METHODSFORANALYSISANDPLANNINGOFMODERNDISTRIBUTIONSYSTEMS By SalemElsaiah ADISSERTATION Submittedto MichiganStateUniversity inpartialoftherequirements forthedegreeof ElectricalEngineering-DoctorofPhilosophy 2015 ABSTRACT METHODSFORANALYSISANDPLANNINGOFMODERNDISTRIBUTION SYSTEMS By SalemElsaiah ThePrincipalcontributionofthisdissertationliesindevelopinganefoptimizationframe- workfordistributionsystemoperationalandplanningstudies.Thepartofthisdissertation introducesanovelpowerwmodel,whichisequallyappropriateforuseatbothdistributionand transmissionlevelsandcanbeextremelyusefulwheneverfast,robust,andrepetitivepowerw solutionsarerequired.WedeveloptheproposedlinearizedACpowerwmodel(LACPF)based onlinearizationofthefullsetofconventionalpowerwequations,andthereforeincludesvoltage magnitudesolutionsandreactivepowerws,unliketraditionallinearizedpowerwmethods. Further,themodelpresentedinthisdissertationisnon-iterative,direct,andinvolvesnoconver- genceissuesevenwithill-conditionsystems.Wetesttheproposedmodelonseveraldistribution systemsandhasfoundtoperformwithspeedandaccuracyappropriateforrepetitivesolutions. Thesecondpartofthisdissertationdevelopsanefoptimizationframeworktohandle severaldistributionsystemoperationalandplanningproblems.Theproposedframeworkuses linearprogramming,becauselinearprogrammingbasedformulationstendtobexible,reliable, andfasterthantheirnonlinearcounterparts.Weconsidervoltagebounds,reactivepowerlimits, andallshuntelementsintheproposedoptimizationmodel.Weusetheproposedoptimization frameworktosolvetheproblemofoptimalsizingandplacementofdistributedgeneration.For thisparticularcase,weuselosssensitivityfactorsandsensitivityanalysistoestimatetheoptimal sizeandpowerfactorofthecandidatedistributedgenerationunits.Wealsoperformexhaustive powerwstudiestoverifythesizesobtainedbytheproposedmethod.Wedemonstratethe effectivenessofthemethodonseveralbenchmarksystemsandprovethatthemethodcouldleadto optimalornear-optimalglobalsolution,whichmakestheproposedmethodverysuitabletousein severaloptimaldistributionsystemplanningstudies. Wesolvetheproblemofoptimaleconomicpowerdispatchofactivedistributionsystems.We proposeapiecewiselinearmodeltoapproximatethecurrentcarryingcapacitiesofdistribution feeders.Thedegreeofapproximationinthismodelcanbeimprovedtothedesiredlevelbyincreas- ingthenumberoflinesegmentsused,withoutsubstantialaffectonthemainroutineandthecom- putationalspeed.Wefurtherdeveloplinearmodelsforcostfunctionsofgeneratingunits,loads, andtotalpowerlosses.Weapplymethods,whicharedevelopedbasedonnonlinearprogramming andconventionallinearprogrammingtoevaluatetheeffectivenessoftheproposedmethod.We showthattheresultsobtainedbytheproposedframeworkcorrespondcloselywiththoseobtained bynonlinearmeans,whilerequiringlowercomputationaleffort. Wedescribeamethodforsolvingthedistributionsystemproblemwithan objectiveofreliabilityimprovement.Frompracticalperspective,distributionsystemsarerecon- radiallyforbestcontrolandcoordinationoftheirprotectivedevices.Therefore,wedevelop agraphtheoreticmethodtopreservethespanningtreestructureofthedistributionsystem.Wefur- therdevelopanintelligentsearchmethodbasedonbinaryparticleswarmoptimizationtechnique, toseekforthebestcombinationsofsectionalizingandtie-switchesthatminimizetheamountof totalpowercurtailment.Sincethetimeandcomputationaleffortspentinevaluatingreliability indicesareofgreatconcerninbothplanningandoperationalstages,weproposeaprobabilistic reliabilityassessmentmethodbasedoneventtreeanalysiswithhigher-ordercontingencyapproxi- mation.Wedemonstratetheeffectivenessoftheproposedmethodonnumerousradialdistribution systemsandshowthattheamountofannualpowercurtailmentofin-serviceconsumerscanbe tremendouslyreducedusingtheproposedmethod. Tomylateparentswithlove,wifeEnas,andbeautifulchildrenAliElsaiahandMuathElsaiah iv ACKNOWLEDGMENTS Iwouldliketoexpressmysincerethankstomyadvisor,Dr.JoydeepMitra,forhiscontinuous advice,encouragement,andunlimitedsupportduringthecourseofmystudyatMichiganState University.Iamtremendouslyfortunatetohavehimasmymentorandadvisor.Hisapproachof teachingandconductingresearchhavebeen,andwillalwaysbe,sourcesofinspirationforme. Withouthisesteemedguidance,noneofthisworkwouldhavebeenpossible. Iwouldliketothankmyguidancecommitteemembersfortheirinterestandcontinuousadvice. IamsogratitudetoDr.BingsenWangforhisconstantsupportandinsightfulguidance.The materialshetaughtmeonRenewableElectricEnergySystemshavepavedtheroadformeto understandseveralrelatedtopics. MyspecialthanksgotoDr.SubirBiswas.Hisperceptiveguidance,encouragement,and continuousadvicehavehelpedme,personallyandprofessionally,throughoutmyPh.D.journey. MyspecialthanksalsogotoDr.ShlomoLeventalfromthedepartmentofProbabilityand Statistics,whokindlyjoiningandservingonmyguidancecommittee.Iamacknowledginghim forhisinterest,insightfulguidance,andunlimitedsupport. Mostofall,Iwishtoexpressmygratitudetomyfamily,whoseloveandencouragementhave supportedmethroughoutmyeducation.IamsogratefultomywifeEnas,whohasstoodbyme throughoutthelongnightsandhardtimes,especiallythemonthsofwritingthisdissertation. Finally,IwouldliketothankallofmyfriendsattheEnergyReliabilityandSecurityResearch laboratoryforthegoodcompanyandtheprofessionalexperiencethattheyhavecreated. v TABLEOFCONTENTS LISTOFTABLES ....................................... ix LISTOFFIGURES ...................................... x Chapter1Introduction .................................. 1 1.1PowerSystemStructure................................1 1.1.1TransmissionLevel..............................2 1.1.2Sub-transmissionLevel............................2 1.1.3DistributionLevel...............................2 1.2DistributionSystemModeling............................3 1.2.1DistributionLineModel...........................3 1.2.2LoadModel..................................4 1.2.3CogeneratorModel..............................5 1.2.4ShuntCapacitorModel............................5 1.2.5SwitchModel.................................5 1.3ThePowerFlowProblem...............................6 1.3.1LiteratureReview...............................6 1.4ReviewofMathematicalProgrammingBasedMethods...............10 1.4.1NonlinearOptimizationMethods.......................11 1.4.2LinearOptimizationMethods.........................12 1.5TheNecessityforNewModelsandMethods.....................14 1.6OverviewofContributions..............................17 1.7ThesisOutline.....................................21 Chapter2AnOptimizationFrameworkDevelopment ............ 23 2.1FullACPowerFlowModel..............................24 2.2DCPowerFlowModel................................25 2.3ProposedPowerFlowModel.............................27 2.4ModelValidationandTestResults..........................32 2.5TestSystemI.....................................33 2.6TestSystemII.....................................36 2.7TestSystemIII....................................37 2.8TestSystemIV....................................39 2.9RobustnessTestoftheProposedModel.......................39 2.10LinearProgrammingBasedOptimization......................42 2.10.1ObjectiveFunctionFormulation.......................42 2.10.2PowerBalanceEquations...........................42 2.10.3RealandReactivePowerConstraints.....................43 2.10.4VoltageConstraints..............................44 2.10.5LineCapacityConstraints..........................44 vi 2.10.6TheProposedOptimizationFramework...................48 2.10.7AdditionalNetworkPerformanceConstraints................50 2.11Summary.......................................50 Chapter3DistributedGenerationSizingandPlacement ................ 53 3.1DistributedGenerationPlacementandSizingProblem................54 3.2OverviewofExistingWork..............................55 3.3DetailedSolutionProcedures.............................56 3.3.1ObjectiveFunctionandConstraints.....................56 3.3.2ofPenetrationLevel......................57 3.3.3SelectionofOptimalLocation:Aprioritylist................58 3.3.4SelectionofOptimalSize...........................58 3.3.5SelectionofOptimalPowerFactor......................61 3.4DemonstrationandDiscussion............................63 3.533BusSystemOptimalLocationsandSizes.....................63 3.5.1AnalyticalMethod..............................63 3.5.2ExhaustivePowerFlow............................66 3.669BusSystemOptimalLocationsandSizes.....................67 3.6.1AnalyticalMethod..............................67 3.6.2ExhaustivePowerFlow............................69 3.7VoltageImprovement.............................70 3.8ComparativeStudy..................................71 3.9Summary.......................................72 Chapter4OptimalEconomicPowerDispatchofActiveDistributionSystems .... 73 4.1OptimalEconomicPowerDispatchProblem.....................74 4.2OverviewofExistingWork..............................74 4.3DevelopmentofModelsandMethods........................75 4.3.1CostFunctionsofGenerators.........................76 4.3.2InclusionofLosses..............................77 4.3.3TheProposedFramework...........................79 4.4DCPFBasedOptimalEconomicDispatch......................80 4.5FACPFBasedOptimalEconomicDispatch......................81 4.6DemonstrationandDiscussion............................82 4.6.1CaseScenarioI................................82 4.6.2CaseScenarioII................................83 4.6.3CaseScenarioIII...............................86 4.7PerformanceAnalysisoftheProposedMethod....................87 4.7.1ComparisonoftheMaximumError.....................88 4.7.2ComparisonoftheOptimalSolution.....................88 4.8Summary.......................................89 vii Chapter5Reliability-constrainedOptimalDistributionSystem .. 92 5.1DistributionSystemProblem....................93 5.1.1ReasonsforDistributionFeeder..............94 5.1.2ReviewofExistingWork...........................94 5.1.3ImportanceofEnhancingDistributionSystemsReliability.........98 5.2DevelopmentofModelsandMethods........................98 5.2.1ProbabilisticReliabilityIndices.......................99 5.2.2TheStateSpace................................99 5.2.3ProbabilisticModelsofComponents.....................100 5.2.4ProbabilisticReliabilityAssessment.....................101 5.2.5Higher-orderContingenciesBasedEventTreeAnalysis...........102 5.2.6ReliabilityEvaluationModel.........................103 5.2.7ImplementationoftheSpanningTreeConstraints..............105 5.2.8CalculationofReliabilityIndices.......................107 5.3FormulationoftheOptimalDistributionSystemProblem.....108 5.3.1ParticleSwarmOptimization.........................109 5.3.2TheSearchSpaceoftheProblem.......................111 5.3.3DetailedSolutionAlgorithm.........................111 5.4DemonstrationandDiscussion............................114 5.4.133BusDistributionSystemTestCase...............114 5.4.1.1CaseScenarioI...........................115 5.4.1.2CaseScenarioII..........................117 5.4.1.3CaseScenarioIII..........................118 5.4.1.4CaseScenarioIV..........................119 5.4.269BusDistributionSystemTestCase...............119 5.4.3118BusDistributionSystemTestCase..............120 5.4.4VoltageImprovement.........................121 5.5Summary.......................................123 Chapter6ConclusionsandFutureWork ........................ 126 6.1Conclusions......................................126 6.2FutureWork......................................130 BIBLIOGRAPHY ................................. 132 viii LISTOFTABLES Table2.1:33BusDistributionSystemPowerFlowResults..............35 Table3.1:ResultsofThe33BusDistributionSystemŒDGTypeI..........67 Table3.2:ResultsofThe33BusDistributionSystemŒDGTypeII..........67 Table3.3:ResultsofThe69BusDistributionSystemŒDGTypeI..........69 Table3.4:ResultsofThe69BusDistributionSystemŒDGTypeII..........70 Table3.5:MinimumBusVoltagesBeforeandAfterDGInstallation.........71 Table3.6:PerformanceComparisonofTheProposedMethod............71 Table4.1:EDResultsofthe30BusSystemŒCaseScenarioI.......83 Table4.2:EDResultsofthe30BusSystemŒCaseScenarioII......86 Table4.3:EDResultsofthe30BusSystemŒCaseScenarioIII......88 Table4.4:ComparisonofTotalCostandTotalGenerationErrorsObtainedbythe ProposedLACPFbasedEDmethodandtheDCPFbasedEDmethod...89 Table4.5:ComparisonofMethodPerformance....................89 Table5.1:ReliabilityDatafortheTestSystems....................115 Table5.2:ParametersoftheBPSO...........................116 Table5.3:ResultsofCaseScenarioI.........................124 Table5.4:ResultsofCaseScenarioII.........................124 Table5.5:ResultsofCaseScenarioIII.........................124 Table5.6:ResultsofCaseScenarioIV.........................124 Table5.7:Resultsof69BusSystem..........................125 Table5.8:Resultsof118BusSystem.........................125 Table5.9:ComparisonofBusVoltagesBeforeandAfter......125 ix LISTOFFIGURES Figure1.1:Powersystemsingle-linediagram.....................1 Figure1.2:Distributionlinemodel...........................3 Figure2.1:Simpletransmissioncircuit.........................24 Figure2.2:Single-linediagramofthe33busdistributionsystem...........34 Figure2.3:Voltageofthe33bussystemobtainedbytheproposedLACPF modelandtheFACPFmodel........................36 Figure2.4:Single-linediagramofthe69busdistributionsystem...........38 Figure2.5:Voltageofthe69bussystemobtainedbytheproposedLACPF modelandtheFACPFmodel........................39 Figure2.6:Single-linediagramofthe118busdistributionsystem...........40 Figure2.7:Piecewiseapproximationoflinecapacityconstraints...........46 Figure2.8:Piecewiseapproximationoflinecapacityconstraints...........46 Figure3.1:Asimpleradialdistributionfeeder.....................60 Figure3.2:Activepowerlosssensitivityfactorsofthe33bussystem.........64 Figure3.3:Activepowerlosssensitivityfactorsofthe69bussystem.........68 Figure4.1:Linearizationofcostfunction........................77 Figure4.2:Transmissionline p model.........................78 Figure4.3:30BusDistributionSystem....................84 Figure5.1:Twostatemodelrepresentation.......................101 Figure5.2:Conceptofeventtreeanalysis........................101 Figure5.3:Agraphwith n = 4verticesand m = 5edges................107 Figure5.4:Flowchartoftheproposedmethod.....................113 x Figure5.5:Single-linediagramofthe33busdistributionsystem......115 Figure5.6:Single-linediagramofthe69busdistributionsystem...........120 Figure5.7:Single-linediagramofthe118buslarge-scaledistributionsystem122 xi Chapter1 Introduction Theelectricpowersystemisaverylargeandcomplexnetworkconsistsofgeneratingunits,trans- missionlines,transformers,switches,loads,etc.Theobjectiveoftheelectricpowersystemis tosupplyconsumerswithelectricdemandinapracticalandreliablemanner.Thischapter discussesthebasicpowersystemstructureanditsmajorparts.Itintroducesthegeneralpowerw problemanddiscussesthemethodspresentedintheliteraturetohandlethepowerwproblem. Inaddition,thischaptersummarizesthecommonmathematicalprogramingbasedmethodsand highlightsthemajorcontributionsofthisthesis. 1.1PowerSystemStructure ThesinglelinediagramofasimplepowersystemisshownbelowinFig.1.1.Thissystemconsists mainlyofthreeregions,whicharethegenerationregion,transmissionregion,anddistribution region.Thereisonevoltagelevelinthegenerationregion,whichisthegenerationvoltagelevel. However,variousvoltagelevelsareusedattheothertworegions. Figure1.1:Powersystemsingle-linediagram 1 1.1.1TransmissionLevel Thetransmissionlevelistypicallydifferentfromdistributionandsub-transmissionlevelsinits characteristicsandoperatingstrategies.Onedistinctivefeatureoftransmissionsystem,forin- stance,isthatitconnectsamixtureofgeneratingunits;e.g.thermal,nuclear,hydraulic,etc. Moreover,thedirectionofpowerwintransmissionsystemscanbeinaforwardorbackward directionandisusuallyreversedtoimposecertainoperationalconstraints.Further,transmission systemsarecharacterizedbyhighX/Rratios,whichcouldattainafactorof10,forhighvoltage networks. 1.1.2Sub-transmissionLevel Thesub-transmissionlevelreceivestheelectricalpowerfromthebulkpowersubstationsandtrans- mitsittothedistributionsubstations.Thetypicalvoltagesinsub-transmissionlevelareusually variesbetween11kVand138kV.Sub-transmissionlevelsaretreatedastransmissionsystemsand, sometimes,asdistributionsystemsdependingontheoperatingconstrainsandregulationsused. 1.1.3DistributionLevel Thedistributionsystemlevelconsistsofradialdistributionfeeders,whicharefedfromdistribution substationsthroughpowertransformers.TwodistributionvoltagelevelsarewidelyusedintheU.S. distributionsystems.Thesevoltagesaretheprimaryvoltage,whichis13 : 2kVandthesecondary voltage,whichis120Volts.Distributionsystemshavecertaindistinctivefeatures.Examples ofthesefeaturesinclude,radialstructurewithweakly-meshedtopology,highR/Xbranchratios, untransposedorrarelytransposeddistributionfeeders,unbalancedloadsalongwithsingle-phase anddouble-phaselaterals,anddispersedgeneration. 2 1.2DistributionSystemModeling Thissectionhighlightssomemodelingaspectsfordistributionsystemcomponents.Wefeltthatit isaquiteappropriatetopresentfewcomponentmodelsinthissection,howeverdetailedcomponent modelscanbefoundin[1,2]. 1.2.1DistributionLineModel Fig.1.2showsathree-phaselinesectionconnectedbetweenbus k andbus m .Thelineparam- eterscanbefoundbythemethoddevelopedbyCarsonandLewis[3].A4 4sizedprimitive matrix,whichtakesintoaccounttheeffectoftheself-and-mutualcouplingbetweenphasescanbe expressedas[4], Figure1.2:Distributionlinemodel However,itisconvenienttorepresent(1.1)asa3 3matrixinsteadofthe4 4matrixby usingKron'smethod[3,4].Theeffectofthegroundconductorisstillincludedintheresultant matrix.Thatis, Z 0 abc = 2 6 6 6 6 6 6 6 6 6 4 Z aa Z ab Z ac Z an Z ba Z bb Z bc Z bn Z ca Z cb Z cc Z cn Z na Z nb Z nc Z nn 3 7 7 7 7 7 7 7 7 7 5 (1.1) 3 Z abc = 2 6 6 6 6 6 4 Z aa n Z ab n Z ac n Z ba n Z bb n Z bc n Z ca n Z cb n Z cc n 3 7 7 7 7 7 5 (1.2) Itshouldbenoted,however,thatsingle-phaseandtwo-phaselinesectionsaremostcommonin distributionnetworks.Hence,inthisthesisforanyphasedoesnotpresent,thecorrespondingrow andcolumnin(1.1)willhavezeroentries. 1.2.2LoadModel Theactiveandreactivepowerloadsondistributionnetworkscanberepresentedasconstantpower, constantcurrent,constantimpedance,oramixtureofthesetypes.Hence,theloadmodelindistri- butionsystemscanbegenerallyrepresentedbyanexponentialfunctionas, P k = P ref V k V ref a (1.3) Thereactivepowercanberepresentedas, Q k = Q ref V k V ref b (1.4) where, V ref isthereferencebusvoltage, V k istheoperatingvoltageatbus k , P ref and Q ref respec- tivelyaretheactiveandreactivepowerconsumptionsatthereferencebus k , a and b areexponents bywhichtheloadcharacteristiccanbedetermined.Thatis,constantpowerloadmodelcanbe foundbysetting a = b = 0.Theconstantpowermodelisoftenusedinpowerwstudiesand hasthereforebeenadaptedinthiswork.Further,constantcurrentmodelandconstantimpedance modelcanbeobtainedbysetting a = b = 1or a = b = 2,respectively. 4 1.2.3CogeneratorModel Cogeneratorsordistributedgeneratorsaresmall-scaleenergysources,whichareusuallyusedat distributionsystemleveltoenhancethereliabilityandsecurityofthesystem.Cogeneratorscanbe modeledeitherasconstantpowernodesorconstantvoltagenodes.InaccordancewiththeIEEE standard1547-2003[5]forinterconnectingdistributedresourceswithelectricpowersystems,dis- tributedgeneratorsarenotrecommendedtoregulatebusvoltages,butstronglyrecommendedtobe modeledasconstantPQnodes.Therefore,thedistributedgeneratorsusedinthisthesisaremod- eledasconstantPQnodes,withnegativeinjections.However,thepowerfactorofthedistributed generationunitiscalculatedaccordingtotheapplicationinwhichthedistributedgeneratorisused. 1.2.4ShuntCapacitorModel Capacitorbanksarewidelyusedindistributionsystemstoregulatebusvoltagesandretainreactive powerlimitsinthedesiredrange.Inthisthesis,capacitorbanksaremodeledasconstantcapaci- tancedevicesandarerepresentedbycurrentinjectionstothenodeatwhichtheyareconnected. 