optimal power flow paper 7: developing line current
TRANSCRIPT
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OptimalPowerFlowPaper7
StaffpaperbyPaulaA.LipkaRichardP.O’NeillShmuelOren
April2013
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DevelopingLineCurrentmagnitudeConstraintsfor
IEEETestProblems
OptimalPowerFlowPaper7
PaulaA.Lipka,RichardP.O’Neill,[email protected];[email protected];[email protected]
April2013
AbstractandExecutiveSummaryAllpowersystemoperatorsensuretheirsystemsadheretothermallimitsontransmissionlinesinordertoavoidlinedeformation.Also,thermallimitsareusedassurrogatesforvoltagestability.TheIEEEtestproblemsdonotincludedataontheselimits.Thepurposeofthispaperistopresentasimplemethodforconstructingcurrentmagnitudeconstraintsandtoreportonthecomputationalpropertiesinsolvingtheresultingproblems.Thispaperfindslimitsonthemaximumallowablecurrentmagnitudethatresultinafeasiblesolutionforthe14‐bus,30‐bus,57‐bus,and118‐busIEEEtestproblems.Foreachtestproblem,onesinglelimitisappliedtoalllinesthatmakestheoptimalsolutionwithouttheselimitsinfeasible.Foreachproblemwedevelopa‘tight’anda‘loose’constraint.Wesolvetheresultingproblemusingthecurrentvoltageformulation.Differenttestproblemsexhibitdifferentcharacteristics.IncludingtheseconstraintsintheACOPFincreasesthesolutiontimebetween2to20timesandcosts objectivefunction upto25percent.
Disclaimer:Theviewspresentedarethepersonalviewsoftheauthorsandnotthe
FederalEnergyRegulatoryCommissionoranyofitsCommissioners.
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TableofContents
1.Introduction....................................................................................................4
2.Notation............................................................................................................4
3.Current‐VoltageACOPFModel................................................................5
4.Methods............................................................................................................6
5.ComputationalAnalysis...............................................................................7
6.Conclusions......................................................................................................17
References............................................................................................................18
Appendix...............................................................................................................21
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1. IntroductionTheamountofcurrentthatcanflowthroughpowersystemtransmission
assets referredtohereaslines islimitedbythermalrestrictions.Thethermalratingsofthetransmissionarefunctionsofthematerialsthatcomposetheassets.Theexcessiveheatcausedbythelinecurrentcandeformanddegradetransmissionlinesandcausethemtosag.Heatlossesareproportionaltocurrentmagnitudesquared.Inaddition,currentmagnitudeconstraintsareoftenusedassurrogatesforotherconstraintssuchasvoltagestability.However,mostIEEEtestproblemsdonotincludecurrentmagnitudelimitsonthetransmissionlineseventhoughtheyareanimportantaspectinmodeltesting.
Intheabsenceoftheseconstraints,oneapproachistoputinconstraintsbasedonphysicalcharacteristics.Often,thereislittleinformationaboutthelines.Ittakesconsiderabletimetodevelopconstraintsbasedonphysicalcharacteristics,andtheresultmaynotbebindingconstraints.
ThepurposeofthispaperistodevelopamethodologyforcreatinglinecurrentmagnitudeconstraintsusingasetoftheIEEEtestproblems.Weareinterestedincreatingbindingconstraintsbasedonmaximumcurrentmagnituderatherthanonapparentpoweronthelinesoronvoltageangledifferences.Theapproachweemployistocreateconstraintsfromtheoptimalsolutionwithouttheseconstraints.Withtheseconstraints,theresultingpowerflowproblemhasafeasiblesolution.Subsequenttestinghelpstounderstandhowconstrainingthelineflowsaffectstheresultingpowerflowsolutionandsolutiontime.
