs.Progressive NumberofPotentialBranches

Buses Lines NormalMIP ProgressiveMIP14 20 21,700 9530 41 862,190 20057 86 37,055,939 425118 186 1,806,641,034 925

Thefollowingprocedureisrepeatedfor4differentapproaches:16and32preprocessedvoltagecutswithlinearandquadraticvoltagestepsizereduction.

1 Usethemodeldescribedinsection4withtheaddedconstraint∑kzk k’wherek’ 1.Thismeansthatinanyanswer,only1transmissionlinecanbeopen.2 Usetheiterativelinearizationmethodtosolvetheproblem.ThisiterationwillsolveaMIPwithN 1branches.

3 Ifalineisopened,fixitasaparameter aka,z 1 .Now,letk’ 2.Notethatsinceonelineisopen,onlyonemorelinecanpossiblybeopened.4 Again,usetheiterativelinearizationmethodtosolvetheproblem.5 Fixallopenlinesasparameters,andincrementk’ k’ 1.6 Repeatsteps4and5untiltheproblemwherek’ 5hasbeensolved.7 Whentheiterativelinearizationmethodhaseitherconvergedorexceededtheiterationlimit,solvethenonlinearACOPFwithtransmissionswitchingasinsection3,butagainthelineconfiguration zk variablesaresetasparametersaccordingtothesolutioninstep2 .ThisisdonetoevaluatewhethertheLIV‐ACOPFsolutionwastrulyACfeasible.

5.1BRUTE‐FORCEVERSUSMIPONELINEAPPROACHRESULTSTobenchmarkthelinearizedACOPFprogram,boththelinearizedand

nonlinearACOPFprogramsweresolvedonafixedconfigurationofopen/closedlinesforallpossibleconfigurationsofonelineopenandallothersclosed theBruteForceMethod .Theobjectivefunctionvaluesofalltheseconfigurationsarecompared,andthe‘best’answeristheproblemwiththelowestobjectivefunctionvalue.ThisBruteForceapproachiscomparedtosolvingtheproblemwiththelinearizedACOPFandallowingthemixedintegersolvertofindthebestnetworkconfigurationwithonelineout MIP .Thenonlinear“bruteforce”methodusesIPOPTtosolveeverypossiblenetworkconfigurationwithonelineout,andtheobjectivevaluegivenistheminimumcostofalloftheseconfigurations.Thenonlinear“MIP”methodusesKNITROtosolvethenonlinearMIPwithoneopenline.

14‐busproblem.Theresultsforthe14busproblemareshowninTables3and4.Theobjectivefunctionvalueswerewithin2%ofthebest‐knownvalue thenonlinearsolution forboththeMIPandBruteForcemethodsinboththetightandloosecurrentconstrainedcases.Inaddition,allofthelinearapproacheswereatleast4timesasfastasthenonlinearcase.Inthetightcurrentconstrainedcase,thesameline 4‐7 wasswitchedoutbyallbutoneofthe8approaches.Thelinearcaseobjectivevaluesareallveryclosetoeachother–thebiggestdifferenceis0.1%.Removingline4‐7washighlybeneficialinthetightlyconstrainedcase;thetotalgenerationcostwasreducedbynearly10%.Intheloosecurrentconstrainedcase,againthesameline 2‐5 wasselectedtobeswitchedbyallbutoneofthe8approaches.However,theobjectivevaluesofthelinearapproachesdiffermorethaninthetightcurrentcase;thebiggestdifferenceis3.2%.Whilethenonlinearcaseshowsasmallimprovementinthetotalgenerationcostbyswitchingoutaline,someofthelinearapproachesshowanimprovementwhilesomedonot.Inthetightlyconstrainedcase,theMIPrunsfasterthanthebruteforcemethod.Surprisingly,inthelooselyconstrainedcase,theMIPrunsslowerthanthebruteforcemethod;thisislikelynotsignificantduetorun‐to‐runtimingvariability.Table3.OneLineResults:14Bus,TightCurrentConstraintStepSizeFunction Linear Quadratic

NonlinearNumberofCuts 16 32 16 32WithoutOpeningLines ObjValue 105.69 106.53 105.69 106.53 107.36MIP ObjValue 95.05 95.04 95.05 95.04 93.77OpeningOneLine LineSwitched 4‐7 4‐7 4‐7 4‐7 4‐7 CPUTime 6.25 6.38 5.18 9.32 54.30BruteForce ObjValue 95.04 94.99 95.00 94.97 94.09OpeningOneLine LineSwitched 4‐7 2‐5 4‐7 4‐7 4‐7 CPUTime 6.81 9.44 10.85 14.59 192.12Table4.OneLineResults:14Bus,LooseCurrentConstraintStepSizeFunction Linear Quadratic

NonlinearNumberofCuts 16 32 16 32WithoutOpeningLines ObjValue 85.94 85.63 85.15 86.13 86.51MIP ObjValue 82.80 85.26 85.54 85.55 84.07OpeningOneLine LineSwitched 2‐5 2‐3 2‐5 2‐5 2‐5 CPUTime 10.89 22.06 5.80 9.80 100.78BruteForce ObjValue 84.94 84.85 85.28 85.29 84.42OpeningOneLine LineSwitched 2‐5 2‐5 2‐5 2‐5 2‐5 CPUTime 8.94 9.47 5.11 6.97 63.52

30busproblem.Theresultsforthe30busproblemareshowninTables5and6.Inboththetightlyandlooselyconstrainedcases,thebiggestdifferencebetweentheobjectivevaluesinthelinearapproachesis2%.Inbothtightandloosecurrentconstraintcases,itappearsthatremovingonelinedoesimprovetheobjectivevalue,althoughthisimprovementisverysmall.TheMIPandBruteForcemethodschoosedifferentlinestoopeninthesameproblem,althougheachmethodchoosesthesamelinewithinthelinearapproaches.However,choosingthedifferentlinedoesnotseemtoaffectthevalueoftheobjectivefunction.Here,theMIPhasaclearadvantageovertheBruteForcemethodinthesolutiontime.TheMIPrunsmorethantwiceasfastastheBruteForcemethodinallapproachesexceptforthelooselyconstrainedcasewithquadraticstepsizeand32cuts.Inthebruteforcemethod,linearapproachessolvedatleast4timesasfastasthenonlinearapproach.SincethereweredifferentresultsondifferentsimulationsfortheMIPwiththenonlinearcase,thesimulationresultofthelastruniscapturedhere.Table5.OneLineResults:30Bus,TightCurrentConstraintStepSizeFunction Linear Quadratic

NonlinearNumberofCuts 16 32 16 32WithoutOpeningLines ObjValue 6.02 6.08 6.02 6.08 6.10MIP ObjValue 5.82 5.91 5.87 5.91 5.74OpeningOneLine LineSwitched 25‐27 25‐27 25‐27 25‐27 25‐27 CPUTime 37.05 40.49 20.71 34.85 536.4BruteForce ObjValue 5.93 5.93 5.92 5.93 5.79OpeningOneLine LineSwitched 6‐28 6‐28 6‐28 6‐28 6‐28 CPUTime 124.8 117.2 93.19 113.6 554.5Table6.OneLineResults:30Bus,LooseCurrentConstraintStepSizeFunction Linear Quadratic

NonlinearNumberofCuts 16 32 16 32WithoutOpeningLines ObjValue 5.96 5.98 5.96 5.98 6.00MIP ObjValue 5.81 5.81 5.81 5.89 5.74OpeningOneLine LineSwitched 25‐27 25‐27 25‐27 25‐27 24‐25 CPUTime 30.81 66.29 21.67 29.90 633.7BruteForce ObjValue 5.92 5.92 5.92 5.92 5.78OpeningOneLine LineSwitched 6‐28 6‐28 6‐28 6‐28 24‐25 CPUTime 75.39 156.0 89.76 53.73 683.1

57busproblem.Theresultsforthe57‐busproblemareshowninTables7and8.8Thelineschosentoopenarenotveryconsistentbetweenthedifferentapproaches,suggestingthattheobjectivefunctionhasseveraltopologieswithsimilarobjectivevalues.Theobjectivefunctionvalueswerewithin0.2%ofthebest‐knownvaluewiththebruteforcemethod.Here,theadvantageoftheMIPovertheBruteForcesolutiontimeincreases;theMIPsolvesmorethan25timesfasterthanthebruteforcemethod.However,theBruteForcemethodappearstofindbettersolutions;allsolutionsofthebruteforcemethodareanimprovementovernotswitchinganylines,andallMIPsolutionshavegreatergeneratorcostthantheBruteForcesolutions.Table7.OneLineResults:57Bus,TightCurrentConstraintStepSizeFunction Linear Quadratic

NonlinearNumberofCuts 16 32 16 32WithoutOpeningLines ObjValue 424.02 422.71 424.07 423.69 433.85MIP ObjValue 424.66 426.98 426.99 426.75 419.08OpeningOneLine LineSwitched 56‐57 38‐48 2‐3 2‐3 9‐13 CPUTime 34.82 54.85 12.15 21.10 3000.1BruteForce ObjValue 421.16 421.65 421.49 421.70 420.71OpeningOneLine LineSwitched 12‐16 38‐49 9‐11 9‐13 48‐49 CPUTime 1,298.15 1,708.27 815.8 1,034.99 10,966.5Table8.OneLineResults:57Bus,LooseCurrentConstraintStepSizeFunction Linear Quadratic

