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Numerical comparison of cold-start linearizations of the power flow equations for the transmission system

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Title:
Numerical comparison of cold-start linearizations of the power flow equations for the transmission system
Creator:
Wiebe, Daniel
Place of Publication:
Denver, CO
Publisher:
University of Colorado Denver
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Language:
English

Thesis/Dissertation Information

Degree:
Master's ( Master of science)
Degree Grantor:
University of Colorado Denver
Degree Divisions:
College of Engineering and Applied Sciences, CU Denver
Degree Disciplines:
Engineering and applied science
Committee Chair:
Mancilla-David, Fernando
Committee Members:
Dey, Satadru
Vahid, Alireza

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Abstract:
The AC Power Flow (PF) equations are central to the proper management and maintenance of the electrical power grid. Due to their inherently non-linear nature, linearizations of the AC PF equations, such as the standard decoupled (DC) PF equations are frequently leveraged for the sake of common optimization problems. These linearizations can be categorized into two sets: (i ) Warm-start models, where the linearization is constructed around a known solution to the AC PF equations, and (ii ) Cold-start models, where the linearization is derived from general approximations to the AC PF equations. Cold-start models are used when there is no known solution to the AC PF equations to construct a warm-start model with, and are useful in security constrained unit commitment problems, FTR-CRR auctions and allocations, and medium- to long-term planning studies. In recent years, several new cold-start linearization models have been presented as improvements on the standard DC PF equations. This paper presents a thorough numerical comparison and analysis of these new cold-start linearization models.

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Copyright Daniel Wiebe. Permission granted to University of Colorado Denver to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.

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NUMERICALCOMPARISONOFCOLD-STARTLINEARIZATIONSOFTHE POWERFLOWEQUATIONSFORTHETRANSMISSIONSYSTEM by DANIELWIEBE B.A.ComputerScience,St.OlafCollege,2010 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfulllment oftherequirementsforthedegreeof MasterofScience EngineeringandAppliedScience 2018

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ThisthesisfortheMasterofSciencedegreeby DanielWiebe hasbeenapprovedforthe EngineeringandAppliedScienceProgram by FernandoMancilla{David,Advisor,Chair SatadruDey AlirezaVahid December4,2018 ii

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Wiebe,DanielM.S.,EngineeringandAppliedScience NumericalComparisonofCold-StartLinearizationsofthePowerFlowEquationsFor theTransmissionSystem ThesisdirectedbyAssociateProfessorFernandoMancilla{David ABSTRACT TheACPowerFlowPFequationsarecentraltothepropermanagementand maintenanceoftheelectricalpowergrid.Duetotheirinherentlynon-linearnature, linearizationsoftheACPFequations,suchasthestandarddecoupledDCPF equationsarefrequentlyleveragedforthesakeofcommonoptimizationproblems. Theselinearizationscanbecategorizedintotwosets: i Warm-startmodels,where thelinearizationisconstructedaroundaknownsolutiontotheACPFequations,and ii Cold-startmodels,wherethelinearizationisderivedfromgeneralapproximations totheACPFequations.Cold-startmodelsareusedwhenthereisnoknownsolution totheACPFequationstoconstructawarm-startmodelwith,andareusefulin securityconstrainedunitcommitmentproblems,FTR-CRRauctionsandallocations, andmedium-tolong-termplanningstudies.Inrecentyears,severalnewcold-start linearizationmodelshavebeenpresentedasimprovementsonthestandardDCPF equations.Thispaperpresentsathoroughnumericalcomparisonandanalysisof thesenewcold-startlinearizationmodels. Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:FernandoMancilla{David iii

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DEDICATION ThisthesisisdedicatedtoGod,whoismyKing,trueFather,andfriend. iv

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ACKNOWLEDGMENT First,Iwouldliketoexpressmythankstomyadvisor,Dr.FernandoMancilla{David, forhisadviceandsupportduringthetheprocessofcompletingmyMaster's. IalsothankmyfellowstudentsCarlosSoriano,HectorRobles,andAmin Bunaiyanfortheirassistanceandsupportalongthejourney. IwouldliketothankDr.SatadruDeyandDr.AlirezaVahidforservingonmy defensecommittee. IalsothankDouglasWaples,mycurrentemployer,forhisexceptionalexibility withmyworkhoursasIcompletemyMaster's. Iwouldliketothankmyfamilyforalwaysbeingthereformeandsupportingme faithfullyinprayer. Andnally,IwouldliketothankGodforalwayswatchingoverme,encouraging me,andgivingmetheenergy,focusanddrivetonish. v

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TABLEOFCONTENTS ListofAbbreviations................................viii Chapter Tables........................................ix 1.Introduction...................................1 2.LinearizationModels..............................6 2.1Introduction...............................6 2.2ACPowerFlow.............................6 2.3DCPowerFlow.............................8 2.4ModiedThetaLinearizationMTLMethod.............10 2.5DecoupledLinearizationDLMethod.................13 2.6NaturalLogLinearizationNLLMethod...............16 3.TestMethodology................................22 3.1SyntheticPowerTestSystems.....................22 3.2SimulationsMethodology........................23 3.2.1RemoveTapChangersandPhaseShifters...........26 3.2.2ScalingLoadLevels.......................26 3.2.3VaryingGeneratorVoltageSetpoints..............27 3.2.4VaryingGeneratorRealPowerSetpoints............27 3.2.5VaryingtheBranchX/RRatios.................28 4.ResultsandAnalysis..............................30 4.1Basecase.................................30 4.1.1VoltageMagnitudes.......................30 4.1.2Voltageangles..........................31 4.1.3Realpowerow..........................31 4.1.4Reactivepowerow.......................35 4.1.5Basecase:Discussion......................35 vi

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4.2RemoveTapChangersandPhaseShifters...............36 4.3ScalingLoadLevels...........................41 4.4VaryingGeneratorVoltageSetpoints..................41 4.5VaryingGeneratorRealPowerSetpoints...............41 4.6VaryingtheBranchX/RRatios....................41 4.7Conclusions...............................46 5.ConcludingRemarks..............................55 5.1FurtherWork..............................55 References ......................................56 vii

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ABBREVIATIONS DC Decoupled DL DecoupledLinear FOT First-orderTaylor FTR-CRR FinancialTransmissionRights,Congestionrevenue rights LP LinearPrograms MILP MixedIntegerLinearPrograms MTL ModiedThetaLinearization OPF OptimalPowerFlow PF PowerFlow SCED SecurityConstrainedEconomicDispatch SCUC SecurityConstrainedUnitCommitment UC UnitCommitment viii

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TABLES Table 3.1TestPowerSystems.............................24 3.2TestPowerSystemscon't.........................25 ix

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1.Introduction TheACpowerowequations,whichexpresstherelationshipbetweenthevoltages inaelectricalpowersystemandtherealandreactivepowerinjectionsatthebuses ofthesystem,arewidelyusedinnearlyeveryaspectofthepowersystemindustry. Theusageoftheseequationsarethecenterofallcomputations,fromthestandard loadowproblemtotheoptimalpowerowanditsvariants,tovariouspowersystem stabilityanalysesandlong-termplanningofthepowergrid.Theequations,intheir standardform,arewellknowntobehighlynon-linear,includingbothtrigonometric andquadraticterms.ThismakesfordicultieswhenusingtheequationsasconstraintsfortheoptimalpowerowOPFproblem,orthesecurityconstrainedunit commitmentSCUCproblem.Forlongtermplanning,itcausesanotherwisedicult mixedintegerprogrammingproblemtobecomeanon-linearmixedintegerprogrammingproblem.Thenon-linearitiesgenerallycausetheoptimizationsdependingon ittohavemultiplelocalminimainan-dimensionalspacewherencouldbeinthe thousandsortenthousandsonaregularbasisandthuscomputationallyintractable. Asaresult,mostimplementationsoftheseproblemsutilizelinearizationsoftheAC powerowequationsinordertoturntheoptimizationsintoLinearProgramsLP orMixedIntegerLinearProgramsMILP,whichcanbesolvedinreasonabletime. Theparticularlinearizationthatismostwellknownandwellusedovertheyearsis thedecoupledpowerowDCPFequations[10].TheseDCPFequationsarewell knowntohavetheirlimitations,astheirimplicitassumptionsareoftennotmet,such ashighX/R[8].Onepaperinliteraturecitesanmaxerrorof173.2%inthepower owsfor62,000bussystem[9]. Inrecentyears,manyattemptshavebeenmadetoimproveontheDCPF equationsinvariousways,whilestillretaininglinearityinthevariablesofinterest[10],[3],[5],[1],[11],[12],[6].Thesenewformulationscanbecategorizedinto threegeneraltypes:warmstartmodels,coldstartmodels,andincrementalmod1

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els[10].Althoughabriefoverviewofwarm-startmodelsshallbeincludedbelow, coldstartmodelsarethefocusofthispaper. Warmstartmodelsarelinearapproximationsbuiltaroundaknownsolutionto theexactACPFequations.Thesemodelsaregenerallydesignedtobehighlyaccurate closetotheknownsolution,andlesssofurtheraway.Aneasilyunderstoodexampleof thisistherstorderTaylorFOTapproximation,wherethelinearizationessentially representsthetangenthyperplanetotheACPFmanifoldatthelocationoftheknown solution[3].Warmstartmodelsaregenerallyusedinshort-termapplications,such assecurityconstrainedeconomicdispatchSCED,runevery15minutes[10].In thiscase,aknownACoperatingpointisavailablefromsolvingthestateestimator problem,anditisexpectedthatthesystemhasnotchangedmuchsincethatproblem wassolved.Italsohasapplicationsinshorttomediumtermplanning.Incremental modelsareessentiallyaformofwarmstartmodelswherethevariablesintheequation arethedeviationsfromanexistingsolutiontotheACPFequations[10].In[10], severalwarmstartformsoftheDCPFarepresented,matching"loss"parameters invariouswaystotheexistingACPFsolution.Thosecasesonlyaccountedforthe voltageanglesandrealpowerows.In[3],awarm-startmodelispresentedbasedon therst-ordertaylorapproximationtothefullACPF,includingvoltagemagnitudes, angles,andcomplexpowerow. Coldstartmodels,ontheotherhand,aregenerallyderivedfromtheACPF equationsusingassumptionscommontothetransmissionsystemasawhole.Cold startmodelsdonotdependonacertainoperatingpoint,andsoareintendedto beagoodapproximationtotheACPFequationsoverawiderangeofpossible operatingconditions.TheDCPFisanexampleofacoldstartmodel,makinguseof assumptionssuchasthelineresistancesbeingnegligibleandthevoltagemagnitudes allbeingapproximatedas1pu.TheDCPFandothercoldstartmodelshavebeen showntobeinferiortowarmstartmodelswhenaknownACsolutionispresent[10]. 2