2. Notation
Variablesandparametersareindexedoverbusesdenotedbysubscriptsnandm.Transmissionlinesareindexedbyterminalbusesnandmandk.Foracomplexvariableorparameter,thesuperscriptrdenotestherealportionandthesuperscriptjdenotestheimaginaryportion.Forexample,ifx a jb,xr a,xj bwherej ‐1 1/2.Variablespn realpowerinjectedatbusnqn reactivepowerinjectedatbusnvrn realpartofthevoltageatbusnvjn imaginarypartofthevoltageatbusnvn thevoltagemagnitudeatbusnirn realpartofthecurrentinjectedatbusnijn imaginarypartofthecurrentinjectedatbusnin thecurrentmagnitudeof injection atbusnirnmk realpartofthecurrentonlinek atbusn tobusm
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ijnmk imaginarypartofthecurrentonlinek atbusn tobusminmk thecurrentmagnitudeonlinek atbusn tobusmParameterscpn pn costfunctionofrealpowerinjectedbyageneratoratbusncqn qn costfunctionofreactivepowerinjectedbyageneratoratbusncpln pn linearcostfunctionofrealpowerbyageneratoratbusncqln qn linearcostfunctionofreactivepowerbyageneratoratbusnbnmk imaginarypartoftheadmittancematrixforlinek betweennandmgnmk realpartoftheadmittancematrixforlinek betweennandmpminn minimumrequiredrealpoweratbusnpmaxn maximumallowedrealpoweratbusnqminn minimumrequiredreactivepoweratbusnqmaxn maximumallowedreactivepoweratbusnvminn minimumrequiredvoltagemagnitudeatbusnvmaxn maximumallowedvoltagemagnitudeatbusnimaxnmk maximumallowedcurrentmagnitudeonlinekfrombusntobusm
3. Current‐VoltageACOPFModel
Thecurrent‐voltage IV ACOPFformulationisusedtofindasetofvoltagesandcurrentsateachbusandcurrentsoneachtransmissionlinethatminimizetheobjectivefunctionintermsofrealandreactivepower.MoredetailcanbefoundinCainetal 2012 andO’Neilletal 2012 .TheIV‐ACOPFmodelis:Minimize∑ncpn pn cqn qn
Subjecttoirnmk gnmk vrn‐vrm ‐bnmk vjn‐vjm
ijnmk bnmk vrn‐vrm gnmk vjn‐vjm
irn ∑mkirnmk
ijn ∑mkijnmk
pn vrnirn vjnijn
pminn pn pmaxn
qn vjnirn‐vrnijn
qminn qn qmaxn
vrn 2 vjn 2 vmaxn 2
vminn 2 vrn 2 vjn 2
irnmk 2 ijnmk 2 imaxnmk 2
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Theobjectivefunctionfortestproblemsisgenerallypresentedasaquadraticfunction,butmanyapplicationsrequirelinearbidfunctions.TheIV‐ACOPFwithlinearizedobjectivefunctionprogramhasthesameconstraintsasthenonlinearmodel,buttheobjectivefunctionisastepfunctionapproximationofthenonlinearobjectivefunction.InsolvingtheACOPFmodels,weallowforaninfeasibleanswerthatispenalizedbytheamountthesystemisinfeasible.WeaddcPenalty vr,vj,ir,ij,p,q tothelinearizedobjectivefunction,wherethequantityx isequaltomax x,0 ,thatis,ifxispositive,x x;ifxisnegative,thenx 0.Forexample,iftherealpowerisgreaterthanthemax,theobjectivefunctionispenalizedbythatquantitytimessomecost;ifitislessthanorequaltothemaximum,thereisnopenalty.
cPenalty vr,vj,ir,ij,p,q ∑nmkcpenn vn‐vminn cpenn vmaxn‐v
cpenn inmk‐imaxnmk cpenn pn‐pmaxn cpenn pminn‐pn
cpenn qminn‐qn cpenn qn‐qmaxn
Thesysteminfeasibilitiescouldpossiblyoccurbecausethevoltageisabovethemaximumorbelowtheminimumlevels,becausethecurrentisabovethemaximumlevel,orbecauserealorreactivepowerviolatemaximumorminimumlimits,asdetailedinthepenaltycost.Herewesetcpenn 105.AproblemisdeclaredinfeasibleifthesolutionsetscPenaltytobegreaterthan10‐3.
4. Methods
Weconsideredtwomethodsforcreatinglinecurrentmagnitudeconstraints.Inboththesemethods,wesolvedthealternatingcurrentoptimalpowerflowmodelwithoutlineconstraintsandextractedtheoptimalcurrentmagnitudesofeachlinefromtheoptimalsolution.