NonlinearNumberofCuts 16 32 16 32WithoutOpeningLines ObjValue 422.60 422.42 423.27 423.47 425.48MIP ObjValue 426.33 426.98 429.01 426.75 418.47OpeningOneLine LineSwitched 38‐48 38‐48 1‐2 2‐3 3‐4 CPUTime 34.09 54.85 9.62 21.10 2982.6BruteForce ObjValue 421.16 421.65 421.49 421.70 420.71OpeningOneLine LineSwitched 12‐16 38‐49 9‐11 9‐13 48‐49 CPUTime 1,292.2 1,668.6 768.5 964.6 9,287.3

118busproblem.Theresultsforthe118‐busproblemareshowninTables9and10.Inthetightcurrentconstraintcase,thelinearapproacheschooseeitherline77‐80orline77‐82toberemoved.Intheloosecurrentcase,thereismoredeviationinwhichlineistakenout.TheMIPisagainmuchfasterthanthebruteforcemethodinbothtightlyandlooselyconstrainedcases .Objectivefunctionvaluesarewithin3.2%ofthenonlinearvaluesinthetightcaseandwithin1.5%oftheloosecase.Inthetightcurrentconstraintcase,theobjectivevaluesfoundbytheMIPandthebruteforcemethodareverysimilar.Intheloosecurrentconstraint,thebruteforcemethodappearstofindbettersolutions;alltheobjectivevaluesinthebruteforcemethodareanadvantageovernotswitchinganylines;mostofthesolutionsfoundbytheMIParemorecostlythannotswitchinganylines.Table9:OneLineResults:118Bus,TightCurrentConstraintStepSizeFunction Linear Quadratic

NonlinearNumberofCuts 16 32 16 32WithoutOpeningLines ObjValue 1,381.15 1,380.4 1,388.3 1,388.3 1,364.9MIP ObjValue 1,379.12 1,378.2 1,385.7 1,385.8 1,344.3OpeningOneLine LineSwitched 77‐82 77‐82 77‐82 77‐82 17‐31 CPUTime 213.1 307.8 256.6 330.1 30,000BruteForce ObjValue 1,379.2 1,378.4 1,384.6 1,384.1 1,368.9OpeningOneLine LineSwitched 77‐80 77‐82 77‐82 77‐80 77‐82 CPUTime 12,303 14,631 6,534.6 16,175 26,359Table10:OneLineResults:118Bus,LooseCurrentConstraintStepSizeFunction Linear Quadratic

NonlinearNumberofCuts 16 32 16 32

WithoutOpeningLines ObjValue 1,307.7 1,311.8 1,310.7 1,314.6 1,300.1MIP ObjValue 1,314.1 1,315.2 1,314.4 1,314.4 1,296.8OpeningOneLine LineSwitched 25‐26 25‐26 25‐26 25‐26 24‐70 CPUTime 123.1 186.0 34.56 61.64 322.6BruteForce ObjValue 1,305.3 1,304.9 1,305.9 1,307.6 1,304.6OpeningOneLine LineSwitched 46‐48 100‐104 45‐46 40‐41 19‐20

CPUTime 22,780 34,747 11,874 16,091 21,236

COMPARISON:ONEMIP,REPEATEDMIP,ANDPROGRESSIVEMIP14Bus,TightlyConstrainedObjectiveValueFunction LP AllthreemethodsofsolvingtheMIPreportedverysimilarobjectivefunctionvalues,exceptfortheprogressiveMIPwithalinearstepsizefunctionand32cuts,wheretheanswerwasinfeasible seeTable11 .Table11.14Bus,TightlyConstrained:LinearObjectiveValue linearobjective,linearconstraints StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 95.02 95.04 95.10 95.04OneMIP 95.25 95.04 95.25 95.04ProgressiveMIP 95.19 INF 95.42 95.04Nolinesopen 105.69 106.53 105.69 106.53

LPvs.NLPObjectiveFunctionValueDifferences TheLPvs.NLPObjectiveValueDifferenceiscomputedastheobjectivevaluesoftheNLP‐LP /LP.Asareminder,theNLPObjectiveValueistheanswertotheproblemwherethetransmissionlinesfoundtobetakenoutintheLParefixedusingthesolverIPOPT,andtheACOPFissolvedwiththeoriginal,nonlinearconstraints butwithlinearobjectivefunction .Inthiscase,asshownintables12and13,allthenonlinearanswersareclosetothelinearobjectivevalue,exceptfortheprogressiveMIPwith32cutsandalinearstepsize.Thislargediscrepancyoccursbecausethelinearmethodfindsaninfeasiblesolutionandthenonlinearmethodfindsafeasiblesolution.

92

93

94

95

96

97

98

99

100

16 32 16 32

Linear Quadratic

Objectiv

e Va

lue (LP)

Repeated MIP

One MIP

Progressive MIP

Table12.14Bus,TightlyConstrained:NLPObjectiveValue nonlinearconstraints,linearobjective StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 95.06 95.07 95.06 95.06OneMIP 95.41 95.06 95.42 95.06ProgressiveMIP 95.43 94.95 95.26 95.06Table13.14Bus,TightlyConstrained:%DifferencebetweenLPandNLPObjectiveValueStepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 0.04 0.04 ‐0.05 0.02OneMIP 0.17 0.02 0.18 0.02ProgressiveMIP 0.26 INF ‐0.17 0.02

CPUTime Table14showsthatusingonlyoneMIPatthebeginningdramaticallyreducesthetimetosolvetheswitchingproblem.TheprogressiveMIPhasaslightadvantageoverusingoneMIP.Table14.14Bus,TightlyConstrained:CPUTime seconds StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 34.68 101.03 26.75 59.09OneMIP 8.36 20.60 15.83 22.56ProgressiveMIP 7.11 15.18 6.24 14.15

‐3%

‐2%

‐1%

0%

1%

2%

3%

16 32 16 32

Linear Quadratic

LP vs. NLP % Differen

ce

Repeated MIP

One MIP

Progressive MIP

14Bus,LooselyConstrainedObjectiveValueFunction LP Thelinearobjectivevaluesfoundbyallmethodsarerelativelyclose,asseeninTable15,excepttherepeatedMIPwithlinearstepsizefunctionappearstodeviatefromalltheothermethods,althoughitstillfindsafeasiblesolution.Table15.14Bus,LooselyConstrained:LinearObjectiveValue linearobjective,linearconstraints StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 82.53 82.10 84.65 84.38OneMIP 84.49 85.08 84.49 85.08ProgressiveMIP 85.01 85.21 85.09 85.36Nolinesopen 85.94 85.63 85.15 86.13

0.00

20.00

40.00

60.00

80.00

100.00

16 32 16 32

Linear Quadratic

CPU Tim

e (s)

Repeated MIP

One MIP

Progressive MIP

81.0081.50

82.0082.5083.00

83.5084.0084.50

85.0085.50

16 32 16 32

Linear Quadratic

Objectiv

e Va

lue (LP)

Repeated MIP

One MIP

Progressive MIP

LPvs.NLPObjectiveFunctionValueDifferences Tables16and17displaythatthenonlinearandlinearobjectivevaluesarewithin1%exceptfortherepeatedMIPmethodwithlinearstepsizereductions.However,thenonlinearobjectivevalueforthiscaseiscloseto85ratherthan82,whichiswhattheoneMIPandprogressiveMIPmethodsfound.Table16.14Bus,LooselyConstrained:NLPObjectiveValue nonlinearconstraints,linearobjective StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 85.20 85.43 85.21 85.20OneMIP 85.11 85.12 85.11 85.11ProgressiveMIP 85.02 85.36 85.10 85.23Table17.14Bus,LooselyConstrained:%DifferencebetweenLPandNLPObjectiveValueStepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 3.13 3.90 0.66 0.97OneMIP 0.73 0.05 0.73 0.04ProgressiveMIP 0.02 0.18 0.01 ‐0.15

‐1%0%1%1%2%2%3%3%4%4%5%

16 32 16 32

Linear Quadratic

LP vs N

LP: %

Differen

ce

Repeated MIP

One MIP

Progressive MIP

CPUTime Table18showsthattheoneMIPmethodsolvedsignificantlyfasterthantherepeatedMIPmethod,andtheprogressiveMIPmethodsolvedsignificantlyfasterthantheprogressiveMIPmethod.However,thereisnotaclearpatternwhichnumberofcutsorstepsizefunctionisfaster.Table18.14Bus,LooselyConstrained:CPUTime seconds StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 258.65 152.38 126.00 165.26OneMIP 22.08 59.40 23.12 40.16ProgressiveMIP 4.21 5.15 9.77 5.14