1.2.5SwitchModel Distributionsystemsareequippedwithtwotypesofswitches;sectionalizingswitchesandtie- switches.Thesectionalizingswitchesarenormallyclosedandareusedtoconnectvariousdistri- butionlinesegments.Thetie-switches,ontheotherhand,arenormallyopenandcanbeusedto transferloadsfromonefeedertoanotherduringabnormalconditions.Inthiswork,bothtypesof switchesaremodeledasbrancheswithzeroimpedance.Thatis,thecurrentwinanybranchor switchinthesystemcanbecomputeddirectlyfromthepowerwsolutionandviceversa. 5 1.3ThePowerFlowProblem Powerwanalysisisacrucialandbasictoolforthesteady-stateanalysisofanypowersystem. Thesolutionofthepowerwproblemaimsatdeterminingthesteady-statevoltagephaseand magnitudeatallbusesaswellasrealandreactivepowerwsineachline,forloading conditions.Nonlinearmodelsandlinearizedmodelsareusedintheliteraturetohandlethepower wproblem.AreviewofthecommonpowerwmethodsispresentedbelowinSection1.3.1. 1.3.1LiteratureReview PowerwanalysishaslongbeenperformedusingfullACpowerw(FACPF).Thefundamental methods,whicharewidelyusedinsolvingpowerwsattransmissionlevelaretheGauss-Seidel method,Newton-Raphsonmethod,andFast-Decoupledmethod[6,7,8].Theideabehindthe Gauss-Seideliterativemethodisverysimple,howevertheiterativesolutionhasaslowconverge properties.Itis,therefore,knownasaslow-iterativeproblemsolvingtechniquesinceitconstantly requiresthesolutionofasetofnonlinearequationswhosecardinalityapproximatelyequalsthe numberofsystembuses.ExperiencewithGauss-Seidelmethodhasshownthatconvergencemay notbeattainedforallsystems[2].However,intherealisticpowersystems,powerwstudies arecarriedoutusingperunitquantitiesandallbusvoltagesarerestrictedtobecomeclosetotheir ratedvalues.Consequently,convergenceisattainedbutwithanextensivecomputationalburden. TheNewton-Raphsonmethodisafastandrobusttangentialapproximationtechniqueinwhich lineparametersandothervariablesarestoredintheJacobianmatrix.IntheNewton-Raphson method,Taylor'sseriesisuseduptothetermandthemethodconvergesveryfast.Themajor drawbackoftheNewton-Raphsonmethod,however,isthattheformulationoftheJacobianmatrix iscomputationallycumbersomeintermsofexecutiontimeandstoragerequirement.Infact,the 6 Jacobianmatrixneedstoberecalculatedandevaluatedineverysingleiteration.More, theJacobianmatrixishighlysparseandtendstobesingularundercertainoperatingconditions. TheFast-Decoupledmethod[9,10,11],whichisaoftheNewton-Raphson method,usesanapproximateandconstantJacobianthatignoresthedependenciesbetween(a) realpowerandvoltagemagnitude,and(b)reactivepowerandvoltageangle.Itis,therefore,fast andeffectiveapproachtosolvepowerwsattransmissionsystemlevel.Nevertheless,forsys- temswithR/Xbranchratiosgreaterthan1,thestandardFast-Decoupledmethoddoesnotconverge well[9].TheFast-Decoupledmethodisfrequentlyusedintheengineeringapplicationsinwhich fastpowerwestimationsarerequired. Aconsiderableamountofresearchhasbeenpresentedintheliteraturetoimprovetheperfor- manceoftheFast-Decoupledmethod.Inthiscontext,thecompensationbasedmethodproposed byRajicicandBose[12]isprobablythemostpopulartechniquethatisbeingusedtoenhance theperformanceoftheFast-Decoupledmethod,inparticular,forsystemswithhighR/Xbranch ratios.Theparallelcompensationofthedecoupledpowerwmethodhasbeenintroducedin [13].Ejebeetal[14],proposedafastcontingencyscreeningmethodforvoltagesecurityanalysis basedonFast-Decoupledmethod.Thecontingencyanalysisisperformedintwostages;which arethescreeningstageandthesolutionstage.Thescreeningstageusesasingleiterationofthe Fast-Decoupledmethod.Ithasbeendemonstratedin[14]thatmostofthecomputationaltimeis utilizedfortheQŒVhalfiteration.Consequently,ascreeningmethodisdevelopedin[14]inorder toreducethecomputationaltime.Thisscreeningmethodwasaccomplishedusingsparsevector techniquesandwasabletosolvetheQŒVhalfiterationonly. Numerouspowerwsolutionmethodswereproposedintheliteraturetohandledistribution systemspowerwduetotheirspecialfeatures,inparticular,thehighR/Xbranchratiosand theweakly-meshedtopologicalstructure.SomemethodshaveusedversionsofNewton- 7 Raphsonmethodanditsdecoupledformwhileothershavebeendevelopedbasedontheback- ward/forwardsweepingmethod.Thelattercategorycanbeascurrentsummationmeth- ods,admittancesummationmethods,andpowersummationmethods. ZimmermanandChiangpresentedafastdecoupledloadwmethodin[15].Inthismethod, asetofnonlinearpowermismatchequationsareformulatedandthensolvedbyNewton-Raphson method.Theadvantageofthismethodisthatitorderedthelateralsinsteadofbuses;hencethe problemsizehasbeenreducedtothenumberofsystem'slaterals.Useoflateralsasvariables insteadofnodesmakesthisalgorithmmoreefforagivensystemtopology,howeveritmay addsomedififthenetworktopologyischangedregularly,whichisthecaseindistribution networksasaresultoftheswitchingoperation.BaranandWu[16]developedamethodforsolving distributionsystempowerwbysolvingthreeequationsrepresentingthevoltagemagnitude, realpower,andreactivepower.Inthismethod,onlysimplealgebraicequationsareutilizedto developtheJacobianmatrixandthepowermismatches.Nevertheless,theformulationofthe Jacobianmatrixineveryiterationtendstobecomputationallydemanding,particularlyforlarge- scaledistributionsystems. Shirmohammadietal.[17]presentedacompensationbasedmethodforpowerwanalysis ofbalanceddistributionsystems.Basically,themethodused KCL and KVL toobtainbranch currentsandbusvoltagesandthenaforward/backwardsweepisappliedtoobtainthepowerw solution.Thismethodhasalsoaccountedforweakly-meshednetworksbybreakingthegiven systemtoanumberofpointsŒbreakpointsŒandhenceasimpleradialnetworkcanbeobtained. Theradialnetworkwasthensolvedbythedirectapplicationof KCL and KVL .Theeffectiveness ofthismethoddiminishesasthenumberofbreakpointsgoesup.Asaresult,theapplication ofthismethodtotheweakly-meshednetworkswaspracticallyrestricted.Analgorithmforthe powerwsolutionofunbalanceddistributionnetworkswasdevelopedin[18]byChengand 8 Shirmohammadi.Thismethodcanbeconsideredasanextensiontotheworkdonein[17],butit hasdealtwiththemodelingofvoltagecontrolbuses,elaboratedonthemodelingaspectsofvarious distributionsystemcomponents,andwassuccessfullyappliedonrealisticdistributionsystems. KerstingandMandive[19,20]suggestedamethodtosolvethedistributionsystempower wproblembasedonLadder-Networktheoryintheiterativeroutine.Thisapproachhasthe advantageofbeingderivative-freeandusesbasiccircuittheorylaws.However,laddernetwork methodassumesconstantimpedanceloads,whichisnotthecaseinmostdistributionnetworks. GoswamiandBasupresentedadirectapproachtoobtainasolutionforthedistributionsystem powerwforbothradialandmeshednetworksin[21].Themethodwasalsoappliedtobalanced andunbalancednetworks.Theofthismethodisthatconvergenceisachievedfora widerangeofrealisticdistributionsystems,includingweakly-mesheddistributionsystems.The maindrawbackofthismethod,ontheotherhand,isthatnonodeinthesystemcanserveasa junctionformorethanthreenetworkbranches,whichlimitsthepracticaluseofthismethod. SeveraldistributionpowerwmethodshavebeenproposedbyTeng[22,23,24].Onemethod [22]isdevelopedbasedontheoptimalorderingschemeandtriangularfactorizationofthebus admittancematrix.ThismethodwasdevelopedbasedontheGauss-Seidelmethodbuthasused factorizationoftheadmittancematrixtoreducethecomputationalburden.Themethodpresented in[23]isdevelopedbasedontheequivalentcurrentinjectiontechnique.Amongsttheadvantages ofthismethodistheconstantJacobianmatrix,whichneedstobeconvertedonlyonce.Distribution feedersreactances'havenotbeentakenintoconsiderationinthismethodbyassumingthatline reactanceismuchsmallerthanitsresistance.However,distributionnetworksarecharacterizedby awiderangeofresistancesandreactances,whichthatthemethodmayfailtoconvergeif linereactances'areaccountedfor.Adirectapproachtoobtainthedistributionpowerwsolution hasbeenproposedin[24].Twomatricesanddirectmatrixmultiplicationareusedtoobtainthe 9 distributionpowerwsolution.Thesolutioninvolvesdirectmatrixmultiplication,andthuslarge memoryspaceisneeded,particularlyifthemethodisusedtohandlepowerwsoflarge-scale distributionsystems.PrakashandSydulu[25]haveintroducedcertaintothemethod presentedin[24]sothatthepowerwsolutionisobtainedwithlowercomputationalburden. However,balancedconditionsareonlyconsidered.Thenetworktopologybasedmethodhasalso beentoincludetheweakly-meshednetworksin[26]. Itisworthnotingherethatalloftheaforementionedpowerwmethodsarenonlinearbased methods.Theyhavebeenusedtocarryoutseveralpowerwproblems.Themaindrawbackof manyofthesenonlinearbasedmethods,however,isthattheyconstantlyconvergeslowly,andthere existsomecaseswhenconvergenceisnotattendedatall.More,inmostcases,when usedinoptimizingtheoperationoftoday'sdistributionsystems,thesenonlinearbasedmethodsare computationallydemanding.Asummaryofthemethodsusedtooptimizetheoperationofdistri- butionsystemsusingnonlinearandlinearoptimizationtechniquesarereviewedinthesubsequent section.Thechallenges,advantages,anddisadvantagesarealsohighlighted. 1.4ReviewofMathematicalProgrammingBasedMethods Optimizationmethods,withthebroadcanbeasconventionaloptimization methods,whicharedevelopedbasedonmathematicalprogramming,andintelligentoptimization methods,whicharedevelopedbasedonswarmintelligenceandevolutionaryprogramming.We havedealtwiththelattercategoryinChapter5.Anoverviewoftheformercategory,however,is presentedinthesubsequentsection. 10 1.4.1NonlinearOptimizationMethods PowersystemsareinherentlynonlinearaswasdiscussedearlierinSection1.3.Consequently, nonlinearoptimizationtechniquescanbeusedtohandlepowersystemoperationandplanning problems.Examplesofthenonlinearoptimizationmethods,whicharewidelyusedintheliterature include,nonlinearprogramming,quadraticprogramming,andmixed-integerprogramming.These methodsarereviewedbelow. Innonlinearprogrammingbasedoptimization,boththeobjectivefunctionandconstraintsare nonlinear.Inordertohandleanonlinearprogrammingproblem,weusuallystartbychoosinga searchdirection,whichisobtainedbythereducedgradientoftheobjectivefunction.The crucialadvantageofusingnonlinearprogrammingtechniquesinpowersystemstudiesispartly attributedtothereabilitytoachievehigheraccuracy.However,themaindisadvantageofnon- linearprogrammingbasedmethodsisthatslowconvergenceratemayoccur,whichmakesthese methodscomputationallyexpensive,especiallyfortheapplicationsinwhichmultiplesolutions arerequired.Inaddition,foratypeofengineeringapplicationstheobjectivefunctionis non-differentiable.Thiscouldlimittheuseofnonlinearprogrammingbasedmethods. Quadraticprogrammingcaninfactbeconsideredasaspecialcaseofnonlinearprogram- mingbasedtechnique.Thatis,inquadraticprogrammingbasedmethodstheobjectivefunction isquadraticwhiletheconstraintsarelinear.Acommonobjectivefunctionusedinpowersystem studiesistominimizethetotalgenerationcostoremission,whichisinherentlyaquadraticfunc- tion.Thequadraticprogramminghandlesthisproblemef,however,atthesametime,the computationalburdenisconsiderablylarge. 1 Moreprominently,thestandardsimpleformofthe quadraticprogrammingisnotquiteoftenusedbecauseconvergenceisnotalwaysguaranteed. 1 InChapter4,wesolvetheoptimaleconomicpowerdispatchproblemofactivedistributionsystems efcientlyandweshowthatthecomputationaleffortcanbetremendouslyreducedusingtheproposed optimizationframework. 11 Nonlinearoptimizationproblemscanalsobeformulatedasamixed-integerprogrammingusing certainintegercontrolvariables.Handlingoptimizationproblemsusingmethodsbasedonmixed- integerprogrammingcanbecomputationaldemanding,particularlyforlarge-scalesystems. 1.4.2LinearOptimizationMethods Linearmethodsareusedtotransformnonlinearoptimizationproblemstolinearproblems.Inthis context,linearprogramming(LP)isprobablythemostpopulartechnique,whichiswidelyusedto handletheengineeringapplicationsthatrequirerepetitive,prompt,andmultiplesolutions.Linear programmingisanoptimizationtechniqueintroducedinthe1930sbysomeeconomiststosolve theproblemofoptimalallocationofresources[27],[28].Unlikenonlinearprogrammingbased methods,boththeobjectivefunctionandconstraintsarelinearfunctionsintheLPmodel.The advantagesofusingLPbasedmethodsindistributionsystemoperationandplanninginclude, 1. Reliabilityoftheoptimizationandxibilityofthesolution. 2. Rapidconvergencecharacteristicsandfastexecutiontime. 3. Nonlinearconvexcurvescanbehandledusingpiecewiselinearmodels. 4. EqualityandinequalityconstraintscanbeequallyhandledinthebasicLProutine. Supposewehave m constraintswith n variables,thestandard maximum linearprogramming problemcanbeformulatedas[29], max(Z) = n å j = 1 c j x j (1.5) 12 Subjecttothefollowingconstraints, n å j = 1 a ij x j d i (1.6) x j 0 (1.7) Inmatrixnotation,thecoefin(1.6)canberepresentedas, A = 0 B B B B B B B B B @ a 11 a 12 ::: a 1 n a 21 a 22 ::: a 2 n . . . . . . . . . . . . a m 1 a m 2 ::: a mn 1 C C C C C C C C C A (1.8) Theright-handsidevector d oftheconstraintsconsistsof m constants, d = d 1 ; d 2 ; ; d m T (1.9) Therowvectoroftheobjectivefunction c consistsof n coef c = c 1 ; c 2 ; ; c n T (1.10) Thedualofthisstandard maximum problemisthestandard minimum problem,whichcanbe formulatedas, min(W) = m å i = 1 b i y i (1.11) Subjecttothefollowingconstraints, n å j = 1 a ij y j c j (1.12) 13 y i 0 (1.13) Ingeneral,thestandardlinearprogrammingcanberepresentedas, max(Z) = CX (1.14) withtheequalityconstraints, AX = d (1.15) X 0 (1.16) Alternatively,wecanusethecompactformtorefertothestandardmaximizationandminimization problems.Thus,thestandardmaximizationproblemcanbestatedas, Maximize: c T x j Ax d ; x 0 (1.17) Theprimal-dualstandardminimizationproblemisstatedas, Minimize: y T d j y T A c T ; y 0 (1.18) where A isacoefmatrixasin(1.6)withdimensions ( i j ) , c 2 R j , x 2 R j , d 2 R i , and y 2 R i ,and R isthesetofrealnumbers. 1.5TheNecessityforNewModelsandMethods Thestructureofdistributionsystemshasbeenrecentlychangedduetocertainenvironmental,eco- nomic,andpoliticalreasons.Thischangehascomethroughtheemergenceofseveralreal-time engineeringapplicationsinbothoperationalandplanningstages.Examplesoftheseapplications 14 includesizingandplacementofdistributedgenerators,economicpowerdispatchofactivedistri- butionsystems,feederforservicerestorationandreliabilityenhancement,andso forth.Itisverywellknownthattheseapplicationsrequireapowerwstudyatthestepofthe solution.Nevertheless,andnotsurprisingly,thevastmajorityoftheseapplicationsrequirerepet- itiveandpromptpowerwsolutions.PerformingfullACpowerw,ononehand,giveshigh calculationprecisionbutrequiresaquiteextensivecomputationalburdenandstoragerequirements. Ontheotherhand,andmoreprominently,thelargestpartoftheaforementionedapplicationsises- sentiallynonlinearcomplexcombinatorialconstrainedoptimizationproblems.Theformulationof thenonlinearproblem,however,tendstobeatedioustaskandcomputationallycumbersomein termsofexecutiontime,storagerequirements,andprogramming.Thesefactscombinedwiththe largenumberofdistributionsystemcomponentswillincontestablyincreasethecomplexityofthe problem.Ithasthereforebecomenecessarytodevelopmorepowerfultoolsforbothplanningand operationalstudiesnotonlytoperformtheaforementionedapplicationsmoreexpeditiouslyand ef,butalsotohandletheothernewtasks,whicharecomingintheimmediatefuture. Thevastmajorityofpowersystemoptimizationproblemsareessentiallyoptimalpowerw problemswithvariousobjectives.Examplesofthecommonpowersystemoptimizationproblems include,totallossminimization,totalgenerationcostminimization,andtotalloadcurtailmentmin- imizationproblem.Thelatterobjectivefunctionisinfactveryimportantobjectivefunctionthatis usedforreliabilityevaluationofpowersystems.Wehavedealtwiththesethreeobjectivesinthis thesisinChapter3,Chapter4,andChapter5,respectively.However,wewouldliketoelaborate hereonthelatterobjectivefunction,whichisconcernedwithreliabilityimprovementofdistribu- tionsystem.Foreverytaskofreliabilityevaluation,apowerwstudyisconstantlyperformed. Towardthisend,threepowerwmodelshavegenerallybeenusedforreliabilityevaluationof powersystems.ThesemodelsarethefullACpowerwmodel,theCapacityFlowmodel,and 15 theDCpowerwmodel.ExperiencewiththeFACPFmodelhaveshown,however,thatwhen thismodelisincorporatedinthereliabilityassessmentframework,thetaskofreliabilityevaluation becomesextremelycomplex,sometimesxible,andoftentimescomputationallyintractable[ ? ]. Also,therequireddataandstoragebothbecomehigh.Ontheotherhand,theCapacityFlowmodel onlyusesthecapacityconstraintsofthetie-lines;andthereby,generallyspeaking,itisnotappli- cableforeveryreliabilitystudy.Inviewofthesereasons,inmanycases,ithasbeenfoundtobe moreconvenienttousetheDCPFmodel. TheDCpowerwmodel,whichwasdevisedmorethan35yearsago,hasbeenwidelyutilized inreliabilityassessmentofpoweranddistributionsystems.ItisdenotedasDCpowerw(DCPF), inanalogytoaDCcircuitfedbyaDCvoltagesource[27,30,31,32].Infact,thismodelisa linearizedversionofthefullACpowerwmodel,howeveritignoresmostoftheaspectsofthe FACPFmodel.TheDCPFmodelisnon-iterative,linear,andabsolutelyconvergent,butwithless accuracythantheFACPFmodel.TheDCPFmodelassumesvoltageatallbusesand losslesstransmissionlines.Itisusuallyusedwheneverfastpowerwsolutionsarerequiredasin optimaleconomicpowerdispatch,contingencyanalysis,andreliabilityandsecurityassessment. TheDCPFmodelisunquestionablyapowerfulcomputationaltool,however,howitsassumptions areinterpretedandhowtheycanbeunderstoodisstillanopenquestion.Asamatteroffact, busvoltagesaremainlydependentonlineparametersaswellastheoperatingconditions,which thatvoltagearenotalwaysguaranteed.More,voltagelimits andreactivepowerwsarevitalconstraintsintherealisticpowersystemsandcannotthusbe neglected.Moreover,forpowersystemoperationalandplanningstages,forinstance,totalreal powerlossescannotbeomittedastheyaretypicallyconsideredasasecondaryobjectiveinseveral studies.EventhoughtheDCPFmodelhassuchshortcomings,itwasapproximatelyinvolvedin severalengineeringapplicationsbecauseofitssimplicityofformulationandimplementation. 16 TheuseoftheDCPFmodelincertainreliabilityevaluationstudiesofdistributionsystemshas beenbasedontheassumptionthatadequateinformationaboutreactivepowerwareunavailable inadvance.Suchanassumptionisforseveralplanningstudiesbecausereactivepoweris usuallysuppliedinaformofcapacitorbanks,whichcanbetreatedasaseparateoptimizationprob- lem.However,frompracticalperspective,theviolationofvoltageboundsandinsufreactive powersupportmayinitiatetriggeringevents,whichcouldinturnleadtoamassiveunder-voltage loadshedding.Therefore,itwouldbeifacertainamountoflinearityisintroducedinto theconventionalpowerwequations,withoutlossofgenerality,sothatafastandxiblesolu- tionsareobtained.Havingdonethis,severaloptimizationproblemsthatrequirerepetitivepower westimationsandoptimalsolutionscanbeformulatedandsolvedmoreexpeditiouslyand,at thesametime,withareasonableengineeringaccuracy. 1.6OverviewofContributions Theprincipalcontributionofthisworkliesindevelopinganefoptimizationframeworkfor distributionsystemoperationalandplanningstudies.Thisframeworkhasbeendevelopedinthe secondchapterofthisthesis.Wedevelopafastandeffectiveformulationforthepowerw problem.TheproposedlinearizedACpowerwmodel(LACPF)isdevelopedbasedonlineariza- tionofthefullsetofconventionalpowerwequations,andthereforeincludesvoltagemagnitude solutionsandreactivepowerws,unliketraditionallinearizedpowerwmodels.Hence,the techniqueproposedinthisthesisisnon-iterative,direct,andinvolvesnoconvergenceissues.More prominently,themodelproposedinthisthesisisequallyappropriateforuseatbothdistribution andtransmissionlevelsandcanbeextremelyusefulwheneverfast,robust,andrepetitivepower wsolutionsarerequired.Further,theincaseofunbalanceddistributionnetworks 17 arestraightforwardandlargelylieincertainelementsinthebusadmittancematrix;andthusthe advantagesobtainedwithbalancedoperationarepreserved.WetesttheproposedLACPFmodel onseveralbalanced,unbalanced,andweakly-mesheddistributionsystemsandfoundtoperform withspeedandaccuracyappropriateforrepetitivesolutions.Weprovideandthoroughlydiscussed theresultsofvarioustestsystems,includingalarge-scaledistributionsystemtestcaseinChap- ter2.Itisworthpointingoutherethatseveraltestsystemswithvariouscharacteristicsandsizes wereutilizedtoverifytheaccuracyoftheproposedpowerwmodel.However,wereportthe resultsofselectedsystemsinChapter2,asthesesystemshavebeenusedlatertodemonstratethe effectivenessoftheproposedoptimizationframework. WedevelopanoptimizationframeworkbasedonlinearprogrammingmethodinChapter2. Optimizationmethods,whicharedevelopedbasedonlinearprogrammingarecompact,xible, reliable,andfasterthantheirnonlinearcounterparts.Theproposedoptimizationframeworkhas theadvantageofbeingsuitableforstudiesthatrequireextensivecomputationalburdensuchas inreliabilityandsecurityassessmentanddistributedgenerationsizingandplacement.Aunique featureofourdevelopmentisthatvoltagelimitsandreactivepowerconstraintshavebothbeen consideredintheproposedmodel.Theseutmostconstraintsareentirelyignoredinthetraditional linearizedmodelsavailableintheliterature.Akeyelementinsolvinganyoptimizationproblem inpowersystemistoconsiderthethermalcapacitiesoftransmissionlines.Inthiscontext,we developnovelmodelstohandlecurrentlimitsconstraints,whichareinvolvedinalmostallofthe realisticpowersystemoptimizationproblems.Towardthisend,weparticularlydevelopageneric piecewiselinearmodelsothattheinequalitycurrentconstraintscanbeapproximatedbyan numberoflinearsegments.Thecrucialadvantageofthismodelisthatthedegreeofapproximation canbeimprovedtoanydesiredlevelbyincreasingthenumberoflinesegmentsinvolved,without substantialaffectonthemainroutineandthecomputationalspeed. 