Inthefirstmethod,weconstrainedthecurrentmagnitudeoneachlinetobeafraction lessthan1 ofitsoptimalcurrentmagnitudeintheunconstrainedproblem.However,thismethoddidnotalwaysreturnfeasiblesolutions.Thisislikelybecausetheunconstrainedoptimizationminimizeslossesbyloweringcurrentmagnitudessubjecttoallotherconstraintsinthemodel.Therefore,itmaynotbefeasibleforalllinecurrentstoberestrictedtosomethinglowerthanintheunconstrainedsolution.
Inthesecondmethod,weconstrainedthecurrentmagnitudeoneachtransmissionlinetosomefractionofthehighestoptimalcurrentmagnitudeoveralllinesintheunconstrainedproblem.Thismethodreturnedlocaloptimalsolutionssincetheproblemisnon‐convex forsomelimits,but,ifthecurrentmagnitudelimitwastoolow,thesolvercouldnotfindafeasiblepoint.Thiscurrentmagnitudelimitlevelvariedwidelydependingonthetestproblem.
Procedure:
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1. SolvetheACOPFwithoutanythermallineconstraints2. Extracttheoptimallinecurrentmagnitudesinmk*forcurrentmagnitudeforlinekatbusntom.
3. Letimax* maximum inmk* overalln,m,andk.4. Solvethethermallyconstrainedproblembyincludingtheconstraintinmk f*imax*forallcombinationsofn,m,andkwherefisaparameter
with0 f 1.5. ComputationalAnalysis
Theinputmodeldatausedisfromthe14,30,57,and118‐busIEEEtestsystemdataathttp://www.ee.washington.edu/research/pstca/index.html.ThegeneratorcostscomefromMATPOWER seeZimmermanet.al.2011 .Thequadraticcostparametersareshownintheappendix.Wheretherearemultipletransmissionlinesbetweentwonodes,thelinesareaggregatedintoanequivalentsinglelinebetweenthetwonodes.Thecurrentmagnitudemeasurementisperunit.Table1summarizesthetestproblemcharacteristics.
Table1:IEEETestSystemDataBuses Lines Generators Demand Vmax Vmin Best
KnownValue*Number TotalCapacity14 20 5 7.72 2.59 1.06 0.94 80.8130 41 6 326.80 42.42 1.10 0.94 5.74557 80 7 326.78 235.26 1.06 0.94 417.4118 186 54 99.66 42.42 1.06 0.94 1297
*QuadraticObjective,NonlinearConstraintsHardware.TheproblemsweresolvedonanIntelXeonE7458serverwith864‐bit2.4GHzprocessorsand64GBmemory.Minordifferencesinsolutiontimeswererecordedwhenthesameproblemswererunatdifferenttimes,butthedifferencesweresmallenoughtobeconsideredbackgroundnoise.Software.TheprocedureswereformulatedinGAMS23.6.2.ThenonlinearsolverusedwasIPOPTversion3.8.Approach.TheIV‐ACOPFformulation,asdescribedabove,wasusedtosolvethetestcasesshownbelow.Foreachtestproblem,afigureshowstheimpactonthelinecurrentmagnitudesontestsystemswereexaminedwiththecurrentmagnitudelevelat75%,80%,90%,and100%ofthemaximumcurrentmagnitudelevelinthetestproblemwithoutlinecurrentmagnitudeconstraints.Theoptimalsolutionmaybeinfeasible,thatis,haveapenaltyfunctiongreaterthanzero.Asecondfigureshowsthelineindexversusthelinecurrentmagnitudewherethelineindexissortedfromlowesttohighestcurrentmagnitudelevel.Theredsolidlineineachgraphisthechosentightcurrentmagnitudelimitandthedashedgreenlineisthe
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chosenloosecurrentmagnitudelimit.Acrosswalkfromlinenumbertothebusesthelineisconnectingisintheappendix.TestingCurrentMagnitudeConstraints.Fromthetestingdescribedintheprevioussections,theuniformrulerestrictingcurrentmagnitudetothesamepercentageofmaximumcurrentmagnitudehasadifferentimpactonthedifferenttestcases.Therefore,wepickeddifferentpercentagesofcurrentmagnitudesforeachcaseandcreatedtwosetsofconstrainedproblems:tightandloose.Thetightcurrentmagnitudelevelisnearthecurrentmagnitudelevelwheretheproblembecomesinfeasible.Theloosecurrentmagnitudelevelisalevelthatconstrainstheproblembutisfartherawayfromthepointofinfeasibility.TheselevelsareshowninTable2.Table2:Tightandloosetestinglevelsofcurrentmagnitudeconstraints 14‐bus 30‐bus 57‐bus 118‐bus