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However,insomecases,particularlysecurityconstrainedunitcommitment,auctions andallocationsrelatedtotransmissionrights,andinmedium-tolong-rangeplanning, noreasonableACsolutionisavailable[10].Thatis,inthesecasesthelinearizationis expectedtoperformsucientlyoveralargerangeofpossibleoperatingconditions, ratherthanjustaroundasingleoperatingpoint. Weshallnowpresenttherecentpapersoncoldstartmodels.Inthemuch-cited [10],theDCPFanditsmorewell-knowncold-startderivativesnetlossdispersaland lossredistributionwerere-presented,andtestsconductedtocomparethemethods. Thesemethodsallneglectedbranchshuntelements,butincludedtapchangersand phaseshifters.ThederivativeswereshowntoimproveupontheDCPF,howeverthe derivativesrequireana-prioriestimateonthelossesinthesystemwithrespectto overalldemand. In[1],acoldstartlinearmodelwhichincludesvoltagemagnitudesandreactive powerowsispresentedthatformulatesalinearexpressionintermsof V 2 and V 2 . Thislinearizationneglectsphaseshifters,tapchangers,andshuntelementsinits eortstowardsalinearexpression.ThislinearizationiscalledtheModiedTheta LinearizationMTLinthispaper.Aquadraticexpressionfortherealpowerows wasalsoformulatedintermsofthesolutionvariables,allowingforthepossibility ofthemethod'suseinoptimizations.Testsweredoneonthepresentedmethodin theformofsolvingthepowerowproblem,comparingtheresultstotheDCPF andtheDCPFwithnetlossdispersal.TheMTLwasshowninafavorablelightin comparisontotheothertwo,butonlytheresultsofthemethodsontheIEEE118 bussystemandthePolish3374bussystemavailablein[14]werepresented.Only theresultsofthevoltageangleerrorandtherealpowerowerrorwerepresented. Therewerealsominimaltestsalteringthebasecasesofthosetwotestsystems,both increasingtheloadlevelsanddecreasingX/Rratio. In[11],theDecoupledLinear"DLmethodispresented,anothernovelcold 3

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startlinearmodelrelatingtheinjectedrealandreactivepowerslinearlytothebus voltagemagnitudesandangles.Thedecoupled"partofthenameissomewhat misleadingwhencomparedtotheDCPF;itactuallyreferstothevoltagemagnitudes andanglesbeingdecoupledfromeachother,ascomparedtotheMTL,wherethey arecoupled.SimilarlytotheMTL,theDLincludesitsownapproximationtothe powerowequations,thoughinthiscasetheyarelinearratherthanquadratic.The DLPFisthencomparedusingthepowerowproblemtotheMTLandtheDCPF on10systems,includingtheIEEE30-,33-,57-,123-,118-,and300-bussystems, amongothers.ThesecomparisonsshowedtheDLtobesuperiortotheMTLand DCPF,althoughtheyonlyshowedresultsforthebaseoperatingconditionsforthe systemstheytestedon.Inonlyonetestwasbasecasealtered:theloadlevelfora singlenodewasboostedtoseehowthelinearizationshandledthechange.Also,only voltagemagnitudeerrorandrealpowerowerrorwereexamined,omittingvoltage angleerrorandreactivepowerowerror. In[12],aseparatecoldstartlinearizationmodelispresentedintermsof V 2 and thevoltageangle.Aquadraticexpressionforthepowerowswasincluded,and testingwasdoneinanumberofIEEEtestsystemsintheformofOPFsimulations. Comparisonsweredonewithanumberoflinearizationsnotmentionedhere,andthis linearizationwasconsideredthebestamongtheonescomparedto. In[5],averydierentcoldstartmodelwaspresented,wherebytheACPFequationswereapproximatedintermsofln V andtheta.Thismethodwillbereferred toastheNaturalLogLinearizationNLLinthispaper.Unfortunately,theNLL containsmultiplicationsbetweenthepowerinjectionsand ln V ,whichonlybecome linearwhenconsideredforthePowerFlowproblem,butnotforanoptimization. Inthepowerowproblem,foreachbusthequadratictermsbecomelinearbecause nomorethanoneofthetwotermsareconsideredunknown.Whereas,inanoptimizationthePFequationsbecomeconstraints,andsothequadratictermdoesnot 4

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disappear.Hence,theNLLcanonlybeutilizedasalinearmethodin2-stagesecurity constrainedSCUCproblems,wherethepowerowproblemisembeddedinoneofthe stages.In[5],comparisonswereconductedusingthepowerowproblembetweenthe MTL,DL,DCPF,theNLL,andthelinearizationexpressedin[12].Thequantitiesof comparisonweretheerrorsinvoltagemagnitude,andrealandreactivepowerows, omittingvoltageanglefromthecomparison.Themethodsweretestedon25systems, allofwhichareprovidedin[14],utilizingthesystemswithoutalteration. Toourknowledge,otherthan[5],nocomprehensivetestinghasbeenconducted acrossrecentlypublishedcold-startlinearizationsoftheACPFequations.Andas mentionedabove,[5]onlyincludestestingofbasecasesofthesystemsofinterest. However,ifalinearizationisutilizedinanoptimizationprocess,amuchlargersection ofthepowerowmanifoldistypicallyexploredthanjustonecase. Inthispaper,wepresentamoredetailednumericalcomparisonoftheDCPF, MTL,DL,andNLLmethodsthaniscurrentlypresentedinliterature.Simulations wereconductedon29systems,bothasisandwithanumberofcontrolledalterations tothesystemcases.Comparisonsbetweenthelinearizationsareconductedforvoltage magnitude,voltageangle,andrealandreactivepowerows.Thesequantitieswere chosenduetotheirusageinconstraintsforsecurityconstrainedoptimizations.In chapter2,thevariousmodelsarederivedandpresentedintheirnalforms,including severalderivationsandpointsofinterestnotcontainedintheirsourcepapers.Then inchapter3thetestingmethodologyisexplained.Chapters4and5thenfollow-up withtheresultsofthesimulations,conclusions,andfuturework. 5

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2.LinearizationModels 2.1Introduction InthischapterweshallcoverabriefoverviewoftheACPFequations,followed byderivationsofthelinearmodels. 2.2ACPowerFlow TheACPowerFlowequationsarewellknown,althoughtheycanbeexpressedin severalequivalentforms.Tosimplifythelateranalysis,weshallpresenttheequations inthisform: P i = n X j =1 G ij V i V j cos ij + n X j =1 B ij V i V j sin ij .1 Q i = )]TJ/F23 7.9701 Tf 18.02 14.944 Td [(n X j =1 B ij V i V j cos ij + n X j =1 G ij V i V j sin ij .2 i =1 ; 2 ; ;n Where P i and Q i aretherealandreactivepowerinjectionsatbus i , G ij and B ij aretherealandimaginarypartsof Y ij whichisinturntheelementattheithrow andjthcolumnofthebusadmittancematrix, V i and V j arethevoltagemagnitudes attheiandjbuses, ij isthevoltageangledierenceacrossthelineconnectingbuses iandj,and n isthenumberofbusesinthenetwork.Wemustnowestablishhow thebusadmittancematrix Y isbuilt.Traditionally,withastandardpimodelofthe transmissionlinesandtransformersseegure2.1a,andignoringfornowtheeects oftapchangerandphaseshiftertransformers,thematrix Y isbuiltlikethis: Y ij = 8 > < > : y ii + n P k =1 ;k 6 = i y ik + y sh ik if j = i )]TJ/F22 11.9552 Tf 9.299 0 Td [(y ij if j 6 = i .3 Where y ii isthebusshuntadmittanceatbus i ,andtherestoftheelementsare labeledingure2.1a. Weshallnowconsidernowtheeectsoftapchangersandphaseshifters.The 6

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aStandardpimodelforatransmissionlineor transformer bAugmentedpimodel,incorporatingtap-changerand phase-shiftertransformermodelingelements Figure2.1: Standardpimodelsforbranchesinanetwork,bothfortransmissionlinesand transformers 7

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mostcommonwayintheliteratureofmodelingthemistouseanaugmentedpimodel forsuchtransformers,asshowningure2.1b.Inthatgure, t ij isthemagnitude ofthevoltagechangeacrossthetransformerterminalsaddedbythetapchangeand ij isthephaseshiftappliedbythephaseshifter.Notethatinthecaseofstandard lines, t ij =1and ij =0,leavinguswiththestandardpimodel. Inordertoaccountforthe t ij e j ij elementintheaugmentedpimodel,thegeneral approachistoalterthebusadmittancematrix,creatinganasymmetrical Y .Since theaugmentedpimodelisasymmetricaloneachline,weshalldenetheset TE i asreferringtoallthebusesthatareattheto-endofalinewithafromendofbus i . Wenowshalldene Y asthefollowing: Y ij = 8 > > > > > > > < > > > > > > > : y ii + P j 2 TE i y ij + y sh ij =t 2 ij + P i 2 TE j y ji + y sh ji if j = i )]TJ/F22 11.9552 Tf 9.299 0 Td [(y ij =t ij e )]TJ/F20 7.9701 Tf 6.586 0 Td [(j ij if j 6 = i and j 2 TE i )]TJ/F22 11.9552 Tf 9.299 0 Td [(y ij =t ij e j ij if j 6 = i and i 2 TE j .4 Notthatinthepresenceofphaseshifters,thematrixformedbytheabove equationsisnotsymmetrical.Thisformulationshallbecalledtheasymmetrical" tap/phasemethod. 2.3DCPowerFlow ThestandarddecoupledDCpowerowformulationforthetransmissionnetworkisbasedonthefollowingassumptions: Voltagemagnitudesareallassumedtobeverycloseto1pu,andsoareapproximatedas1pu Angledierencesacrosslinesareassumedtobecloseto0,andsosin ij ij andcos ij 0 Lineresistancesareconsideredtobenegligible r ij 0, G ij 0 Shuntelementsareconsideredtobenegligible 8