18 AswasmentionedearlierinSection1.1.3,distributionsystemsarethemostextensivepartin theentirepowersystemduetotheirspanningtreestructureandthehighR/Xbranchratios.There- fore,weproposeananalyticalmethodforoptimalplacementandsizingofdistributedgeneration unitsondistributionsystemsinChapter3.Theobjectiveoftheanalyticalmethodpresentedin Chapter3istominimizethedistributionsystemlosses.Analyticalmethodsarereliable,compu- tationallyefandaresuitableforplanningstudiessuchasdistributedgenerationplanning. Furthermore,analyticalapproachescouldleadtoanoptimalornear-optimalglobalsolution.We identifythepenetrationlevelofthedistributedgenerationunits.Then,wedevelopapriority listbasedonlosssensitivityfactorstodeterminetheoptimallocationsofthecandidatedistributed generationunits.Weperformsensitivityanalysisbasedontherealpowerinjectionofthedis- tributedgenerationunitstoestimatetheoptimalsizeandpowerfactorofthecandidatedistributed generationunits.Wedealwithvarioustypesofdistributedgeneratorsandalsoproposeviable solutionstoreducetotalsystemlosses.Wevalidatetheeffectivenessoftheproposedmethodby applyingitonthesamebenchmarksystemsusedbeforeinChapter2,inparticularthe33bus andthe69busdistributionsystems,sincebothsystemshavebeenextensivelyusedasexamples insolvingtheplacementandsizingproblemofdistributedgenerators.Wealsoperformexhaus- tivepowerwroutinestoverifythesizesobtainedbytheanalyticalmethod.Wevalidatethe optimallocationsandsizesobtainedbytheproposedanalyticalmethodbycomparingthemwith someotheranalyticalmethodsavailableintheliterature.Thetestresultsshowthattheproposed analyticalmethodcouldleadtoanoptimalornear-optimalglobalsolution,whilerequiringlower computationaleffort. Weproposeamethodtosolvetheoptimaleconomicpowerdispatchproblemofactivedistri- butionsystemsinChapter4.Nonlinearprogrammingandlinearprogrammingbasedmethodsare widelyusedintheliteraturetosolvetheoptimaleconomicdispatchproblem.Nevertheless,the 19 vastmajorityofthelinearprogrammingbasedmethodsweredevelopedbasedontheDCPFmodel, whichhasseveraldrawbacksthatwediscussedearlierinSection1.5.InChapter4,inadditionto thepiecewiselinearmodelwehavedevelopedearlierinChapter2tohandlethethermalcapacities oftransmissionlines,wedeveloppiecewiselinearmodelstodealwiththeloads,costcurvesof generatingunits,andtotalpowerlosses.Wetaketheeffectofdistributedgenerationunitsinsev- eralcasescenariosbyconsideringdifferentpenetrationlevels.Wedemonstratetheeffectivenessof theproposedmethodbyperformingnumerouscasestudies.Wewereabletoshowthattheresults obtainedbytheproposedmethodcorrespondcloselywiththoseobtainedbynonlinearmeansand isappropriateforseveralplanningstudies. Weproposeamethodtosolvethedistributionsystemproblemwithanobjective ofreliabilitymaximizationinChapter5.Reliabilityenhancementofdistributionsystemsthrough feederisnotwellstudiedintheliterature.InthepartofChapter5weintroduce acompleteoptimizationframeworktohandlethereliabilitymaximizationproblem.Then,we proceedbydiscussingbasicreliabilityconceptsandintroducedprobabilisticreliabilitymodels. Sincethetimeandcomputationaleffortspentinevaluatingreliabilityindicesareofgreatconcern inbothplanningandoperationalstages,weuseaprobabilisticreliabilityassessmentmethodbased oneventtreeanalysiswithhigher-ordercontingencyapproximation.Therefore,theeffectofthe higher-ordercontingenciesislimitedand,atthesametime,thecomputationalburdenisimproved. Theobjectivefunctionoftheoptimizationframeworkusedinthischapteristominimizethetotal loadcurtailment.Wechoosetheexpectedunservedenergy(EUE)astheenergyindexthatneedsto beminimized.However,toknowhowmuchreliablethesystemis,weintroduceanotherreliability measure,whichistheenergyindexofunreliability(EIUR).Frompracticalperspective,theradial topologicalstructurehasbeentakenasanecessaryconditionduringtherealizationofthiswork. Therefore,wedevelopanotherconstraintsbasedontheoreticalgraphtopreservethespanningtree 20 structureofthedistributionsystem. InthesecondpartofthisChapter5,weformulatethedistributionsystemprob- lemforreliabilitymaximization.Inthiscontext,weproposeanintelligentsearchmethodbased onparticleswarmoptimizationtechnique(PSO).Particleswarmoptimizationisameta-heuristic optimizationmethodinspiredbythesocialbehaviorofofbirdsorschoolsofwhichis introducedin1995.Theadvantagesofusingparticleswarmoptimizationinhandlingthedistri- butionsystemproblemaremanifold.Forinstance,thestatusofsectionalizingand tie-switchesindistributionsystemscanbeeasilyrepresentedasbinarynumbersof(0,1).More- over,particleswarmoptimizationbasedmethodshaveconsiderablyfastconvergencecharacteris- ticsand,generallyspeaking,havefewparameterstotuneupcomparedtosomeothermeta-heuristic approaches.Moreprominently,particleswarmoptimizationhastwomainparameters,whichare thepersonalbestandthegroupbest.Everyparticleintheswarmremembersitsownpersonalbest andatthesametimeitsgroupbest.Consequently,PSObasedmethodshaveconsiderablymore memorycapabilitythansomeotherswarmintelligencebasedmethods. Wedemonstratetheeffectivenessoftheproposedmethodonseveraldistributionsystemsand showthattheamountoftheannualunservedenergycanbereducedusingtheproposedmethod. 1.7ThesisOutline Thisthesispresentsafast,xible,andreliableoptimizationframeworkfordistributionsystem operationalandplanningstudies.ThebasicoptimizationframeworkispresentedinChapter2. Chapter3,Chapter4,andChapter5,discussindetailtheapplicationsoftheproposedmodels andmethodsinmoderndistributionsystems.Eachchapterpresentsthemainconceptof theapplication,andthenadescriptionofthemethodusedinthisapplicationisexploitedindetail. 21 Followingthedescriptionofthemethod,commentsandassumptionsregardingtheimplementation ofthemethodareprovided. Chapter3presentsamethodforlossreductionindistributionsystemsusingdistributedgen- erationunits.Thestructureofthischapterisasfollows:itprovidesadescriptionaboutthebasic concepts,followedbydescriptionofthemethodused,andcommentsontheimplementation.Con- cludingremarksareprovidedattheendofthischapter. Chapter4proposesaneffectivemethodtosolvetheoptimaleconomicpowerdispatchproblem ofpowerdistributionsystems.ThemainstructureofthischapterissimilartothepatternofChapter 3,discussingbasicconcepts,followedbyadescriptionofthemethodused,andcommentsonthe implementation.Conclusionremarksarealsoprovidedattheendofthischapter. Chapter5introducesaxibleandrobustmethodfordistributionsystemusing particleswarmoptimizationbasedmethod.Themainstructureofthischapterissimilartothe patternsofChapter3andChapter4.Then,thischapterpresentsthereliabilityevaluationmethod andtheproposedintelligentsearchmethod.Themethodofimplementation,discussion,andcon- clusionsarealsoprovidedinthischapter. Chapter6highlightstheconclusionsdrawnfromthepresentedworkandsummarizesthemain contributions.Possibleareasoffutureresearchareoutlinedinthischapter. 22 Chapter2 AnOptimizationFramework Development Inthepartofthischapter,wedevelopafast,robust,andeffectiveformulationforthepower wproblem.WedeveloptheproposedlinearizedACpowerwmodelbasedonlinearization ofthefullsetofconventionalpowerwequations,andthereforeincludesvoltagemagnitude solutionsandreactivepowerws,unliketraditionallinearizedpowerwmodels.Wetestthe proposedLACPFmodelonseveraldistributionsystems,includingsystemswithvarioussizes, complexities,characteristics,andsystemswithweakly-meshedtopologies,andfoundtoperform withspeedandaccuracyappropriateforrepetitivesolutions. Inthesecondpartofthischapter,wedevelopanefoptimizationframeworktosolvesev- eraldistributionsystemsoperationalandplanningstudies.Inparticular,wedevelopformulations forpowerbalanceequations,activeandreactivepowerconstraints,andvoltagelimits.Wedevelop apiecewiselinearmodeltohandlethecurrentcapacityconstraints.Weintroducethemaincon- straints,whichareusuallyusedinsolvinganytypicaloptimizationproblem.Wealsodiscussthe inclusionofadditionalconstraints,whichwearegoingtouseinthesubsequentpartsofthisthesis. However,beforeproceedingtodescribetheproposedpowerwmodel,itisimperativeto touchupontheFACPFmodelandtheDCPFmodel,whichhavebothbeenusedinthischapterto validatetheeffectivenessoftheproposedLACPFmodel. 23 2.1FullACPowerFlowModel Formulationofthepowerwproblemrequirestheconsiderationoffourvariablesateachbusin thesystem.Thatis,atanybus k inthesystem,thesevariablesaretheactivepowerinjection P k , thereactivepowerinjection Q k ,voltagemagnitude V k ,andvoltageangle d k .Therealandreactive powerwsandpowerlossescanalsobedeterminedfromthepowerwsolution. LetusconsiderthesimpletransmissioncircuitshowninFig.2.1.Thecurrentinjectedtobus k canbecalculatedas, I k = å m 2 Y k Y km E m (2.1) where Y k isthesetofbusesadjacenttobus k , I k isthecurrentinjectedtobus k , E m isthevectorof busvoltages,and Y km isthebusadmittancematrixofthesystem. Figure2.1:Simpletransmissioncircuit Letususetherectangularandpolarcoordinatesforadmittancesandbusvoltages, Y km = G km + jB km E k = V k e j d k E m = V m e j d m (2.2) 24 Therefore,(2.1)canalternativelybeexpressedas, I k = å m 2 Y k ( G km + jB km ) ( V m e j d m ) (2.3) where V k and V m arethevoltagemagnitudesatbus k andbus m ,respectively. G km istherealelement ( k ; m ) ofthebusadmittancematrix, B km istheimaginaryelement ( k ; m ) ofthebusadmittance matrix,and d km isthevoltageangledifferencebetweenbus k andbus m ,respectively.Thecomplex powerinjectedatbus k canbeexpressedas, S k = E k I k =( V k e j d k ) å m 2 Y k ( G km jB km ) ( V m e j d m ) (2.4) Therefore,therealandreactivepowerinjectionsatbus k canbeexpressedas[8], P k = V k å m 2 Y k V m ( G km cos d km + B km sin d km ) (2.5) Q k = V k å m 2 Y k V m ( G km sin d km B km cos d km ) (2.6) Here(2.5)and(2.6)arethegenericpowerwequations.Ascanbeseenfromthisformulation, inordertogetasolutiontothepowerwequations,twoofthefourvariablesmustbeknown inadvance.Duetothenonlineartermsinthepowerwequations,thesolutionfollowsiterative processinwhichconvergenceisnotalwaysguaranteed. 2.2DCPowerFlowModel AswasdiscussedearlierinChapter1,Section1.4,someengineeringapplicationsrequireprompt andfastpowerwestimations.TheDCPFmodeliscommonlyusedtopreformtheseapplications. 25 TheDCPFmodelcanbeformulatedbasedonthefollowingassumptions: 1. Losslesstransmissionlinemodel. 2. Flatvoltageatallbuses. 3. Voltageangledifferences(lateralorsub-lateral)areassumedtobesmall. Letusnowconsiderthegenericpowerwequationgivenby(2.5)andassumethatlineresis- tanceisverysmallcomparedtolinereactance,thatis G km = 0,magnitudesofbusvoltagesareall setequalto1 : 0perunit,thatis V k = V m = 1 : 0perunit,andsin d km ˇ d km .Sinceactivepowerin- jectionsareknowninadvance,theDCPFmodelcanthereforeberepresentedas[27, ? ,30,31,32], P k = N b å m = 1 B km ( d k d m ) (2.7) Invectornotation,theDCPFmodelcanalternativelyberepresentedas[ ? ], P G + ‹ B d = P D (2.8) F =( b ‹ A ) d (2.9) where N b isthenumberofsystembuses, N f isthenumberofdistributionfeeders, b isadiago- nalmatrixofdistributionlinessusceptances N f N f , ‹ A istheelement-nodeincidencematrix N f N b , P G isthevectorofrealpowergeneration N f 1 , F isthevectorofwcapacitiesof distributionlines N f 1 . 26 2.3ProposedPowerFlowModel ThissectionintroducestheproposedlinearizedACpowerwformulation.Thismodeliscollab- orativelydevelopedbyJ.Mitra,S.Elsaiah,andN.Cai.Wewillusethisformulationlateronto developanoptimizationframeworktosolveseveraloptimaldistributionsystemsstudies. Recalltherealandreactivepowerinjectionsatbus k expressedearlierby(2.5)and(2.6).That is, P k = V k å m 2 Y k V m ( G km cos d km + B km sin d km ) (2.10) Q k = V k å m 2 Y k V m ( G km sin d km B km cos d km ) (2.11) where Y k isthesetofthebusesadjacenttobus k . P k and Q k aretherealandreactivepower injectionsatbus k . V k and V m arethevoltagemagnitudesatbus k andbus m ,respectively.Also, G km istherealelement ( k ; m ) ofthebusadmittancematrix, B km istheimaginaryelement ( k ; m ) ofthebusadmittancematrix,and d km isthevoltageangledifferencebetweenbus k andbus m , respectively. Inpractice,wekeepbusvoltagesaround1.0p.u.,withavalue(usually 5%). Therefore,thevoltagemagnitudeatbus k andbus m canalternativelyberepresentedas, V k = 1 : 0 D V k V m = 1 : 0 D V m (2.12) where D V k and D V m arebothexpectedtobesmallquantities. Inordertoapproximatethepowerinjectionatbus k ,letusignorethesmallportion D V k atbus k in(2.12).Itisimportanttonotethatthisisonlyan approximation thatenablesthelinearization; itis not an assumption thatthevoltagemagnitudeequals1.0p.u.Therefore,(2.10)and(2.11)can 27 bewrittenas, P k ˇ å m 2 Y k V m ( G km cos d km + B km sin d km ) (2.13) Q k ˇ å m 2 Y k V m ( G km sin d km B km cos d km ) (2.14) Further,letusassumethatthephaseangledifferencebetweenbus(lateralorsub-lateral) k andbus m issmall;(2.13)and(2.14)canbenowexpressedas, P k ˇ å m 2 Y k V m ( G km + B km d km ) (2.15) Q k ˇ å m 2 Y k V m ( G km d km B km ) (2.16) Eqs.(2.15)and(2.16)canbeexpressedas, P k ˇ å m 2 Y k ( V m G km + V m B km d km ) (2.17) Q k ˇ å m 2 Y k ( V m G km d km V m B km ) (2.18) Asbefore,letusmakeafurtherapproximation, V m ˇ 1 : 0p.u.,inthesecondtermof(2.17).This impliesthat, P k ˇ å m 2 Y k ( V m G km + B km d km ) (2.19) Letusexpand(2.19)asthefollowing, P k ˇ å m 2 Y k V m G km + å m 2 Y k B km d km (2.20) 28 Alternatively,(2.19)canberewrittenas, P k ˇ å m 2 Y k V m G km + å m 2 Y k B km ( d k d m ) (2.21) Now,(2.21)canbefurtherbrokentotwopartsasfollows, P k ˇ P ku + P kv (2.22) where P ku = å m 2 Y k V m G km (2.23) and P kv = å m 2 Y k B km d k å m 2 Y k B km d m (2.24) where B km = 8 > < > : å b km + b kk for m = k b km for m 6 = k (2.25) Here b kk isthetotalsusceptanceoftheshuntelementsconnectedatbus k .Itisevidentfrom(2.25) thatsummingthe B km termsforall m 2 y k yields, å m 2 Y k B km = b k 1 b k 2 ::: +( å m 6 = k b km + b kk ) ::: b kN = b kk (2.26) Hence,(2.24)canbewrittenas, P kv = å m 6 = k B km d m ( B kk b kk ) d k (2.27) 29 Therefore,(2.22)willhavethefollowingform, P k ˇ å m 2 Y k V m G km å m 6 = k B km d m ( B kk b kk ) d k (2.28) Thereactivepowerequationcanbeexpressedas, Q k ˇ å m 2 Y k V m B km + å m 2 Y k G km ( d k d m ) (2.29) Now,inasimilarfashiontowhatwedidwiththerealpowerterm,(2.29)canbefurtherbrokento, Q k ˇ Q ku + Q kv (2.30) where Q ku = å m 2 Y k V m B km (2.31) and Q kv = å m 2 Y k G km ( d k d m ) (2.32) Now, Q kv isobtainedas, Q kv = å m 6 = k G km d m ( G kk g kk ) d k (2.33) where G km = 8 > > > > > < > > > > > : å g km + g kk for m = k g km for m 6 = k (2.34) 30 Here g kk isthetotalconductanceoftheshuntelementsconnectedatbus k .Itisevidentfrom(2.34) thatsummingthe G km termsforall m 2 Y k yields, å m 2 Y k G km = g k 1 g k 2 ::: +( å m 6 = k g km + g kk ) ::: g kN = g kk (2.35) Consequently,(2.30)willhavethefollowingform, Q k ˇ å m 6 = k G km d m ( G kk g kk ) d k å m 2 Y k V m B km (2.36) Inmatrixnotation,theproposedLACPFmodelingeneralformcanbeexpressedas, 2 6 4 P K Q K 3 7 5 = 2 6 4 B 0 G G 0 B 3 7 5 2 6 4 d K V K 3 7 5 (2.37) where B 0 isasusceptancematrix, G 0 isaconductancematrix, G isthecon- ductancematrix,and G isthesusceptancematrix.Thesematrices,whichcomprisetheproposed powerwsolution,arerespectivelyas, B 0 = 2 6 6 6 6 6 6 6 6 6 4 ( B 11 b 11 ) B 12 B 1 N B 21 ( B 22 b 22 ) B 2 N . . . . . . . . . . . . B N 1 B N 2 ( B NN b NN ) 3 7 7 7 7 7 7 7 7 7 5 (2.38) 31 G 0 = 2 6 6 6 6 6 6 6 6 6 4 ( G 11 g 11 ) G 12 G 1 N G 21 ( G 22 g 22 ) G 2 N . . . . . . . . . . . . G N 1 G N 2 ( G NN g NN ) 3 7 7 7 7 7 7 7 7 7 5 (2.39) G = 2 6 6 6 6 6 6 6 6 6 4 G 11 G 12 G 1 N G 21 G 22 G 2 N . . . . . . . . . . . . G N 1 G N 2 ::: G NN 3 7 7 7 7 7 7 7 7 7 5 (2.40) B = 2 6 6 6 6 6 6 6 6 6 4 B 11 B 12 B 1 N B 21 B 22 B 2 N . . . . . . . . . . . . B N 1 B N 2 B NN 3 7 7 7 7 7 7 7 7 7 5 (2.41) 2.4ModelValidationandTestResults Wetesttheproposedlinearizedpowerwmodelonseveralknowndistributionnetworks;includ- ingrealisticandlarge-scaledistributionsystems.Weimplementtheproposedmodelon33bus, 69bus,and118busdistributionsystems.Itisappropriatetopointoutherethatthesenetworks havebeenselecteddeliberatelyastheyarebeingbroadlyusedasexamplesinsolvingnumerous problemsintheareaofoptimaldistributionsystemoperationandplanning.Inaddition,weim- plementtheproposedpowerwmethodonapractical13busunbalancednetwork[24]anda 32 76busunbalanceddistributionnetwork[35].Eventhoughweimplementtheproposedmodelon numeroussystems,wewillfocusthe33bussystem,69bussystem,and118bussystemsincewe usethemthroughouttherestofthisthesis. 