LowLevel TightConstraint :%ofMaximumOptimalCurrentmagnitude: 18.5 93.3 76.2 24.3HighLevel LooseConstraint :%ofMaximumOptimalCurrentmagnitude: 51.1 95.4 77.0 71.9Low TightConstraint CurrentmagnitudeLevel 0.2264 0.3092 1.4027 0.9294High LooseConstraint CurrentmagnitudeLevel 0.6246 0.3162 1.4168 2.7536
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14‐BusProblem.Inthe14‐bussystemwhencurrentisnotconstrained,onelinehasnearlytwicethecurrentmagnitudeasthenexthighestcurrentline.Asthecurrentmagnitudeconstraintisdecreased,theeffectonotherlinesisshowninfigures1and2.Thecurrentmagnitudesthatchangetheleastunderrestrictionarethelineswithcomparativelylowercurrentmagnitudes.Forthelooseconstraint,onlyonecurrenthasabindingconstraint.Forthetightconstraint,fourconstraintsarebindingornearbinding.
Figure1.‘Optimal’14‐BusCurrentMagnitudesperLineunderPercentRestrictions
Figure2.14‐BusOptimalCurrentMagnitudesundertheLooseandTightConstraints.
00.10.20.30.40.50.60.70.80.91
1.11.21.3
0 2 4 6 8 10 12 14 16 18 20
LineCurrentMagnitudePerUnit
LineIndex
14‐busLineCurrentMagnitudeasaFunctionof%RestrictionontheLine
100%MaxCurrent
90%MaxCurrent
80%MaxCurrent
75%MaxCurrent
00.10.20.30.40.50.60.70.80.91
1.11.21.3
0 5 10 15 20
LineCurrentMagnitudePerUnit
LineIndex
14‐busLineCurrentMagnitudewithno,loosely,andtightlyconstrictingcurrentlimits
NoLimit
LooseLimit
TightLimit
LooseConstraint
TightConstraint
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30‐BusProblem.Forthe30‐bussystem,thecurrentlevelcouldnotberestrictedmuchlowerthanthemaximumoptimalcurrentlevel 0.3314 beforethesystembecameinfeasible.Restrictingthecurrentshiftsthelinecurrentmagnitudesmorethaninthe14‐busproblem;therearemanylineswherecurrentactuallyincreases.Whilethedifferencebetweenthetight 0.3092 andloose 0.3162 constraintsissmall,theserestrictionsdohavealargeimpactontheresultingcurrentthroughsomeofthelines.Inbothcases,onlytwolineshavebindingconstraints
Figure3.‘Optimal’30‐BusCurrentMagnitudesperLineunderPercentRestrictions
Figure4.30‐BusOptimalCurrentMagnitudesundertheLooseAndTightConstraints.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42
CurrentM
agnitude
LineIndex
30‐busCurrentperLineasaFunctionof%RestrictionontheLine
75%,80%,and90%areInfeasible100%MaxCurrent90%MaxCurrent80%MaxCurrent75%MaxCurrent
0
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30‐busLineCurrentMagnitudewithno,loosely,andtightlyconstrictingcurrentlimits
NoLimitLooseLimitTightLimitLooseConstraintTightConstraint
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57‐BusProblem.Forthe57‐bussystem,theoptimalsolutionhasmostofcurrentmagnitudesofhalforlessthehighestcurrentmagnitude,similartothe14‐busproblem.Whileabouthalfofthelinesexhibitminimalchange,theotherhalfhavecurrentsthatchangesignificantlywiththerestriction.Likethe30‐busproblem,thetwocurrentrestrictionsareclosetogether,with1.4027forthelooselimitand1.4168forthetightlimit.Inbothcases,onlytwolineshavebindingconstraints.However,eventhissmalldifferenceinlimitsgreatlyimpactsthecurrentonsomelines.