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Withtheseapproximationsinhand,equation2.1canbeapproximatedas: P i = n X j =1 B ij ij .5 = n X j =1 B ij i )]TJ/F22 11.9552 Tf 11.955 0 Td [( j .6 P i = n X j =1 ;j 6 = i B ij i )]TJ/F23 7.9701 Tf 24.591 14.944 Td [(n X j =1 ;j 6 = i B ij j .7 Equation2.2isneglectedinitsentirety,asreactivepowerisneglectedasaderivativeassumptionfromtheaboveassumptions.Onceagain,wemustdecidehowto handlethetapchangersandphaseshifters.ThetraditionalformulationoftheDC PowerFlowequationsomitsthetapchangersandphaseshifters,givingthefollowing equation: P i = )]TJ/F22 11.9552 Tf 9.298 0 Td [(B ii i )]TJ/F23 7.9701 Tf 24.591 14.944 Td [(n X j =1 ;j 6 = i B ij j .8 Notethattheapproximationofnoshuntelementswasalsoincludedinthisstep oftheprocess. Inordertoincorporatetapchangersandphaseshiftersintotheformulation,we followthemethodoutlinedin[13],suchthatthetapchangeisincorporatedinto thebusadmittancematrix,andthephaseshiftisincorporatedasadditionalpower injections.Thebusadmittancematrixwouldbe: B ij = 8 > > < > > : P j 6 = i b ij =t ij if j = i )]TJ/F22 11.9552 Tf 9.298 0 Td [(b ij =t ij if j 6 = i .9 Where b ij =Im y ij .Consideringthefactthat b ij isalmostalwaysnegative,the phaseshiftsaremodeledasapowerinjectionatthefrombusandapowerextraction atthetobus: P i;ij = )]TJ/F22 11.9552 Tf 9.298 0 Td [(b ij ij .10 P j;ij = b ij ij .11 9

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Where P i;ij istheadditionalpowertobeinjectedatthe i bustoaccountforthe phaseshiftforline ij ,and P j;ij istheadditionalpowertobeinjectedatthe j bus. Thisformulationofhandlingtapchangersandphaseshiftersshallbereferredtoas thesymmetrical"tap/phasemethod. 2.4ModiedThetaLinearizationMTLMethod TheMTLmethodlinearizationseekstoapproximatetheACPFequationsas linearexpressionswithrespectto V 2 and V 2 asfollows: P = G y V s )]TJ/F22 11.9552 Tf 11.955 0 Td [(B z 0 Q = )]TJ/F22 11.9552 Tf 11.955 0 Td [(B y V s )]TJ/F22 11.9552 Tf 11.955 0 Td [(G z 0 .12 Where V s isthevectorof V 2 i valuesforthebusesinthesystem, 0 isthevector of V 2 i i quantitiesforeachbus, P and Q arethevectorsofpowerinjections,and thefour G and B matricesshallbedenedlater.Oncethisequationissolvedwith respectto V 2 and V 2 ,thecomplexvoltagesinthesystemcanbereconstructed fromthesolutionvariables.Weshallderive.12dierentlythanitisdonein[1]as follows.Startingwith.1,theapproximationsspeciedin[1]canbeexpressedin thefollowingsteps: P i = n X j =1 G ij V i V j cos ij + n X j =1 B ij V i V j sin ij .13 n X j =1 G ij V i V j + n X j =1 B ij V i V j i )]TJ/F22 11.9552 Tf 11.955 0 Td [( j .14 = n X j =1 G ij V i V j + n X j =1 B ij V i V j i )]TJ/F22 11.9552 Tf 11.955 0 Td [(V i V j j .15 P i n X j =1 G ij V 2 j + n X j =1 B ij V 2 i i )]TJ/F22 11.9552 Tf 11.955 0 Td [(V 2 j j .16 10

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Similarily,with.2: Q i = )]TJ/F23 7.9701 Tf 18.021 14.944 Td [(n X j =1 B ij V i V j cos ij + n X j =1 G ij V i V j sin ij .17 )]TJ/F23 7.9701 Tf 29.976 14.944 Td [(n X j =1 B ij V i V j + n X j =1 G ij V i V j i )]TJ/F22 11.9552 Tf 11.955 0 Td [( j .18 = )]TJ/F23 7.9701 Tf 18.021 14.944 Td [(n X j =1 B ij V i V j + n X j =1 G ij V i V j i )]TJ/F22 11.9552 Tf 11.955 0 Td [(V i V j j .19 Q i )]TJ/F23 7.9701 Tf 29.975 14.944 Td [(n X j =1 B ij V 2 j + n X j =1 G ij V 2 i i )]TJ/F22 11.9552 Tf 11.955 0 Td [(V 2 j j .20 Therstapproximationinthederivationsaboveconsiderstheangledierence acrossaline ij tobecloseenoughtozerotoconsidercos ij 1andsin ij ij .At thesametime,thetuningfactor isintroduced,whichwasempiricallyfoundin[1] tocreatetheclosestapproximationtotheACPFequationsatavalueof0.95.The secondapproximationintheabovederivationstakesintoaccountthefactthatthe voltagemagnitudesoneithersideofalinewillbeveryclosetoeachother,andso V i V j V 2 i V 2 j . Atthisstepwewillalsodeneanewvariable,referredtoasthemodiedtheta": V 2 i i = 0 i .21 Introducingthisvariableandaftersomemanipulations,wecanarriveatthestandard matrixformofthelinearization: P i = n X j =1 G ij V 2 j + n X j =1 B ij 0 i )]TJ/F22 11.9552 Tf 11.955 0 Td [( 0 j .22 = n X j =1 G ij V 2 j + n X j =1 ;j 6 = i B ij 0 i )]TJ/F23 7.9701 Tf 24.592 14.944 Td [(n X j =1 ;j 6 = i B ij 0 j .23 P = G y V s )]TJ/F22 11.9552 Tf 11.955 0 Td [(B z 0 .24 .25 11

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Q i = )]TJ/F23 7.9701 Tf 18.02 14.944 Td [(n X j =1 B ij V 2 j + n X j =1 G ij 0 i )]TJ/F22 11.9552 Tf 11.955 0 Td [( 0 j .26 = )]TJ/F23 7.9701 Tf 18.02 14.944 Td [(n X j =1 B ij V 2 j + n X j =1 ;j 6 = i G ij 0 i )]TJ/F23 7.9701 Tf 24.591 14.944 Td [(n X j =1 ;j 6 = i G ij 0 j .27 Q = )]TJ/F22 11.9552 Tf 11.955 0 Td [(B y V s )]TJ/F22 11.9552 Tf 11.955 0 Td [(G z 0 .28 where G y = G .29 B y = B .30 G z = 8 > > < > > : P n j =1 ;j 6 = i G ij i = j G ij i 6 = j .31 B z = 8 > > < > > : P n j =1 ;j 6 = i B ij i = j B ij i 6 = j .32 Itshouldbenotedherethatinthederivationof[1], G y = G z and B y = B z .This isonlypossibleifthenegativeofthesumoftheo-diagonalelementsisequaltothe diagonalelementforeachrow.SoweassumethattheMTLmethodincludesthe followingadditionalapproximations: Neglectpi-modelshuntadmittancesandbusshuntadmittances Neglecttapchangersandphaseshifters Notethattheseapproximationsareassumedbybasedonthediagramsusedin[1] andtheexpressionsusedfortherealpowerows-thepaperdoesnotexplicitlystate howtheyhandleshuntadmittances,tapchangers,andphaseshifters.Thisisnotthe sameassumptionthat[5]usedwhentryingtoreproducetheresultsof[1],butwe feelitistheclosestestimationofwhatwasdoneintheoriginalpaper.Indeed,the expressionusedin[1]toapproximatetherealpowerowwithrespecttothesolution 12

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variablesis: P ij g ij 2 i V 2 i )]TJ/F22 11.9552 Tf 11.955 0 Td [( j V 2 j 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(b ij i V 2 i )]TJ/F22 11.9552 Tf 11.956 0 Td [( j V 2 j .33 Usingthesamederivationtheyused,wecanndthereactivepowerowtobe: Q ij )]TJ/F22 11.9552 Tf 23.113 8.088 Td [(b ij 2 i V 2 i )]TJ/F22 11.9552 Tf 11.955 0 Td [( j V 2 j 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(g ij i V 2 i )]TJ/F22 11.9552 Tf 11.955 0 Td [( j V 2 j .34 2.5DecoupledLinearizationDLMethod TheDecoupledLinearizationmethodexpressesanlinearizationoftheACPF equationswithrespectto V and directly[11]: P = GV )]TJ/F22 11.9552 Tf 11.955 0 Td [(B 0 Q = )]TJ/F22 11.9552 Tf 9.299 0 Td [(BV )]TJ/F22 11.9552 Tf 11.955 0 Td [(G 0 .35 Where V and arevectorsofthebusvoltagemagnitudesandvoltageangles, respectively, G and B aretherealandimaginarypartsof Y ,and G 0 and B 0 shallbe denedlater.Theequationsaredecoupled"inthesensetherearenomultiplications between V and . Inordertounderstandthismethodproperly,weshallre-derivethesameequations usedin[11]toderivetheirbasicequations,expandingtheirequationsforthesakeof clarity.TheDLPFutilizestheasymmetricalmethodseesection2.2forhandling tapchangersandphaseshifters,andsoweshallincorporatethisintothederivation: P i = n X j =1 G ij V i V j cos ij + n X j =1 B ij V i V j sin ij = g ii V 2 i + X j 2 TE i g ij t 2 ij V 2 i )]TJ/F22 11.9552 Tf 11.955 0 Td [(g 0 ij V i V j cos ij )]TJ/F22 11.9552 Tf 11.955 0 Td [(b 0 ij V i V j sin ij + X i 2 TE j )]TJ/F22 11.9552 Tf 5.479 -9.683 Td [(g ij V 2 i )]TJ/F22 11.9552 Tf 11.955 0 Td [(g 00 ij V i V j cos ij )]TJ/F22 11.9552 Tf 11.955 0 Td [(b 00 ij V i V j sin ij g ii V i + X j 2 TE i g ij t 2 ij V i )]TJ/F22 11.9552 Tf 11.955 0 Td [(g 0 ij V j )]TJ/F22 11.9552 Tf 11.955 0 Td [(b 0 ij i )]TJ/F22 11.9552 Tf 11.955 0 Td [( j 13