30Bus,TightlyConstrainedObjectiveValueFunction LP Theresultsofsolvingthe30bussystemwithtightcurrentconstraintsaregiveninTable19.Allmethodsfoundafeasibleanswerforthisproblem.Likethepreviousproblem,therepeatedMIPmethodwithlinearstepsizefunctionseemedtofindanswersthatdeviatedfromalltheothermethods.Table19.30Bus,TightlyConstrained:LinearObjectiveValue linearobjective,linearconstraints StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 5.83 5.83 5.96 5.93OneMIP 5.94 5.97 5.96 5.97ProgressiveMIP 5.89 5.90 5.93 5.93Nolinesopen 6.02 6.08 6.02 6.08

0306090120150180210240270

16 32 16 32

Linear Quadratic

CPU Tim

e (s)

Repeated MIP

One MIP

Progressive MIP

LPvs.NLPObjectiveFunctionValueDifferences Tables20and21revealthattheNLPsolutionwasveryclosetotheLPsolutionintheOneMIPandProgressiveMIPapproaches;however,thenetworkconfigurationfoundbytheLPintherepeatedMIPcases exceptforthelinearstepsizewith16cuts wasinfeasibleonceallthenonlinearitieswereconsidered.Table20.30Bus,TightlyConstrained:NLPObjectiveValue nonlinearconstraints,linearobjective StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 5.97 INF INF INFOneMIP 5.99 5.98 5.99 6.01ProgressiveMIP 5.94 5.98 5.95 6.07Table21.30Bus,TightlyConstrained:%DifferencebetweenLPandNLPObjectiveValueStepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 2.30 INF INF INFOneMIP 0.81 0.25 0.54 0.67ProgressiveMIP 0.80 1.31 0.40 2.20

5.75

5.80

5.85

5.90

5.95

6.00

16 32 16 32

Linear Quadratic

Objectiv

e Va

lue (LP)

Repeated MIP

One MIP

Progressive MIP

CPUTime FromTable22,weseethattheprogressiveMIPapproachsolvedsignificantlyfasterthantheoneMIPapproachthatsolvedtheproblemmuchfasterthantherepeatedMIPapproach.Table22.30Bus,TightlyConstrained:CPUTime seconds StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 3908.01 6886.65 1718.02 2409.33OneMIP 763.19 1001.37 806.55 1000.99ProgressiveMIP 24.85 34.15 20.71 41.44

0%2%4%6%8%10%12%14%16%18%20%

16 32 16 32

Linear Quadratic

LP vs. NLP: %Diffe

rence

Repeated MIP

One MIP

Progressive MIP

0

1000

2000

3000

4000

5000

6000

7000

16 32 16 32

Linear Quadratic

CPU Tim

e (s)

Repeated MIP

One MIP

Progressive MIP

30Bus,LooselyConstrainedObjectiveValueFunction LP Table23showsthattherepeatedMIPapproach exceptforthe16cuts,quadraticstepsizecase seemedtofindsignificantlydifferentanswersthanthosefoundbyusingOneMIPandtheProgressiveMIP.Table23.30Bus,LooselyConstrained:LinearObjectiveValue linearobjective,linearconstraints StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 5.82 5.83 5.93 5.81OneMIP 5.94 5.89 5.95 5.90ProgressiveMIP 5.89 5.90 5.90 5.91

LPvs.NLPObjectiveFunctionValueDifferences Thereisonlyonecasewherethenetworkconfigurationfoundbythelinearapproachwasnotfeasible–therepeatedMIP,quadraticstepsize,16cutscaseasseeninTables24and25.Otherwise,thedifferencebetweentheanswersfromtheNLPandtheLPare3.1%orless.Table24.30Bus,LooselyConstrained:NLPObjectiveValue nonlinearconstraints,linearobjective StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 5.94 5.97 INF 6.00OneMIP 5.98 5.94 5.99 5.94ProgressiveMIP 5.98 5.95 5.95 5.98Nolinesopen 5.96 5.98 5.96 5.98

5.70

5.75

5.80

5.85

5.90

5.95

6.00

16 32 16 32

Linear Quadratic

Objectie

Value

(LP)

Repeated MIP

One MIP

Progressive MIP

Table25.30Bus,LooselyConstrained:%DifferencebetweenLPandNLPObjectiveValueStepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 2.01 2.36 INF 3.09OneMIP 0.67 0.82 0.77 0.70ProgressiveMIP 1.43 0.87 0.80 1.16

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

16 32 16 32

Linear Quadratic

LP vs. NLP: %Diffe

rence

Repeated MIP

One MIP

Progressive MIP

CPUTime Again,asseeninTable26,theprogressiveMIPwasmuchfasterthanoneMIPthatwasmuchfasterthantheprogressiveMIP.TherepeatedMIPseemedtorunmuchfasterwhenaquadraticstepsizewasused.Table26.30Bus,LooselyConstrained:CPUTime seconds StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 8834.50 7083.96 2871.75 1831.51OneMIP 1000.54 1002.15 1000.51 1000.98ProgressiveMIP 25.99 29.75 20.17 28.86

57Bus,TightlyConstrainedObjectiveValueFunction LP Inthiscase,Table27showsthat5outofthe12approachesfoundinfeasiblesolutions.OnlytherepeatedMIPconsistentlyfoundafeasiblesolution.TheoptimalobjectivevaluefoundbytherepeatedMIPdifferedgreatlybetweenthelinearandquadraticcase.Table27.57Bus,TightlyConstrained:LinearObjectiveValue linearobjective,linearconstraints StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 417.41 417.39 426.67 425.75OneMIP INF INF INF INFProgressiveMIP INF 428.56 426.15 427.96Nolinesopen 424.02 422.71 424.1 423.7

0100020003000400050006000700080009000

16 32 16 32

Linear Quadratic

CPU Tim

e (s)

Repeated MIP

One MIP

Progressive MIP

LPvs.NLPObjectiveFunctionValueDifferences TheconfigurationsfoundbytheLPsolutionwereinfeasiblein9outofthe12differentapproachesasdisplayedinTables28and29.Thissuggeststhatthe57busproblemwithatightcurrentconstraintislikelytobecomeinfeasibleduringswitching.Aconstraintenforcingthevoltageminimumwouldneedtobeaddedtoavoidtheseinfeasibilities.Table28.57Bus,TightlyConstrained:NLPObjectiveValue nonlinearconstraints,linearobjective StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 426.93 427.25 INF INFOneMIP INF INF INF INFProgressiveMIP INF INF 427.62 INFTable29.57Bus,TightlyConstrained:%DifferencebetweenLPandNLPObjectiveValueStepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 2.23 2.31 99.98 99.99OneMIP 99.93 99.68 95.29 99.77ProgressiveMIP 99.95 85.03 0.34 100.00

360

380

400

420

440

460

480

500

16 32 16 32

Linear Quadratic

Objectiv

e Va

lue (LP)

Repeated MIP

One MIP

Progressive MIP

CPUTime Table30revealsthatonceagain,theprogressiveMIPwasmuchfasterthanoneMIPwasmuchfasterthantherepeatedMIPapproach.Table30.57Bus,TightlyConstrained:CPUTime seconds StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 4001.22 6003.74 4528.23 3115.72OneMIP 1015.75 1020.67 1009.96 1011.68ProgressiveMIP 138.03 182.29 80.39 139.19

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

16 32 16 32

Linear Quadratic

LP vs. NLP: %Diffe

rence

Repeated MIP

One MIP

Progressive MIP

0

800

1600

2400

3200

4000

4800

5600

6400

16 32 16 32

Linear Quadratic

CPU Tim

e (s)

Repeated MIP

One MIP

Progressive MIP

57Bus,LooselyConstrainedObjectiveValueFunction LP Table31showsthat6ofthe12approachescouldnotfindafeasiblesolutiontothelooselyconstrained57busproblem.Table31.57Bus,LooselyConstrained:LinearObjectiveValue linearobjective,linearconstraints StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 419.36 416.56 INF 426.51OneMIP INF INF INF INFProgressiveMIP INF 426.41 425.79 424.63Nolinesopen 422.6 422.4 423.3 423.5

LPvs.NLPObjectiveFunctionValueDifferences AsseeninTables32and33,9ofthe12approachescouldnotfindafeasiblesolutionwiththe57busproblemwithloosecurrentconstraint.Table32.57Bus,LooselyConstrained:NLPObjectiveValue nonlinearconstraints,linearobjective StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP INF 427.47 INF INFOneMIP INF INF INF INFProgressiveMIP INF INF 428.93 430.18

050100150200250300350400450500

16 32 16 32

Linear Quadratic

Objectiv

e Va

lue (LP)

Repeated MIP

One MIP

Progressive MIP

Table33.57Bus,LooselyConstrained:%DifferencebetweenLPandNLPObjectiveValueStepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP n/a 2.55 n/a n/aOneMIP n/a n/a n/a n/aProgressiveMIP n/a n/a 0.73 1.29