Figure5.‘Optimal’57‐BusCurrentMagnitudesperLineunderPercentRestrictions
Figure6.57‐BusOptimalCurrentMagnitudesundertheLooseandTightConstraints.
0
0.5
1
1.5
2
0 20 40 60 80
LineCurrentMagnitudePerUnit
LineIndex
57‐busCurrentperLineasaFunctionof%RestrictionontheLine
75%isInfeasible
100%MaxCurrent
90%MaxCurrent
80%MaxCurrent
75%MaxCurrent
0
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LineCurrentMagnitudePerUnit
LineIndex
57‐busLineCurrentMagnitudewithno,loosely,andtightlyconstrictingcurrentlimits
NoLimit
LooseLimit
TightLimit
LooseConstraint
TightConstraint
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118‐BusProblem.The118‐bussystemhasseverallineswithhighercurrents,similartothe30‐bussystem.Thecurrentcanberestrictedgreatlybeforetheproblembecomesinfeasible.Reducingcurrentbyupto75%ofitsoriginaloptimalmaximumdoesnothavemuchimpactonlinecurrentsexceptthehighestones,asseeninFigure7.Constrainingthecurrentwiththelooselimitimpactsthecurrentsslightly;however,constrainingthecurrentwiththetightlimithasamajorimpactonthecurrentmagnitudes,asinFigure8.Underthelooseconstraint,onlyonelineconstraintisbinding.Underthetightcurrentmagnitudes,22ofthelineshavethehighestpermissiblecurrentunderthisrestriction,andevenlineswithlowercurrentsexhibitbigdifferencesfromtheunrestrictedandloosecurrentcases,bothhigherandlowerthanbefore.
Figure7.‘Optimal’118‐BusCurrentMagnitudesperLineunderPercentRestrictions
Figure8.118‐BusOptimalCurrentMagnitudesunderLooseandTightConstraints.
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 50 100 150
LineCurrentMagnitudePerUnit
LineIndex
118‐busCurrentperLineasaFunctionof%RestrictionontheLine
100%MaxCurrent
90%MaxCurrent
80%MaxCurrent
75%MaxCurrent
00.51
1.52
2.53
3.54
4.5
0 50 100 150
LineCurrentMagnitudePer
Unit
LineIndex
118‐busLineCurrentMagnitudewithno,loosely,andtightlyconstrictingcurrentlimits
NoLimitLooseLimitTightLimitLooseConstraintTightConstraint
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NormalizedObjectiveFunctionValueasaFunctionoftheCurrentMagnitudeLimitWhenthecurrentmagnitudeisrestricted,thislimitisbindingonatleastone
line.AsshowninFigure9,thedifferentIEEEtestcasesexhibitinfeasibilityaccordingtothesolveratdifferentpercentagesoftheirmaximumcurrentmagnitude.Thedenominatorofthenormalizedobjectivefunctionistheobjectivefunctionwithoutconstraints.
Figure9.NormalizedObjectiveFunctionversusCurrentMagnitudeLimitNormalizedSolutionTimeasaFunctionoftheCurrentMagnitudeLimit
AsshowninFigure10,thecomputationaltimetosolve findalocaloptimalsolutionordeclaretheprobleminfeasible theconstrainedproblemonlyseemstogrowgreatlywhenthecurrentmagnitudeisrestrictedto5‐15%ofitsoriginalvalue.Whenitisrestrictedto20%ormoreofitsoriginalvalue,theaddedconstraintsdonotseemtoincreaseordecreasethecomputationaltimepredictably.
1.00
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0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Fraction Restriction on Maximum Optimal Line Current
No
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Ob
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Fu
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14 Bus System
30 Bus System
57 bus System
118 Bus System
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Figure10:ImpactofCurrentmagnitudeConstraintsonCPUTime
EffectofCurrentMagnitudeConstraintsontheOptimalValue.Whenbindingcurrentmagnitudeconstraintsareaddedtothemodel,thefeasibleregionshrinksandtheobjectivefunctionvalueincreases,asshowninFigures11and12andTable3.Theobjectivefunctionvalueofthe14‐busproblemwiththetightconstraintis30percentgreater.Forotherproblems,theincreasesinobjectivefunctionvaluewithcurrentmagnitudeconstraintswere6percentorless.