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+ X i 2 TE j )]TJ/F22 11.9552 Tf 5.479 -9.684 Td [(g ij V i )]TJ/F22 11.9552 Tf 11.955 0 Td [(g 00 ij V j )]TJ/F22 11.9552 Tf 11.955 0 Td [(b 00 ij i )]TJ/F22 11.9552 Tf 11.955 0 Td [( j P i = g ii V i + X j 2 TE i g ij t 2 ij V i )]TJ/F22 11.9552 Tf 11.955 0 Td [(g 0 ij V j + X i 2 TE j )]TJ/F22 11.9552 Tf 5.479 -9.684 Td [(g ij V i )]TJ/F22 11.9552 Tf 11.955 0 Td [(g 00 ij V j )]TJ/F28 11.9552 Tf 11.291 24.03 Td [(0 @ i X j 2 TE i b 0 ij + X j 2 TE i )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [()]TJ/F22 11.9552 Tf 9.299 0 Td [(b 0 ij j 1 A )]TJ/F28 11.9552 Tf 11.291 24.03 Td [(0 @ i X i 2 TE j b 00 ij + X i 2 TE j )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [()]TJ/F22 11.9552 Tf 9.299 0 Td [(b 00 ij j 1 A = n X j =1 G ij V j )]TJ/F23 7.9701 Tf 18.02 14.944 Td [(n X j =1 B 0 ij j .36 where g 0 ij = g ij t ij cos ij )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(b ij t ij sin ij .37 g 00 ij = g ij t ij cos ij + b ij t ij sin ij .38 b 0 ij = g ij t ij sin ij + b ij t ij cos ij .39 b 00 ij = )]TJ/F22 11.9552 Tf 10.494 8.088 Td [(g ij t ij sin ij + b ij t ij cos ij .40 Theapproximationsbeingappliedinthethirdstepabovebeing: sin ij ij cos ij 1 V i 1foroneofthe V i 'sineachterminvolving g ii and g ij V i V j 1fortheterminvolving b ij Notethatthequantitiesdenedin.37{.40areessentiallythenegativeoftheo diagonalelementsofthe G and B matrices,respectively.Hence, B 0 wouldbedened 14

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as: B 0 ij = 8 > > < > > : n P j =1 ;j 6 = i )]TJ/F22 11.9552 Tf 9.299 0 Td [(B ij if j = i B ij if j 6 = i .41 Itshouldbenotedthatduetotheasymmetryof Y ,thesumintheaboveexpressionisoverthetermsintherow i ,andisnotvalidifthesumisconducteddownthe column i instead.Ifwerepeatthesameprocesswiththeequationforthereactive power,wewouldgetthis: Q i = )]TJ/F23 7.9701 Tf 17.357 14.944 Td [(n X j =1 B ij V i V j cos ij + n X j =1 G ij V i V j sin ij = )]TJ/F22 11.9552 Tf 9.299 0 Td [(b ii V 2 i + X j 2 TE i )]TJ/F22 11.9552 Tf 10.494 8.088 Td [(b ij t 2 ij V 2 i )]TJ/F22 11.9552 Tf 14.687 9.168 Td [(b sh ij 2 t 2 ij V 2 i + b 0 ij V i V j cos ij )]TJ/F22 11.9552 Tf 11.955 0 Td [(g 0 ij V i V j sin ij ! + X i 2 TE j )]TJ/F22 11.9552 Tf 9.299 0 Td [(b ij V 2 i )]TJ/F22 11.9552 Tf 13.15 9.167 Td [(b sh ij 2 V 2 i + b 00 ij V i V j cos ij )]TJ/F22 11.9552 Tf 11.955 0 Td [(g 00 ij V i V j sin ij ! )]TJ/F22 11.9552 Tf 21.088 0 Td [(b ii V i + X j 2 TE i )]TJ/F22 11.9552 Tf 10.494 8.088 Td [(b ij t 2 ij V i )]TJ/F22 11.9552 Tf 14.688 9.168 Td [(b sh ij 2 t 2 ij V i + b 0 ij V j )]TJ/F22 11.9552 Tf 11.955 0 Td [(g 0 ij i )]TJ/F22 11.9552 Tf 11.955 0 Td [( j ! + X i 2 TE j )]TJ/F22 11.9552 Tf 9.299 0 Td [(b ij V i )]TJ/F22 11.9552 Tf 13.151 9.167 Td [(b sh ij 2 V i + b 00 ij V j )]TJ/F22 11.9552 Tf 11.955 0 Td [(g 00 ij i )]TJ/F22 11.9552 Tf 11.956 0 Td [( j ! = )]TJ/F22 11.9552 Tf 9.299 0 Td [(b ii V i + X j 2 TE i )]TJ/F22 11.9552 Tf 10.494 8.087 Td [(b ij t 2 ij V i )]TJ/F22 11.9552 Tf 14.688 9.168 Td [(b sh ij 2 t 2 ij V i + b 0 ij V j ! + X i 2 TE j )]TJ/F22 11.9552 Tf 9.298 0 Td [(b ij V i )]TJ/F22 11.9552 Tf 13.151 9.168 Td [(b sh ij 2 V i + b 00 ij V j ! )]TJ/F28 11.9552 Tf 11.291 24.03 Td [(0 @ i X j 2 TE i g 0 ij + X j 2 TE i )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [()]TJ/F22 11.9552 Tf 9.299 0 Td [(g 0 ij j 1 A )]TJ/F28 11.9552 Tf 11.291 24.031 Td [(0 @ i X i 2 TE j g 00 ij + X i 2 TE j )]TJ/F25 11.9552 Tf 5.479 -9.684 Td [()]TJ/F22 11.9552 Tf 9.299 0 Td [(g 00 ij j 1 A Q i = )]TJ/F23 7.9701 Tf 17.356 14.944 Td [(n X j =1 B ij V j )]TJ/F23 7.9701 Tf 18.02 14.944 Td [(n X j =1 G 0 ij j .42 In[11],itisassumedthat G 0 ij G ij duetotheshuntconductancebeingsmall.In additiontothis,thereisahiddenassumption-thisalsoassumesthat g ij =t 2 g 0 ij 15

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g 00 ij .Thisadditionalassumptionisquitevalidinmostcases,asinmostcasestheX/R ratiofortapchangersandphaseshiftersisreasonablylarge. Finally,theapproximatebranchowequationsfrom[11],restatedhere: P ij g ij t ij V i t ij )]TJ/F22 11.9552 Tf 11.955 0 Td [(V j )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(b ij t ij ij )]TJ/F22 11.9552 Tf 11.955 0 Td [( ij .43 Q ij )]TJ/F22 11.9552 Tf 25.106 8.088 Td [(b ij t ij V i t ij )]TJ/F22 11.9552 Tf 11.955 0 Td [(V j )]TJ/F22 11.9552 Tf 13.15 8.088 Td [(g ij t ij ij )]TJ/F22 11.9552 Tf 11.955 0 Td [( ij .44 Notethat[11]doesnotactuallyhavetheequationfor Q ij ,butitcanbetrivially derivedusingthesameapproach. 2.6NaturalLogLinearizationNLLMethod Initssimplestform,theNLLmethodexpressesitslinearizationintermsofln V and asfollows: P i )]TJ/F15 11.9552 Tf 11.955 0 Td [(ln V i X j [ G ij +ln V j + B ij ij ].45 Q i )]TJ/F15 11.9552 Tf 11.955 0 Td [(2ln V i X j [ )]TJ/F22 11.9552 Tf 9.299 0 Td [(B ij )]TJ/F15 11.9552 Tf 11.955 0 Td [(ln V i +ln V j + G ij ij ].46 Itshouldbeexpressedthattheaboveequationsareonlytrulylinearifconsidered inthecontextofsolvingthePFproblem.Thatis,whensolvingthePFproblem,for eachbusinthesystemeither P i or V i orbothisknown,allowingthequadraticterm ontheleftsideof2.45tobecomelinear.Thesamethingoccursforthequadratic termin.46. Equations.45-.46requirealterationsinordertoincludetapchangersand phaseshifters.Theoriginalpaperaddstapchangersandphaseshiftersinthemanner oftheasymmetricalmethodmentionedinsection2.2.Thederivationincludingthe tapchangersandphaseshiftersislengthy,andonlythepartsnecessaryforunderstandingtheimplementationshallbeincludedhere.Inparticular,thederivationof thematrixformofthelinearizationisnotgivenin[5],andsohencemustbeincluded here. 16

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Asshouldbeobviousfromtheequations.45-.46,theequationfor P is signicantlydierentfromtheonefor Q .Asdescribedin[5],thestartingpointfor thefullequationsfor P areexpressedlikeso: S 0 i = y sh 0 i u i + X j y 0 ij u ij +j ij ; .47 where u i =ln V i .48 u ij =ln V i )]TJ/F15 11.9552 Tf 11.955 0 Td [(ln V j .49 S 0 i = S i )]TJ/F28 11.9552 Tf 18.758 11.357 Td [(X j 2 TE i y sh ij )]TJ/F22 11.9552 Tf 11.955 0 Td [(t 0 ij )]TJ/F22 11.9552 Tf 11.955 0 Td [(y ij t 0 ij +j ij =t ij )]TJ/F28 11.9552 Tf 18.462 11.357 Td [(X i 2 TE k y sh ki + y ki t 0 ki +j ki )]TJ/F22 11.9552 Tf 11.955 0 Td [(y ii .50 t 0 ij =ln t ij .51 y sh 0 i = S i + y ii + X j 2 TE i y sh ij =t ij + X i 2 TE k y sh ki .52 y 0 ij = 8 > > > > < > > > > : y ij =t ij ; if j 2 TE i y ji ; if i 2 TE j 0 ; otherwise .53 where S i istheinjectedcomplexpoweratbus i .Itmustbenotedthatalthoughthe quantities S i , y ij ,andothersappearintheequationsabove,theunitsforeachofthese quantitiesisactuallypuamperes.Everyadditivetermintheaboveequationsexcept for and ij ,ofcoursecarriestheunitsofpuamperes,althoughthequantitiesare astheyappearintheaboveequations,intheirexpectedpu-scaledvalues.Seethe derivationoftheaboveequationsin[5]fordetails. Leavingtheunitsissueaside,forsimplicityweshallre-write.50and.52as thefollowing: S 0 i = S i )]TJ/F22 11.9552 Tf 11.956 0 Td [(m i .54 17

