Optimal Power flow Paper 7: Developing Line Current ......bus, 30‐bus, 57‐bus, and 118‐bus...

OptimalPowerFlowPaper7

StaffpaperbyPaulaA.LipkaRichardP.O’NeillShmuelOren

April2013

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.

TableofContents

1.Introduction....................................................................................................4

2.Notation............................................................................................................4

3.Current‐VoltageACOPFModel................................................................5

4.Methods............................................................................................................6

5.ComputationalAnalysis...............................................................................7

6.Conclusions......................................................................................................17

References............................................................................................................18

Appendix...............................................................................................................21

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

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

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:

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

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

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


LineIndex

14‐busLineCurrentMagnitudewithno,loosely,andtightlyconstrictingcurrentlimits

NoLimit

LooseLimit

TightLimit

LooseConstraint

TightConstraint

30‐BusProblem.Forthe30‐bussystem,thecurrentlevelcouldnotberestrictedmuchlowerthanthemaximumoptimalcurrentlevel 0.3314 beforethesystembecameinfeasible.Restrictingthecurrentshiftsthelinecurrentmagnitudesmorethaninthe14‐busproblem;therearemanylineswherecurrentactuallyincreases.Whilethedifferencebetweenthetight 0.3092 andloose 0.3162 constraintsissmall,theserestrictionsdohavealargeimpactontheresultingcurrentthroughsomeofthelines.Inbothcases,onlytwolineshavebindingconstraints


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

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 42LineCurrentMagnitudePerUnit

LineIndex


NoLimitLooseLimitTightLimitLooseConstraintTightConstraint

57‐BusProblem.Forthe57‐bussystem,theoptimalsolutionhasmostofcurrentmagnitudesofhalforlessthehighestcurrentmagnitude,similartothe14‐busproblem.Whileabouthalfofthelinesexhibitminimalchange,theotherhalfhavecurrentsthatchangesignificantlywiththerestriction.Likethe30‐busproblem,thetwocurrentrestrictionsareclosetogether,with1.4027forthelooselimitand1.4168forthetightlimit.Inbothcases,onlytwolineshavebindingconstraints.However,eventhissmalldifferenceinlimitsgreatlyimpactsthecurrentonsomelines.


Figure6.57‐BusOptimalCurrentMagnitudesundertheLooseandTightConstraints.

0

0.5

1

1.5

2

0 20 40 60 80


LineIndex


75%isInfeasible

100%MaxCurrent

90%MaxCurrent

80%MaxCurrent

75%MaxCurrent

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 20 40 60 80


LineIndex


NoLimit

LooseLimit

TightLimit

LooseConstraint

TightConstraint

118‐BusProblem.The118‐bussystemhasseverallineswithhighercurrents,similartothe30‐bussystem.Thecurrentcanberestrictedgreatlybeforetheproblembecomesinfeasible.Reducingcurrentbyupto75%ofitsoriginaloptimalmaximumdoesnothavemuchimpactonlinecurrentsexceptthehighestones,asseeninFigure7.Constrainingthecurrentwiththelooselimitimpactsthecurrentsslightly;however,constrainingthecurrentwiththetightlimithasamajorimpactonthecurrentmagnitudes,asinFigure8.Underthelooseconstraint,onlyonelineconstraintisbinding.Underthetightcurrentmagnitudes,22ofthelineshavethehighestpermissiblecurrentunderthisrestriction,andevenlineswithlowercurrentsexhibitbigdifferencesfromtheunrestrictedandloosecurrentcases,bothhigherandlowerthanbefore.


Figure8.118‐BusOptimalCurrentMagnitudesunderLooseandTightConstraints.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 50 100 150


LineIndex


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


NoLimitLooseLimitTightLimitLooseConstraintTightConstraint

NormalizedObjectiveFunctionValueasaFunctionoftheCurrentMagnitudeLimitWhenthecurrentmagnitudeisrestricted,thislimitisbindingonatleastone

line.AsshowninFigure9,thedifferentIEEEtestcasesexhibitinfeasibilityaccordingtothesolveratdifferentpercentagesoftheirmaximumcurrentmagnitude.Thedenominatorofthenormalizedobjectivefunctionistheobjectivefunctionwithoutconstraints.