Figure11.Valueofthequadraticobjectivefunctionwithcurrentmagnitudeconstraints
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Fraction Restriction on Maximum Optimal Line Current
No
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U T
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14 Bus
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118 Bus
Effect on Objective Value of Current Line Constraints on Fully Nonlinear IV ACOPF
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14 Bus 30 Bus 57 Bus 118 Bus
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Loose Constraint
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Figure12.Valueofthelinearobjectivefunctionwithcurrentmagnitudeconstraints
Table3:ObjectivefunctionvalueforeachtestsystemandconstrainttypeProblemType Constrainttype 14‐bus 30‐bus 57‐bus 118‐bus
QuadraticObjectiveNonlinearConstraints
TightConstraint 105.4 5.89 421.5 1364.9LooseConstraint 85.3 5.79 419.2 1300.1Unconstrained 80.8 5.75 417.4 1296.6
LinearObjectiveNonlinearConstraints
TightConstraint 107.4 6.10 432.2 1388.4LooseConstraint 86.5 5.98 425.5 1315.5Unconstrained 82.8 5.92 423.8 1311.5
EffectofCurrentMagnitudeConstraintsontheSolutionTime.CurrentmagnitudeconstraintsalsoimpactthesolutiontimeofeachtestsystemasshowninFigures13and14andTable4.Addingconstraintsatleastdoublesthesolutiontimeoftheproblem.Forthe30‐busand118‐busproblems,themoretightlyconstrainedproblemsolvesfasterthanthelessconstrainedproblem.Forthe14‐busand118‐busproblems,thelessconstrainedproblemsolvesfasterthanthemoreconstrainedproblem.The57‐busand118‐bussolutiontimetooksignificantlylongerthanthe14‐busand30‐bussystems,butsolutiontimeisnotsolelybasedonthenumberofbuses.Thelinearobjectivefunctionrequiredconsiderablymoretimetosolve.
Effect on Objective Value of Current Line Constraints on Nonlinear IV ACOPF with Linear Objective Function
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Figure13.CPUTimewithcurrentmagnitudeconstraintsofquadraticobjective
Figure14.CPUTimeofthecurrentmagnitudeconstrainedlinearobjectiveTable4.CPUTimeforallbussystems,constrainttypes,andmodeltypesProblemType Constrainttype 14‐bus 30‐bus 57‐bus 118‐bus
QuadraticObjectiveNonlinearConstraints
TightConstraint 0.72 1.80 8.34 28.95LooseConstraint 0.58 1.94 7.77 33.33Unconstrained 0.28 0.78 1.95 8.13
LinearObjectiveNonlinearConstraints
TightConstraint 1.16 1.55 8.42 32.20LooseConstraint 0.91 2.88 12.05 38.75Unconstrained 0.31 0.48 0.70 2.61
Effect on CPU Time of Current Line Constraints on the Fully Nonlinear IV ACOPF
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Effect on CPU Time of Current Line Constraints on the Nonlinear IV ACOPF with Linear Objective Function
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6. ConclusionsForeachtestproblem,onesinglelimitisappliedtoalllinesthatmakesthe
optimalsolutionwithouttheselimitsinfeasible.WesolvetheresultingproblemusingtheIV‐ACOPFformulation.Foreachproblemwedevelopa‘tight’anda‘loose’constraint.
Forthe14,30,57and118‐busproblems,creatinglinecurrentmagnitudeconstraintsfortheACOPFproblemcanresultininfeasibleproblems.Asonetightensthecurrentmagnitudeconstraints,theobjectivefunctionincreasesgraduallyatfirst,thenincreasesexponentiallynearthepointofinfeasibility.Differenttestproblemsexhibitdifferentcharacteristicsinthelinecurrentmagnitudedistributionandatwhatcurrentmagnitudelevelconstrainttheproblembecomesinfeasible.