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y sh 0 i = S i + n i .55 with m i = X j 2 TE i y sh ij )]TJ/F22 11.9552 Tf 11.955 0 Td [(t 0 ij )]TJ/F22 11.9552 Tf 11.956 0 Td [(y ij t 0 ij +j ij =t ij + X i 2 TE k y sh ki + y ki t 0 ki +j ki + y ii ; .56 n i = y ii + X j 2 TE i y sh ij =t ij + X i 2 TE k y sh ki .57 Andtriviallytakingtherealpartoftheabove: P 0 i = P i )]TJ/F15 11.9552 Tf 11.955 0 Td [(Re m i .58 g sh 0 i = P i +Re n i .59 Withthesethingsinplace,takingtherealpartof.47andwithsomere-arranging, weobtain: P 0 i = g sh 0 i + X j 6 = i g 0 ij u i + X j 6 = i )]TJ/F22 11.9552 Tf 9.298 0 Td [(g 0 ij u j )]TJ/F15 11.9552 Tf 11.955 0 Td [( X j 6 = i b 0 ij i )]TJ/F28 11.9552 Tf 11.955 11.358 Td [(X j 6 = i )]TJ/F22 11.9552 Tf 9.299 0 Td [(b 0 ij j .60 Re-organizinginvectorform,andsubstitutingtheequation.59,wehave: P 0 i = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 )]TJ/F22 11.9552 Tf 9.298 0 Td [(g 0 i 1 )]TJ/F22 11.9552 Tf 9.298 0 Td [(g 0 i 2 . . . P i + < n i + P j 6 = i g 0 ij . . . )]TJ/F22 11.9552 Tf 9.298 0 Td [(g 0 in 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 T u )]TJ/F28 11.9552 Tf 11.955 77.829 Td [(2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 )]TJ/F22 11.9552 Tf 9.298 0 Td [(b 0 i 1 )]TJ/F22 11.9552 Tf 9.298 0 Td [(b 0 i 2 . . . P j 6 = i b 0 ij . . . )]TJ/F22 11.9552 Tf 9.298 0 Td [(b 0 in 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 T .61 with u beingthevectorof u i valuesforeachbus.Now,deningtherowvectorsin theaboveequationasrowsinthematrices G y and B y ,andalsosplittingthevectors intothepartspertainingtotheslack V ,non-slackgenerator PV ,andload PQ 18

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buses,wehave: P 0 i = G y iV G y iPV G y iPQ 2 6 6 6 6 4 u V u PV u PQ 3 7 7 7 7 5 )]TJ/F28 11.9552 Tf 11.955 16.857 Td [( B y iV B y iPV B y iPQ 2 6 6 6 6 4 V PV PQ 3 7 7 7 7 5 .62 Ofimportantnotehereisthefactthatthematrix G y isnotsquare.Sinceitcontainsin thedenitionforeachrowtheinjectedrealpower,onlythelineswheretheinjected realpowerisknownareusablewhensolvingthePFproblem.Inconsiderationof thestandardPFproblem,wenowre-organizetheequationtoseparatetheknown quantitiesfromtheunknownquantities: P 0 i = )]TJ/F22 11.9552 Tf 9.299 0 Td [(B y iV G y iV G y iPV 2 6 6 6 6 4 V u V u PV 3 7 7 7 7 5 + )]TJ/F22 11.9552 Tf 9.298 0 Td [(B y iPV )]TJ/F22 11.9552 Tf 9.298 0 Td [(B y iPQ G y iPQ 2 6 6 6 6 4 PV PQ u PQ 3 7 7 7 7 5 .63 InordertosolvethestandardPFproblem,wewouldneed.63foreachPVbus andeachPQbus.Beforeweconsolidatethisintofullmatrixform,however,wewill nowaddresstheQequations.Thestartingpoint,asreferencedin[5]ishere: S 00 i =2 S i u i + X j y 00 ij u ij +j ij .64 where S 00 i = S i )]TJ/F28 11.9552 Tf 18.758 11.357 Td [(X j 2 TE i y sh ij )]TJ/F22 11.9552 Tf 11.956 0 Td [(y ij t 0 ij +j ij =t 2 ij )]TJ/F28 11.9552 Tf 19.127 11.357 Td [(X i 2 TE k y sh ki + y ki t 0 ki +j ki )]TJ/F22 11.9552 Tf 11.955 0 Td [(y ii .65 y 00 ij = 8 > > > > < > > > > : y ij =t 2 ij ; if j 2 TE i y ji ; if i 2 TE j 0 ; otherwise .66 Onceagain,wewillsimplify.65asfollows: S 00 i = S i )]TJ/F22 11.9552 Tf 11.955 0 Td [(o i .67 19

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where o i = X j 2 TE i y sh ij )]TJ/F22 11.9552 Tf 11.955 0 Td [(y ij t 0 ij +j ij =t 2 ij + X i 2 TE k y sh ki + y ki t 0 ki +j ki + y ii Andtrivially,theimaginarypartof.67is: Q 00 i = Q i +Im o i .68 Continuingon,takingtheimaginarypartof.64andre-arrangingsome,wehave: )]TJ/F22 11.9552 Tf 9.298 0 Td [(Q 00 i = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 Q i u i + X j )]TJ/F22 11.9552 Tf 5.48 -9.684 Td [(b 00 ij u i )]TJ/F22 11.9552 Tf 11.955 0 Td [(u j + g 00 ij i )]TJ/F22 11.9552 Tf 11.955 0 Td [( j .69 Q 00 i = )]TJ/F28 11.9552 Tf 11.291 20.444 Td [( )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 Q i + X j 6 = i b 00 ij ! u i )]TJ/F28 11.9552 Tf 11.955 11.358 Td [(X j 6 = i )]TJ/F22 11.9552 Tf 9.299 0 Td [(b 00 ij u j )]TJ/F22 11.9552 Tf 11.956 0 Td [( i X j 6 = i g 00 ij ! )]TJ/F28 11.9552 Tf 11.955 11.358 Td [(X j 6 = i )]TJ/F22 11.9552 Tf 9.299 0 Td [(g 00 ij j .70 Now,re-workingtheequationsinvectorform: Q 00 i = )]TJ/F28 11.9552 Tf 11.291 77.829 Td [(2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 )]TJ/F22 11.9552 Tf 9.298 0 Td [(b 00 i 1 )]TJ/F22 11.9552 Tf 9.298 0 Td [(b 00 i 2 . . . )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 Q i + P j 6 = i b 00 ij . . . )]TJ/F22 11.9552 Tf 9.299 0 Td [(b 00 in bus 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 T u )]TJ/F28 11.9552 Tf 11.955 77.829 Td [(2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 )]TJ/F22 11.9552 Tf 9.298 0 Td [(g 00 i 1 )]TJ/F22 11.9552 Tf 9.298 0 Td [(g 00 i 2 . . . P j 6 = i g 00 ij . . . )]TJ/F22 11.9552 Tf 9.299 0 Td [(g 00 in bus 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 T .71 Splittingupthevectorsintermsofthedierentbustypes,anddeningthevectors aboveaspartsofthematrices B yy and G yy : Q 00 i = )]TJ/F28 11.9552 Tf 11.291 16.857 Td [( B yy iV B yy iPV B yy iPQ 2 6 6 6 6 4 u V u PV u PQ 3 7 7 7 7 5 )]TJ/F28 11.9552 Tf 11.955 16.857 Td [( G yy iV G yy iPV G yy iPQ 2 6 6 6 6 4 V PV PQ 3 7 7 7 7 5 .72 Andnally,re-organizingtoseparatetheknownvariablesfromtheunknownvariables: Q 00 i = )]TJ/F22 11.9552 Tf 9.299 0 Td [(G yy iV )]TJ/F22 11.9552 Tf 9.299 0 Td [(B yy iV )]TJ/F22 11.9552 Tf 9.298 0 Td [(B yy iPV 2 6 6 6 6 4 V u V u PV 3 7 7 7 7 5 + )]TJ/F22 11.9552 Tf 9.299 0 Td [(G yy iPV )]TJ/F22 11.9552 Tf 9.298 0 Td [(G yy iPQ )]TJ/F22 11.9552 Tf 9.299 0 Td [(B yy iPQ 2 6 6 6 6 4 PV PQ u PQ 3 7 7 7 7 5 .73 20

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Andofcourseitshouldbestatedthatweneedoneoftheaboveequationsforevery PQbus. Now,combiningequations.63and.73inmatrixform,wehave: 2 6 6 6 6 4 P 0 PQ P 0 PV Q 00 PQ 3 7 7 7 7 5 = 2 6 6 6 6 4 )]TJ/F22 11.9552 Tf 9.299 0 Td [(B y PV;V G y PV;V G y PV;PV )]TJ/F22 11.9552 Tf 9.299 0 Td [(B y PQ;V G y PQ;V G y PQ;PV )]TJ/F22 11.9552 Tf 9.298 0 Td [(G yy PQ;V )]TJ/F22 11.9552 Tf 9.298 0 Td [(B yy PQ;V )]TJ/F22 11.9552 Tf 9.299 0 Td [(B yy PQ;PV 3 7 7 7 7 5 2 6 6 6 6 4 V u V u PV 3 7 7 7 7 5 + 2 6 6 6 6 4 )]TJ/F22 11.9552 Tf 9.299 0 Td [(B y PV;PV )]TJ/F22 11.9552 Tf 9.298 0 Td [(B y PV;PQ G y PV;PQ )]TJ/F22 11.9552 Tf 9.299 0 Td [(B y PQ;PV )]TJ/F22 11.9552 Tf 9.299 0 Td [(B y PQ;PQ G y PQ;PQ )]TJ/F22 11.9552 Tf 9.298 0 Td [(G yy PQ;PV )]TJ/F22 11.9552 Tf 9.299 0 Td [(G yy PQ;PQ )]TJ/F22 11.9552 Tf 9.298 0 Td [(B yy PQ;PQ 3 7 7 7 7 5 2 6 6 6 6 4 PV PQ u PQ 3 7 7 7 7 5 .74 2 6 6 6 6 4 P 0 PV P 0 PQ Q 00 PQ 3 7 7 7 7 5 + 2 6 6 6 6 4 B y PV;V )]TJ/F22 11.9552 Tf 9.299 0 Td [(G y PV;V )]TJ/F22 11.9552 Tf 9.299 0 Td [(G y PV;PV B y PQ;V )]TJ/F22 11.9552 Tf 9.298 0 Td [(G y PQ;V )]TJ/F22 11.9552 Tf 9.298 0 Td [(G y PQ;PV G yy PQ;V B yy PQ;V B yy PQ;PV 3 7 7 7 7 5 2 6 6 6 6 4 V u V u PV 3 7 7 7 7 5 = )]TJ/F28 11.9552 Tf 11.291 38.377 Td [(2 6 6 6 6 4 B y PV;PV B y PV;PQ )]TJ/F22 11.9552 Tf 9.298 0 Td [(G y PV;PQ B y PQ;PV B y PQ;PQ )]TJ/F22 11.9552 Tf 9.298 0 Td [(G y PQ;PQ G yy PQ;PV G yy PQ;PQ B yy PQ;PQ 3 7 7 7 7 5 2 6 6 6 6 4 PV PQ u PQ 3 7 7 7 7 5 .75 Inthislastform,theleftsideoftheequationisallknownquantities,withthevector ontherightbeingtheunknownquantitiesofinterest.Thisformsthelinearexpression inordertosolvethePFproblemfortheNLL. Finally,thefrom"powerowequations,expressedintermsofthevariablesof interest,is: P ij = g ij u ij )]TJ/F22 11.9552 Tf 11.955 0 Td [(t 0 ij )]TJ/F22 11.9552 Tf 11.955 0 Td [(b ij ij )]TJ/F22 11.9552 Tf 11.955 0 Td [( ij + g ij 2 h u ij )]TJ/F22 11.9552 Tf 11.955 0 Td [(t 0 ij 2 + ij )]TJ/F22 11.9552 Tf 11.955 0 Td [( ij 2 i .76 Q ij = )]TJ/F22 11.9552 Tf 11.955 0 Td [(b ij u ij )]TJ/F22 11.9552 Tf 11.955 0 Td [(t 0 ij )]TJ/F22 11.9552 Tf 11.955 0 Td [(g ij ij )]TJ/F22 11.9552 Tf 11.955 0 Td [( ij )]TJ/F22 11.9552 Tf 13.151 8.087 Td [(b ij 2 h u ij )]TJ/F22 11.9552 Tf 11.955 0 Td [(t 0 ij 2 + ij )]TJ/F22 11.9552 Tf 11.955 0 Td [( ij 2 i )]TJ/F22 11.9552 Tf 11.955 0 Td [(b sh ij + u i )]TJ/F22 11.9552 Tf 11.955 0 Td [(t 2 .77 Notethattheequationfor Q ij hasanaddedtermontheendtoaccountfortheshunt admittanceforthebranch.Thistermwasnotpresentin[5],andappearstobean omission.Theto"powerowequationscanbederivedsimilarly,butweshallnot dosohere. 21