2.5TestSystemI Thetestsystemconsideredinthiscaseisa33busmediumvoltageradialdistributionnetwork[16]. Thesinglelinediagramof33bussystemisdepictedinFig.2.2.The33bussystemconsistsof33 busesand32branches.Thetotalrealandreactivepowerloadsonthissystemare3715kWand 2300kVAR,respectively.Thebasevaluesarechosentobe12 : 66kVand100kVA.The33bus distributionsystemhasR/Xbranchratiosof3. WeusetheproposedLACPFmodeltoobtainthepowerwsolutionofthe33bussystemand comparetheresultswiththoseobtainedbytheFACPFmodel.Thevoltageobtainedbythe proposedLACPFmodelisdepictedinFig.2.3.Wealsopresentdetailedresultsofvoltagemag- nitudesandvoltageanglesinTable2.1.Forvalidationpurposes,wecalculatethevoltagevector obtainedbytheproposedLACPFmodelandthatobtainedbytheFACPFmodelandcomputethe voltagevectorerrorateverybusas, D V = j V FACPF V LACPF j j V FACPF j 100% (2.42) where V FACPF and V LACPF arethevectorsofbusvoltagesobtainedbytheFACPFmodelandthe LACPFmodel,respectively. AscanbeseenfromFig.2.3andTable2.1,thepowerwresultsobtainedbytheproposed modelcorrespondcloselywiththoseobtainedbytheFACPFmodel.Itisindispensabletohighlight 33 herethattheerrorinvoltageobtainedbytheproposedLACPFmodelwaslessthan1%atallbuses. Themaximumvoltageerrorobtainedbyperformingtheproposedmodelwas ( 0 : 008 0 : 001 i ) p.u.or0 : 843%andhadoccurredatbus18,thefarthestbusfromthesubstation.Ontheotherhand, thevoltageerrorobtainedbythetraditionalDCPFmodelatbus18isfoundtobe9 : 773%.The proposedpowerwmodeloutperformsthetraditionalDCPFmodel.Accordingly,themethod reportedherecouldhandlepowerwsofdistributionsystemswithhighR/Xbranchratiosmore expeditiouslywithasmall,butacceptable,intheengineeringaccuracy. Figure2.2:Single-linediagramofthe33busdistributionsystem 34 Table2.1:33BusDistributionSystemPowerFlowResults FACPF LACPF Error Bus j V j Angle j V j Angle Vector Absolute (p.u.) (radian) (p.u.) (radian) (p.u.) (%) 1 1 0 1 0 0 0 2 0.9970 0.0002 0.9972 0.0002 -0.000+0.000i 0.020 3 0.9828 0.0017 0.9839 0.0015 -0.001+0.000i 0.102 4 0.9753 0.0028 0.9769 0.0025 -0.002+0.000i 0.155 5 0.9679 0.0040 0.9700 0.0035 -0.002+0.000i 0.219 6 0.9494 0.0024 0.9530 0.0020 -0.004+0.000i 0.369 7 0.9459 -0.0017 0.9498 -0.0014 -0.004-0.000i 0.412 8 0.9322 -0.0044 0.9372 -0.0036 -0.005-0.000i 0.526 9 0.9259 -0.0057 0.9314 -0.0046 -0.006-0.000i 0.605 10 0.920 -0.0068 0.9261 -0.0055 -0.007-0.000i 0.653 11 0.9192 -0.0067 0.9253 -0.0054 -0.007-0.000i 0.664 12 0.9177 -0.0065 0.9239 -0.0052 -0.007-0.000i 0.676 13 0.9115 -0.0081 0.9183 -0.0065 -0.008-0.001i 0.747 14 0.9092 -0.0095 0.9162 -0.0075 -0.008-0.001i 0.782 15 0.9078 -0.0102 0.9149 -0.0080 -0.008-0.001i 0.795 16 0.9064 -0.0106 0.9137 -0.0083 -0.008-0.001i 0.807 17 0.9043 -0.0119 0.9118 -0.0093 -0.008-0.001i 0.842 18 0.9037 -0.0121 0.9113 -0.0094 -0.008-0.001i 0.843 19 0.9964 0.0001 0.9966 0.0000 -0.000+0.000i 0.010 20 0.9929 -0.0011 0.9931 -0.0011 -0.000-0.000i 0.020 21 0.9922 -0.0015 0.9924 -0.0015 -0.000+0.000i 0.023 22 0.9915 -0.0018 0.9918 -0.0018 -0.000-0.000i 0.020 23 0.9793 0.0011 0.9804 0.0010 -0.001+0.000i 0.113 24 0.9726 -0.0004 0.9739 -0.0005 -0.001+0.000i 0.134 25 0.9693 -0.0012 0.9707 -0.0012 -0.001+0.000i 0.145 26 0.9475 0.0031 0.9512 0.0026 -0.004+0.000i 0.391 27 0.9449 0.0040 0.9488 0.0034 -0.004+0.000i 0.403 28 0.9335 0.0055 0.9383 0.0045 -0.005+0.000i 0.515 29 0.9253 0.0068 0.9307 0.0056 -0.006+0.000i 0.585 30 0.9217 0.0087 0.9274 0.0070 -0.006+0.001i 0.620 31 0.9175 0.0072 0.9236 0.0059 -0.007+0.000i 0.655 32 0.9166 0.0068 0.9228 0.0055 -0.008+0.000i 0.666 33 0.9163 0.0067 0.9225 0.0054 -0.008+0.000i 0.666 35 Figure2.3:Voltageofthe33bussystemobtainedbytheproposedLACPFmodelandthe FACPFmodel 2.6TestSystemII Thetestsystemconsideredforthiscasestudyisa69bus,12 : 66KVmediumvoltageradialdistri- butionsystem[33].Thesinglelinediagramofthe69busdistributionsystemisshowninFig.2.4. The69bussystemiswidelyusedintheliteratureandhasbeenconsideredasarelativelylarge- scaledistributionnetwork.Thetotalrealandreactivepowerloadsonthissystem,respectively,are 3802 : 19KWand2694 : 06KVAR,buttheR/Xratiosforsomenetwork'sbranchesaregreaterthan 3.Wechoosethebasevaluesas12 : 66kVand100kVA. WeusetheproposedLACPFmodeltoobtainthepowerwsolutionofthe69bussystem andcomparetheresultswiththoseobtainedbytheFACPFmodel.Thevoltageobtained bytheproposedLACPFmodelandtheFACPFmodelisshowninFig.2.5.FromFig.2.5,it isevidentthatthevoltageobtainedbytheproposedLACPFmodelisalmostsimilarto thatobtainedbytheFACPFmodel.Wecalculatethevoltageerrorateachbususing(2.42).It isinterestingtohighlightherethattheerrorinbusvoltagewaslessthan1%atallbuses,and 36 onlyfewbuses(exactly10buses)hadanerrorbetween0 : 27%and0 : 936%.Themaximumvector errorwas ( 0 : 009 + 0 : 004 i ) or0 : 936%andhadoccurredatbus65,whichisquitefarfromthe substationbus.Ontheotherhand,thevoltageerrorobtainedbythetraditionalDCPFmodelat bus65isfoundtobe10 : 01%.TheproposedpowerwmodeloutperformsthetraditionalDCPF modelandcanthereforebeusedfornumerousengineeringapplicationsinwhichfast,reliable,and repetitivepowerwestimationsarerequired. 2.7TestSystemIII Wetesttheproposedpowerwmodelonamorecomplicatedandrealisticdistributionsystemin ordertovalidateitsfeasibilityintheseconditions.Thesystemusedforthiscasestudyisan11kV, 118buslarge-scaleradialdistributionsystem[34].Thissystemisextensivelyusedintheliterature asalarge-scalebenchmarksystem.Thesingle-linediagramofthe118busdistributionsystemis depictedinFig.2.6.Thetotalrealandreactivepowerloadsonthe118busdistributionsystemare 22709 : 7kWand17041 : 1kVAR,respectively.Distributionfeedersdata,loadingconditions,and otherofthe118bussystemcanbefoundin[34]. Thevoltageerrorforthiscasestudywaslessthan1%atallbusesandthemaximumvoltage errorobtainedbytheproposedLACPFmethodwas0.89%,whichhadoccurredatbus77.Itis appropriatetomentionherethatthe118buslarge-scaledistributionsystemhas15tie-switches,in additiontothesectionalizingswitches.TheproposedLACPFmodelhasgivensatisfactoryresults evenwhenweclosealltieswitches,unliketheFACPFmodel,whichtendstodivergewhena fewloopshavetakenplaceinthesystem.Thismakestheproposedmodelverysuitableforthe engineeringapplicationsinwhichrigidandreliablepowerwestimationsarerequired. 37 Figure2.4:Single-linediagramofthe69busdistributionsystem 38 Figure2.5:Voltageofthe69bussystemobtainedbytheproposedLACPFmodelandthe FACPFmodel 2.8TestSystemIV WetesttheproposedLACPFmodelona76busdistributionsystemwithunbalancedloads[35]. Branchdataandloadingconditionsofthe76busdistributionsystemcanbefoundin[16,35].This systemhasR/Xbranchratiosof3.ThemaximumvoltageerrorobtainedbytheproposedLACPF method,foreachphase,wereasfollows:0 : 80%forPhaseA,0 : 86%forPhaseB,and0 : 93%for PhaseC.Thisamountofvoltageerrorcanbeconsideredgoodenoughforcertainoperationaland planningengineeringapplicationssincethissystemhasquitehighR/Xbranchratios. 2.9RobustnessTestoftheProposedModel Divergenceofthesolutionofthepowerwmethodsusuallyoccurswhencertainill-conditioned casesarepresentedinthesystem.AswasmentionedearlierinChapter1,Section1.1.3,distribution systemswithhighR/Xbranchratiosareinherentlyill-conditionedpowernetworks.Ill-conditioned 39 Figure2.6:Single-linediagramofthe118busdistributionsystem 40 casesoftentakeplacewhenthesysteminvolvesshort-lines,orevenlonglines,whichisusually thecaseinmostrealisticandruraldistributionsystems.Inordertoshowtheeffectivenessofthe proposedLACPFmodelinhandlingsuchconditions,therangeoftheimpedancesinthe33bus distributionsystem,whichisconsideredasanexampleinthiscase,havebeenwidelychanged tohaveanill-conditionedrepresentation.Amongsttheselectedimpedances,somearedividedby afactorof g ,tohaveashortdistributionlinerepresentation,whileothersaremultipliedbythe samefactortohavealonglinerepresentation.Whenwehavechanged g from1to10,thetest resultshaverevealed,infact,thattheFACPFdivergedforvaluesof g 3.Forthesecases,some othermethodssuchastheonereportedin[17,18],whichareverycomputationallydemanding, needtobeused.Ontheotherhand,theproposedLACPFmodelcanhandlethepowerwsof ill-conditioneddistributionsystemsmoreexpeditiouslyand,atthesametime,withareasonable engineeringaccuracy. Itisworthpointingoutherethattheproposedlinearizedpowerwmodelisnotintended toreplacethepowerfulnonlinearpowerwmethodssuchasNewton-Raphsonmethod[8,7], Gauss-Seidelmethod[8,7],Fast-decoupledmethod[9,10,11]anditsversions[12,14], orthebackward/forwardsweepingmethod[17,18].Itratherallowsustogetfast,xible,and reliablepowerwestimations,whicharehighlysoughtinmanyoftoday'sengineeringappli- cations.Further,wetesttheproposedlinearizedACpowerwmodelonseveraltestsystems, includingtransmissionsystems,balancedandunbalanceddistributionsystems,weakly-meshed distributionsystems,anddistributionsystemswithdistributedgenerationunits,andundervarious loadingconditions.Inalltestcases,theobtainedresultswerepromising. 41 2.10LinearProgrammingBasedOptimization Thelinearprogrammingproblemisastheproblemofmaximizingorminimizingalinear functionsubjecttolinearconstraints.Theproblemcanbeofequalityorinequalityconstraints sinceinequalityconstraintscanbetransformedintoequalityconstraintsbyintroducingslackor surplusvariables.Aswasdiscussedearlier,thisworkaimsatdevelopinganefoptimization frameworkbasedonlinearprogrammingmethod.Theproceduresofdevelopingtheoptimization frameworkarediscussedbelow. 2.10.1ObjectiveFunctionFormulation Wetheminimizationprobleminthepresentedworkas, Minimize: F ( h ; z ; u ) (2.43) Subjectto: v ( h ; z ; u )= 0 (2.44) w ( h ; z ; u ) 0 (2.45) where h representsthenetworkconstraintsintermsofpowerw, z representsthenetworkparam- eters,and u representsthecontrolvariables. 2.10.2PowerBalanceEquations Powerbalanceequationsareequalityconstraintsrepresentthesumofthepoweratacertainbus. Basically,balanceequationsarederivedfromthelinearizedpowerwmodel,whichisdeveloped 42 earlierinSection2.3.Mathematically,thepowerbalanceequationscanbeposedas, B 0 d GV + P G = P D G 0 d + BV + Q G = Q D (2.46) Where, d isthevectorofbusvoltageangles, V isthevectormagnitudesofbusvoltages, P G and Q G ,respectively,arethevectorsoftherealandreactivepowerofthegenerators,and P D and Q D arethevectorsoftherealandreactivepowerofloadbuses,respectively. Ascanbeseenfrom(2.46),reactivepowerconstraintsandbusvoltagelimitshavebothbeen takenintoaccountintheproposedoptimizationframework.Thisisacrucialadvantageofthe proposedoptimizationframeworkwhencomparedtosomeothertraditionalmodelsavailablein theliterature,inwhichthesevitalconstraintsareentirelyignored. 2.10.3RealandReactivePowerConstraints Therealpowerlimitsofgeneratingunitsareconstantlyboundedbytheavailablemaximumand minimumpower.Theworkpresentedhereassumesthattheoutputpowerofthegeneratingunits isadjustedinasmoothandinstantaneousmanner.Further,thisworkboundsthereactivepowerof eachgeneratingunitwithitsupperandlowerlimits.Itisappropriatetopointouthere,however, thatthecapabilitycurveofeachgeneratingunitcanbelinearizedaroundtheoperatingpointand usedtodeterminethereactivepowerconstraints.However,withseveralandvariousgenerating units,thisapproachseemsnotonlyawkwardtoimplementbutalsoimpractical.Inthiswork,the 43 realandreactivepowerconstraintsofthegeneratingunitsareexpressedas, P min G P G P max G Q min G Q G Q max G (2.47) Ascanbeobviouslyseenfrom(2.47),reactivepowerconstraintshavebeenincludedinthepro- posedoptimizationframework. 2.10.4VoltageConstraints Voltageconstraintsareprimarilydependentonthereactivepowerinjectiontothegeneratorbus. Conventionally,voltagelimitsareusuallykeptwithinarangeof 5%.Forpractical reasons,thisworkboundstheupperandlowervoltagelimitsto 5%sothattheupperandlower voltagelimitsaresetequalto1 : 05p.u.and0 : 95p.u.,respectively.Therefore,atanybus k ,the followinginequalityvoltageconstraintholds, j V min k j V k j V max k j (2.48) 2.10.5LineCapacityConstraints Linecapacitiesofdistributionfeedersareveryimportantconstraintsthatshouldbeconsideredin solvinganyoptimizationproblem.Inparticular,linecapacityconstraintsareusedtokeepvarious systemcomponentsoperatewithintheirratingandnominalvalues.Thisworkproposesapiecewise linearmodeltocalculatetheseinequalityconstraints.Toformulatethecurrentcapacityconstraints, letusstartbycalculatingthecurrentwingoutfrombus k towardsbus m ofthetransmission 44 circuitshowninFig.2.1.Thatis, b I km = b V k b V m b Z km (2.49) Inrectangularcoordinates, b I km canbeexpressedas, b I km = ( V k cos d k V m cos d m ) b Z km + i ( V k sin d k V m sin d m ) b Z km (2.50) Letusmaketheapproximationthatvoltageangledifferenceissmallandlineresistanceisless thanitsreactance.Thisassumptionwillnotpreventlineresistancefrombeingincludedinline constraints,ratheritwouldhelpinmakinglinearization.Thus,(2.50)canbewrittenas, j I km jˇ r h ( d k d m ) 2 +( V k V m ) 2 i = j Z km j (2.51) ˇ q I p km 2 + I q km 2 (2.52) Therealcomponentofthelinecurrentisexpressedas, I p km = ( d k d m ) j Z km j (2.53) Theimaginarycomponentofthelinecurrentisexpressedas, I q km = ( V k V m ) j Z km j (2.54) Here I p km and I q km aretherealandimaginarycomponentsofthelinecurrent,respectively.As canbeobviouslyseenfrom(2.51)and(2.52)thelinecurrentcanbeexpressedbytwocomponents, however,themagnitudeofthiscurrentisstillnonlinear. 45 Now,letusconsiderthecircleofradius I max km formedbythelocusoftherealandimaginary componentsexpressedby(2.53)and(2.54).Thetwocomponentsofthelinecurrentcanbeap- proximatedbynumberoflinearsegments.Thedegreeofapproximationcanabsolutelybe improvedtoanydesiredlevelbyincreasingthenumberoflinesegments. Figure2.7:Piecewiseapproximationoflinecapacityconstraints Fig. ?? showsthelocusoftherealandreactivecomponentsofthelinecurrent I p km and I q km , respectively.LetusnowdrawourattentiontothestraightlinesegmentshowninFig.2.8.The straightlineequationjoinsthetwopoints ( 0 ; x ) and ( y ; 0 ) canbesimplyexpressedas, Figure2.8:Piecewiseapproximationoflinecapacityconstraints I q km = c 1 I p km + c 2 (2.55) FromFig.2.8,itisobviousthatthediameter I max km isperpendiculartothestraightlineconnects 46 ( 0 ; y ) and ( x ; 0 ) .Therefore,(2.55)canbeexpressedas, I q km = 1 tan q l + c 2 (2.56) Also, c 2 = 1 tan q l ( I max km cos q l + I 0 km ) (2.57) where I 0 km = I max km sin q l tan q l (2.58) Therefore,theintendedstraightlineequationcanbeformulatedas, I q km = 1 tan q l I p km + I max km ( cos q l + sin q l tan q l ) (2.59) Letusmultiplybothsidesof(2.55)bytan q l andrearrange.Thecurrentcarryingcapacityofany feederinthedistributionsystemcanbeformulatedas, I max km = I p kml cos q l + I q kml sin q l (2.60) wherel=1,2,...,m Thecurrentlimitconstraintsintheproposedoptimizationframeworkcanbeexpressedas, bA 0 d + bA 00 V I max f bA 0 d bA 00 V I max r (2.61) 47 where b isadiagonalmatrixofthedistributionfeedersadmittances N f N f , V isthevectorof busvoltagemagnitudes ( N b 1 ) , d isthevectorofnodevoltageangles ( N b 1 ) , A istheelement- nodeincidencematrix N f N b and, A 0 = Acos q l A 00 = Asin q l (2.62) Itisimportanttopointoutherethatpowerwsareconventionallyusedtospecifythecurrent capacityconstraints.Nevertheless,indistributionsystemsthecurrentlimitsareconstantlyrepre- sentedintermsofAmperes.Thisclearlyouruseofcurrentinsteadofpower.Technically, thelinecurrentlimitsareoftendenotedascurrentcarryingcapacitiesofdistributionfeeders. 2.10.6TheProposedOptimizationFramework Theentireoptimizationframeworkispresentedinthissection.SupposeFisthefunctiontobe minimized.Mathematically,theminimizationproblemcanbeposedas, C = min N b å i = 1 F i ! (2.63) Subjectto 1. RealandReactivePowerInjections B 0 d GV + P G = P D G 0 d + BV + Q G = Q D (2.64) 48 2. RealandReactivePowerConstraints P min G P G P max G Q min G Q G Q max G (2.65) 3. FeederCapacityConstraints bA 0 d + bA 00 V I max f bA 0 d bA 00 V I max r (2.66) 4. VoltageBoundConstraints j V min k j V k j V max k j (2.67) 5. AngleConstraints d isunrestricted (2.68) where N b isnumberofbuses,and N f isnumberofdistributionfeeders B 0 , B , G 0 and G areas giveninChapter2,Section2.3withdimensionsof ( N b N b ) , d isthevectorofnodevoltage angles ( N b 1 ) , V isthevectorofbusvoltagemagnitudes ( N b 1 ) , P G and Q G arethevectorsof realandreactivepowergeneration ( N b 1 ) , P C and Q C arethevectorsofrealandreactiveload curtailments ( N b 1 ) , P D and Q D arethevectorsofbusrealandreactiveloads ( N b 1 ) , P max G and Q max G arethevectorsofmaximumavailablerealpowergeneration ( N b 1 ) , P min G and Q min G are thevectorsofminimumavailablerealpowergeneration ( N b 1 ) , I max f and I max r arethevectorsof forwardandreversewcapacitiesofdistributionlines N f 1 , V max and V min arethevectors ofmaximumandminimumallowablevoltages ( N b 1 ) , b isadiagonalmatrixofthedistribution feederadmittances N f N f ,and A isthenode-branchincidencematrix N f N b . 49 2.10.7AdditionalNetworkPerformanceConstraints Wewillutilizetheproposedoptimizationframeworktohandledifferentapplicationsinmodern distributionsystems.Thus,wewillconsiderdifferentandnumerouspracticalconstraintsforeach problem.Forinstance,iftheproblemaimsattheoptimallocationsandsizesofdistributed generationunits,aswillbeseenlaterinChapter4,otherconstraints,whichconsidersthestep sizesofthedistributedgenerationunitswillbeaddedtothesolutionframework.However,ifthe problemaimsatmaximizingthereliabilityofthedistributionsystemthroughtheminimization oftotalloadcurtailmentoftheconsumers,whichistheaimofChapter5,additionalnetwork constraintssuchastheloadcurtailmentswillbeconsideredinthepowerbalanceinjectionsandthe constraints.Moreover,otherconstraints,whicharedevelopedbasedontheoreticalgraphwillbe includedintheoptimizationframework.Tosummarize,wewillintroduceappropriateadditional networkconstraintsaccordingtotheapplicationwearedealingwithandtheproblemonhand. 2.11Summary Inthepartofthischapterwehavedevelopedafastandeffectivemodelforthepowerw problem.Wehavedevelopedthismodelbasedonlinearizationofthefullsetofconventional powerwequations,andthereforeitincludesvoltagemagnitudesolutionsandreactivepower ws,unliketraditionallinearizedpowerwmodels.Thetechniqueproposedinthischapteris non-iterative,direct,andinvolvesnoconvergenceissues.Further,theproposedpowerwmodel isequallyappropriateforuseatbothdistributionandtransmissionlevelsandcanbeextremely usefulwheneverfast,robust,andrepetitivepowerwsolutionsarerequired.Wehavetested theproposedLACPFmodelonnumerousbalanced,unbalanced,andweakly-mesheddistribution systemsandfoundtoperformwithspeedandaccuracyappropriateforrepetitivesolutions.We 50 havereportedseveraltestcasesanddiscussedtheresultsinthepartofthischapter. Inthesecondpartofthischapterwehaveintroducedanoptimizationframeworkbasedon theproposedpowerwmodelandlinearprogramming.Linearprogrammingbasedmethods aremorecompact,xible,reliable,andfasterthantheirnonlinearcounterparts.Theproposed optimizationframeworkdevelopedinthischapterisveryrobustandcanthereforebeusedwhen- everrepetitiveandpromptsolutionsarerequired.Theoptimizationframeworkdevelopedinthis chapterhasthefollowingmainfeatures: 1. WehavedevelopedthisframeworkbasedonanovellinearizedACpowerwmodel,in whichthecouplingbetweenactivepowerandvoltagemagnitudeaswellasthecoupling betweenreactivepowerandvoltageangleismaintained. 2. Theproposedlinearizedmodelcanhandlepowerwsofawiderangeofdistributionsys- temsincluding,balanced,weakly-meshed,andunbalanceddistributionsystems.Themod- incaseofunbalanceddistributionnetworksarestraightforwardandlargelyliein certainelementsinthebusadmittancematrix;thustheadvantagesobtainedwithbalanced operationarepreserved. 3. Wehavetakenintoconsiderationbothvoltagelimitsandreactivepowerconstraints,which havebeentotallyneglectedinthetraditionallinearizedmethodsavailableintheliterature suchastheDCPFmodel,forinstance. 4. Wehaveintroducedapiecewiselinearmodeltohandlethecurrentcarryingcapacitiesofdis- tributionfeeders.Theproposedlinearmodelallowsustoapproximatethecurrentcapacity constraintsandincludethemintheoptimizationframework.Akeyfeatureoftheproposed piecewisemodelisthatwecouldimprovethedegreeofapproximationtothedesireddesired 51 levelbyincreasingthenumberoflinesegmentsused,withoutsubstantialaffectonthemain routineandcomputationalspeed. 52 Chapter3 DistributedGenerationSizingand Placement Thischapterintroducesananalyticalmethodforplacementandsizingofdistributedgeneration unitsondistributionsystems.Theobjectiveoftheanalyticalmethodpresentedinthischapter istominimizethedistributionsystemlosses.Analyticalmethodsarereliable,computationally efandaresuitableforoff-linedistributionsystemplanningstudies.Moreprominently,ana- lyticalmethodscouldleadtoanoptimalornear-optimalglobalsolution.Inthischapter,westartby determiningthepenetrationlevelofthedistributedgenerationunits.Then,wedevelopapriority listbasedonlosssensitivityfactorstodeterminetheoptimallocationsofthecandidatedistributed generationunits.Weperformsensitivityanalysisbasedontherealpowerinjectionofthedis- tributedgenerationunitstoestimatetheoptimalsizeandpowerfactorofthecandidatedistributed generationunits.Inthischapter,wehavedealtwithvarioustypesofdistributedgeneratorsand havealsoproposedviablesolutionstoreducetotalsystemlosses.Wevalidatetheeffectivenessof theproposedmethodbyapplyingiton33busand69busdistributionsystems,whichareexten- sivelyusedasexamplesinsolvingtheproblemofplacementandsizingofdistributedgenerators. Weperformexhaustivepowerwroutinestoverifythesizesobtainedbytheanalyticalmethod. Wevalidatetheoptimalsizesandlocationsobtainedbytheanalyticalmethodbycomparingthem withsomeotheranalyticalmethodsavailableintheliterature.Wealsoreportontheeffectofdis- 53 tributedgenerationunitsontheoverallvoltageofthedistributionsystem.Thetestresults showthattotallossescanbereducedandvoltagecanbetremendouslyimprovedbythe properplacementandsizingofdistributedgenerationunits.More,thetestresultsre- vealthattheproposedanalyticalmethodcouldleadtoanoptimalornear-optimalglobalsolution, andisveryappropriatetouseforseveraldistributionsystemsplanningstudies. 