Thecurrentmagnitudeconstraintsalsoincreasethesolutiontime,althoughstricterconstraintsdonotnecessarilyincreasethesolutiontimemorethanlooserconstraints.Forproblemslikethiscasewheretheproblembecomesinfeasiblequickly,itmayworkwelltorestrictonlysomeofthelinecurrentmagnitudesratherthanallofthem.
IncludingtheseconstraintsintheACOPFincreasesthesolutiontimebetween2to20timesandobjectivefunctionupto25percent.
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ReferencesMaryB.Cain,RichardP.O’Neill,AnyaCastillo,“HistoryofOptimalPowerFlowandFormulations,”FERCStaffTechnicalPaper,December2012http://www.ferc.gov/industries/electric/indus‐act/market‐planning/opf‐papers/acopf‐1‐history‐formulation‐testing.pdfRichardPO’Neill,AnyaCastilloandMaryCain,“TheIVFormulationandLinearApproximationsoftheACOptimalPowerFlowProblem,”FERCStaffTechnicalPaper,December2012http://www.ferc.gov/industries/electric/indus‐act/market‐planning/opf‐papers/acopf‐2‐iv‐linearization.pdfUniversityofWashington,Dept.ofElectricalEngineering,“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,”PowerSystems,IEEETransactionson,vol.26,no.1,pp.12‐19,Feb2011.
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Appendix Thegeneratorcoststakeontheformcost aP bP2 .0001*|Q|,wherePistherealpowerand|Q|isthemagnitudeofreactivepowerq.Welistallgeneratorcostsusedforeachtestsystem.
TableA1:GeneratorCostCoefficientsforthe14,30and57‐busProblems14‐bus CostCoefficient 30‐bus CostCoefficient 57‐bus CostCoefficient
Generator a b Generator a b Generator a b1 0.04303 20 1 0.02000 2.00 1 0.07758 202 0.25000 20 2 0.01750 1.75 2 0.01000 403 0.01000 40 13 0.02500 3.00 3 0.25000 206 0.01000 40 22 0.06250 1.00 6 0.01000 408 0.01000 40 23 0.02500 3.00 8 0.02222 20 27 0.00834 3.25 9 0.01000 40 12 0.03226 20
TableA2:118‐busGeneratorCosts
Generator CostCoefficient
Generator CostCoefficient
Generator CostCoefficient
a b a b r a b1 0.01000 40 46 0.52632 20 87 2.50000 204 0.01000 40 49 0.04902 20 89 0.01647 206 0.01000 40 54 0.20833 20 90 0.01000 408 0.01000 40 55 0.01000 40 91 0.01000 4010 0.02222 20 56 0.01000 40 92 0.01000 4012 0.11765 20 59 0.06452 20 99 0.01000 4015 0.01000 40 61 0.06250 20 100 0.03968 2018 0.01000 40 62 0.01000 40 103 0.25000 2019 0.01000 40 65 0.02558 20 104 0.01000 4024 0.01000 40 66 0.02551 20 105 0.01000 4025 0.04545 20 69 0.01936 20 107 0.01000 4026 0.03185 20 70 0.01000 40 110 0.01000 4027 0.01000 40 72 0.01000 40 111 0.27778 2031 1.42857 20 73 0.01000 40 112 0.01000 4032 0.01000 40 74 0.01000 40 113 0.01000 4034 0.01000 40 76 0.01000 40 116 0.01000 4036 0.01000 40 77 0.01000 40 40 0.01000 40 80 0.02096 20 42 0.01000 40 85 0.01000 40
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TableA3:14and30‐busLineIndexMapping14‐bus 30‐bus
LineIndex
BusesConnected LineIndex
BusesConnectedTo From To From
1 1 2 1 6 82 1 5 2 12 133 2 3 3 21 224 4 5 4 2 65 2 4 5 4 66 5 6 6 1 27 2 5 7 1 38 7 9 8 2 49 4 7 9 3 410 6 13 10 5 711 4 9 11 2 512 3 4 12 4 1213 9 14 13 27 2814 7 8 14 25 2715 6 12 15 15 2316 6 11 16 6 717 9 10 17 9 1018 13 14 18 6 919 10 11 19 8 2820 12 13 20 10 20