CPUTime Table34showsthattheprogressiveMIPissignificantlyfasterthantheoneMIPapproach,whichisusuallymuchfasterthandoingarepeatedMIP.Inthiscase,usingtheprogressiveMIPisboththefastestmethodandthemostlikelytofindafeasiblesolution.Table34.57Bus,LooselyConstrained:CPUTime seconds StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 20007.06 7005.45 1031.57 4388.51OneMIP 1016.32 1019.00 1010.00 1010.06ProgressiveMIP 138.00 182.26 71.79 122.51

‐20%

‐15%

‐10%

‐5%

0%

5%

10%

15%

20%

1 2 3 4

LP vs. NLP: %Diffe

rence

Repeated MIP

One MIP

Progressive MIP

0

5000

10000

15000

20000

16 32 16 32

Linear Quadratic

CPU TIm

e (s)

Repeated MIP

One MIP

Progressive MIP

118Bus,TightlyConstrainedObjectiveValueFunction LP AsseeninTable35,thereisonlyoneapproach oneMIP,linearstepsize,16cuts wherethelinearsolvercannotfindafeasiblesolution.Table35.118Bus,TightlyConstrained:LinearObjectiveValue linearobjective,linearconstraints StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 1381.32 1377.03 1385.04 1380.17OneMIP INF 1389.79 1385.13 1390.99ProgressiveMIP 1377.49 1377.63 1381.14 1382.52Nolinesopen 1381.2 1380.4 1388.3 1388.3

LPvs.NLPObjectiveFunctionValueDifferences Tables36and37showthattheNLPwasabletofindafeasiblesolutiontothesameconfigurationoflinesastheLPwheretheLPapproachcouldnotfindafeasiblesolution.Inallothercases,thenonlinearsolutionwaswithin1%ofthelinearsolution.Table36.118Bus,TightlyConstrained:NLPObjectiveValue nonlinearconstraints,linearobjective StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 1388.74 1386.37 1387.64 1384.41OneMIP 1385.19 1391.23 1385.19 1391.23

13501355136013651370137513801385139013951400

16 32 16 32

Linear Quadratic

Objectiv

e Va

lue (LP)

Repeated MIP

One MIP

Progressive MIP

ProgressiveMIP 1383.35 1383.37 1382.15 1382.82Table37.118Bus,TightlyConstrained:%DifferencebetweenLPandNLPObjectiveValueStepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 0.53 0.67 0.19 0.31OneMIP INF 0.10 0.00 0.02ProgressiveMIP 0.42 0.42 0.07 0.02

CPUTime WhiletheprogressiveMIPwasfasterthanoneMIPwhichwasfasterthantherepeatedMIP,theprogressiveMIPapproachwascomparativelymuchfasterinthe57buscasesthanthiscase.Table38showstheprogressiveMIPspeedingupsolutiontimesbetween1.2and2.4x;in57buscase,theprogressiveMIPspedupsolutiontimesbyatleast5xversustheoneMIPapproach.Table38.118Bus,TightlyConstrained:CPUTime seconds StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 7008.08 3010.86 4089.92 4014.06OneMIP 1106.94 1149.24 1033.84 1043.54ProgressiveMIP 591.78 936.41 422.64 652.28

‐20%

‐15%

‐10%

‐5%

0%

5%

10%

15%

20%

16 32 16 32

Linear Quadratic

LP vs. NLP: % Differen

ce

Repeated MIP

One MIP

Progressive MIP

118Bus,LooselyConstrainedObjectiveValueFunction LP AsseeninTable39,the118bussolutionsmainlydifferinthelinearstepsize,repeatedMIPapproachversusalloftheotherapproaches.Table39.118Bus,LooselyConstrained:LinearObjectiveValue linearobjective,linearconstraints StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 1291.18 1301.09 1314.15 1316.88OneMIP 1313.85 1311.76 1313.47 1314.60ProgressiveMIP 1306.34 1307.86 1314.62 1315.33Nolinesopen 1307.7 1311.8 1310.7 1314.6

0

1000

2000

3000

4000

5000

6000

7000

16 32 16 32

Linear Quadratic

Repeated MIP

One MIP

Progressive MIP

1280

1285

1290

1295

1300

1305

1310

1315

1320

16 32 16 32

Linear Quadratic

Objectiv

e Va

lue (LP)

Repeated MIP

One MIP

Progressive MIP

LPvs.NLPObjectiveFunctionValueDifferences Therewasonelineconfigurationoutthe12approacheswherethenonlinearsolvercouldnotfindafeasiblesolution,seeninTables40and41.Otherwise,thenonlinearanswerswerewithin2%orlessofthelinearanswers.Likeinsomeoftheothercases,thenonlinearsolutionfortheconfigurationfoundwiththerepeatedMIP,linearstepsizeismoreconsistentwiththeotherapproachesthanthelinearsolution.Ifweexcludethelinearstepsize,repeatedMIPapproachandtheinfeasibleanswer,thenonlinearsolutioniswithin1%ofthelinearsolution.Table40.118Bus,LooselyConstrained:NLPObjectiveValue nonlinearconstraints,linearobjective StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 1317.14 1319.32 1322.53 INFOneMIP 1318.49 1315.89 1318.77 1315.55ProgressiveMIP 1317.29 1317.29 1325.20 1322.33Table41.118Bus,LooselyConstrained:%DifferencebetweenLPandNLPObjectiveValuesStepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 1.97 1.38 0.63 99.99OneMIP 0.35 0.31 0.40 0.07ProgressiveMIP 0.83 0.72 0.80 0.53

CPUTime AsdisplayedinTable42,boththeoneMIPandprogressiveMIPapproachesrunmuchfasterthantherepeatedMIPapproaches;However,theprogressiveMIPisslowerthantheoneMIPapproachwhenthelinearstepsizeisused.

0%2%4%6%8%10%12%14%16%18%20%

16 32 16 32

Linear Quadratic

LP vs. NLP: % Differen

ce

Repeated MIP

One MIP

Progressive MIP

Table42.118Bus,LooselyConstrained:CPUTime seconds StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32RepeatedMIP 13009.89 13033.29 5003.83 6012.06OneMIP 1075.83 1134.47 1014.72 1038.70ProgressiveMIP 1159.56 2642.57 666.49 818.48

010002000300040005000600070008000900010000110001200013000

16 32 16 32

Linear Quadratic

CPU Tim

e (s)

Repeated MIP

One MIP

Progressive MIP

6. SummaryFromthisstudy,weseethattransmissionswitchingprovidescostbenefitsin

ACsystems.Tables43and44showthatthemagnitudeofthiscostsavingsisverydependentonthecurrentconstraints;oneobtainsmoresavingswithlowerlinelimits.Wefindthatthelinearapproximationgenerallyachievesaboutthesameamountofsavingsastheoriginalnonlinearprogram;however,therearecaseswherethelinearprogramfindsabetteranswer likethe14buscasewithloosecurrentlimits oraworseanswer likethe14buscasewithtightcurrentlimits .Inthiscase,openingthefirstlineprovidesthemostofthebenefits.TheconflationoftheMIPplusthechangingfixedpointsoccasionallyleadstoabetteranswerwithallowingonelineopenratherthanfivelinesopen.Thisisoftenduetothefivelinesopenproblemchoosinganetworkconfigurationthatisbetterinthefirstiterationthanopeningonelinebutworsewhentheproblemhasfinallyconverged.Table43.%SavingswithuptoonelineopenCurrentConstraint Method 14 30 57 118Loose Linear 3% 3% 0% 0% IPOPT 2% 4% 1% 0%Tight Linear 11% 3% 0% 0% IPOPT 13% 5% 3% 0%Table44.%SavingswithuptofivelinesopenCurrentConstraint Method 14 30 57 118Loose Linear 4% 2% 1% 1% IPOPT 2% 1% 0% 0%Tight Linear 10% 3% 1% 0% IPOPT 12% 3% 1% 0%7. Conclusions

TheoneMIPapproachsolvesthe14and30busproblemsquickerthantherepeatedMIPapproachandwithin4%orlessoftherepeatedMIPobjectivevalue.However,whiletheoneMIPapproachhasfastperformanceinthe57and118busproblems,boththeoneandrepeatedMIPapproachesfindworsesolutionsthannotswitchinganylinesinmanyofthecases.Inthelargernetworks,itappearsthatthereareeithermultipleoptimalsolutionsormultiplenetworkconfigurationsthathaveaverysimilarcost. ThelinearizedACOPFismuchfasterthantheunmodifiedACOPFandusuallyfindssolutionswithin1%oftheACOPF.TheprogressiveandoneMIPapproachesaremuchfasterthantherepeatedMIPapproachandgenerallygiveanswerswithin1%oftherepeatedMIPapproach.However,thereareinstanceswherethe

linearizedACOPFfindsanoptimalnetworkconfigurationthatisnotfeasibleintheunmodifiedACOPF,duetothelinearizedACOPFfindingasolutionwithvoltagesthatarebelowtheminimumthreshold.