3.1DistributedGenerationPlacementandSizingProblem Distributionsystemshavebeenoperatedinaverticalandcentralizedmannerformanyyears, forbestcontrolandcoordinationoftheirprotectivedevices.Inaddition,distributionsystems arecharacterizedbyhighR/Xbranchratioswithradialorweakly-meshedtopologicalstructure [1,4,35,36].Infact,theradialtopologicalstructuremakesdistributionsystemsthemostextensive partintheentirepowersystem.Thepoorvoltageregulationandthehighlineresistancebothplay aroleinincreasingtotalpowerlossesofdistributionsystems.Inthiscontext,minimiza- tionofpowerlossesofdistributionsystemsisconstantlyachievedbyfeederbased techniques[37,38,39,40,41,42].However,distributedgenerators(DG)[43]havebeenrecently proposedintheliteraturetominimizedistributionsystempowerlosses.Thepotentialof distributedgeneratorsinstallationondistributionnetworksincludetotalsystemlossesreduction, voltageimprovement,peakloadshaving,andreliabilityenhancement[43,44,45].Given thesetremendousadvantages,distributedgeneratorscanplayvitalroleinreducinglossesandim- provingvoltageandtherebyincreasingthereliabilityandsecurityofdistributionsystems, iftheyareproperlylocated,sized,andtheirpenetrationlevelisalso 54 3.2OverviewofExistingWork Theproblemofplacementandsizingofdistributiongenerationisessentiallyanonlinearcomplex mathematicaloptimizationproblem.Multiplesolutions,withvariousscenarios,areconstantly soughtwhilehandlingtheproblemofplacementandsizingofdistributedgenerators.Agreatva- rietyofsolutiontechniquesareproposedintheliteraturetohandletheproblemofplacementand sizingofdistributedgeneratorsondistributionsystems.Thesesolutiontechniquescanbebroadly aspopulationbasedoptimizationmethodsorheuristicsandanalyticalbasedtechniques. Populationbasedoptimizationmethodsmayincludegeneticalgorithms[46,47],bee colonyalgorithm[48],tabusearch[49],particleswarmoptimization[50],andevolutionarypro- gramming[51].Populationbasedoptimizationmethodsarewidelyadaptedinbothoperational andplanningstudiesandhavegivensatisfactoryresultsovertheyears. Overthepastyears,therehasbeengreatinterestinusinganalyticalapproachestohandlethe placementandsizingproblemofdistributedgenerators[52,53,54,55].Acommonobjective functionusedintheseapproachesisdistributionsystemlossesreductionandvoltageim- provement.Thevastmajorityoftheanalyticalmethodsavailableintheliteratureweredeveloped basedontheexactlossformuladevelopedbyElgerd[6].Theexactlossformulaisanequation relatingvoltagemagnitudeandvoltageangleatabuswiththeactivepowerandreactivepower injectionstothatbusinahighlynonlinearfashion.Oneofthemajordrawbacksofusingtheexact lossformulaistheprocessofupdatingthepowerlossconstants,whichareverynonlinearcom- plexfunctions.Asamatteroffact,analyticalapproachesarelesscomplicatedthansomeofthe heuristictechniquesmentionedabove.However,exhaustivepowerwsarestillbeingperformed inthesolutionprocedures.Thenumberofpowerwsperformedindistributedgeneratorsplace- mentandsizingbasedonanalyticalapproaches,forinstance,couldpossiblyattain( n s 1)fora 55 radialsystemwith n s buses.Usingnonlinearbasedmethods,ononehand,giveshighcalculation precisionbutrequiresaquiteextensivecomputationalburdenandstorage.Ontheotherhand,with thedistinctivepropertiesofdistributionsystemssuchashighR/Xbranchratios,thereisagood chancethatthepowerwsolutionmightfailtoconverge.Nevertheless,themethodproposed inthischapterdoesnotencounterconvergenceproblemsandallowsustogetfastandreliable powerwestimations,whicharehighlysoughtinnumerousapplicationsthatrequirerepetitive andreliablesolutionssuchasoptimaldistributedgenerationunitsplacementandsizingproblem. 3.3DetailedSolutionProcedures Thissectionpresentstheobjectivefunctionandconstraints.Itdevelopsaprioritylistbasedon losssensitivityfactorstodeterminethecandidatebusesforDGplacement.Further,DGinjection basedsensitivityanalysisisperformedtodeterminetheoptimalsizesoftheDGunits.Amethod toselectthecorrespondingoptimalpowerfactorisalsoprovidedinthissection. 3.3.1ObjectiveFunctionandConstraints Theobjectivefunctionoftheproblemaimsatminimizingtherealpowerlossandimprovingthe voltageatallsystembuses.Mathematically,theproblemcanbeposedas, TotalLoss = min N b å i = 1 j I 2 k j R k ! (3.1) Subjectto 56 1. RealandReactivePowerInjections B 0 d GV + P G = P D G 0 d + BV + Q G = Q D (3.2) 2. FeederCapacityConstraints bA 0 d + bA 00 V I max f bA 0 d bA 00 V I max r (3.3) 3. VoltageBoundConstraints j V min k j V k j V max k j (3.4) 4. DistributedGeneratorsSizes S min DG S DG S max DG (3.5) where N b isnumberofbuses, I k isthecurrentwingoutofbranch k , R k istheresistanceof branch k . S min DG and S max DG representtheavailablerealandreactivepowercapacitiesofthedistributed generationunits.OtherabbreviationsareasearlierinChapter2,Section2.10. 3.3.2ofPenetrationLevel Inthiswork,wethepenetrationlevelofthedistributedgenerationas, PenetrationLevel = S DG S TD 100% (3.6) 57 where S DG and S TD aretheoutputpowerofthedistributedgenerationunitandthetotalsystem demand,respectively. 3.3.3SelectionofOptimalLocation:Aprioritylist Inthiswork,theactivepowerlosssensitivityfactor l p hasbeenandusedtodetermine theoptimallocationsforthedistributedgenerationunitsfortotalpowerlossreduction.Toestimate thissensitivityfactor,letusconsiderthesimpleradialdistributionfeedershowninFig.3.1.From Fig.3.1,thelinepowerlossescanbecalculatedas, P loss = ( P 2 Lk ; eff + Q 2 Lk ; eff ) j V 2 k j R k (3.7) where P Lk ; eff and Q Lk ; eff aretheeffectiverealandreactivepowerloadsbeyondbus k . Wetheactivepowerlosssensitivityfactor l p as[56], l p = ¶ P loss ¶ P Lk ; eff = 2 P Lk ; eff R k j V 2 k j (3.8) Theactivepowerlosssensitivityfactoriscalculatedusing(3.8)forallbuses,andthenthecalcu- latedvaluesarearrangedinaprioritylistwithadescendingordersothatbuseswithhigh l p are consideredforallocatingtheavailabledistributedgenerationunits. 3.3.4SelectionofOptimalSize Todeterminetheoptimalsizeofthecandidatedistributedgenerationunit,letusstartbyconsid- eringthedistributionfeederdepictedinFig.3.1.AsshowninFig.3.1,adistributedgeneration unitofsize P DG and Q DG isarbitrarilyplacedatbus k ofthesystem.Beforetheinstallationof 58 thisdistributedgenerationunit,thepowerlossesinthelinesection k ; k 1isestimatedasin(3.7). Thatis, P L = ( P 2 Lk ; eff + Q 2 Lk ; eff ) j V 2 k j R k (3.9) However,aftertheDGisinstalledatbus k ,powerlossesinthelinesection k ; k 1canbeestimated as, P + L = " ( P DG P Lk ; eff ) 2 j V 2 k j + ( Q DG Q Lk ; eff ) 2 j V 2 k j # R k (3.10) Alternatively,(3.10)canberepresentedas, P + L = " P 2 DG 2 P DG P Lk ; eff + P 2 Lk ; eff j V 2 k j + Q 2 DG 2 Q DG Q Lk ; eff + Q 2 Lk ; eff j V 2 k j # R k (3.11) Consequently,thedifferenceinpowerlossesbeforeandaftertheinstallationoftheDGunitatbus k canbeestimatedas, D P L = P + L P L (3.12) Alternatively,(3.12)canbewrittenas, D P L = " P 2 DG + Q 2 DG 2 P DG P Lk ; eff 2 Q DG Q Lk ; eff j V 2 k j # R k (3.13) Therefore,forthetotalrealpowerlossestobeminimuminthefeedersection( k 1 ; k ),the derivativeof(3.12)withrespecttotheactivepowerinjectedbythedistributedgenerationunitat thisparticularbusshouldbedriventozero.Thatis, ¶ D P L ¶ P DG = 0 (3.14) 59 Figure3.1:Asimpleradialdistributionfeeder Hence,afterperformingthepartialderivativesof(3.13)andrearranging,theoptimalsizeofthe DGunitcanbewrittenas, S DG = q P 2 DG + Q 2 DG (3.15) TheoptimalDGsizeinWattisgivenbelowas, P DG = P Lk ; eff + a Q Lk ; eff 1 + a 2 (3.16) TheoptimalDGsizeinVarisgivenas, Q DG = P Lk ; eff + a Q Lk ; eff a + b (3.17) where a and b arerespectivelyas, a = tan q = Q DG P DG (3.18) b = cot q = P DG Q DG (3.19) where q isthepowerfactorangleofthecandidateDGunit. From(3.15)Œ(3.17)theoptimalDGsizesareobtained.Conventionally,theDGsizeisrepre- 60 sentedinVAandthishasbeengivenby(3.15).Asaruleofthumb,itwasdeterminedthatthe totallossesofthesystemareminimumwhenthesizeofthedistributedgenerationunitmatches theeffectiveloadconnectedtothatbus.Infact,thisconcurswithourderivationsin(3.16)and (3.17).Forinstance,iftheselecteddistributedgenerationunitwasaphotovoltaicthatsuppliesreal poweronlyi.e.thepowerfactorofthedistributedgenerationunitisunity,accordingto(3.16)the optimalsizeofthedistributedgenerationunitinWatt( P DG )willbeequaltothatofthetotaleffec- tiveloadconnectedtothatbus.TheoptimalsizeofthedistributedgenerationunitinVar( Q DG ) willbeequaltozerointhiscase.Ontheotherhand,iftheselecteddistributedgenerationunitwas asynchronouscondenserthatregulatesthebusvoltagebyinjectingreactivepoweronly,according to(3.17),theoptimalsizeofthisdistributedgenerationunitinVar( Q DG )willbeequaltothatof thetotaleffectiveloadconnectedtothatbus.However,theoptimalsizeofthisdistributedgener- ationunitinWatt( P DG )willbeequaltozerointhiscase.Thesameprincipleappliesforvarious distributedgenerationunitsascanbeseenfrom(3.15)Œ(3.17). 3.3.5SelectionofOptimalPowerFactor Ascanbeclearlyseenfortheformulationgivenby(3.15)Œ(3.17),thepowerfactorofthedis- tributedgenerationunitplaysaroleindeterminingitsoptimalsize.However,esti- matingtheoptimalpowerfactorthatcontributestominimizingthetotalsystemlossesisnotan easytask.Forinstance,distributedgenerationunitsshouldoperateatpracticalpowerfactorsto maintaintheupperandlowervoltageconstraints;andtherebycontributeinareductionofthetotal realpowerlosses.However,themajorityoftheDGunitsusedatdistributionsystemlevelare notdispatchedandnotwell-controlledsincetheirdispatchcouldpossiblyleadtocertainopera- tionalproblemsintheprotectionsystemofthedistributionnetwork.Moreimportantly,according totheIEEEregulationsandstandards,inparticularstandard1547Œ2003[5]forinterconnecting 61 distributedresourceswithelectricpowersystems,distributedgeneratorsarenotrecommendedto regulatebusvoltagesattheinstallationnode.Thestrategythatiswidelyusedtokeepbusvoltages withinthepermissiblerangeisthroughcapacitorbanks.Nonetheless,evenwhencapacitorbanks arepresented,violationsofvoltagelimitshappencommonlyindistributionsystems. Thevastmajorityofthemethodsusedtoestimatetheoptimalpowerfactorofdistributedgen- eratorsaredevelopedbasedonexpertsystems.Forinstance,theoptimalpowerfactorofthedis- tributedgeneratorshasbeenassumedtobeequaltothatoftheloadconnectedtothesamebuswith reverseoperatingstrategyin[48].Further,theoptimalpowerfactorofthedistributedgenerators wasassumedtobeequaltothatofthetotaldownstreamloadatwhichthedistributedgeneration unitisconnectedin[54].Ithasbeendemonstratedin[54]thatthelossesofthedistributionsystem areminimumwhenthepowerfactorofthedistributedgeneratorisselectedtobeequaltothatofthe totalloaddownstream.Infact,thisassumptionseemstobemorerealistic,andhence,itisadapted inthiswork.Dependingontheircapabilitiesofinjectingactiveandreactivepower,distributed generatorscanbeas[54], 1. DGTypeI:ThisDGiscapableofinjectingactiveandreactivepowersuchassynchronous generators. 2. DGTypeII:ThisDGiscapableofinjectingactivepowerbutabsorbsreactivepowerfrom thesystemsuchasinductiongenerators. 3. DGTypeIII:ThisDGiscapableofinjectingactivepoweronlysuchasphotovoltaic. 4. DGTypeIV:ThisDGiscapableofinjectingreactivepoweronlysuchassynchronouscon- densers. Thesignofthepowerfactorhasalsobeenacquiredfrom[54].Thatis, a willbeconsidered positiveiftheDGinjectsreactivepower,asinTypeIandTypeII,forinstance.Thus,ifaDGunit 62 isgoingtobeinstalledatbus k ofthedistributionfeedershowninFig.1,thepowerfactorofthis unitiscalculatedas, PF DG = P Lk ; eff q P 2 Lk ; eff + Q 2 Lk ; eff (3.20) 3.4DemonstrationandDiscussion Inordertodemonstratetheeffectivenessoftheproposedanalyticalmethod,wetestitondifferent distributionsystems.Weprovidetheresultsofthe33busdistributionsystemandthe69bus distributionsysteminthissection. 3.533BusSystemOptimalLocationsandSizes TheoptimalresultsofplacementandsizingoftheDGunitsonthe33busdistributionsystem usingtheproposedanalyticalsolutionarepresentedinthissection.Theseresultsarealsov byperformingexhaustivepowerwroutines. 3.5.1AnalyticalMethod WeutilizetheLACPFmodel,whichisdevelopedearlierinChapter2todeterminetheoptimal location,size,andpowerfactorofthecandidateDGunitsforthe33busdistributionsystem.We performsensitivityanalysisbasedonrealpowerlosses;andtherebywethelosssensitivity factors l p ,ateachbususing(3.8),todeterminetheoptimallocationsofthecandidateDGunits. TheresultsofthelosssensitivityfactorsaredepictedinFig.3.2.AscanbeseenfromFig.3.2,the optimallocationfortheDGunitisfoundtobeatbus6.Thepowerfactorofthedownstream loadatbus6isalsoobtainedfromthepowerwsolutionandwasfoundtobeequalto0.8907 63 lagging.Eventhoughthisworkisconcernedwithactivepowerlossesminimization,thefourDG typesmentionedearlierinSection4.3.5havebeensizedaccordingto(3.15)Œ(3.17)andinstalled atbus6,oneatatime,toselecttheDGtypewhichhasmuchimpactontotallossreduction. Asexpected,DGTypeIandDGTypeIIcontributedtolossesreductionnotonly becausebothoftheminjectactivepowertothesystem,butalsotheDGlocationswereselectedin accordancetoactivepowerlossesreduction.ThenumberofDGunitsusedforlossreductionwill belimitedtotwodistributedgenerationunitsofTypeIandTypeIIandtheresultsofthesetypes willbepresentedanddiscussedinthesubsequentsection. Figure3.2:Activepowerlosssensitivityfactorsofthe33bussystem TheresultsobtainedbytheproposedmethodarepresentedinTable3.1andTable3.2,respec- tively.Weperformexhaustivepowerwstoverifythesizesobtainedbytheanalyticalmethod. AllsimulationsarecarriedoutusingMatlab7.9onanIntelcore,4GB,800MHzcomputer. Table3.2showstheresultsobtainedbybothmethodsforthecaseofDGTypeI.ThisDG typeinjectsbothrealandreactivepowertothebusatwhichitisconnected.WhenasingleDG isselectedtobeinstalledatbus6,theoptimalsizeobtainedbytheproposedanalyticalapproach is2486kVA.Thereductioninpowerlossesduetothisinjectionwas62.08%.Asindicatedprevi- 64 ously,weareinterestedinlocatingandsizingoftwoDGunitsinthepresentedwork,thusanother DGofthesametypeisconnectedatbus28,whichisfromsensitivityanalysisasan optimallocationforthesecondDGunitascanbeseenfromFig.3.2.TheDGsizeforthiscase isgivenbelowinTable3.1.Thetotalreductionintotalpowerlossesachievedbyinstallingthe secondDGis69.32%.ItisworthmentioningherethatoncewedeterminethesizeoftheDGunit fromtheanalyticalsolution,wechangetheobtainedDGsizeinastepwisefashionandthenselect thesizethatyieldstototalminimumpowerlosses.Itisalsoimportanttopointoutherethatwhile thesizeoftheDGsizewealmostgotthenearestintegersizeobtainedbytheanalytical methodaswehavechosensmallstepsizetochangethesizeoftheDGunits.Duringtheentire searchprocess,ifanyoftheconstraintslistedinSection4.3.1isviolated,weconsiderthenext availableDGlocationforfurtherevaluation. Table3.2showstheresultsobtainedbytheanalyticalmethodandtheexhaustivepowerw routinesforthecaseofTypeIIDG,whichinjectsrealpoweronly.Asexpected,thereductionin powerlosseswhenthisDGtypeisusedislessthanthatobtainedbydeployingTypeIDG.Using theproposedanalyticalmethod,thetotallossreductionwhenasingleDGisutilizedis43.79%, whilethatobtainedbyusingtwoDGunitsis45.46%. Thegeneralsolutionproceduresoftheanalyticalmethodaresummarizedas[57], 1. EnterthenumberandthetypeofcandidateDGunits. 2. ObtaintheinitialsystemlossusingtheLACPF. 3. Usethelosssensitivityfactorsandsetupaprioritylisttodeterminetheoptimallocationsof thecandidateDGunits. 4. Use(3.15)Œ(3.19)andtheLACPFtodeterminetheoptimalDGsizeandthecorresponding powerfactor. 65 5. ChangetheoptimalDGsizeobtainedinstep(4)insmallstepwisefashiontoestimatethe suitablesize.Choosethesizethatgivesminimumlosses. 6. Checkconstraints.IfanyoftheconstraintsgiveninSection4.3.1isviolated,considerthe nextavailableDGlocation. 3.5.2ExhaustivePowerFlow ToverifythesizesoftheDGunitsobtainedbytheproposedanalyticalmethod,wecomparethem withthoseobtainedbyperformingexhaustivepowerwsolutions.Inthiscase,weidentify thepenetrationleveloftheDGunitsusing(3.6).Weassumeapenetrationlevelof15%Œ60%. Accordingly,theallowableminimumandmaximumDGoutputpowerforthe33bussystemvaries between556kVAand2620kVA.SinceDGsizesaregivenindiscretevalues,wechangetheDG sizesinsmallstepsandinstallthecandidateDGunitsatbus6,whichisfromsensitivity analysisasanoptimallocationforDGinstallation.Weperformpowerwsolutionseveraltimes anddeterminetheDGsizethatgivesminimumlosses.Thissizeisfoundtobe2485kVA,which givesareductionintotallossesof62.03%.ThesameproceduresarerepeatedforthesecondDG (TypeII),whichisconnectedatbus28accordingtothesensitivityanalysis.TheDGsizeobtained forthiscaseisalsogivenbelowinTable3.1.Further,thetotallossreductionachievedinthiscase is69.17%. AscanalsobeseenfromTable3.1andTable3.2,theresultsobtainedbytheproposedanalytical methodcorrespondcloselywiththoseobtainedbyperformingexhaustivepowerwsolutions.It isworthnotingherethatwhiletheresultsobtainedfrombothmethodswerealmostsimilar,the proposedanalyticalmethodisshowntobeconsiderablyfasterthantheexhaustivepowerwasit requiresfewpowerwsolutionstodeterminetheoptimalsize. 66 Table3.1:ResultsofThe33BusDistributionSystemŒDGTypeI Method AnalyticalMethod ExhaustivePowerFlow Item SingleDG TwoDG SingleDG TwoDG OptimalLocation 6 28 6 28 S DG (KVA) 2487 1204 2485 1200 PowerLoss(KW) 69.01 55.85 69.00 55.73 %LossReduction 62.08 69.32 62.03 69.17 Table3.2:ResultsofThe33BusDistributionSystemŒDGTypeII Method AnalyticalMethod ExhaustivePowerFlow Item SingleDG TwoDG SingleDG TwoDG OptimalLocation 6 28 6 28 P DG (kW) 2215 785 2214 785 PowerLoss(kW) 102 99 102 97 %LossReduction 43.79 45.46 43.8 45.4 3.669BusSystemOptimalLocationsandSizes TheoptimalresultsofplacementandsizingoftheDGunitsonthe69busdistributionsystem usingtheproposedanalyticalsolutionarepresentedinthissection.Theseresultsarealsov byperformingexhaustivepowerwroutines. 3.6.1AnalyticalMethod WeusetheproposedpowerwmodeltodeterminetheoptimallocationsandsizesofDGunits forthe69busdistributionsystem.Then,weperformsensitivityanalysis;andfromwhichwe thelosssensitivityfactors l p ,ateachbususing(3.8),todeterminetheoptimallocationsofthe candidateDGunits.TheresultsofthelosssensitivityfactorsaredepictedinFig.3.3.Theoptimal locationfortheDGunitisfoundtobeatbus61.Thepowerfactorofthedownstreamloadat bus61isalsoobtainedfromthepowerwsolutionandwasfoundtobeequalto0.82lagging. Table3.3showstheresultsobtainedbytheproposedanalyticalmethodandthoseobtainedby performingexhaustivepowerwsolutionsforthecaseofDGTypeI.ThisDGtypeinjectsboth 67 realandreactivepowertothebusatwhichitisconnected.AscanbeseenfromTable3.3,the DGsizesobtainedbytheproposedanalyticalmethodcorrespondcloselytothoseobtainedbythe exhaustivepowerwroutines,whilerequiringlowercomputationaleffort. Figure3.3:Activepowerlosssensitivityfactorsofthe69bussystem AscanbeseenfromTable3.3,whenasingleDGisconcerned,thepercentagelossreduction achievedbytheanalyticalmethodis86.86%.Thepercentagereductioninlossesbyperforming exhaustivepowerwsisalmostthesame.TheDGsizesforthiscasestudyarealsogiveninTable 3.3.Now,asecondDGoftypeIandofsize945kVAisconnectedatbus49,whichis fromthesensitivityanalysisasanoptimallocationforDGinstallation.Thetotallossreduction obtainedbytheanalyticalapproachis88.10%. Table3.4showstheresultsobtainedbybothmethodsforthecaseofTypeIIDG,whichinjects realpoweronly.Asexpected,totallossreductionwhensuchDGtypehasbeenusedislessthan thatobtainedbydeployingTypeIDG.ThetotallossreductionwhenasingleDGisutilizedis 57.06%.Ontheotherhand,whentwoDGunitsareinstalledatthebuses,thetotalloss reductionobtainedbytheproposedanalyticalmethodis87.91%. 68 Table3.3:ResultsofThe69BusDistributionSystemŒDGTypeI Method AnalyticalMethod ExhaustivePowerFlow Item SingleDG TwoDG SingleDG TwoDG OptimalLocation 61 49 61 49 S DG (kW) 1918 945 1916 945 PowerLoss(kW) 23.92 21.63 23.90 21.63 %LossReduction 86.86 88.10 86.87 88.09 3.6.2ExhaustivePowerFlow InordertoverifythesizesoftheDGunitsobtainedbytheproposedanalyticalexpressions,we havecomparedthemwiththoseobtainedbyperformingexhaustivepowerwsolutions.Inthis case,weidentifythepenetrationleveloftheDGunitsusing(3.6).Weassumeapenetration levelof15%Œ60%forthiscasestudy.Accordingly,theallowableminimumandmaximumDG outputpowerforthe69bussystemisvaryingbetween700kVAand2795kVA.SinceDGsizes aregivenindiscretevalues,wechangetheDGsizesinsmallstepsandinstalledthecandidate DGunitsatbus61,whichisasanoptimallocationforDGinstallationfromsensitivity analysis.Weperformpowerwsolutionseveraltimesanddeterminethesizethatgivesminimum loss.Thissizeisfoundtobe1918kVA,whichgivesareductionintotallossesof86.86%.The sameproceduresarerepeatedforthesecondDG(TypeII),whichisconnectedatbus49according tothesensitivityanalysis.Theoptimallocationsandsizesofthedistributedgenerationunitsfor thiscasestudyaregiveninTable3.4.Itisworthtonoteherethatifanyoftheconstraintslisted inSection3.3.1isviolatedduringthesearchfortheoptimalDGsize,thisDGunitisomittedfrom thelistandthefollowingDGunitlocationisconsideredforfurtherevaluation. 