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3.TestMethodology Inthissection,weshallpresentthetestmethodologyforcomparingthefour presentedmethods.Weshallstartwithdescribingthetransmission-leveltestsystems thatweareusing,andthentheactualsimulationsmethodology. 3.1SyntheticPowerTestSystems Anumberofsynthetictransmission-levelpowertestsystemshavebeencreated overtheyearstoserveastestbedsforalgorithmssolvingthePF,OPF,andother similarproblems.ThetestsystemsweareusingarelistedinTables3.1-3.2.The majorityofthesystemsareincludedinthecodeforMatPower[14],butsomealso havebeenincludedfromTexasA&MUniversity's"ElectricGridTestCaseRepository",whichwerecreatedasper[2].Inthetable,thenumberofvariouscomponents isincluded,alongwiththenumberoftapchangersandphaseshifterslabeledas "Tap"and"Phase"respectively.Alsoofinterestisthetotalrealandreactivepower demandedineachsystem,whichisnotnecessarilycorrelatedwiththenumberof busesinthesystem.EachofthedatasetsinTables3.1-3.2representsnapshotsin time,withtheloadlevelsandgeneratorsettingsrepresentingthestateofthesystem ataninstantduringadayofnormaloperation.Afewofthesystemsfrom[2]have syntheticloaddatathroughtime,althoughitislimitedtoaltering8dierentgroups ofbusesupanddown,ratherthanalteringeachbusonanindividualbasis. MostoftheIEEEtestcasesarequiteold,suchasthe30,57,and118bussystems, whichdatebacktothe1960's,andthe300bussystem,whichwascreatedin1993. ThePolishsystemswerecreatedbasedondatafrom1999to2008,withthelarger systemsbeingmorerecentthanthesmallersystems.ThePEGASEsystemsare syntheticrepresentationsofthesizeandbasicstructureoftheEuropeanhighvoltage powersystem.Finally,theACTIVSgsystemsarefrom[2],andrepresentdierent sectionsoftheUSpowergrid,whetherIllinois,SouthCarolina,Texas,thewestern US,oralargeportionofthenortheasternUS.Hence,thesystemsavailabletoteston 22

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arebiasedtowardstheEuropeanpowergrid.AndoftheNorthAmericansystems, whichincludeanumberoftheIEEEsystemsandtheACTIVSgsystems,manyareso oldastopossiblynotproperlyrepresentthepowergridoperationtoday.Although thisisnotanidealsituationwithregardstodata,itissomeofthebestthatis currentlyavailablepublicly. 3.2SimulationsMethodology ThebasicformofthetestmethodologyistosolvethePFproblemforavariety oftestsystemsinourcase,tables3.1-3.2,usingboththeACPFNewton-Raphson methodandeachofthefourlinearizations.Aftersolvingeachsystem,theresultant quantities V , , P flow ,and Q flow arecomparedwiththeirexact"known"valuesfrom theACPFsolution.Thesefourquantitiesarechosenspecicallybecausetheyareall usedinconstraintsinsecurityconstrainedoptimizationproblemsandalsoinmedium tolongtermplanning. P flow and Q flow inthesecasesaretheowscalculatedbythe method-specicequationsforthepurposeofuseasconstraintsinanoptimization problem.Whencomparingtheresultsoftwomethodsonaspecicsystem,the medianandmaxerrorforeachquantityshallbecompared,calculatingthevalue acrossallthebuses/branchesinthesystem.Themedianvalueischosenduetoits relativeinsensitivitytooutliers,asopposedtotheaverageortheRMSerror.Inour analysis,theerrordistributionofthevariousquantitiesappearstobeclosertoan exponentialdistribution,withsomefairlyextremeoutliers.Hence,themeanerror wouldnotbeanaccuraterepresentationofthemiddle"valueinthedataset. ItshouldbenotedthattherearemanysolutionstotheACformofthePF problem,andtheNewton-Raphsonmethodonlyndsoneofthem.Itisgenerally assumedthattheonefoundbytheNewton-Raphsonmethodisthe"correct"one,or rathertheonethatthepowersystemwouldexistinwheninstableoperation.Several papershavebeenpublishedworkingontheproblemofndingallthesolutionstothe ACPFproblemforcertainsmallsystems[7],[4].Fromdataobtainedfromthe 23

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Table3.1: TestPowerSystems No.NameBusGen.BranchTapPhaseTotal P D Total Q D 1IEEERTS243338502850580 2IEEE30-bus3064100189107 3IEEE57-bus577801501251336 4IEEE118-bus118541869042421438 5IEEE145-bus1455045352028305178700 6ACTIVSg20020038245001476421 7IEEE300-bus30069411620235267788 8ACTIVSg500500565970077512067 9PEGASE13541354260199123437306013401 10French1888188829025314051591102271 11French19511951366259648618065717680 12ACTIVSg200020004323206006710919014 13Polish2383238332728961705245588144 14Polish2736273627032691711180755340 15Polish2737273721932691731112673953 16Polish2746wop274643133071720189625534 17Polish2746wp274645632791710248737147 18French284828485113776558352562170 19French28682868561380860637882614729 20PEGASE286928695104582496513243729008 21Polish30123012385357220102717010201 22Polish3120312029836932060211818723 24

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Table3.2: TestPowerSystemscon't No.NameBusGen.BranchTapPhaseTotal P D Total Q D 23Polish33753374479416138304836319527 24French64686468399900013196852975691 25French647064707619005133359659213178 26French6495649568090191359610391614874 27French6515651568490371367510726417414 28PEGASE9241924114451604913192731235473582 29ACTIVSg10k10000193712706970015091739962 authorsof[4],allbutoneoftheknownsolutionstothe14-bussystemarehighly unrealistic,withvoltageswellbelow0.9orwellabove1.1pu.Andsothereisatleast someevidencethattheNewton-Raphsonsolutionisindeedtheonlystablesolution closeto1puvoltage.Weareimplicitlyassuminginthispaperthatthisresultcan beextendedtolargersystems.Withthisinhand,wecanhopethatcomparingthe ACPFNewton-Raphsonsolutionstothelinearsolutionsarepropercomparisons. WehaveconductedallourtestinginMatlab2018b,usingMatPower6.0.2[14] forrunningtheACPFandtheDCPF.Wehavenotfullyoptimizedthecodeforthe newlinearizations,andsoacomparisonofruntimefordierentsystemsisconsidered spurious. Inourtesting,weshallstartwithrunningeachofthebasecasesinTables3.13.2.Thebasecases,however,onlyeachrepresentasinglesnapshotintime,asingle possiblestateofthesystem.Whenrunninganoptimizationoraunitcommitment problem,amuchlargerspaceofpossiblestatesareexplored.Inanattempttoexploremorepossiblestatesofthesystem,wehavedevelopedasetoftestingscenarios wherebythebasecasesarealteredinspecicways.Thesealterationsshallbede25

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scribedbelow.Notethatforthebelowsimulations,weshallomitthe2largest systemsinTables3.1-3.2duetotheirsizemakinglargenumbersofsimulationson themcomputationallyprohibitive. 3.2.1RemoveTapChangersandPhaseShifters Ashasbeendiscussedpreviously,theMTLmethoddoesnotincludeaformulation oftapchangersorphaseshifters.Itisthusappropriatetotestthecasewhereall thetapchangersandphaseshiftersareremovedentirelyfromthesystemsofinterest, whilekeepingallelsethesame. 3.2.2ScalingLoadLevels Onlyafewsystemsinourtestcaseshavesyntheticloaddatathroughtime,and whatdatathereisisnotverygranular,andsowechosetotakeamoregeneralapproachtoscalingtheloadlevelsinourtests.Giventhelackofinformationaboutthe natureofthedierentbusesresidential,commercial,variouslevelsofindustrial,etc thereisnowaytomatchtheloadprolespublishedbyvariouselectricalcompanies tospecicbusesinthesystem.Hence,inourtestsweshallscalealltheloadsinthe systemupordownbythesameamount.Therealandreactivepowerloadsshallbe scaledbythesameamount,asitiswellknownthatthepowerfactorsforloadstend tostayconstantacrossdierentloadlevels. WeknowthatthedierentbasecasesinTables3.1-3.2representawiderange ofpossibleloadlevelsforthesystems.Forexample,thePolish2737systemisa testcaserepresentingasummero-peaksnapshotoftheloadsandgenerationlevels. Hence,insteadofsimplyscalingalltheloadsupanddownby25%,forexample, wedecidedtoscaletheloadswithreferencetothetotalrealpoweravailablefor generationineachsystem,whichispresentasdatainthecases.Thisscalevalue couldbeconsideredaroughproxyforhowloadedthesystemis.Amoreaccurate metricfortherelativeloadlevelofthesystemisbeyondthescopeofthispaper;such wouldrequireexaminationofhowloadedthelinesinthesystemwereandavoltage stabilityanalysis.Inanycase,theexistingbasecaseshaveloadswithreferenceto 26