21 15 1822 10 2123 10 1724 12 1525 24 2526 10 2227 12 1628 27 3029 27 2930 19 2031 6 2832 12 1433 6 1034 22 2435 23 2436 18 1937 25 2638 29 3039 16 1740 14 1541 9 1135 23 2436 18 1937 25 2638 29 3039 16 1740 14 1541 9 11
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TableA4:57‐busLineIndexMapping57‐bus 57‐bus
LineIndexBusesConnected
LineIndexBusesConnected
To From To From1 8 9 40 50 512 14 46 41 38 443 10 51 42 35 364 24 26 43 24 255 7 8 44 4 56 7 29 45 6 77 15 45 46 41 438 1 15 47 52 539 9 55 48 54 5510 3 15 49 11 4111 1 2 50 41 4212 1 17 51 38 4913 13 49 52 12 1614 9 11 53 3 415 46 47 54 25 3016 14 15 55 49 5017 6 8 56 34 3518 2 3 57 53 5419 9 10 58 32 3420 9 13 59 41 5621 1 16 60 10 1222 4 18 61 12 1723 28 29 62 18 1924 11 43 63 48 4925 27 28 64 30 3126 5 6 65 36 4027 44 45 66 20 2128 4 6 67 32 3329 12 13 68 37 3930 37 38 69 22 3831 38 48 70 22 2332 11 13 71 56 5733 13 14 72 23 2434 9 12 73 39 5735 36 37 74 42 5636 26 27 75 40 5637 29 52 76 19 2038 47 48 77 31 3239 13 15 78 21 22
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TableA5:118‐busLineIndexMapping118‐bus 118‐bus 118‐bus 118‐bus
LineIndex
Buses LineIndex
Buses LineIndex
Buses LineIndex
BusesTo From To From To From To From
1 9 10 46 77 78 91 51 52 136 51 582 8 9 47 66 67 92 104 105 137 105 1063 5 8 48 45 49 93 93 94 138 100 1014 37 38 49 76 77 94 2 12 139 34 435 17 30 50 22 23 95 103 104 140 105 1076 49 66 51 59 61 96 75 77 141 29 317 26 30 52 49 50 97 55 59 142 32 1148 89 92 53 47 69 98 20 21 143 70 719 68 116 54 100 106 99 34 36 144 19 2010 68 69 55 56 59 100 82 83 145 70 7511 64 65 56 94 95 101 37 40 146 27 3212 59 63 57 100 104 102 99 100 147 43 4413 89 90 58 75 118 103 62 67 148 106 10714 23 25 59 85 88 104 68 81 149 7 1215 80 81 60 74 75 105 1 3 150 60 6216 63 64 61 94 100 106 54 59 151 40 4217 38 65 62 92 93 107 101 102 152 52 5318 65 66 63 59 60 108 40 41 153 3 1219 25 26 64 11 12 109 78 79 154 90 9120 25 27 65 21 22 110 12 117 155 12 1621 77 80 66 62 66 111 23 24 156 15 3322 4 5 67 48 49 112 17 31 157 32 11323 60 61 68 80 97 113 27 115 158 105 10824 61 64 69 110 111 114 80 99 159 56 5825 69 75 70 49 69 115 96 97 160 24 7226 69 70 71 37 39 116 53 54 161 86 8727 34 37 72 92 94 117 16 17 162 55 5628 88 89 73 45 46 118 41 42 163 71 7329 15 17 74 11 13 119 47 49 164 98 10030 42 49 75 110 112 120 91 92 165 39 4031 5 6 76 103 110 121 82 96 166 13 1532 23 32 77 27 28 122 77 82 167 1 233 100 103 78 103 105 123 85 86 168 108 10934 69 77 79 35 37 124 70 74 169 71 7235 65 68 80 31 32 125 83 84 170 114 11536 8 30 81 80 98 126 56 57 171 15 1937 5 11 82 50 57 127 33 37 172 54 5638 30 38 83 6 7 128 95 96 173 35 3639 49 54 84 61 62 129 17 113 174 19 3440 17 18 85 44 45 130 94 96 175 109 11041 79 80 86 83 85 131 18 19 176 76 11842 85 89 87 80 96 132 92 100 177 24 7043 4 11 88 92 102 133 12 14 178 54 5544 49 51 89 46 47 134 46 48 179 14 1545 3 5 90 84 85 135 28 29