69 Table3.4:ResultsofThe69BusDistributionSystemŒDGTypeII Method AnalyticalMethod ExhaustivePowerFlow Item SingleDG TwoDG SingleDG TwoDG OptimalLocation 61 49 61 49 P DG (kW) 1580 760 1580 760 PowerLoss(kW) 78 22 78 22 %LossReduction 57.06 87.91 57.06 87.91 3.7VoltagePrImprovement Amongstthepotentialadvantagesofinstallingdistributedgenerationunitsondistributionsys- temsistheirabilitytoimprovetheoverallvoltageofthesystem.Inthissection,westudy theofthedistributedgenerationunitsonthesystem.Table3.5depictedbelowshows theminimumbusvoltagesobtainedbeforeandaftertheinstallationofthedistributedgeneration units,forthe33bussystemandthe69bussystem,respectively.Inthecontextofvoltage improvement,theminimumbusvoltageofthe33bussystembeforetheintroductionofthedis- tributedgenerationunitswas0 : 9113p.u.andhadoccurredatbus18.Themaximumbusvoltage wasequal1.0p.u.,whichisthesubstationbusvoltage.Theminimumbusvoltageofthe33bus distributionsystemhasbeenboostedto0.9509p.u.whenasingleDGofTypeIisinstalled.How- ever,theminimumbusvoltagehasbeenboostedto0.9693p.u.whentwoDGunitsofTypeIare installed.Theseminimumbusvoltageshadoccurredatbus18forbothcases.Infact,this thattheimprovementintheoverallvoltageofthe33bussystemisabout4.35%whena singledistributedgenerationunitisinstalledandalmost6.36%,whentwodistributedgeneration unitshavebeeninstalledonthe33busdistributionsystem. Forthe69busdistributionsystem,theminimumbusvoltagewas0.9170p.u.andhadoccurred atbus65whilethemaximumbusvoltagewasequalto1.0p.u.,whichisthesubstationbusvolt- age.Aftertheinstallationofasingledistributedgenerationunit,andtwoDGunitsofTypeI,the 70 minimumbusvoltageofthe69busdistributionsystembecomes0.9728p.u.Thismeanstheim- provementintheoverallvoltageofthe69bussystemisalmost6.09%.Ontheotherhand, themaximumbusvoltageaftertheinstallationoftheseDGunitsbecomes1.011p.u.andoccurs atbus27.Table3.5showsdetailedresultsforthesecasestudies. Table3.5:MinimumBusVoltagesBeforeandAfterDGInstallation Item DGTypeI DGTypeII TestSystem SingleDG TwoDG SingleDG TwoDG 33BusDistributionSystem 0.9509 0.9693 0.9411 0.9516 VoltageImprovement(%) 4.35 6.36 3.27 4.42 69BusDistributionSystem 0.9728 0.9729 0.9695 0.9697 VoltageImprovement(%) 6.09 6.10 5.73 5.75 3.8ComparativeStudy Totesttheeffectivenessoftheproposedanalyticalmethod,wecomparetheperformanceofthe proposedanalyticalmethodwithsomeotheranalyticalmethodspresentedintheliterature.We considerthe33bussystemforthiscomparativestudy.Thecomparisonresultsareshownbelow inTable3.6.AscanbeclearlyseenfromTable3.6,theproposedanalyticalmethodhasledtoa globaloptimalsolution,whilerequiringlowercomputationaleffort. Table3.6:PerformanceComparisonofTheProposedMethod Methodand Method Method ProposedAnalyticalMethod Item Ref.[53] Ref.[54] BasedFACPF BasedLACPF OptimalLocation 6 6 6 6 OptimalSize(kW) 2490 2601 2236 2215 %LossReduction 47.33 47.39 46.30 43.79 Iterations/Powerw 5 5 5 Œ NET(ms)/Powerw 290.52 287.50 295.87 51.23 71 3.9Summary Thischapterhasproposedananalyticalmethodforplacementandsizingofdistributedgenerators onpowerdistributionsystemswithanobjectiveoflossreduction.Analyticalapproachescould leadtooptimalornear-optimalglobalsolutionandareveryappropriateforoptimaldistribution systemplanningstudies.Wehavedeterminedthepenetrationlevelofthecandidatedistributed generationunits.Then,wehaveusedthepowerwmodel,whichwedevelopedearlierinChapter 2,toperformsensitivityanalysistodeterminetheoptimallocation,size,andpowerfactorof thecandidatedistributedgeneratorunit.Themainfeaturesoftheworkpresentedinthischapter includethefollowing, 1. Theuseofsensitivityanalysistoselecttheappropriatelocationsofthecandidatedistributed generationunits;andtherebyreducingthesearchspaceandcomputationalburden. 2. Thisworkisdifferentfromthatpresentedintheliteratureinthesensethatitproposesdirect andunderstandableexpressionstoestimatetheoptimal,ornear-optimal,sizeofthedis- tributedgenerationunits.Theworkpresentedhereindoesnotutilizethecomplicatedexact lossformula. 3. Thisworkconsidersvarioustypesofdistributedgeneratorsandincorporatestheoptimal powerfactorinthesizingproblem.Thevastmajorityoftheworkpresentedintheliterature assumesunitypowerfactorforthecandidatedistributedgeneratorunits. 4. Theproposedanalyticalmethodcouldleadtooptimalornear-optimalglobalsolution,while requiringlowercomputationaleffort.Themethodreportedhereissuitableforseveraldistri- butionsystemplanningstudies. 72 Chapter4 OptimalEconomicPowerDispatchof ActiveDistributionSystems Thischapterdescribesamethodforsolvingtheoptimaleconomicpowerdispatchproblemofactive distributionsystems.Inthischapter,wedeveloppiecewiselinearmodelstodealwiththecost curvesofgeneratingunitsandtotalpowerlosses.Weconsiderdistributedgeneratorsofconstant powertypeandunitypowerfactorsimilartothosewhichhavebeendealtwithinChapter3, however,weassumevariouspenetrationlevels.Toshowtheeffectivenessoftheproposedmethod, wecompareitsperformancewithtwoothermethods,whicharewidelyusedintheliteratureto solvetheoptimaleconomicpowerdispatchproblem.Namely,thesemethodsarethefullACpower wbasedeconomicdispatchandtheDCpowerwbasedeconomicdispatch.Weshowthrough simulationsthattheproposedmethodoutperformstheDCpowerwbasedeconomicdispatch methodandtheresultsobtainedbytheproposedmethodcorrespondcloselywiththoseobtained bynonlinearmeans,whilerequiringlowercomputationaleffort.Further,Weholdacomparison withsomeothermethods,whichareavailableintheliterature,tovalidatetheconsistencyofthe proposedmethodintheoptimalornear-optimalglobalsolution.Theresultsshowthatthe proposedmethodcouldleadtooptimalornear-optimalglobalsolution,andisappropriatetouse forseveralpowersystemoperationalandplanningstudies. 73 4.1OptimalEconomicPowerDispatchProblem Forabouthalfacentury,theoptimaleconomicpowerdispatchproblemhasbeenthesubjectof intensiveresearchallaroundtheworld.Itisacrucialtoolintheoperationandplanningstages ofanypowersystem.Theoptimalpowerwproblemwasformulatedandintroducedasa networkconstrainedeconomicdispatchproblem(ED)in1962byCarpentier[58]andwas laterasanoptimalpowerwproblembyDommelandTinney[59].Thetaskofperforming optimalpowerwaimsessentiallyatdeterminingtheoptimalsettingsofthegivenpowernetwork byoptimizingcertainfunctions,e.g.minimizationoftotalgenerationcost,totalpowerlosses,or totalemission,whilesatisfyingasetofoperatingandtechnicalconstraints.Numerouscontrol variablesthatincludegenerators'realpower,transformers'tapchangersandphaseshifters,static anddynamicVARcompensatorsarealsoinvolvedinoptimizingtheobjectivefunction. 4.2OverviewofExistingWork Thesolutionoftheoptimaleconomicpowerdispatchproblemisveryimportantforseveralpower systemoperationalandplanningstudies.ThesolutiontheoptimalEDproblemwasmainlyde- votedtotransmissionsystems.Nevertheless,duetotheliberalizationofenergymarketsandthe introductionofnon-utilitygeneration,thesolutionofEDproblematdistributionsystemlevelhas increasinglybecomeofgreatinterestinrecentyears.Agreatvarietyofsolutiontechniqueshave beenproposedintheliteraturetosolvetheoptimaleconomicdispatchproblemsinceitsincep- tion.Momohetal.presentedareviewofoptimalpowerwmethodsselectedin[60,61].The challengestooptimalpowerwsarereportedbyMomohetal.in[62].Examplesofsomecom- monmethods,whichareusedintheliteraturetosolvetheoptimalpowerwprobleminclude, nonlinearprogramming[63,64,65,66],quadraticprogramming[67],Newton'smethod[68,69], 74 heuristicandswarmintelligence[70,71,72,73],interiorpointmethod[74,75],andlinearpro- gramming(LP)basedmethods[76,77,78,79].Amongstthesemethods,LPbasedmethodsare recognizedasviableandpromisingtoolsinsolvingtheconstrainedeconomicdispatchproblem. TheearliestversionsoftheLPbasedeconomicpowerdispatchweredevelopedbasedonthepure DCpowerwmodel[8,30,80].Lately,theconstraintsintheLPmodelswerelinearizedand treatedbynonlinearpowerwsinordertoimposethemprecisely.Theinherentfeaturesofthe LPbasedeconomicpowerdispatchmethodsincludereliabilityofoptimization,xibilityofthe solution,rapidconvergencecharacteristics,andfastexecutiontime[29]. Severalapproacheshavebeenrecentlyproposedintheliteraturetosolvetheeconomicdispatch probleminthepresenceofdistributedgenerators.Aheuristicmethodbasedanoptimaldynamic powerdispatchwithrenewableenergysourcesispresentedin[71].Theproblemofeconomic powerdispatchinthepresenceofintermittentwindenergysourcesiscarriedoutusinglinearpro- grammingin[81].Theproblemofoptimaleconomicpowerdispatchinthepresenceofintermittent windandsolargeneratorsiscarriedoutusingsequentialquadraticprogrammingin[82].Sequen- tialquadraticprogrammingisabletohandletheconvexcostcurvesofgeneratingunitsef sincethelatterisinherentlyaquadraticfunction.Nonetheless,quadraticprogrammingrequires considerablecomputationaleffort.More,thestandardsimpleformofthequadratic programmingisnotoftenusedbecauseconvergenceisnotalwaysguaranteed.Theoptimalpower dispatchinthepresenceofdistributedgeneratorshasalsobeenproposedin[83,84,85]. 4.3DevelopmentofModelsandMethods Thissectionreviewssomemodelingaspectstohandletheoptimaleconomicpowerdispatchprob- lem.Itdevelopspiecewiselinearmodelsforthecostfunctionsofgenerators.Italsodevelops 75 linearmodelfortransmissionlinelossessothattheycanbeincludedintheproposedmethod. 4.3.1CostFunctionsofGenerators Thecostfunctionofaconventionalgenerator i isusuallyexpressedasasecond-orderquadratic polynomialas, F i ( P Gi )= a i + b i P Gi + g i P 2 Gi (4.1) Theworkadaptedthelinearizationmodeldevelopedin[30].Thus,thenonlinearcostfunction canbeapproximatedbyaseriesofstraight-linesegmentsasdepictedinFig.4.1.Thethreelinear segmentsshowninFig.4.1arerepresentedas P i 1 , P i 2 ,and P i 3 .Further,theslopeofeachlinear segmentisgivenas m 1 , m 2 ,and m 3 ,respectively. Thecostfunctioncanthereforeberepresentedas[30], F i ( P G i )= F i ( P G min i )+ m 1 P i 1 + m 2 P i 2 + m 3 P i 3 (4.2) with, 0 P ih P + ih , h = 1 ; 2 ; 3 (4.3) Thecostfunctionisobtainedbyincludingallthelinearsegmentsinthe P ih .Thatis, P G i = P G min i + P i 1 + P i 2 + P i 3 (4.4) 76 Figure4.1:Linearizationofcostfunction 4.3.2InclusionofLosses TheproposedLACPFmodel,whichisdevelopedearlierinSec.2.3,islosslessmodelascan beseenfrom(2.38).Thissectionpresentsapiecewiselinearmodeltoincludethelossesinthe proposedframework.LetusconsiderthesimplecircuitshowninFig.4.2.Therealandreactive powerwingoutfrombus k towardsbus m canbecalculatedas[8], P km = V 2 k g km V k V m g km cos d km V k V m b km cos d km (4.5) Q km = V 2 k b km + V k V m b km cos d km V k V m g km sin d km (4.6) Therealandreactivepowerwingoutfrombus m towardsbus k canbeobtainedinasimilar manner.Thatis, P mk = V 2 m g km V k V m g km cos d km V k V m b km cos d km (4.7) Q mk = V 2 m b km + V k V m b km cos d km V k V m g km sin d km (4.8) 77 Therefore,therealandreactivepowerlossescanbeobtainedas, P L = g km ( V 2 k + V 2 m 2 V k V m cos d km ) (4.9) Q L = b km ( V 2 k + V 2 m 2 V k V m cos d km ) (4.10) Letususetheapproximations V k ˇ 1 : 0p.u., V m ˇ 1 : 0p.u.,andcos d km ˇ 1 d 2 km 2 ,theapproximate realandreactivepowerlossesformulascanbewrittenas, P 4 L = g km d 2 km (4.11) Q 4 L = b km d 2 km (4.12) Eqs.(4.11)and(4.12)indicatethattheactiveandreactivepowerlossesarequadraticformsof d km . Therefore,inordertoincludethelossesintheproposedmethod,theangles d k and d m areobtained fromthebasepowerwsolutionandtheinitiallossesarecomputed.Theselossesarethenadded tosystemloadsandapowerwsolutionisexecutedtoobtaintheexactlosses. Figure4.2:Transmissionline p model 78 4.3.3TheProposedFramework Inthissection,wepresenttheentireoptimizationframeworkforsolvingtheoptimalEDprob- lem.Theobjectivefunctionusedhereistominimisethetotalgenerationcost.Theminimisation problemisformulatedas, TotalCost = min N g å i = 1 F i ( P Gi ) ! (4.13) Subjectto 1. RealandReactivePowerInjections B 0 d GV + P G = P D G 0 d + BV + Q G = Q D (4.14) 2. RealandReactivePowerConstraints P min G P G P max G Q min G Q G Q max G (4.15) 3. LineCapacityConstraints bA 0 d + bA 00 V I max f bA 0 d bA 00 V I max r (4.16) 4. VoltageBoundConstraints j V min jj V jj V max j (4.17) 5. AngleConstraints d isunrestricted (4.18) 79 4.4DCPFBasedOptimalEconomicDispatch TodemonstratetheeffectivenessoftheproposedoptimalEDmethod,wecompareitsperformance withsomeothermethods,whichhavebeenextensivelyusedintheliterature.Wecomparethe proposedoptimalEDmethodwiththetheDCPFbasedeconomicdispatchandthefullACpower wbasedeconomicdispatchmethod.WeimplementtheDCPFbasedeconomicdispatchmethod anduseMatpowersimulatorpackage[87]totheresultsofthefullACpowerwbased economicdispatchmethod.Forthesakeofcompleteness,thesetwomethodsarereviewed inthesubsequentsection. ThetraditionalDCpowerwbasedeconomicdispatchcanbeformulatedas, TotalCost = min N g å i = 1 F i ( P Gi ) ! (4.19) Subjectto 1. RealPowerInjections B 0 d + P G = P D (4.20) 2. RealPowerConstraints P min G P G P max G (4.21) 3. FeederCapacityConstraints b ‹ A d I max f b ‹ A d I max r (4.22) 4. AngleConstraints d isunrestricted (4.23) 80 4.5FACPFBasedOptimalEconomicDispatch ThefullACpowerwbasedeconomicdispatchmethodusedinthisworkisformulatedas, TotalCost = min N g å i = 1 F i ( P Gi ) ! (4.24) Subjectto 1. NonlinearPowerFlowInjections g ( h ; z )= 0 (4.25) 2. RealandReactivePowerConstraints P min G P G P max G Q min G Q G Q max G (4.26) 3. FeederCapacityConstraints I f I max f I r I max r (4.27) 4. VoltageBoundConstraints j V min jj V jj V max j (4.28) 5. AngleConstraints d isunrestricted (4.29) where g ( h ; z ) representsthenonlinearpowerwequations.Otherabbreviationsareasin Section4.4 81 4.6DemonstrationandDiscussion Theeffectivenessoftheproposedoptimaleconomicpowerdispatchmethodisdemonstratedona 30bussystem[86].The30bussystemconsistsof30buses,41transmission lines,and2shuntcapacitors,whichareplacedatbus5andbus24,respectively.Thissystemhas sixgenerators,whichareplacedatbus1,bus2,bus13,bus22,bus23,andbus27,respectively. Thetotalrealandreactivepowerpeakloadsonthe30systemare189 : 2MWand107 : 2 MVar,respectively.WeusefourmethodstosolvetheoptimalEDproblem.Thesemethodsare theFACPFbasedED(FACPFbasedED),LACPFbasedED(LACPFbasedED)includinglosses, LACPFbasedED(LACPFbasedED)losslessmodel,andtheDCPFbasedED(DCPFbasedED). TheresultsoftheFACPFbasedEDareobtainedusingMatpowersimulationpackage[87].Three casescenariosareperformedanddiscussedindetailinthesubsequentsection. 4.6.1CaseScenarioI Thepurposeofthiscasescenarioistovalidatethemethodsandmodelsdevelopedinthischapter. Therefore,nodistributedgeneratorsareconsideredinthiscasescenario.Therealpowerand voltagemagnitudesofthegeneratorsofthe30bussystem,whichareobtainedusing theproposedmethodare,respectively,presentedinTable4.1.AscanbeseenfromTable4.1, theresultsoftheoptimaleconomicpowerdispatchofthe6generators,whichareobtainedby theLACPFbasedED(lossymodel),correspondcloselywiththoseobtainedbytheFACPFbased ED.Furthermore,voltagemagnitudesobtainedbytheLACPFbasedED(lossymodel)arealso correspondcloselywiththoseobtainedbytheFACPFbasedED.Inthiscontext,thepercentageof voltageerrorsobtainedbytheproposedmethodatbuses1,2,13,22,23,and27,respectively,are 0.00%,0.88%,4.39%,0.88%,0.58%,and3.91%,whicharelessthan5%atallbuses. 82 Thetotalgenerationcost,totalgeneration,andtotallossesofthe30bussystemare alsogiveninTable4.1.AscanbeseenfromTable4.1,theresultsobtainedbytheproposed LACPFbasedED(lossymodel)correspondcloselywiththoseobtainedbytheFACPFbasedED. Theresultsofthetotalgenerationcost,totalgeneration,andtotallosses,whichareobtainedby bothmethodswerealmostsimilar.Theresultspresentedinthiscasestudyshowthattheproposed methodcanhandletheoptimalEDproblemmoreefthansomeothertraditionallinearized methods,whicharedevelopedbasedontheDCPFmodel. Table4.1:EDResultsofthe30BusSystemŒCaseScenarioI Power(MW) FACPF LACPFBasedED DCPF Voltage(p.u.) BasedED ModelI ModelII BasedED P G 1 39.92 45 45 45 P G 2 53.03 48.958 47.20 47.884 P G 13 14.96 15 15 15 P G 22 22.46 21 21 21 P G 23 16.06 16 16 16 P G 27 45.58 45 45 44.316 V 1 1.050 1.050 1.050 1.0 V 2 1.028 1.037 1.036 1.0 V 13 1.048 1.002 1.003 1.0 V 22 1.019 1.010 1.011 1.0 V 23 1.026 1.020 1.022 1.0 V 27 1.050 1.009 1.008 1.0 T.Cost($/hr) 576.92 579.72 572.95 572.95 T.Gen.(MW) 192.06 190.96 189.20 189.20 T.Losses(MW) 2.86 2.84 0 0 4.6.2CaseScenarioII Inthiscasescenario,wesolvetheoptimalEDprobleminthepresenceofdistributedgenerators. AshasbeendiscussedinChapter3,theadvantagesofdistributedgenerationsourcesincludetotal lossreduction,voltageimprovement,peakloadshaving,andreliabilityandsecurityen- hancement.ThedistributedgeneratorsusedinthisworkaremodeledasconstantPQnodeswith 83 Figure4.3:30BusDistributionSystem negativeinjections.ThisassumptionconcurswiththeIEEEstandard1547-2003[5]forinter- connectingdistributedresourceswithelectricpowersystems,whichemphasizesthatdistributed generatorsarenotrecommendedtoregulatebusvoltages;andtherebytheyshouldbemodeledas PQnodesnotPVnodes.Further,distributedgeneratorsoperatedatunitypowerfactorhavebeen assumedinthiswork. Whendealingwithdistributedgenerators,itiscustomarytotheirpenetrationlevelinthe 84 distributionsystem.Therefore,wethepenetrationlevelofthedistributedgenerators( h p )in thesystemas[81], h p = 1 P D N r å m = 1 P m (4.30) where P D representsthetotalsystemdemand, P m representstheoutputpowerofthedistributed generators,and N r representsthenumberofthedistributedgenerators. Itisnoteworthytohighlightherethatweonlyconsiderthepenetrationlevelofthedistributed generationunitsinthiswork.Theproblemofdeterminingtheoptimallocationsandsizesof distributedgenerationunitsisnotgermanetothepresentedwork.InaccordancetotheElectric PowerResearchPowerInstitute(EPRI)andotherstudies,thepenetrationlevelofthedistributed generatorsindistributionsystemswasinthevicinityof20%in2010.Inthiswork,weconsidertwo penetrationlevelsofdistributedgenerators.Inthecase,whichisdiscussedhere,weassumea penetrationlevelof10%,whichisequivalenttoapproximately18 : 92MW.Weplacesixdistributed generationunitsatbuses4,7,15,21,24,and31,respectively.Theselocationsareselectedbasedon thepeakloaddensities.Eachdistributedgenerationunitisratedatapproximately3.15MW. Therealpowerandvoltagemagnitudesofthegeneratorsofthe30bussystemare, respectively,presentedinTable4.2.AscanbeseenfromTable4.2,theresultsoftheoptimal economicpowerdispatchofthegenerators,whichareobtainedbytheLACPFbasedED(lossy model),correspondcloselywiththoseobtainedbytheFACPFbasedED.Furthermore,voltage magnitudesobtainedbytheLACPFbasedED(lossymodel)arealsocorrespondcloselywith thoseobtainedbytheFACPFbasedED.Thiscasestudyhasshownthattheproposedmethodis appropriatetohandletheoptimalEDproblemofactivedistributionsystemsefandcanbe usedforoptimaldistributionsystemplanningstudies. Thetotalgenerationcostandtotalgenerationofthe30bussystemaregiveninTable 85 4.2.AscanbeseenfromTable4.2,theresultsobtainedbytheproposedLACPFbasedED(lossy model)correspondcloselywiththoseobtainedbytheFACPFbasedED.Theresultsofthetotal generationcost,totalgeneration,andtotallosses,whichareobtainedbybothmethodswerealmost similar.Theresultspresentedinthiscasestudyshowthattheproposedmethodcanhandlethe optimalEDproblemmoreefthansomeothertraditionallinearizedmethods,whichare developedbasedontheDCPFmodel. Table4.2:EDResultsofthe30BusSystemŒCaseScenarioII Power(MW) FACPF LACPFBasedED DCPF Voltage(p.u.) BasedED ModelI ModelII BasedED P G 1 39.30 45 45 45 P G 2 51.64 40 40 40 P G 13 13.46 15 15 15 P G 22 21.56 21 21 21 P G 23 13.56 16 16 16 P G 27 34.33 34.625 33.3 33.3 V 1 0.968 1.050 1.050 1.0 V 2 0.965 1.036 1.036 1.0 V 13 1.055 0.997 0.997 1.0 V 22 1.008 1.011 1.011 1.0 V 23 1.016 1.023 1.023 1.0 V 27 1.049 1.009 1.008 1.0 T.Cost($/hr) 505.13 505.1568 500.468 500.468 T.Gen.