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theavailablegenerationfrom0.43to0.9.Intheloadscalingtestspresentedhere,the loadswerescaledfrom0.5to0.8ofthemaximumavailablegenerationinincrements of0.1.Thedispatchedgenerationbythesameamount.Inthisway,theloadlevels areapproximately"normalized"withreferencetotheavailablegenerationacrossthe systems. Foreachsystem-method-quantitysete.g.system1,MTL,voltagemagnitude, asinglemedianandmaxerrorshallbecalculated,aggregatingtheresultsofthe4 levelsintotwonumbers. 3.2.3VaryingGeneratorVoltageSetpoints Inthissetoftests,weshallvarythegeneratorvoltagesetpointsfromtheiroriginal values.Foreachtest,thegeneratorvoltagesetpointsotherthantheslackbus,shall beincreasedordecreasedbyarandomvaluebetween-0.06and0.06pu.Twentyof thesetestsshallbeconductedforeachsystem,andthemedianandmaxerrorfor eachquantity V , ,etcbecomputedforeachlinearization,aggregatingacrossall 20tests. Thereisapossibilitythatrandomizingthevoltagelevelsinthiswaywillcause fairlyextremecases,duetothepossibilityoftwogeneratorsthatarenexttoeachother beingsettoverydierentvoltagelevels.Hence,thissetoftestscanbeconsidereda relativelyextremetestcase,expectedtostressthelinearizationstoafairlyheavily. 3.2.4VaryingGeneratorRealPowerSetpoints Inordertoproperlyvarythegeneratorrealpowersetpoints,thebasecasedispatchedpowermustbere-distributedrandomlyacrossthevariousgenerators.Our methodfordoingthisisstraightforward: Separateallthenon-slackgeneratorsintotwolistsofgenerators,wherethesum ofthegeneratorsetpointsforeachgroup P sum;i areapproximatelyequalto eachother Randomlyswitchanumberofthegeneratorsbetweenthetwogroups,making 27

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surethat0 : 7


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shallbechangedlikeso: R new = R old r +1 2 r .1 X new = X old r +1 2 .2 ThiswillresultinachangeintheactualX/Rratioof r .Thelineshuntadmittance valuesshallbeleftconstant. 29

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4.ResultsandAnalysis Inthissection,theresultsofthemethodologiesdiscussedinsection3.2arepresented,inthesameorderthatwaspresentedinthatsection. 4.1Basecase Theerrorsinthelinearizationsforthebasecasearedisplayedingures4.1-4.4. Onalltheplots,thesystemnumber,asindicatedintables3.1-3.2,isplacedonthe x-axis.They-axisquantitiesaredisplayedintermsofalogscale,inordertomore clearlyportraytheerrorsofinterest. 4.1.1VoltageMagnitudes Intheplotsofthevoltageerror,theerrorintheDCPFwasincludedprimarily asasanitycheck.TheDCPFassumesallthevoltagestobe1pu.Hence,allthe newlinearizations,allofwhichsolveforthevoltagemagnitude,shouldbeingeneral betterthanthecaseofjustassuming1puforallthevoltage.Thisindeedappearsto bethecase,otherthanforacouplesystems. LookingatthemedianerrorplotFig.4.1a,theerrorforallthenewlinearizationsmostofthetimeremainsbelow0.02pu.Inastandardpowergrid,voltagesare typicallyconstrainedtobeingbetween0.9and1.1pu.Withanaverageerrorof0.02 orless,thevoltageconstraintinanoptimizationislikelytonotcausemuchdeviation fromthetrueACoptimum.However,somecareisneededinsomecases,asthemax errorforsomesystemscanriseabove0.1pu,evenashighas0.5puFig.4.1b, particularlyforsystems18-19and24-27.Thesesystemsofgreatesterrorarethe Frenchsystems,whichindicatesthattheremaybesomestructuralaspecttothose systemsthatmakesthemspecicallydicultforcertainlinearizationstomodel.On examination,theFrenchsystemshappentohavesomeofthehighestnumbersoftap changersandalsothehighestnumbersoflineswithnegativeXvaluesassociated withmodeling3-windingtransformers.Thesefactors,andothers,maybemaking foradditionaldiculty.Indeed,lookingaheadtotheplotofthemaxVerrorswithout 30

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tapchangersandphaseshiftersFig.4.5b,itappearsthatforsystems18and19, theespeciallyhigherrorsareattenuated.Also,thevoltagesetpointsforgeneratorsin systems24-27varybetween0.91and1.17,withonesetpointoddlyat0.5pu.These valuesarefairlyextreme,andcouldresultintheapproximationofvoltagesbeing closeto1pubreakingdown. Overall,theDLandNLLmethodsappeartodothebestonaverage,andthe NLLmethoddoesbestatattenuatingthemaxerrorpresent. 4.1.2Voltageangles Evenmoresothanthevoltagemagnitudes,theerrorsinthevoltageanglesFig. 4.2appeartobefairlyextreme.Themedianerrorsaremostlywithinreasonable limits,stayingbelow3degreeserrorforthemostpart.However,whenlookingat themaxerrors,thevaluesarecommonlyabove10degreeserror,andinafewlimited casesabove30degreeserror.Itshouldbenoted,however,thatfortheworstcases, theDCPFdoesn'tdomuchbetterthantheothermethods,indicatingthatthereis perhapssomethingwrongwiththesystemitself,orthatthepowersystemmanifoldis simplytoocurvedtoadmitanaccuratecold-startlinearization.System5hasresults thatareparticularlypoor,althoughthereasonsforthatcouldpossiblybethatthe systemwasdesignedfortestingvoltagestabilityunderfaultconditions.Hence,the systemmayalreadybeclosetoitsstabilitylimits. Overall,thereisnoparticularlybettermethodthananyotherasfarascalculating thevoltageanglesareconcerned-theresultsappeartobeverysystem-dependent. 4.1.3Realpowerow Themedianrealpowererrorsareingeneralwellwithinreason,beingmostly wellbelow10MWinerror.Andforsystemsthatwithdemandsfrom10,000MWto 300,000MW,thisissortoferrorisminimal.Themaxerrors,however,cansometimes bequitelarge,especiallyfortheMTLandDLmethods,wheresomeoftheerrors aregreaterthan5000MW.Interestingly,itisonceagaintheFrenchsystems-11, 31

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aMedianvoltagemagnitudeerror bMaxvoltagemagnitudeerror Figure4.1: Voltagemagnitudeerrorpu,basecase 32

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aMedianvoltageangleerror bMaxvoltageangleerror Figure4.2: Voltageangleerrordegrees,basecase 33

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18-19,and24-27thatallhaveanorderofmagnitudeworsemaxerrorsthanforany othersystems. Incomparingthedierentlinearizations,theDLmethoddoesthebestonaverage forrealpowerows,withtheDCPFclosebehind.Inthemaxerror,however,the NLLseemstodothebest,withtheDCPFsecond. aMedianrealpowerowerror bMaxrealpowerowerror Figure4.3: RealpowerowerrorMW,basecase 4.1.4Reactivepowerow 34

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ThereactivepowerowFig.4.4showsmanyofthesametrendsasthereal powerow.Particularly,theDLmethodhasextremeerrorsfortheFrenchsystem, whiledoingwellinaverage.TheNLmethoddoesnearlyaswellastheDLinaverage, whilehavingmuchlowermaxerrors. aMedianreactivepowerowerror bMaxreactivepowerowerror Figure4.4: ReactivepowerowerrorMVAR,basecase 4.1.5Basecase:Discussion Lookingonlyatthebasecase,itwouldappearthattheDLmethodandtheNLL methodperformthebest,withtheformerbetterinaverageandthelatterbetterat 35

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attenuatinghigherrors.TheDCPFstilldoesquitewellforthequantitiesthatit measures,butothermethodsoutperformitonthewhole.Therelativeaccuracyof thedierentmethodsseemsquitesystemdependent,however,andnomethodsdid wellatcalculatingvoltageanglecorrectly. 4.2RemoveTapChangersandPhaseShifters Inthissetofsimulations,thetapchangersandphaseshifterswereremovedand allthecalculationsrunagain.Figures4.5-4.8containtheresultsofthecalculations. Thegapsintheplotsfromsystem24-28indicatecaseswheretheACPFsolution failedtoconverge.ApparentlyforthelargerFrenchsystems,thetapchangersand phaseshiftersareneededinordertogetareasonablesolutionintherstplace. Intermsofchangefromthebasecase,theMTLimprovessignicantlyinterms ofvoltagemagnitude,butdoesn'tseemtochangemuchinothermetrics.Thisimprovementisexpected,astheMTLdoesnotmodeltapchangersorphaseshifters, andtherearefarmoretapchangersinthetestsystemsthanphaseshifters.This improvementismostseenintheFrenchsystems,althoughseveralothersalsoshow improvement.TheDLmethod'smaxerrorisimprovedforvoltagemagnitudes,and forthepowerows.TheNLLseemsoveralltoimproveslightlyforsomesystems,but beunchangedforothers. 36

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aMedianvoltagemagnitudeerror bMaxvoltagemagnitudeerror Figure4.5: Voltagemagnitudeerrorpu,basecasew/outtapchangersorphaseshifters 37

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aMedianvoltageangleerror bMaxvoltageangleerror Figure4.6: Voltageangleerrordegrees,basecasew/outtapchangersorphaseshifters 38

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aMedianrealpowerowerror bMaxrealpowerowerror Figure4.7: RealpowerowerrorMW,basecasew/outtapchangersorphaseshifters 39

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aMedianreactivepowerowerror bMaxreactivepowerowerror Figure4.8: ReactivepowerowerrorMW,basecasew/outtapchangersorphaseshifters 40