Figure9.NormalizedObjectiveFunctionversusCurrentMagnitudeLimitNormalizedSolutionTimeasaFunctionoftheCurrentMagnitudeLimit

AsshowninFigure10,thecomputationaltimetosolve findalocaloptimalsolutionordeclaretheprobleminfeasible theconstrainedproblemonlyseemstogrowgreatlywhenthecurrentmagnitudeisrestrictedto5‐15%ofitsoriginalvalue.Whenitisrestrictedto20%ormoreofitsoriginalvalue,theaddedconstraintsdonotseemtoincreaseordecreasethecomputationaltimepredictably.

1.00

1.05

1.10

1.15

1.20

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

rmal

ized

Ob

ject

ive

Fu

nct

ion

14 Bus System

30 Bus System

57 bus System

118 Bus System

Figure10:ImpactofCurrentmagnitudeConstraintsonCPUTime

EffectofCurrentMagnitudeConstraintsontheOptimalValue.Whenbindingcurrentmagnitudeconstraintsareaddedtothemodel,thefeasibleregionshrinksandtheobjectivefunctionvalueincreases,asshowninFigures11and12andTable3.Theobjectivefunctionvalueofthe14‐busproblemwiththetightconstraintis30percentgreater.Forotherproblems,theincreasesinobjectivefunctionvaluewithcurrentmagnitudeconstraintswere6percentorless.

Figure11.Valueofthequadraticobjectivefunctionwithcurrentmagnitudeconstraints

0.1

1

10

100

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

Fraction Restriction on Maximum Optimal Line Current

No

rmal

ized

CP

U T

ime

14 Bus

30 Bus

57 Bus

118 Bus

Effect on Objective Value of Current Line Constraints on Fully Nonlinear IV ACOPF

1.00

1.04

1.08

1.12

1.16

1.20

1.24

1.28

1.32

14 Bus 30 Bus 57 Bus 118 Bus

Rat

io o

f O

bje

ctiv

e Funct

ion to U

nco

nst

rain

ed

obje

ctiv

e Funct

ion

Tight Constraint

Loose Constraint

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


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

1.00

1.05

1.10

1.15

1.20

1.25

1.30


Rat

io o

f O

bje

ctiv

e Funct

ion to U

nco

nst

rain

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Obje

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e Funct

ion

Tight Constraint

Loose Constraint

Figure13.CPUTimewithcurrentmagnitudeconstraintsofquadraticobjective

Figure14.CPUTimeofthecurrentmagnitudeconstrainedlinearobjectiveTable4.CPUTimeforallbussystems,constrainttypes,andmodeltypesProblemType Constrainttype 14‐bus 30‐bus 57‐bus 118‐bus

QuadraticObjectiveNonlinearConstraints


LinearObjectiveNonlinearConstraints


Effect on CPU Time of Current Line Constraints on the Fully Nonlinear IV ACOPF

1

1.5

2

2.5

3

3.5

4

4.5


Rat

io o

f C

PU

Tim

e to

Un

con

stra

ined

CP

U T

ime

Tight Constraint

Loose Constraint

Effect on CPU Time of Current Line Constraints on the Nonlinear IV ACOPF with Linear Objective Function

123456789

101112131415161718


Rat

io o

f C

PU

Tim

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Unco

nst

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ed C

PU

Tim

e

Tight Constraint

Loose Constraint

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.

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.

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



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

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

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

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

Optimal Power flow Paper 7: Developing Line Current ......bus, 30‐bus, 57‐bus, and 118‐bus...

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Transcript of Optimal Power flow Paper 7: Developing Line Current ......bus, 30‐bus, 57‐bus, and 118‐bus...