REFERENCESM.B.Cain,R.P.O’Neill,A.Castillo.“HistoryofACOptimalPowerFlowProblems,”FERCStaffPaper,2012.C.Campaigne,P.A.Lipka,R.P.O’Neill,andS.S.Oren.“TestingStepSizeLimitsforSolvingtheLinearizedCurrentVoltageACOptimalPowerFlow,”FERCStaffPaper,2013.E.B.Fisher,R.P.O’Neill,andM.C.Ferris.“Optimaltransmissionswitching,”IEEETrans.PowerSyst,vol.23,no.3,pp.1346‐1355,Aug.2008.J.D.Fuller,R.Ramasra,andA.Cha,“Fastheuristicsfortransmissionlineswitching,”IEEETrans.PowerSyst.,vol.27,no.3,pp.1377–1386,Aug.2012.K.W.Hedman,R.P.O’Neill,E.B.Fisher,andS.S.Oren,“Optimaltransmissionswitching–sensitivityanalysisandextensions,”IEEETrans.PowerSyst,vol.23,no.3,pp.1469‐1479,Aug.2008.K.W.Hedman,R.P.O’Neill,E.B.Fisher,andS.S.Oren,“Optimaltransmissionswitchingwithcontingencyanalysis,”IEEETrans.PowerSyst,vol.24,no.3,pp.1577‐1586,Aug.2009.P.A.Lipka,R.P.O’Neill,andShmuelOren,“DevelopingLineCurrentmagnitudeConstraintsforIEEETestProblems,”FERCStaffPaper,2012.R.P.O’Neill,AnyaCastillo,M.B.Cain.“FormulationtotheACOptimalPowerFlowProblemandLinearizations,”FERCStaffPaper,2012.J.Ostrowski,J.Wang,andC.Liu,“Anti‐IslandinginTransmissionSwitching”,2012inreview .Exploitingsymmetryintransmissionlinesfortransmissionswitching,”IEEETrans.PowerSyst.,vol.27,no.3,pp.1708–1709,Aug.2012.MehrdadPirnia,RichardP.O’Neill,PaulaA.Lipka,ClayCampaigne,“AComputationalStudyofLPApproximationtotheIVACOPFFormulation,”FERCStaffPaper,2012.

T.PotluriandK.W.Hedman,“ImpactsofTopologyControlontheACOPF,”inIEEEPESGeneralMeeting2011,SanDiego,CA,July2012.Ruiz,P.A.,Foster,J.M.;Rudkevich,A.;Caramanis,M.C.,“TractableTransmissionTopologyControlUsingSensitivityAnalysis,”IEEETrans.PowerSyst.,vol.27,No.3,pp.1550‐1559,Aug.2012UniversityofWashington,DeptofElectricalEngineering,“PowerSystemTestCaseArchive,”May2010.Available:http://www.ee.washington.edu/research/pstca/index.html.R.D.Zimmerman,C.E.Murillo‐Sánchez,andR.J.Thomas,“MATPOWRStead‐StateOperations,PlanningandAnalysisToolsforPowerSystemsResearchandEducation,”IEEETrans.PowerSyst.,vol.26,no.1,pp.12‐19,Feb.2011.

Appendix:DetailedNumericalResultsA.1FULLREPEATEDMIPVERSUSMIPONLYONFIRSTPASSRESULTS14busproblem.TablesA.1andA.2showhowtherepeatedMIPcomparestotheapproachofusingonlyoneMIP.Inthetightlyconstrainedcase,theobjectivevaluesareverysimilarbetweenbothapproaches,differingatthemostby0.25%.UsingoneMIPhasasignificanttimeadvantage.Thelinesopeneddifferslightly,withtherepeatedMIPchoosingtoopentwolinesinsomechaseswhiletheapproachusingoneMIPonlyresultsinonelinebeingopened.ThelooselyconstrainedcasedisplaysmorevariabilitybetweentheapproachesusingoneMIPandtherepeatedMIP,althoughthereisalsomorevariabilitybetweenstepsize/numberofcutsapproaches.However,thebiggestdifferenceinobjectivefunctionvaluesis3.5%.TheapproachusingoneMIPisatleasttwiceasfastastherepeatedMIPapproach.ThelinesopenvarybetweenoneMIPandrepeatedMIPwiththerepeatedMIPapproachopeningline3‐4butneveropeningline6‐12,whichtheoneMIPapproachopensineverycase.However,bothcasesopenline2‐5,andthelinesopenedintheoneMIPapproachexceptforline6‐12areopenedinoneoftherepeatedMIPapproaches.TableA.1.Repeatedvs.OneMIP:14Bus,TightCurrentConstraint StepSizeFunction Linear Quadratic NumberofCuts 16 32 16 32 #LinesOpen

ObjectiveValueNoLinesOpen 0 105.69 106.53 105.69 106.53RepeatedMIP 5 95.02 95.04 95.10 95.04

OneMIP 5 95.25 95.04 95.25 95.04CPUTime RepeatedMIP 5 34.7 101.0 26.7 59.1 OneMIP 5 8.4 20.6 15.8 22.6

No.ofLinesOpenedRepeatedMIP 5 1 1 1 1OneMIP 5 2 1 2 1

LinesOpened RepeatedMIP 5 4‐7 4‐7 4‐7 4‐7 OneMIP 5 2‐5 4‐7 2‐5 4‐7 4‐7 4‐7

TableA.2:Repeatedvs.OneMIP:14Bus,LooseCurrentConstraint StepSizeFunction Linear Quadratic NumberofCuts 16 32 16 32 #LinesOpen

ObjectiveValueNoOpenLines 0 85.94 85.63 85.15 86.13RepeatedMIP 5 82.53 82.10 84.65 84.38

OneMIP 5 84.49 85.08 84.49 85.08CPUTime RepeatedMIP 5 258.6 152.4 126.0 165.3 OneMIP 5 22.1 59.4 23.1 40.2

No.ofLinesOpenedRepeatedMIP 5 2 5 5 5OneMIP 5 5 5 5 5

LinesOpenedRepeatedMIP 5 2‐5 2‐5 2‐5 2‐5 3‐4 3‐4 3‐4 3‐4

4‐9 4‐5 4‐5 9‐14 4‐7 4‐7 12‐13 4‐9 4‐9 OneMIP 2‐5 2‐5 2‐5 2‐5 4‐5 4‐5 4‐5 4‐5 4‐7 4‐7 4‐7 4‐7 4‐9 4‐9 4‐9 4‐9 6‐12 6‐12 6‐12 6‐1230busproblem.TableA.3showsthatthetightcurrentconstraintcasehasuptoa2.3%gapbetweentheoneandrepeatedMIPapproaches.Thereisaclearadvantagetoopeninglines.UsingoneMIPisatleasttwiceasfastasusingaMIPateverystep.Inbothcases,line6‐28isopened,buttheotherfourlinesarecompletelydifferentinthecaseoflinearandquadraticstepsizewith16cuts;3linesaredifferentforthelinearandquadraticstepsizewith32cuts.InTableA.4,theloosecurrentconstraintcasehasuptoa2%gapbetweentheoneandrepeatedMIPapproaches.However,theoneMIPapproachalwaysfindsamorecostlyanswerthantherepeatedMIPapproach;insomecases,openinglineshasaminorimpact.Again,theoneMIPapproachismuchfasterthantherepeatedMIPapproach.Inbothtightandloosecases,thequadraticstepsizewithrepeatedMIPappearstorunmuchfasterthanthelinearstepsize.Allcasesopenthelineconnectingbuses6and28;theotherlinesopenedvarygreatlybetweentheoneandrepeatedMIPapproachesaswellasthedifferentstepsizeandnumberofcutsapproaches.

TableA.3:Repeatedvs.OneMIP:30Bus,TightCurrentConstraint StepSizeFunction Linear Quadratic NumberofCuts 16 32 16 32 #LinesOpen

ObjectiveValueWithoutOpeningLines 0 6.02 6.08 6.02 6.08

RepeatedMIP 5 5.83 5.83 5.96 5.93 OneMIP 5 5.94 5.97 5.96 5.97CPUTime RepeatedMIP 5 3908.0 6886.7 1718.0 2409.3 OneMIP 5 763.2 1001.4 806.5 1001.0#ofLinesOpened RepeatedMIP 5 5 5 5 3 OneMIP 5 5 5 5 5LinesOpened RepeatedMIP 5 6‐28 6‐28 6‐28 6‐28 9‐11 9‐11 9‐11 9‐11 10‐21 10‐21 10‐21 25‐27 10‐22 10‐22 10‐22 23‐24 16‐17 23‐24 OneMIP 5 6‐28 6‐28 6‐28 6‐28 10‐20 10‐20 10‐20 10‐20 12‐15 12‐15 12‐15 12‐15 14‐15 16‐17 14‐15 16‐17 25‐27 25‐27 25‐27 25‐27TableA.4.Repeatedvs.OneMIP:30Bus,LooseCurrentConstraint StepSizeFunction Linear Quadratic NumberofCuts 16 32 16 32 #LinesOpen