(MW) 172.84 171.526 170.3 170.3 4.6.3CaseScenarioIII Inthiscasescenariothepenetrationlevelofthewindturbinegeneratorsisassumedtobe20%.This penetrationlevelisapproximatelyequivalentto37 : 84MW.SimilartoCaseScenarionII,weplace sixdistributedgenerationunitsatbuses4,7,15,21,24,and31,respectively.Theselocationshave beenselectedbasedonthepeakloaddensities.Inthiscasescenario,eachdistributedgeneration unitisassumedtoberatedatapproximately6.13MW. 86 Therealpowerandvoltagemagnitudesofthegeneratorsofthe30bussystemare, respectively,presentedinTable4.3.AscanbeseenfromTable4.3,allbusvoltagesarewithinthe permissiblerange.Theresultsoftheoptimaleconomicpowerdispatchofthe6generators,which areobtainedbytheLACPFbasedED(lossymodel),correspondcloselywiththoseobtainedby theFACPFbasedED.Furthermore,voltagemagnitudesobtainedbytheLACPFbasedED(lossy model)arealsocorrespondcloselywiththoseobtainedbytheFACPFbasedED.Thiscasestudy hasshownthattheproposedmethodisappropriatetohandletheoptimaleDproblemofactive distributionsystemsefandcanbeusedforoptimaldistributionsystemplanningstudies. Thetotalgenerationcostandtotallossesofthe30bussystemaregiveninTable 4.3.AscanbeseenfromTable4.3,theresultsobtainedbytheproposedLACPFbasedED(lossy model)correspondcloselywiththoseobtainedbytheFACPFbasedED.Theresultsofthetotal generationcostandtotallosses,whichareobtainedbybothmethodswerealmostsimilar.The resultspresentedinthiscasestudyshowthattheproposedmethodcanhandletheoptimalED problemmoreefthansomeothertraditionallinearizedmethods,whicharedeveloped basedontheDCPFmodel. 4.7PerformanceAnalysisoftheProposedMethod Inthissection,wecomparetheperformanceoftheproposedoptimalEDmethodwithsomeother methodsavailableintheliterature.Firstly,wecomparethemaximumerrorinthecomputedtotal costandtotalgenerationcostobtainedbytheproposedmethodwiththoseobtainedbythefull ACbasedEDandtheDCPFbasedEDmethods.Then,wedemonstratetheeffectivenessand theconsistencyoftheproposedmethodintheoptimalornear-optimalglobalsolutionby comparingitsperformancewithtwoothermethodsavailableintheliterature. 87 Table4.3:EDResultsofthe30BusSystemŒCaseScenarioIII Power(MW) FACPF LACPFBasedED DCPF Voltage(p.u.) BasedED ModelI ModelII BasedED P G 1 34.61 45 45 45 P G 2 47.33 40 40 40 P G 13 10.58 10.356 10 10 P G 22 20.40 21 21 21 P G 23 11.01 12 12 12 P G 27 33.78 24 23.4 23.4 V 1 0.964 1.050 1.050 1.0 V 2 0.961 1.036 1.036 1.0 V 13 1.063 1.000 1.000 1.0 V 22 1.014 1.010 1.010 1.0 V 23 1.021 1.021 1.021 1.0 V 27 1.045 1.014 1.014 1.0 T.Cost($/hr) 451.55 433.2579 429.8162 429.8162 T.Gen.(MW) 157.72 152.356 151.4 151.4 4.7.1ComparisonoftheMaximumError Theerrorsinthetotalgenerationandtotalgenerationcostobtainedbythestudiedcasescenar- iosarepresentedinthissectionandtheresultsaresummarizedinTable4.4.Ascanbenoticed fromTable4.4,thepercentageerrorsinthetotalgenerationandtotalgenerationcostobtainedby theproposedLACPFbasedEDmethodaremuchlowerthanthoseobtainedbytheDCPFbased EDmethod.ThismakestheproposedmethodverysuitablenotonlytohandletheoptimalED problemofactivedistributionsystems,butalsotosolveseveralotheroptimaldistributionsystem operationalandplanningproblems,inwhichrepetitiveoptimalsolutionsarehighlysought. 4.7.2ComparisonoftheOptimalSolution TotesttheconsistencyoftheproposedLACPFbasedEDmethodintheoptimalornear optimalglobalsolution,ithasbeencomparedwithsomeothermethods,whicharepresentedin theliterature.Weuseamethod,whichisdevelopedbasedonhybridparticleswamoptimization 88 Table4.4:ComparisonofTotalCostandTotalGenerationErrorsObtainedbytheProposed LACPFbasedEDmethodandtheDCPFbasedEDmethod System = Model DCPFbasedED LACPFbasedED Case CostError Gen.Error CostError Gen.Error Scenarios (%) (%) (%) (%) CaseScenarioI 4.45 1.61 0.04 0.19 CaseScenarioII 1.46 0.92 0.55 0.25 CaseScenarioIII 1.94 1.28 0.08 0.14 [72],andalsoamethod,whichisdevelopedbasedonSQPusingMatpowersimulationpackage [87].Weconsiderthe30bussysteminthiscomparativestudy.Thecomparisonresults areshowninTable4.5.AscanbeseenfromTable4.5,theproposedLACPFbasedEDmethod couldleadtooptimalornear-optimalglobalsolutionsimilartothatsolution,whichisobtained usingnonlinearmeans. Table4.5:ComparisonofMethodPerformance Comparison Method Method Proposed Item Ref.[72] Ref.[87] Method TotalCost($/hr) 575.41 578.29 579.72 TotalGen.(MW) 191.85 192.01 190.96 TotalLosses(MW) 2.86 2.81 2.84 4.8Summary Thischapterhaspresentedamethodforsolvingtheoptimaleconomicpowerdispatchproblemof activedistributionsystems.Themethodpresentedinthispaperisdevelopedbasedonlinearpro- grammingandalinearizednetworkmodelinwhichvoltagemagnitudesandreactivepowerws havebothbeenaccountedfor,unliketraditionallinearizedpowerwmethods.Somefeaturesof theworkpresentedinthischapterincludethefollowing, 1. Wehavedevelopedpiecewiselinearmodelstohandlethethermalcapacitiesoftransmission 89 lines,loads,costcurvesofgeneratingunits,andlinelosses.Weshowtheeffectivenessof theproposedmethodona30bussystem. 2. Wehaveconsideredthreecasescenariosinthischapter.Inthecasescenario,wehave assumednodistributedgenerationunitsarepresentinthesame.Theobjectiveofthiscase scenarioistovalidatetheaccuracyoftheproposedmethodsandmodelsinhandlingthe optimalEDproblem. 3. WehaveconsidereddifferentpenetrationlevelsinCaseScenarioIIandCaseScenarioIII. Wehaveassumedtwopenetrationlevelsofdistributedgenerators.InCaseScenarioII, weassumeapenetrationlevelof10%,whichisapproximatelyequivalentto18 : 92MW.We haveplacedsixdistributedgenerationunitsatbuses4,7,15,21,24,and31,respectively.These locationsareselectedbasedonthepeakloaddensities.Eachdistributedgenerationunitis approximatelyratedat3.15MW. 4. Wehavealsoassumedapenetrationlevelofdistributiongenerationunitsof20%CaseSce- narioIII.Thispenetrationlevelisapproximatelyequivalentto18 : 92MW.SimilartoCase ScenarioII,wehaveplacedsixdistributedgenerationunitsatbuses4,7,15,21,24,and31, respectively.Theselocationsareselectedbasedonthepeakloaddensities.Inthiscase scenario,eachdistributedgenerationunitisassumedtoberatedatapproximately6.13MW. 5. WehavecomparedtheperformanceoftheproposedoptimalEDmethodwithsomeother methodsavailableintheliterature.Further,inorderTotesttheconsistencyoftheproposed LACPFbasedEDmethodintheoptimalornearoptimalglobalsolution,wehave compareditsperformancewithsomeothermethodspresentedintheliterature.Weusea method,whichisdevelopedbasedonhybridparticleswamoptimization[72],andalsoa method,whichisdevelopedbasedonSQPusingMatpowersimulationpackage[87]. 90 6. WeshowthattheproposedLACPFbasedEDmethodcouldleadtooptimalornear-optimal globalsolutionsimilartothatsolution,whichisobtainedusingnonlinearmeans,andcanbe usedtohandleseveraloptimalpowersystemplanningandoperationalstudies. 91 Chapter5 Reliability-constrainedOptimal DistributionSystem Thischapterpresentsamethodtosolvethedistributionsystemproblemwithan objectiveofreliabilityenhancement.Inthepartofthischapter,weintroducethebasicrelia- bilityconcepts,identifythestatespaceoftheproblem,andreportonreliabilitymeasures.Wethen discussprobabilisticreliabilitymodelsandintroduceaprobabilisticreliabilityassessmentmethod basedoneventtreeanalysis.Frompracticalperspective,itisunnecessarytoenumerateallevents tocomputetheexactvaluesofprobabilities;andtherebycalculatethereliabilityindices.There- fore,weimprovethecomputationalperformanceoftheprobabilisticreliabilityassessmentmethod byintroducingacriterionsothattheeffectofthehigher-ordercontingenciesislimitedandthe timeandcomputationaleffortspentinevaluatingthereliabilityindicesaregreatlyreduced. Theobjectivefunctionofthemethodproposedinthischapteristominimizethetotalload curtailment.Wechoosetheexpectedunservedenergy(EUE),whichisalternativelyknownas thelossofenergyexpectation(LOEE),asthereliabilityindexthatneedstobeminimized.We introduceaformulatocalculateEUEreliabilityindexbasedontheexpecteddemandnotsupplied (EDNS)index,whichrepresentsthetotalloadcurtailmentinthiscase.Weutilizetheenergyindex ofunreliability(EIUR)toevaluatetheoverallreliabilityofthegivendistributionsystem. Inthesecondpartofthischapter,weformulatethedistributionsystemprob- 92 lemforreliabilityenhancement.Weproposeanintelligentsearchmethodbasedonparticleswarm optimizationmethod(PSO)toseekallpossiblecombinationsoftheswitchesthatimprovethedis- tributionsystemreliability.Particleswarmoptimizationisameta-heuristicoptimizationmethod inspiredbythesocialbehaviorofofbirdsorschoolsofwhichisintroducedin1995by [88,89].Theadvantagesofusingparticleswarmoptimizationinhandlingthedistributionsystem problemaremanifold.Forinstance,thestatusofsectionalizingandtie-switches indistributionsystemscanbeeasilyrepresentedasbinarynumbersof(0,1).Moreover,particle swarmoptimizationbasedmethodshaveconsiderablyfastconvergencecharacteristicsand,gener- allyspeaking,havefewparameterstotuneupcomparedtosomeothermeta-heuristicapproaches. Wedemonstratetheeffectivenessoftheproposedreliabilityenhancementmethodonthe33bus system,69bussystem,and118buslarge-scaledistributionsystem,whichhavebeendealtwith earlierinChapter2,asthesesystemsareextensivelyusedintheliteratureasexamplesinsolving thedistributionfeederproblem.Weconsidertheeffectthevoltagesecuritylimits innumerouscasescenarios.Thetestresultsshowthatamountoftheannualunservedenergyand customersinterruptionscanbereducedusingtheproposedreliability-basedoptimal distributionsystemmethod. 5.1DistributionSystemProblem InthissectionwediscussthereasonsfordistributionsystemWereviewand discusstheworkpresentedintheliteraturetosolvethisproblemanddiscussthe ofconsideringreliabilityconstraintswhilethedistributionnetwork. 93 5.1.1ReasonsforDistributionFeeder Thevastmajorityofpowerdistributionsystemsarecharacterizedbyradialtopologicalstructure andpoorvoltageregulation.Theradialtopologyisnecessaryinordertofacilitatethecontrol andcoordinationoftheprotectivedevicesusedatthedistributionsystemlevel.However,with thatradialstructure,thefailureofanysinglecomponentbetweentheloadpointandthesource nodewouldcauseserviceinterruptionsandmayresultindisconnectingseveralloadpoints.Dis- tributionsystemsareequippedwithtwotypesofswitches,whicharesectionalizingswitchesand tie-switches.Thesectionalizingswitchesarenormallyclosedandareusedtoconnectvariousdis- tributionlinesegments.Thetie-switches,ontheotherhand,arenormallyopenandcanbeused totransferloadsfromonefeedertoanotherduringabnormalandemergencyconditions.Distri- butionfeederisoneoftheseveraloperationaltasksthatareperformedfrequently ondistributionsystems.Basically,itdenotestotheprocessofchangingthetopologicalstructure ofdistributionnetworksbyalteringtheopen/closestatusofthesectionalizingandtie-switches toachievecertainobjectives.Oftheseobjectives,reliabilityandsecurityenhancement,voltage improvement,peakloadshaving,andlossminimizationareofmostconcern. 5.1.2ReviewofExistingWork Theapproachtosolvethedistributionsystemproblemwasintroducedby MerlinandBack[37]in1975.Theobjectiveoftheworkpresentedin[37]wastosearchfor theminimumlossoperatingspanningtreeTheworkpresentedin[37]startsby closingallswitches(tie-switchesandsectionalizingswitches)sothatthedistributionsystemis convertedtoameshednetwork.Then,basedoncurrentindices,bothsectionalizing andtie-switchesareopeninasubsequentfashioninordertorestoretheoriginalradialtopology 94 ofthedistributionsystem.Sincethen,severalmethodsareproposedintheliteraturetosolvethe distributionfeederproblem.ShirmohammadiandHong[38]proposedaheuristic methodtominimizetheresistivelossesofdistributionfeeders.Inthemethodreportedin[38],an optimalpowerwpatternisusedtosolvetheoptimaldistributionfeederprob- lem.BaranandWu[33]presentedanetworkmethodwithanobjectiveofloss reductionandloadbalancing.In[33],twoapproximatepowerwsolutionsandaloadindex wereproposedtotheminimumlossofthedistributionsystem.Methodsbased onswarmintelligencehavealsobeenrecentlyproposedintheliteraturetohandlethedistribution feederproblem[39,40,90,91,92].Despitesuchagreatvarietyofsolutiontech- niques,thevastmajorityoftheworkpresentedintheliteraturewasdevotedtodistributionsystem lossreductionandvoltageimprovement.Surprisingly,otherobjectivessuchas reliabilityandsecurityenhancementandservicerestorationofin-serviceconsumershavegotless attentionintheliterature,andhavenotthusbeenfullyaddressed. Inrecentyears,therehasbeenanincreasinginterestinimprovingdistributionsystemreliabil- ityusingdistributionsystemTheexpectedenergynotsupplied(EENS),expected demandnotsupplied(EDNS),andtheexpectedoutagecost(ECOST)aresomeexamplesofthe reliabilitymeasuresusedintheliterature.In[93],areliabilityworthenhancementbaseddistribu- tionsystemisproposed.Analyticalandheuristicmethodshavebeenproposedfor reliabilityworthenhancement.Areliabilitycost-worthmodelofthedistributionsystemisbuilt upandfromwhichtheECOSTandtheEENSareobtained.Theinterruptedenergyassessment rate(IEAR),whichisproposedbyGoelandBillinton[94],hasalsobeenusedin[93]torelatethe interruptioncostandtheexpectedenergynotsuppliedforeveryfeasibleAdistribu- tionsystemforreliabilityworthanalysisbasedonsimulatedannealingisproposed in[95].Theworkreportedin[95]usedthesamereliabilityindicesandthesamecustomerdamage 95 functionasin[93].However,severalconclusionsaboutthesystemaredrawn astheECOSTandtheEENSwerebothconsideredasobjectivesintheoptimizationproblem. Reliabilityimprovementofpowerdistributionsystemsusingdistributedgenerationhasalsobeen proposedintheliterature[96,97,98]. Swarmintelligencebasedoptimizationmethodsormeta-heuristicmethodshavebeenrecently proposedintheliteraturetohandlethedistributionsystemproblemforreliability enhancement.Amongstthesemethods,particleswarmoptimizationbasedmethodsarerecognized asviabletoolsinsolvingthedistributionsystemproblemforreliabilityimprove- ment.Chakrabartietal.presentedareliabilitybaseddistributionsystemmethod in[99].Theobjectiveoftheworkpresentedin[99]wastominimizethelossofloadexpectation (LOLE)andthelossofenergyexpectation(LOEE).Inaddition,MonteCarlosimulationandpar- ticleswarmoptimizationhavebothbeenusedin[99].Amanullaelal[100]extendedthework presentedin[99]andproposedacutsetbasedanalyticalmethodtoimprovetheservicereliability ofdistributionsystems.Itisworthmentioningherethatinallthestudies,whicharecarriedoutin [99,100,101],thenumberofsectionalizingswitcheshavebeenrestrictedandprioryassignedin ordertospeedupthecomputationalburdenandreducethesearchspace.Inaddition, cut-setshaveonlybeenconsideredintheworkproposedin[99,100,101]. Elsaiahetal.[102,103,104]recentlyproposedmethodsforsolvingtheoptimaldistribution feederforreliabilityimprovement.In[102,103],afastmethodforreliability improvementofpowerdistributionsystemviafeederisproposed.Theworkpre- sentedin[102,103]isdevelopedbasedonalinearizednetworkmodelintheformofDCpower wandlinearprogrammingmodel,inwhichcurrentcarryingcapacitiesofdistributionfeed- ersandrealpowerconstraintshavebeenconsidered.Theoptimalstatusofsectionalizingand tie-switchesareusinganintelligentbinaryparticleswarmoptimizationbasedsearch 96 method.Theprobabilisticreliabilityassessmentisconductedusingamethodbasedonhighprob- abilityorderapproximation.Severalcasestudiesarecarriedoutin[102,103]onasmall33bus radialdistributionsystem,whichisextensivelyusedasanexampleinsolvingthedistributionsys- temproblem.Theeffectofembeddedgenerationhasalsobeenconsideredinone casescenario.Theresultsshowthattheproposedmethodyieldsatremendousreductioninthe EDNSreliabilityindexandservicesinterruptionsofin-serviceconsumers.Theworkpresentedin [102,103]hasbeenextendedin[104]toimprovethereliabilityofMicrogrids,whichisbasically adistributionsystemorpartthereof.Theprobabilisticreliabilitywasconductedusinga setbasedmethod.Moreover,agraphtheoreticmethodisdevelopedin[104]topreservetheradial topologicalstructureoftheMicrogrid.Thepenetrationofthedistributiongeneratorsisobtained usingahybridoptimizationmodel,whichisdevelopedbasedonHomersoftwarepackage[105]. Themethodistestedonalarge-scaleMicrogridwithrenewableenergyresources,inparticular windturbineinductiongenerators,anddifferentloadscenarioareconsideredin[104]. Theproblemofoptimalswitchplacementinpowerdistributionsystemsisconductedusing trinaryparticleswarmoptimizationin[106].Theuseofantcolonyoptimizationforplacementof sectionalizingswitchesindistributionsystemsispresentedin[107].Theworkpresentedin[107] onlyconsideredthefailureratesofthedistributionfeedersandassumedthattheotherdistribution systemcomponentssuchastransformers,circuitbreakers,etcaretobeperfectlyreliable,which isnotthecaseinrealisticdistributionsystems.Further,theworkpresentedin[107]hasjust consideredtheplacementofasingleswitchonthedistributionsystemandonecasescenariois performedinthisstudy. 97 5.1.3ImportanceofEnhancingDistributionSystemsReliability Themajorityoftoday'sdistributionsystemsarebeingstressedandoperatedatheavyloading conditionsduetotherapidincreaseinelectricitydemandaswellascertaineconomicandenviron- mentalconstraints.Moreprominently,severalofthepresentpowerdistributionsystemsaresubject tonumerouspowerqualityconstraintssuchasthemitigationofvoltagedips,andservice interruptions,whichhavenotbeenoffullconsiderationintheolddays.Inthiscontext,statistics haveshownthatthevastmajorityofserviceoutageshavetakenplaceatdistributionsystemlevel [108,109].AstudyconductedbythePowerSystemOutageTaskForce(PSOTF)[110]afterthe massiveblackoutoftheUSandCanadatookplaceonAugust14th-15th2003,hasindicatedthat stressedandunsecuredsystemscanbeamajorsourceforblackoutsthathappenedinrecentyears andcouldhappeninfuture.Brown[109]hasreportedthatdistributionsystemscontributeforup to90%ofoverallconsumersreliabilityproblems.Theannualcostoftheseserviceinterruptions couldattainbillionofdollarsaccordingtotheElectricPowerResearchInstitute(EPRI)andtheUS DepartmentofEnergy(DOE)[111].Theseconcernscombinedwiththecomplexitiesofthemod- ernpowerdistributionsystemshasmotivatedustoconsiderthereliabilityandsecurityconstraints whilehandlingtheproblemofoptimaldistributionfeeder 5.2DevelopmentofModelsandMethods Thissectionpresentsprobabilisticmodelsforvariousdistributionsystemcomponents.Italso providesaprobabilisticreliabilityassessmentmethodbasedonhigher-ordercontingencyapproxi- mation,thereliabilityindicesutilizedinthiswork,andthecompleteoptimizationframework. 98 5.2.1ProbabilisticReliabilityIndices Reliabilityofdistributionsystemcanbesimplyastheabilityofthedistributionsystemto satisfyitsconsumersloaddemandundercertainoperatingconditions[112].Thisworkusesthe expecteddemandnotsuppliedandtheexpectedunservedenergytoevaluatethereliabilityofthe distributionsystem.Weusethelossofloadprobabilityreliabilityindex(LOLP)tocalculatethe amountofthetotalloadcurtailment.Inaddition,wehaveusedtheenergyindexofunreliabilityto knowhowsecurethesystemis.Wethereliabilityindicesusedinthisworkas[108], 1. ExpectedUnservedEnergy(EUE):Theexpectednumberofmegawatthoursperyearthatthe systemcannotsupplytoconsumers.Alternatively,itisknownastheexpectedenergynot supplied(EENS). 2. LossofLoadprobability(LOLP):Theprobabilitythatthesystemwillnotbeabletosupply theloaddemandsundercertainoperatingscenarios. 3. EnergyIndexofUnreliability(EIUR):Thefractionoftheenergydemandthatthesystemis notabletomeet. 5.2.2TheStateSpace Thestatespaceoftheproblempresentedinthischapterisasthesetofallpossiblecom- binationsofgenerators N g ,distributionfeeders N f ,sectionalizingswitches N ss ,tie-switches N ts , buses N b ,circuitbreakers N cb ,anddistributiontransformers N tr .Consequently,thedimension ofthestatespacecanbeexpressedas F s = N g + N f + N ss + N ts + N b + N cb + N tr . 1 Probabilistic modelsoftheaforementionedcomponentsareaddressedinthesubsequentsection. 1 Frompracticalperspective,theradialtopologicalstructureofthedistributionsystemistakenasanec- essaryconditionduringtherealizationofthepresentedwork.Therefore,thesubstationbusisassumedto beperfectlyreliableforallcasescenariosperformedherein. 99 5.2.3ProbabilisticModelsofComponents Distributionsystemmodelingaimsattranslatingthephysicalnetworkintoareliabilitynetwork basedonseries,parallel,oracombinationofseries/parallelcomponentconnections.Probabilistic modelingtechniquesareusedtorepresentvarioussystemcomponents.Hence,everysinglecom- ponentinthedistributionsystemisassignedaprobabilityofbeingavailable ( P ) orunavailable ( Q ) sothat P + Q = 1.Theavailabilityofanycomponent i inthesystemcanberepresentedas[108], P i = 1 = l i 1 = l i + 1 = m i = m i m i + l i (5.1) where l i isthefailurerateofcomponent i and m i istherepairrateofcomponent i ,respectively. Eq.