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4.3ScalingLoadLevels Theresultsofnormalizingtheloadlevelsisseenin4.9-4.12.Themostobvious changesfromthebasecaseappearinthevoltagemagnitudeplots,wheretheDL andNLLimprovesome.ThechangesappearthestrongestonceagainintheFrench systems.ThismayindicatethattheDLandNLLworkslightlybetterinsystems thatarenotheavilyloaded,particularlywhentherearemanytapchangers.Butmore workwouldhavetobedonetoconrmthis.Alltheotherplotshaveonlyminimal changefromthebasecase. 4.4VaryingGeneratorVoltageSetpoints Forthetestsofvaryingthegeneratorvoltagesetpoints,theresultswereingeneral muchworsethanthebasecase,buttherewasnoclearchangeintherelativequality ofthedierentlinearizations.Theplotshavebeenomittedforthesakeofbrevity. 4.5VaryingGeneratorRealPowerSetpoints Inthesetests,therewasverylittlechangeatallinthesolutionsfromthebase case.Ahighervariationintherealpowerdispatchmightberequiredinordertosee anysignicantchange. 4.6VaryingtheBranchX/RRatios Forthesakeofbrevity,wehaveonlyincludedtheplotsfortheX/Rratioscaled by0.4Figures4.13-4.16andby1.4Figures4.17-4.20.Therestoftheplotsshow asmoothtrendbetweenthebasecaseand0.4,orthebasecaseand1.4,respectively, andsohavebeenomitted. Ingeneral,thetrendindicatesthatforvoltagemagnitudes,theDLandNLL getbetterasX/Rratiogoesdown,whileMTLbecomesworse.Forvoltageangles, theNLLclearlyimprovesasX/Rratiogoesdown,becomingthebestmethodfor calculatingvoltageanglesat0.4.DLsometimesgetsbetter,sometimesworseforthe 41

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aMedianvoltagemagnitudeerror bMaxvoltagemagnitudeerror Figure4.9: Voltagemagnitudeerrorpu,loadsscaledto0.5-0.8ofavailablegeneration 42

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aMedianvoltageangleerror bMaxvoltageangleerror Figure4.10: Voltageangleerrordegrees,loadsscaledto0.5-0.8ofavailablegeneration 43

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aMedianrealpowerowerror bMaxrealpowerowerror Figure4.11: RealpowerowerrorMW,loadsscaledto0.5-0.8ofavailablegeneration 44

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aMedianreactivepowerowerror bMaxreactivepowerowerror Figure4.12: ReactivepowerowerrorMW,loadsscaledto0.5-0.8ofavailablegeneration 45

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voltageanglesasX/Rgoesdown.Aswouldbeexpected,theDCPFgetsworsein voltageangleandrealpowerowasX/Rgoesdown.Forpowerow,theDLand MTLmethodsgetworseasX/Rratiogoesdown,whileNLLhasinconsistentresults. Thesamerankingsremainforpowerowfromthebasecase. Inconclusion,itseemsthattheNLLandDLoverallgivebetterresultsasthe X/Rratiogoesdown.Infact,inlowX/Rcases,overalltheNLLmethoddoesthe best,evenifitsmedianerrorisnotasgoodfortherealpowerows. 4.7Conclusions Ingeneral,thetestingshowsthattheaccuracyofthepresentedmethodsishighly systemspecic.Theexactnatureofthedierencesbetweenthesystemsthatcause thisvariationisnotentirelyclearfromthetests,althoughhighloadlevelsandlarge numbersoftapchangersseemtobeapotentialcauseaspertheFrenchsystemsin ourtestset. Oftheformulationsthatcanbeusedinoptimizationsasconstraints,theDL formulationseemstoimprovebestontheDCPF,evengiventhefactthatoncertain systemsitcanhaveverypoormaxerrors.TheMTLdoesnotdoaswellingeneral,and evendoesworsethantheDCPFcommonly,althoughitdoesmodelmorequantities thantheDCPF.TheNLLdoesbestwhentheX/Rratioislowinthesystem, whichmayindicatethatitwouldbeagoodformulationtoconsiderfordistribution systems,wheretheX/Rratioisinherentlylow.Ontheotherhand,though,the unitcommitmentproblemisnotgenerallyformulatedforthedistributionsystem, andthatwouldbetheonlyknownoptimizationthattheNLLcouldbeusedwith currently.Overall,theseresultsseemtobefairlyinsensitivetoloadlevelshiftsand PVbusvoltageandrealpowersetpoints.Hence,ifaformulationworkswellfora systeminacoupleofrepresentativecases,itmaybeexpectedtoworkwellinmost cases.However,moretestingmaybeusefultoconrmthis. 46

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aMedianvoltagemagnitudeerror bMaxvoltagemagnitudeerror Figure4.13: Voltagemagnitudeerrorpu,X/Rratiosscaledby0.4 47

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aMedianvoltageangleerror bMaxvoltageangleerror Figure4.14: Voltageangleerrordegrees,X/Rratiosscaledby0.4 48

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aMedianrealpowerowerror bMaxrealpowerowerror Figure4.15: RealpowerowerrorMW,X/Rratiosscaledby0.4 49

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aMedianreactivepowerowerror bMaxreactivepowerowerror Figure4.16: ReactivepowerowerrorMW,X/Rratiosscaledby0.4 50

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aMedianvoltagemagnitudeerror bMaxvoltagemagnitudeerror Figure4.17: Voltagemagnitudeerrorpu,X/Rratiosscaledby1.4 51

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aMedianvoltageangleerror bMaxvoltageangleerror Figure4.18: Voltageangleerrordegrees,X/Rratiosscaledby1.4 52

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aMedianrealpowerowerror bMaxrealpowerowerror Figure4.19: RealpowerowerrorMW,X/Rratiosscaledby1.4 53

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aMedianreactivepowerowerror bMaxreactivepowerowerror Figure4.20: ReactivepowerowerrorMW,X/Rratiosscaledby1.4 54

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5.ConcludingRemarks Inthisthesis,threerecentlypublishedcold-startlinearizationsofthepowerow werecomparedtotheDCPFformulationandtoeachothernumerically.Thebasic formulationsofthelinearizationswerepresented,alongwiththeirrelevantassumptionsandlimitations.Numericaltestswereconductedonavarietyofsystems,which variedgreatlyinsize,structure,andloadlevel.Additionaltestswereconductedundervariationsonthetestsystems,includingalteringtheloadlevel,X/Rlineratios, existenceoftapchangersandphaseshifters,andthegeneratorsetpoints.Overall,itappearsthattheDecoupledLinearformulation,asfoundin[11],isthebest formulationforthepurposesofoptimization. 5.1FurtherWork Thereismuchroomforimprovementintheareaofnumericalstudiesofthese linearizations.Forone,therearemorelinearizationsavailableforcomparison-the DCPFwithlosscompensationforexample,ortheformulationof[12].Furtherwork inexaminingthereasonsforthefairlyextremeerrorsinsomesystemstheMatPower Frenchsystems,forinstancecouldrevealinsightsthatwouldallowforimprovements inthelinearformulations.Testingcouldalsobeconductedonindividuallinesin asystem-closerexaminingtherelationshipsbetweenthelineparametersandthe errorsintheapproximations.Thesetestscanalsobeextendedtorunningactual optimizations,suchasSCUC,ormediumtolongtermplanningsimulations. 55

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REFERENCES [1]S.AbediandS.M.Fatemi.Introducinganoveldcpowerowmethodwithreactivepowerconsiderations. IEEETransactionsonPowerSystems ,30:3012{ 3023,Dec.2014. [2]A.Bircheld,T.Xu,K.M.Gegner,K.S.Shetye,andT.J.Overbye.Grid structuralcharacteristicsasvalidationcriteriaforsyntheticnetworks. IEEE TransactionsonPowerSystems ,32:3258{3265,July2017. [3]S.BolognaniandF.Drer.Fastpowersystemanalysisviaimplicitlinearizationofthepowerowmanifold.In 201553rdAnnualAllertonConferenceon Communication,Control,andComputingAllerton ,Sept2015. [4]B.LesieutreandD.Wu.Anecientmethodtolocatealltheloadowsolutionsrevisited.In 201553rdAnnualAllertonConferenceonCommunication,Control, andComputingAllerton ,Sept2015. [5]Z.Li,J.Yu,andQ.H.Wu.Approximatelinearpowerowusinglogarithmic transformofvoltagemagnitudeswithreactivepowerandtransmissionlossconsideration. IEEETransactionsonPowerSystems ,33:4593{4603,Nov.2017. [6]Y.Liu,N.Zhang,Y.Wang,JYang,andC.Kang.Data-drivenpowerow linearization:Aregressionapproach. IEEETransactionsonSmartGrid ,Early Access,Feb.2018. [7]W.MaandJ.S.Thorp.Anecientalgorithmtolocateallthepowerow solutions. IEEETransactionsonPowerSystems ,8:1077{1083,Aug.1993. [8]K.Purchala,L.Meeus,D.VanDommelen,andR.Belmans.Usefulnessofdc powerowforactivepowerowanalysis.In IEEEPowerEngineeringSociety GeneralMeeting ,September2012. [9]Y.Qi,D.Shi,andD.Tylavsky.Impactofassumptionsondcpowerowmodel accuracy.In 2012NorthAmericanPowerSymposiumNAPS ,June2005. [10]B.Stott,J.Jardim,andO.Alsac.Dcpowerowrevisited. IEEETransactions onPowerSystems ,24:1290{1300,May2009. [11]J.Yang,N.Zhang,C.Kang,andQ.Xia.Astate-independentlinearpowerow modelwithaccurateestimationofvoltagemagnitude. IEEETransactionson PowerSystems ,32:3607{3617,Dec.2016. [12]Z.Yang,H.Zhong,andQ.Xia.Anovelnetworkmodelforoptimalpower owwithreactivepowerandnetworklosses. ElectricPowerSystemsResearch , 144:63{71,Nov.2016. 56

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[13]R.D.ZimmermanandC.E.Murillo-Snchez.Matpower6.0user'smanual,Dec.2016.[Online]Available: http://www.pserc.cornell.edu/matpower/MATPOWER-manual.pdf. [14]R.D.Zimmerman,C.E.Murillo-Snchez,andR.J.Thomas.Matpower:Steadystateoperations,planningandanalysistoolsforpowersystemsresearchand education. IEEETransactionsonPowerSystems ,26:12{19,Feb.2011. 57