RepeatedMIP 5 5.82 5.83 5.93 5.81 OneMIP 5 5.94 5.89 5.95 5.90CPUTime RepeatedMIP 5 8834.5 7084.0 2871.7 1831.5 OneMIP 5 1000.5 1002.1 1000.5 1001.0#ofLinesOpened RepeatedMIP 5 5 5 5 4 OneMIP 5 5 4 5 4LinesOpened RepeatedMIP 5 4‐12 6‐28 6‐9 6‐9 6‐28 9‐11 6‐28 6‐28 16‐17 10‐21 9‐10 25‐27 18‐19 10‐22 9‐11 27‐19 23‐24 23‐24 25‐27 OneMIP 5 6‐28 6‐28 6‐28 6‐28 12‐15 10‐21 12‐15 10‐21 14‐15 14‐15 14‐15 14‐15 19‐20 15‐18 19‐20 15‐18 25‐27 25‐27

57busproblem.Inthetightlyconstrainedcase,therepeatedMIPfindsafeasiblesolutionwhiletheoneMIPapproachdoesnot,displayedinTableA.5.Thelineschosentoopenareverydifferentbetweenthetwocases.Intheboththetightlyandlooselyconstrainedcases TableA.6 ,theoneMIPapproachdoesnotfindafeasiblesolution,whiletherepeatedMIPfindsafeasiblesolutionin3ofthe4approaches.ThereasontheoneMIPapproachfindsinfeasiblesolutionsisthatthenetworkconfigurationitchoosesinthefirstiterationisinfeasible,butinthefirststage,moreinfeasibleareaisinthelinearprogramthatiscutoffbyiterativevoltageconstraintslater.TableA.5.Repeatedvs.OneMIP:57Bus,TightCurrentConstraint StepSizeFunction Linear Quadratic NumberofCuts 16 32 16 32 #LinesOpen ObjectiveValue WithoutOpeningLines 0 424.02 422.71 424.1 423.7 RepeatedMIP 5 417.4 417.4 426.7 425.8 OneMIP 5 INF1 INF INF INFCPUTime RepeatedMIP 5 4001 6003 4528 3116 OneMIP 5 10167 1021 1010 1012#ofLinesOpened RepeatedMIP 5 5 4 3 5 OneMIP 5 5 5 5 5LinesOpened RepeatedMIP 5 1‐15 3‐4 9‐10 2‐3 2‐3 3‐15 9‐13 3‐4 11‐13 22‐38 11‐13 3‐15 13‐15 53‐54 22‐38 53‐54 54‐55 OneMIP 5 9‐10 9‐13 9‐10 9‐13 9‐55 10‐51 9‐55 10‐51 11‐43 13‐49 11‐43 13‐49 14‐46 14‐46 14‐46 14‐46 35‐36 38‐44 35‐36 38‐44

TableA.6.Repeatedvs.OneMIP:57Bus,LooseCurrentConstraint StepSizeFunction Linear Quadratic NumberofCuts 16 32 16 32 #LinesOpen ObjectiveValue

NoLinesOpen 0 422.6 422.4 423.3 423.5RepeatedMIP 5 419.4 416.6 INF 426.5

OneMIP 5 INF INF INF INFCPUTime RepeatedMIP 5 20007 7006 1032 4389 OneMIP 5 1016 1019 1010 1010No.ofLinesOpened RepeatedMIP 5 5 5 5 5

OneMIP 5 5 5 5 5LinesOpened RepeatedMIP 5 3‐4 6‐7 8‐9 1‐2

14‐15 9‐11 13‐49 1‐15 19‐20 9‐13 14‐46 11‐41 34‐35 12‐13 15‐45 12‐17 49‐50 12‐16 49‐50 54‐55 OneMIP 5 9‐10 9‐10 9‐10 9‐10 9‐55 9‐13 9‐55 9‐13 14‐46 10‐51 14‐46 10‐51 25‐30 11‐13 25‐30 11‐13 44‐45 14‐46 44‐45 14‐46

118busproblem.Inthetightlyconstrainedcase TableA.7 ,therepeatedandoneMIPapproachesfindsolutionswithin1%ofeachotherifthecasewheretheoneMIPapproachfindsaninfeasiblesolutionisnotincluded.TheoneMIPapproachismorethan2.5timesfasterthantherepeatedMIPapproach.ThereisalargevarianceinwhichlinesareselectedtobeopenbetweentherepeatedandoneMIPapproaches,andbetweenthedifferentstepsizeandnumberofcutsapproaches.TheoneMIPapproachfindshighercostssolutionsthattherepeatedMIPapproach,andintwocasesfindsworsesolutionsthanwhennolinesareopen.Inthelooselyconstrainedcase TableA.8 ,feasiblesolutionsarefoundinbothrepeatedandoneMIPapproaches,andtheanswersarewithin2%ofeachother.However,boththerepeatedMIPandoneMIPapproachesfindsomesolutionsthatareworsethannotopeninganylines.TheoneMIPapproachrunsfivetimesfasterormorethantherepeatedMIPapproach.Thelinesopenedvarygreatlybetweenthedifferentapproaches,althoughsomesamelinesshowupinseveraldifferentapproaches.TableA.7.Repeatedvs.OneMIP:118Bus,TightCurrentConstraint

StepSizeFunction Linear Quadratic

NumberofCuts 16 32 16 32

#LinesOpen

ObjectiveValue WithoutOpeningLines 0 1381.2 1380.4 1388.3 1388.3RepeatedMIP 5 1381.3 1377.0 1385.0 1380.2

OneMIP 5 INF 1389.8 1385.1 1391CPUTime RepeatedMIP 5 7008 3011 4090 4014 OneMIP 5 1107 1149 1034 1044No.ofLinesOpened RepeatedMIP 5 5 5 5 5

OneMIP 5 5 5 5 5LinesOpened RepeatedMIP 5 60‐61 5‐6 34‐36 42‐49

61‐62 6‐7 35‐36 60‐62 61‐64 47‐49 42‐49 61‐62 75‐77 48‐49 77‐82 92‐94 77‐82 77‐80 82‐83 93‐94 OneMIP 5 32‐113 8‐30 32‐113 8‐30 77‐82 47‐49 77‐82 47‐49 92‐94 48‐49 92‐94 48‐49 92‐100 92‐93 92‐100 92‐93 109‐110 94‐100 109‐110 94‐100

TableA.8.Repeatedvs.OneMIP:118Bus,LooseCurrentConstraint

StepSizeFunction Linear Quadratic

NumberofCuts 16 32 16 32

#LinesOpen


RepeatedMIP 5 1291.2 1301.1 1314.2 1316.9 OneMIP 5 1313.9 1311.8 1313.5 1314.6CPUTime RepeatedMIP 5 13010 13033 5004 6012 OneMIP 5 1075.8 1134.5 1014.7 1038.7#ofLinesOpened RepeatedMIP 5 5 5 5 4 OneMIP 5 5 0 5 0LinesOpened RepeatedMIP 5 1‐2 1‐2 41‐42 65‐68 8‐30 54‐55 42‐49 82‐83 11‐12 64‐65 49‐66 99‐100 65‐68 76‐118 68‐81 110‐112 94‐100 80‐99 110‐112 OneMIP 5 61‐64 n/a 61‐64 n/a 75‐118 75‐118 77‐80 77‐80 92‐102 92‐102 100‐103 100‐103A.2PROGRESSIVEMIPIntheTablesthatfollow,theLinearObjValueistheobjectivevalueofthelineariterativesolver.TheNLPObjValueisthevalueofthenonlinearsolutionwhenthenetworkisfixedastheoptimalnetworkfoundinthelineariterativesolver.TheMIPgapisthegapbetweenthebest‐knownlinearsolutionandthefinalMIPanswer.NewOpenedLinedesignateswhichlinewasopenedintheproblem.Forexample,thefirstproblemonlyallowsonelineopen.Then,thenextproblemfixesthatlineasopenandseesifthereisanotherlinethatwouldbebeneficialtoopen,andsoforth.14bus.Thelinearandnonlinearsolutionsinthe14buscase–bothforthetightandlooseconstraints A.9andA.10 ‐arewithin0.6%ofeachotherexceptforonecasewherethelinearsolverfoundaninfeasiblesolution.Liketherepeated/oneMIPapproaches,thelinesopenedinthetightcurrentconstraintcasewere4‐7and2‐5.Theloosecurrentconstrainedcaseconsistentlyopenslines2‐5,3‐4,and4‐7,butdiffersonthe4thand5thlinesopened anda5thlineisonlyopenedinonecase .Thelinearsolutionwasveryclosetothenonlinearsolutionforthecorresponding

network.ThetotalCPUtimeforallthelinearcasesupto5linesopenisclosetotheCPUtimeofusingoneMIP.TableA.9.ProgressiveMIP:14Bus,TightCurrentConstraint

StepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32NumberofLines

Open ObjValue 1 95.05 95.07 95.06 95.04 2 95.04 95.06 95.06 94.98 3 95.33 95.07 95.41 94.99 4 95.14 95.06 95.41 95.00 5 95.19 INF 95.42 95.04NLPObjValue 1 95.06 95.04 95.05 95.07 IPOPT 2 95.06 95.00 95.03 95.06 3 95.41 95.05 95.40 95.06 4 95.41 94.99 95.28 95.06 5 95.43 94.95 95.26 95.06CPUTime 1 1.3 1.9 1.0 1.9 2 1.1 1.6 0.7 1.4 3 1.7 2.6 1.5 2.9 4 1.4 4.1 1.7 3.7 5 1.7 5.0 1.4 4.3NLPCPUTime 1 1.1 1.1 0.9 3.2 IPOPT 2 1.3 1.3 0.9 1.0 3 2.2 1.3 2.4 1.2 4 4.5 1.6 2.1 2.1 5 5.7 2.0 6.7 1.4LinearIterations 1 4 4 4 4 2 4 3 4 3 3 2 4 3 3 4 3 3 4 2 5 2 3 2 4MIPGap 1 1.5E‐15 2.7E‐15 1.5E‐15 2.7E‐15 2 2.8E‐04 9.5E‐15 1.0E‐14 8.4E‐15 3 4.4E‐15 1.3E‐05 3.9E‐15 2.8E‐14 4 1.3E‐14 9.5E‐04 4.2E‐04 9.6E‐04 5 2.2E‐14 8.5E‐04 1.4E‐14 3.9E‐14NewOpenedLine

1 4‐7 4‐7 4‐7 4‐72 None None None None

3 2‐5 None 2‐5 None 4 None None None None 5 None None None None

TableA.10.ProgressiveMIP:14Bus,LooseCurrentConstraint StepSizeFunction Linear Quadratic

NumberofCuts 16 32 16 32NumberofLines

Open ObjValue 1 85.17 85.17 85.28 85.61 2 85.03 84.97 85.11 85.20 3 85.04 85.08 85.03 85.09 4 84.98 85.24 85.08 85.36 5 85.01 85.21 85.09 85.36NLPObjValue 1 85.61 85.61 85.61 85.29 IPOPT 2 85.25 85.20 85.20 85.01 3 85.10 85.10 85.09 85.10 4 85.09 85.36 85.10 85.20 5 85.02 85.36 85.10 85.23CPUTime 1 0.9 1.3 0.9 1.2 2 0.8 1.3 2.1 1.3 3 0.8 1.0 2.0 1.0 4 0.6 0.8 2.1 0.8 5 1.2 0.7 2.6 0.8NLPCPUTime 1 1.0 1.0 0.8 1.2 IPOPT 2 1.0 1.9 1.2 1.8 3 3.4 4.1 0.7 1.7 4 1.8 3.5 0.7 2.6 5 3.2 4.8 0.6 1.4LinearIterations 1 6 6 4 4 2 4 4 5 4 3 3 3 3 3 4 2 2 4 3 5 3 3 3 2MIPGap 1 3.4E‐15 8.5E‐16 3.4E‐15 8.5E‐16 2 3.0E‐14 3.1E‐15 2.8E‐14 8.6E‐16 3 1.4E‐15 1.0E‐15 1.5E‐15 3.6E‐15 4 4.6E‐15 4.8E‐15 2.9E‐15 1.9E‐14 5 1.4E‐15 3.7E‐14 4.5E‐15 1.1E‐14NewOpenedLine 1 2‐5 2‐5 2‐5 2‐5 2 3‐4 3‐4 3‐4 3‐4 3 4‐7 4‐7 4‐7 4‐7 4 None 9‐14 12‐23 9‐14 5 4‐5,4‐9 None None None

30bus.Inthe30buscasewithtightcurrentconstraints A.11 ,thelinearobjectivevalueiswithin3%oftheobjectivevalueexceptfortwocases–onewheretheydifferby12%andonewherethesolutionisinfeasible.Withloosecurrentconstraints A.12 ,thelinearobjectivevalueiswithin2%ofthenonlinearobjectivevalue.TherepeatedandoneMIPapproachessharesomeofthesamelinesasopenedbytheprogressiveMIPapproach,butthelinesopenedarenotexactlythesamebetweentheapproachesorwithinthedifferentstepsizefunctionsandnumberofcutswithinthesameapproach.Inthe30busproblem,onlyonesolutionisfoundtobeinfeasible–whenlines6‐28,10‐21,and9‐11areopenedinthetightlyconstrainedcasewithquadraticstepsize,32cuts,andupto4linesopen.TableA.11.ProgressiveMIP:30Bus,TightCurrentConstraint


Open ObjValue 1 5.92 5.93 5.92 5.93 2 5.88 5.89 5.93 5.89

3 5.89 5.90 5.94 5.89 4 5.90 5.90 5.92 5.94 5 5.89 5.90 5.93 5.93NLPObjValue 1 5.93 6.10 5.97 6.76 IPOPT 2 5.94 5.94 5.97 5.94 3 5.94 5.96 5.94 5.94 4 5.95 5.95 5.95 INF 5 5.94 5.98 5.95 6.07CPUTime 1 6.4 8.8 6.7 9.1 2 5.4 7.7 3.6 6.5

3 4.7 6.9 3.8 5.2 4 4.6 5.5 3.3 19.1 5 3.7 5.3 3.2 1.5NLPCPUTime 1 3.7 2.2 2.4 1.9 IPOPT 2 11.8 8.2 7.2 6.0 3 7.5 5.6 7.3 6.8 4 4.8 4.6 6.6 2.1 5 6.6 4.9 7.4 1.8LinearIterations 1 4 3 4 3 2 2 4 2 4 3 4 2 4 2 4 2 2 3 9

5 2 2 3 2MIPGap 1 1.7E‐07 1.7E‐07 1.7E‐07 1.7E‐07 2 1.7E‐07 1.7E‐07 1.7E‐07 1.7E‐07 3 1.7E‐07 1.7E‐07 1.7E‐07 1.7E‐07 4 1.7E‐07 1.7E‐07 1.7E‐07 1.7E‐07 5 1.8E‐07 1.7E‐07 1.7E‐07 7.6E‐09NewOpenedLine 1 6‐28 6‐28 6‐28 6‐28 2 10‐21 10‐21 10‐17 10‐21 3 14‐15 15‐18 25‐27 None 4 15‐18 14‐15 18‐19 9‐11 5 12‐15 None 12‐14 15‐18TableA.12.ProgressiveMIP:30Bus,LooseCurrentConstraint


Open ObjValue 1 5.92 5.92 5.92 5.92 2 5.92 5.92 5.93 5.92

3 5.90 5.89 5.89 5.89 4 5.90 5.90 5.90 5.90 5 5.89 5.90 5.90 5.91NLPObjValue 1 5.94 5.92 5.93 6.04 IPOPT 2 5.93 5.92 5.94 5.92 3 5.94 5.94 5.96 5.94 4 5.96 5.95 5.95 5.97

5 5.98 5.95 5.95 5.98CPUTime 1 4.6 5.5 4.5 6.2 2 5.5 7.7 3.5 6.4

3 4.8 6.0 4.5 6.2 4 5.1 5.7 3.7 5.5 5 5.9 4.8 3.9 4.5NLPCPUTime 1 6.8 6.6 8.5 2.8 IPOPT 2 3.7 3.8 8.9 4.0 3 18.5 3.2 8.5 2.8 4 3.5 6.5 5.3 11.7

5 5.7 7.6 8.1 3.6LinearIterations 1 4 2 4 2 2 2 2 2 2 3 4 2 4 2 4 10 2 4 2 5 3 2 4 2

MIPGap 1 1.7E‐07 1.7E‐07 1.7E‐07 1.7E‐07 2 1.7E‐07 1.7E‐07 1.7E‐07 1.7E‐07 3 1.7E‐07 1.7E‐07 1.7E‐07 1.7E‐07

4 1.7E‐07 1.7E‐07 1.7E‐07 1.7E‐07 5 1.7E‐07 1.7E‐07 1.7E‐07 1.7E‐07NewOpenedLine 1 6‐28 24‐25 6‐28 24‐25 2 25‐27 6‐28 19‐20 6‐28 3 10‐21 10‐21 10‐21 10‐21 4 10‐20 14‐15 10‐20

527‐30,18‐19 12‐14

57bus.The57busproblem A.13andA.14 exhibitssomepeculiarities.Therearemanycaseswherethelineariterativesolverfindswhatitbelievestobeafeasiblesolution,butthenonlinearsolverrevealsthattheconfigurationisnotfeasible usuallyduetoviolatingtheminimumvoltageconstraint .Outofthe20differenttestsperformedforthetightcurrentcase,only8findafeasiblesolution.However,thisapproachismuchfasterthaneventheoneMIPapproachanddoesnotperformanyworsethantheoneMIPapproach.Thetightlyconstrainedproblemappearstobeverysensitivetowhatapproachisused.Thefirstthreeproblems,whereupto1,then2,then3linesareallowedopenallhavethesamenetworkconfiguration;however,thenonlinearsolverfindssomesolutionsthatarefeasibleandsomethatareinfeasibleforthe2and3linesopencases.Thisislikelyduetodifferentstartingpointsbeingusedinthenonlinearsolver.Inaddition,itissurprisingthatsomeoftheapproachesrecommendopeningadditionallinesevenwhenitappearstoincreasethetotalcost seethelinearstepsizefunctionwith32cuts;eachadditionallineopenincreasesthecost,althoughthesolverhastheoptionnottoopenanyadditionallines0.Intheloosecurrentcase,only14ofthe20testsfindafeasiblesolution.Thelooselyconstrainedcasealsoappearstobesensitivetowhatstepsizeandnumberofcutsapproachisused.Italsoopensthesamethreelinesinallapproaches,butforupto2or3linesopen,thenonlinearsolvercanfindafeasiblesolutionwiththegivenconfigurationinsomeapproachesandnotinothers,likelyduetothestartingpointused.Inaddition,thelinearapproachalsoopenslinesinsomecaseswhenitincreasesthetotalcost seethequadraticstepsizefunctionwith16cuts,forexample .

TableA.13.ProgressiveMIP:57Bus,TightCurrentConstraintStepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32NumberofLines

Open ObjValue 1 423.10 421.42 423.95 424.04 2 423.18 422.85 422.79 422.84

3 422.14 424.44 423.66 424.45 4 427.76 424.84 425.20 423.44 5 INF 428.56 426.15 427.96NLPObjValue 1 429.56 429.56 429.56 429.56 IPOPT 2 INF INF 430.28 430.30 3 427.43 INF INF INF 4 INF INF INF 426.17

5 INF INF 427.62 INFCPUTime 1 37.1 60.1 35.6 52.4 2 20.5 36.5 9.4 12.7


5 7.3 62.4 32.1 9.8LinearIterations 1 4 7 4 4 2 4 10 3 3 3 4 20 4 5 4 20 3 5 3 5 20 3 3 20MIPGap 1 2.5E‐15 5.1E‐14 2.5E‐15 5.1E‐14 2 1.5E‐14 1.4E‐14 7.3E‐04 1.0E‐03 3 2.0E‐14 1.9E‐15 9.6E‐16 1.9E‐15

4 9.6E‐16 3.3E‐15 3.6E‐15 4.7E‐04 5 3.4E‐13 1.9E‐15 7.9E‐04 6.8E‐16NewOpenedLine 1 3‐4 3‐4 3‐4 3‐4 2 54‐55 54‐55 53‐54 53‐54 3 22‐38 22‐38 22‐38 22‐38 4 32‐34 1‐15 1‐15 14‐15 5 20‐21 3‐15 12‐13 13‐14

TableA.14.ProgressiveMIP:57Bus,LooseCurrentConstraintStepSizeFunction Linear QuadraticNumberofCuts 16 32 16 32NumberofLines

Open ObjValue 1 421.36 422.93 423.09 423.27 2 422.19 422.92 423.42 422.75

3 421.99 421.78 423.61 423.34 4 428.17 422.95 424.81 424.44 5 INF 426.41 425.79 424.63NLPObjValue 1 425.14 425.14 425.14 425.14 IPOPT 2 426.43 426.43 426.44 425.10 3 426.21 INF 426.51 INF 4 INF 432.88 INF 426.55 5 INF INF 428.93 430.18CPUTime 1 33.1 48.0 31.3 48.7 2 29.2 26.8 8.6 14.1

3 18.2 37.1 13.3 23.5 4 31.5 32.1 10.8 17.2 5 26.1 38.3 7.8 19.1NLPCPUTime 1 26.8 19.9 21.6 27.5 IPOPT 2 53.0 22.2 53.1 29.3 3 28.8 19.6 39.5 79.5 4 19.0 47.9 10.6 27.0 5 11.8 17.9 70.3 59.3LinearIterations 1 4 4 4 4 2 10 3 4 3 3 6 10 4 4 4 20 11 5 3 5 20 20 3 3MIPGap 1 9.4E‐15 4.9E‐13 9.4E‐15 4.9E‐13 2 1.4E‐14 6.0E‐15 7.4E‐04 7.4E‐04 3 2.7E‐16 5.7E‐15 5.1E‐15 3.6E‐15 4 1.8E‐15 1.8E‐15 0.0E 00 2.7E‐15 5 1.4E‐12 9.5E‐04 8.1E‐04 4.1E‐16NewOpenedLine 1 3‐4 3‐4 3‐4 3‐4 2 54‐55 54‐55 54‐55 54‐55 3 22‐38 22‐38 22‐38 22‐38 4 32‐34 14‐15 1‐15 1‐15 5 21‐22 13‐14 12‐13 9‐11

118bus.Inthetightlyconstrainedproblem A.15 ,thelinearobjectivevalueiswithin1%orlessofthenonlinearobjectivevalueexceptforonecase upto2linesopen,32cuts,quadraticstepsize .Thefirstlineopenedisthesameinallcases.The2nd,3rd,and4thlinesopenedareconsistentwithinthesamestep‐sizeapproach.However,thenonlinearprogramcannotfindafeasiblesolutioninthe2linesopen,32cuts,quadraticstepsizecasewhileitcanfindafeasiblesolutioninthe2linesopen,16cuts,quadraticstepsizecasewiththesamenetworkconfiguration;thisdifferenceislikelyduetothedifferenceinstartingpoints.Inthelooselyconstrainedproblem A.16 ,thedifferencebetweentheobjectivevaluethelineariterativeprogramfoundandthenonlinearprogramfoundislessthan1%,andtherewerenotanyinfeasiblesolutions.Thefirstlineopenedisconsistentinallapproaches,whiletheotherlinesthatareopeneddifferbetweenapproaches.TableA.15.ProgressiveMIP:57Bus,TightCurrentConstraint


Open ObjValue 1 1379.12 1378.16 1385.68 1385.79 2 1377.59 1378.33 1385.31 1384.70

3 1377.37 1376.77 1381.49 1383.92 4 1376.41 1376.80 1382.34 1382.95 5 1377.49 1377.63 1381.14 1382.52NLPObjValue 1 1385.82 1385.82 1385.82 1385.97 IPOPT 2 1384.13 1384.13 1385.63 INF 3 1383.26 1383.27 1384.40 1383.95 4 1383.05 1383.05 1383.15 1383.63

5 1383.35 1383.37 1382.15 1382.82CPUTime 1 182.7 279.3 186.3 339.5 2 108.0 181.2 26.6 46.0


5 125.5 8.0 178.0 125.5

LinearIterations 1 3 3 9 15 2 3 3 2 4 3 3 3 3 15 4 3 3 5 4

5 3 3 6 6MIPGap 1 7.2E‐15 3.0E‐14 7.2E‐15 3.0E‐14 2 9.9E‐04 7.0E‐04 8.4E‐04 1.0E‐03 3 7.4E‐04 9.2E‐04 3.4E‐04 7.5E‐04

4 5.9E‐04 4.5E‐04 9.3E‐04 8.2E‐04 5 8.8E‐04 9.1E‐05 9.8E‐04 8.8E‐04NewOpenedLine 1 77‐82 77‐82 77‐82 77‐82 2 82‐96 82‐96 60‐62 60‐62 3 47‐49 47‐49 77‐82 77‐82 4 48‐49 48‐49 82‐96 82‐96 5 45‐49 45‐49 61‐62 93‐94TableA.16.ProgressiveMIP:118Bus,LooseCurrentConstraint


Open ObjValue 1 1310.10 1309.91 1315.61 1313.40 2 1309.08 1310.11 1315.14 1315.23

3 1308.54 1311.02 1316.49 1312.51 4 1310.71 1308.90 1312.58 1318.96 5 1306.34 1307.86 1314.62 1315.33NLPObjValue 1 1316.32 1316.32 1316.42 1316.38 IPOPT 2 1317.05 1317.11 1321.49 1317.02 3 1317.17 1317.32 1317.02 1316.68 4 1317.06 1317.43 1317.26 1320.89

5 1317.29 1317.29 1325.20 1322.33CPUTime 1 470 1597 437 1506 2 192 303 42 74

3 173 247 54 50 4 201 250 34 336 5 123 245 99 88NLPCPUTime 1 168 120 91 136 IPOPT 2 130 120 179 155 3 115 119 149 254 4 232 177 123 101

5 200 117 171 173LinearIterations 1 11 11 7 5

2 11 13 6 6 3 10 11 8 2 4 11 10 4 5 5 2 9 5 2MIPGap 1 3.9E‐14 4.0E‐14 3.9E‐14 4.0E‐14 2 9.7E‐04 2.2E‐11 1.0E‐03 7.3E‐04 3 2.2E‐11 2.2E‐11 9.8E‐04 8.5E‐04

4 4.3E‐09 3.3E‐04 9.6E‐04 4.4E‐04 5 2.2E‐11 7.8E‐04 8.4E‐04 4.0E‐04NewOpenedLine 1 80‐81 80‐81 80‐81 80‐81 2 69‐77 69‐77 8‐30 61‐64 3 61‐64 23‐24 94‐100 23‐24 4 76‐118 61‐64 47‐49 59‐63 5 75‐77 76‐118 65‐68 94‐100

optimal power flow paper 10: using the iv-acopf linearization · 2020. 5. 12. · mixed integer...

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