(5.1)canalternativelyberepresentedas, P i = MTTF i MTTF i + MTTR i (5.2) whereMTTF i andMTTR i aretheMeanTimeToFailureandtheMeanTimeToRepairofcompo- nent i ,respectively. Inthiswork,weassumethateachcomponentinthedistributionsystemcanonlyresideinan up-state(available)ordown-state(unavailable).Therefore,thetwo-stateMarkovianmodelshown inFig.5.1isusedtomodelthetransformers,circuitbreakers,switches,andbuses.However,for distributionsystemfeeders,adiscreteprobabilitydensityfunctionhasbeenconstructedforevery distributionline.Ifadistributionfeederistrippedforcertainsystemstate,thedistributionfeeder isremovedfromthebusadmittancematrixanditscapacityissetequaltozero. 100 Figure5.1:Twostatemodelrepresentation 5.2.4ProbabilisticReliabilityAssessment Sincethetimeandcomputationaleffortspentinevaluatingsystemindicesareofgreatconcernin bothplanningandoperationalstages,theworkreportedinthischapterusesaprobabilisticrelia- bilityassessmentmethodbasedontheeventtreeanalysis[108,115,116].Basically,eventtree analysiscanbeconsideredasabinaryformofadecisiontree,whichisutilizedtoobtaintheprob- abilitiesofthedifferentpossibleoutcomesofthesystemafteraneventtakesplace.Amongstthe advantagesofusingeventtreeanalysisinthereliabilityevaluationofdistributionsystemsisat- tributedtoitsabilitytomodelcomplexsystems,suchasdistributionsystems,inanunderstandable manner[115]. Now,letusconsiderFig.5.2[116],whichdepictsaneventtreewithvarioussuccessand failureprobabilities.Theprobabilityofoccurrenceofeachpathisobtainedbymultiplyingthe probabilitiesofoccurrenceoftheeventsformingthatpath.Thatis[112], Figure5.2:Conceptofeventtreeanalysis 101 SystemReliability = å probabilitiesofallsuccessfulpaths (5.3) TheprobabilityofsuccessoftheeventtreeshownaboveinFig.5.2iscalculatedas, P = 1 ( P A ) [( P B ) ( P C )]+[( P A ) ( P B ) ( P C )] +[( P A ) ( P B ) ( P D )]+[( P B ) ( P C ) ( P D )] [( P B ) ( P D )] [( P A ) ( P B ) ( P C ) ( P D )] (5.4) Ascanbeseenfrom(5.4)theprobabilityofsuccessoftheeventtreeshownaboveinFig.5.2canbe representedwithvariousindividualprobabilities P A , P B , P C ,and P D ,uptoforth-orderprobability. 5.2.5Higher-orderContingenciesBasedEventTreeAnalysis Enumeratingalleventsusing(5.4)tocomputetheexactvaluesofprobabilities;andtherebythe reliabilityindices,cansometimesbeunnecessaryandimpractical.Inpractice,probabilitiesof occurrenceofvariouseventsareapproximateduptocertainorder.Thisworkusestheeventtree analysismethodwithhigher-ordercontingencyapproximationtocalculatethereliabilityindices ofthedistributionsystem.Thebasicconceptbehindsuchanapproximationisthatifthecontin- gencyleveldoesnothavemuchimpactonthesystem'sreliability,wecanignorethiscontingency levelwithoutsubstantialeffectontherequiredprecision[113,114].Now,supposewehavethree events E A , E B ,and E C withindividualprobabilitiesof P A , P B ,and P C ,respectively.Therefore, thecompoundeventprobabilitycanalwaysbeexpressedasapolynomialcomprisingthethree probabilities P A , P B ,and P C .Forinstance,letusconsiderthecompoundprobabilityoftheevent ( E A \ E B ) [ E C ,whichcanbeexpressedas, ( E A \ E B ) [ E C =( P C )+( P A ) ( P B ) ( P A ) ( P B ) ( P C ) (5.5) 102 Ascanbeseenfrom(5.5),thiscompoundprobabilityconsistsofthreeterms: ( P C ) witha orderprobability, [( P A ) ( P B )] withasecondorderprobability,and [( P A ) ( P B ) ( P C )] withathird orderprobability.Ifweassumethattheindividualprobabilities P A , P B ,and P C have,approximately, sameordermagnitude,theorderofmagnitudeofeachproducttermwilldependonhowmany probabilitiesareinvolvedineachproductterm.Itisworthpointingoutherethattheprobabilityof failureofmostpowersystemcomponentsisquitesmall,typically 1%peryear[114].Withthat beingsaid,theeffectofhigher-ordereventssuchassecond-orderandabovecanbelimitedusing certainprobabilityorfrequencycriterion. Therefore,intheexamplegivenabove,iftheindividualprobabilities P A , P B ,and P C arevery small,thentheevent ( E A \ E B ) [ E C canbeapproximatedwith [( P C )+( P A ) ( P B )] orevenwith ( P C ) [114].Duringtherealizationofthepresentedwork,theeffectofanycontingencylevelwith afrequencyofoccurrencelessthan10 10 hasbeenneglected. 5.2.6ReliabilityEvaluationModel Inthissection,wepresentanoptimizationframeworktosolvethedistributionsystem rationproblem.Theobjectiveofthisoptimizationframeworkistominimizethetotalloadcur- tailment.Theconstraintsintheproposedmodelarethespanningtreeconstraints,voltagebounds, realandreactivepowerlimits,generationcapacityconstraints,anddistributionfeedersthermal capacities. Thetotalloadcurtailmentminimizationproblemcanbeposedas, LossofLoad = min N b å i = 1 P Ci ! (5.6) Subjectto: 103 1. RealandReactivePowerInjections B 0 d GV + P G + P C = P D (5.7) G 0 d + BV + Q G + Q C = Q D (5.8) 2. RealandReactivePowerConstraints P min G P G P max G (5.9) Q min G Q G Q max G (5.10) 3. FeederCapacityConstraints bA 0 d + bA 00 V I max f (5.11) bA 0 d bA 00 V I max r (5.12) 4. LoadCurtailmentConstraints 0 P C P D (5.13) 0 Q C Q D (5.14) 5. VoltageBoundConstraints j V min jj V jj V max j (5.15) 6. AngleConstraints d unrestricted (5.16) 104 7. SpanningTreeConstraints j ( l )= 0 (5.17) where P C isthevectorofrealloadcurtailments ( N b 1 ) and Q C isthevectorofreactiveload curtailments ( N b 1 ) .OtherabbreviationsareaspreviouslyinChapter2. Itisworthmentioningherethatallnetworkconstraintsareconsideredin(5.6).Further,(5.7) and(5.8)areaugmentedbygenerators,whichareequivalenttotherequiredloadcurtail- ment.Moreover,inordertoobtainafeasiblesolutionfor(5.6),weassumethatoneofthebus anglesequalszerointheconstraints. 5.2.7ImplementationoftheSpanningTreeConstraints Oneofthepracticalaspectsofthepresentedworkisthatitretainstheradialstructureofthe distributionsystem.Wedevelopthisconditionbasedonatheoreticgraphmethod[117, ? ]and imposeitbyaddingconstraint(5.17)totheoptimizationframework.Theproceduresofformu- latingthisconstraintareexplainedasfollows:supposewehaveagraph G ( V ; E ) with n vertices (nodes) V = f v 1 ; v 2 ; ; v n g and m edges(branches) E = f e 1 ; e 2 ; ; e m g .Wethe ( n m ) vertex-edgeincidencematrix A a ij ofthisgraphas, A = 8 > > > > > < > > > > > : 1 ; ifanedgerunsfromvertex i tovertex j 1 ; ifanedgerunsfromvertex i tovertex j 0 ; ifvertex i isnotconnectedtovertex j (5.18) From(5.18),itisevidentthatthereareexactlytwoonesineverycolumnof A .Consequently,we couldobtainanyrowof A fromtheremainingrows,whichthat A islinearlydependent.If wetakeoutanyrowfrom A ,therankoftheresultantmatrixshouldbe n ,which,inotherwords, 105 meansthattheresultantmatrixislinearlyindependent.Suchmatrixisdenotedasthereduced incidencematrix A r ,andwhosedimensionsof ( n 1 ) m andarankof n 1.Thealgorithmused inthisworkisdevelopedbasedonthefollowingtheorem[117,118]. Theorem: Anysub-matrixin A r withdimensions ( n 1 ) ( n 1 ) ,isnon-singularifandonly ifthe n 1edgescorrespondingtothe n 1columnsofthismatrixconstitutesaspanningtree. Therefore,foreverypossibleweevaluatethedeterminantofthereducedinci- dencematrix,oranysub-matrixinit,sothat,ifdet [ A r ]= 1,thismeansthattheradialtopologyis preserved.Ontheotherhand,ifdet [ A r ]= 0,thismeansthatthesystemhasatleastoneloopand, therefore,theisinfeasible.Weobtainthenumberofloopsinevery j ( l ) ,formEuler'sformula[117]as, NumberofLoops = 1 NumberofVertices + NumberofEdges (5.19) Tobetterunderstandthespanningtreealgorithmdevelopedherein,letusconsiderthegraph showninFig.5.3.Thisgraphconsistsof4verticesand5edges.Theincidencematrixdescribed thisgraphhasonecolumnforeachvertexandonerowforeachedgeofthegraph.Theincidence matrixofthegraphdepictedinFig.5.3canthusberepresentedas, A = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1100 0 110 1010 1001 00 11 3 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (5.20) Letusdrawourattentiontotheloopformsbythevertices V 1 , V 2 ,and V 3 andtheedges E 1 , E 2 , 106 Figure5.3:Agraphwith n = 4verticesand m = 5edges and E 3 ,respectively.Thematrixdescribestheseverticesandedgescanbeexpressedas, A = 2 6 6 6 6 6 4 1100 0 110 1010 3 7 7 7 7 7 5 (5.21) Ascanbeseenfrom(5.21),thesummationofthetworowsyieldstothethirdrow,which indicatesthatthematrixdescribedby(5.21)islinearlydependent.Thenumberofloopsinthe graphshowninFig.5.3canbedeterminedusing(5.19),whichisinthiscaseequalto2. 5.2.8CalculationofReliabilityIndices Fromtheformulationgivenabovein(5.6)Œ(5.16),itisevidentthat(5.7)and(5.8)areaugmented bygenerators,whichisequivalenttotherequiredcurtailmentsincetheproblemaimsat minimizingthetotalloadcurtailment.Further,inordertogetafeasiblesolutionforthisstandard minimizationproblem,oneofthebusanglesisassumedtobeequaltozero.Fromapracticalpoint ofview,wetaketheradialnetworktopologyistakenasanecessaryconstraintduringtherealiza- 107 tionofthepresentedwork.WethusapplythealgorithmpresentedinSection3.2.7toimposethe radialtopologyconstraintsonallpossible Usingtheformulationgivingin(5.6),theexpecteddemandnotsuppliediscalculatedas, EDNS = Nc å h = 1 LOL ( h ) Prob ( h ) (5.22) Theexpectedunservedenergyoveraperiodofoneyeariscalculatedas, EUE = 8760 EDNS (5.23) Theenergyindexofunreliability(EIU)canthereforebeestimatedas[96,97], EIUR = LOEE E T (5.24) where T istheperiodofstudyinhours, LOL ( h ) istheloadcurtailmentofstate h , Prob ( h ) isthe probabilityofstate h , N c isnumberofcontingencies,and E T isthetotalenergydemand. 5.3FormulationoftheOptimalDistributionSystem urationProblem AswasmentionedearlierinSection5.1,thisworkproposesamethodbasedonparticleswarmop- timizationtosearchforthebestsetofthesectionalizingandtie-switchesthatmaximizestheservice reliabilityofthedistributionnetwork.Nevertheless,beforeproceedingtodescribetheproposed intelligentsearchmethod,itisindispensabletotouchuponthefundamentalsoftheparticleswarm 108 optimizationtechnique,itspotentialadvantages,disadvantages,andthereasonsbehindchoosing itasasearchingtoolduringtherealizationofthepresentedwork. 5.3.1ParticleSwarmOptimization Particleswarmoptimizationisapopulation-basedoptimizationtechniqueinspiredbythesocial behaviorofofbirdsorschoolsofwhichisdescribedindetailbyKennedyandEber- hartin[88,89].Inparticleswarmoptimization,thepositionsandvelocitiesoftheparticlesare initializedwithapopulationofrandomfeasiblesolutionsandsearchforoptimabyupdatinggen- erations.Theadvantagesofusingparticleswarmoptimizationtechniqueindistributionsystem studiesincludethefollowing: 1. Distributionsystemsaregenerallyequippedwithtwokindsofswitches,whicharethesec- tionalizingswitchesandtie-switches.Thesectionalizingswitchesarenormallyclosedand usedtoconnectvariousdistributionfeederssegments.Ontheotherhand,thetie-switchesare normallyopenandusedtotransferloadsduringabnormalandemergencyconditions.These switchescanbebetterrepresentedbydigitalnumbers0and1,whichareeasytoimplement usingbinaryparticleswarmoptimization. 2. Particleswarmoptimizationhastwomainparameters,whicharethepersonalbestandthe groupbest.Everyparticleintheswarmremembersitsownpersonalbestandatthesame timethegroupbest.Consequently,PSObasedmethodshaveconsiderablymorememory capabilitythansomeotherswarmintelligencebasedmethods. 3. Particleswarmbasedoptimizationonlyneedsfewparameterstotuneup,unlikesomeother swarmintelligence. 4. Particleswarmoptimizationcanhandleawiderangeofnonlinearandnon-differentiable 109 functionsinanefandeffectivemanner. 5. Unlikesomeotherswarmintelligencebasedmethods,particleswarmoptimizationhasbetter convergencecharacteristics,whichisnotlargelyaffectedbytheproblemdimensions,non- linearity,andsize. 6. Particleswarmoptimizationcouldleadtooptimalorsemi-optimalglobalsolutionforawide rangeofpracticalproblems. Inparticleswarmoptimization,themovementoftheparticles,whichrepresentthepotential solutions,isgovernedbytheweightingfactors,theindividualbest,andthegroupbest.Using thesethreecomponents,avectorthatdeterminesthedirectionandmagnitudeofeachparticlein theswarmcanthereforeberepresentedas, v i = v i 1 + j 1 rand ( Ppbest i x i )+ j 2 rand ( Pgbest x i ) (5.25) where rand isauniformlydistributedrandomnumberbetween[0,1], Ppbest i istheparticlebest positionfromtheprobabilityofastateprospectiveparticle i haseverencountered, Pgbest isthe groupofparticlesbestpositionfromtheprobabilityofastateprospectivethegrouphasever encountered.Further, j 1 and j 2 areaccelerationfactors.Theseaccelerationfactorsareusually chosensothat j 1 + j 2 = 4,with j 1 = j 2 = 2. Thechangeinparticlespositionscanbebyasigmoidlimitingtransformationfunction andauniformlydistributedrandomnumberin[0,1]asthefollowing, x id = 8 > < > : 1 ; rand(0,1) < S ( v id ) 0 ; otherwise (5.26) 110 where, x id isthe d th componentofparticle i and S ( v id ) isthesigmoidfunctionof d th 'scomponent ofparticle i whichcanbeexpressedas, S ( v id )= 1 1 + e ( v id ) (5.27) 5.3.2TheSearchSpaceoftheProblem Inthepresenteddistributionsystemproblemforservicereliabilityimprovement, thesolutionspacebeingsearchedbytheintelligentBPSOmethodisasthespaceof allpossiblenetworkSupposethenumberofsectionalizingswitchesis N ssn and thenumberofandtie-switchesis N tsn ,respectively.Thesolutionspacewouldbeoftheform W s = f N ss 1 ; N ss 2 ; ; N ssn g [ f N ts 1 ; N ts 2 ; ; N tsn g .Here, N ssi and N tsi describetheopen/closestatus ofthesectionalizingswitch,ortie-switch, i ,respectively. 5.3.3DetailedSolutionAlgorithm Thesolutionalgorithmexplainsthewoftheproceduresofevaluatingthereliabilityindicesand theoptimalThestepsofthereliabilityassessmentandtheoptimal distributionsystemaresummarizedasfollows, 1. Initializethepositionsandvelocitiesoftheparticles, x i and v i respectively.Particle'spo- sitionsareinitializedusinguniformlydistributedrandomnumbers;switchesthatareinthe closedstatearerepresentedby1'sandswitchesthatareintheopenstatearerepresentedby 0's.Thelengthofavectordimensionequalsthenumberofsystemswitches.Oneofthepar- ticlesischosentorepresenttheoriginalofthesystem,thatis,alltie-switches areinopenstate. 111 2. Checkifthereareidenticalparticles.Ifso,discardtheidenticalonesandsavetherestofthe particlesinatemporaryarrayvectorbyconvertingthebinarynumberstodecimalnumbers. 3. Checkifthereareparticlesalreadyexistinthedatabase,ifso,setloadcurtailmentsoftheex- istingparticlestothesystempeakloadtodecreasethechanceofvisitingthese again.Then,savetherestinthedatabaseandgotothenextstep. 4. Checktheradialityconditionforeachparticle,iftheradialityconditionismet,gotonext step,otherwisesetloadcurtailmentsoftheparticlesthatrepresentinfeasible tothesystempeakloadtodecreasethechanceofvisitingtheseagain. 5. Setsystemparametersandupdatethestatusofthesystemcomponents(sectionalizingwitches, tie-switches,circuitbreakers,etc.),foreveryparticle. 6. Performreliabilityevaluationforeachparticlebysolvingthelinearprogrammingoptimiza- tionproblemforeverysystemstatetocalculatetheexpectedloadcurtailmentforeachparti- cle. 7. Determineandupdatepersonalbestandglobalbestbasedonminimumload curtailments. 8. Checkforconvergence.Afterfewiterations,ifnonewbetterwerediscovered, terminatethealgorithm. 9. Updateparticle'svelocitiesusing(5.25)andupdateparticle'spositionsusing(5.26)and (5.27),andgotostep3. Thesolutionproceduresofreliabilityevaluationandtheoptimalusing theproposedBPSObasedsearchmethodaredepictedinthewchartshowninFig5.4. 112 Figure5.4:Flowchartoftheproposedmethod 113 5.4DemonstrationandDiscussion Inordertodemonstratetheeffectivenessoftheproposedmethodandthesearchmethodofoptimal distributionsystemwetesttheproposedmethodonthreedifferentdistribution systems.Weprovidetheresults,assumptions,anddiscussioninthesubsequentsection. 5.4.133BusDistributionSystemTestCase The33bussystemisa12 : 66kVradialdistributionsystem,whichhasbeenwidelyusedasa benchmarksysteminsolvingthedistributionsystemproblem[16].Thesingle- linediagramofthe33bussystemisdepictedinFig.5.5.Thetotalrealandreactivepowerloads onthissystemare3715KWand2300KVAR,respectively.AsshowninFig.5.5,thissystem consistsof33buses,includingthesubstationbus,32branches,3laterals,and5tie-lines.Forthe initialthenormallyopenswitches(tie-lines)are f S33,S34,S35,S36,S37 g ,which arerepresentedbydottedlines.ThenormallyclosedswitchesaredenotedasS1toS32andare representedbysolidlines. Thereliabilitydatausedinthisworkareacquiredfrom[101]andaregiveninTable5.1.The reliabilityindicesusedinthispaperaretheexpectedunservedenergyEUEandtheenergyindex ofunreliabilityEIUR.TheparametersoftheproposedBPSObasedsearchmethodaregivenin Table5.2.Weconductseveralcasestudiesonthe33bussystemandothersystems,whichare providedhereafter.Itisimperativetomentionherethatweconsidervariousloadingconditions thatincludelightloadingconditionsinwhichthetotalloadisreducedby50%,ratedornominal loadingconditionsinwhichpeakloadsareconsidered,andheavyloadingconditionsinwhichthe totalloadisincreasedby50%.Theresultsofthenominal(peak)loadingconditionsarediscussed inthesubsequentsections. 114 Figure5.5:Single-linediagramofthe33busdistributionsystem Table5.1:ReliabilityDatafortheTestSystems Component Failurerate RepairRate Name (failure/year) (hr) Transformer 0.05882 144 Bus 0.0045 24 CircuitBreaker 0.1 20 DistributionLine 0.13 5 SectionalizingSwitch 0.2 5 5.4.1.1CaseScenarioI Inthiscasescenario,circuitbreakers,switches,andthedistributiontransformerareassumedto beperfectlyreliable.Thefailureratesofdistributionlinesareonlyconsideredinthiscasestudy. Ithasbeenassumedthateverydistributionlinehasasectionalizingswitch.Theoptimalstatusof thesectionalizingandtie-switchesisobtainedusingtheBPSOgiveninSection3.3.4.Asdepicted inTable5.2,theparametersoftheBPSObasedsearchmethodhavebeenchosensothat:number 115 ofparticles = 50,accelerationfactors j 1 = 2,and j 2 = 2,with j 1 + j 2 = 4,andthemaximum numberofiterations = 1000.Itisworthpointingoutherethattheseparameterswereobtainedafter carryingoutseveralindependentrunsandfoundtoprovidebestperformanceintermsofexecution timeandsolution. Table5.2:ParametersoftheBPSO Parameter ProposedValue NumberofParticles 50 SocialConstant ( j 1 ) 2 SocialConstant ( j 1 ) 2 Max.Numberofiterations 1000 Forcomparisonpurposes,theproposedmethodofdistributionsystemforre- liabilitymaximizationhasbeenappliedontheinitialshowninFig.5.5.The optimalsetofthesectionalizingandtie-switchesthatyieldminimumloadcurtailmentwere f S6, S10,S13,S27,S36 g .TheresultsofthiscasestudyaresummarizedbelowinTable5.8. Forthenominalloadscenario,ascanbeseenfromTable5.8,theEUEisreducedfrom17521 : 3 kWh/yearfortheinitialto13536 : 6kWh/yearafterThisisabout 22 : 74%reductionintheannualexpectedunservedenergy. AccordingtotheEnergyDataviaWorldEnergyCouncil(enerdata)[119,120],thetypical householdpowerconsumptionintheUSisabout11,700kWheachyear.Thisisforanaverage householdsizecloserto2.5people.Further,accordingto[119,120]eachAmericanusesabout 4,500kWhperyearinhishome.Weusetheseheretocalculatetheaveragenumber oftheaffectedhouseholdsbeforeandafterTherefore,forCaseScenarioI,for nominalloadingconditions,theaveragenumberoftheaffectedhouseholdsbefore isestimatedtobe1.5households.Thenumberoftheaffectedconsumersbefore was3.89consumers.However,thesehavebeenrespectivelyreducedto1.16and3,after 116 thesystemhasbeenThishasbeenthecaseforthelightandheavyloadingconditions ascanbeclearlyseenfromTable5.8. AsanexampleoftheeffectivenessoftheproposedmethodinminimizingtheEIU,forthecase ofnominalload,theEIUhasbeenreducedfrom0.00054beforeto0.00042after thesystemhasbeen 5.4.1.2CaseScenarioII Inthiscasescenario,wetakethefailureratesofalldistributionsystemcomponentssuchascircuit breakers,switches,buses,distributionlines,andtransformerintoconsideration.Inaddition,we assumethateverydistributionlinehasasectionalizingswitch.Theoptimalsetofthesectional- izingandtie-switchesthatyieldminimumloadcurtailmentare f S6,S10,S13,S27,S36 g .For comparisonpurposes,theproposedmethodofdistributionsystemforreliability maximizationhasbeenappliedtotheinitialshowninFig.5.5.Theresultsof thiscasescenarioaresummarizedbelowinTable5.9. AscanbeobviouslyseenfromTable5.9,theEUEhasbeenreducedfrom85692 : 2kWh/year fortheinitialto74235 : 6kWh/yearafterThisisabout13 : 4%re- ductionintheexpectedunservedenergy.Moreover,inCaseScenarioII,fornominalloading conditions,theaveragenumberoftheaffectedhouseholdsbeforeisestimatedto be7.32households.Thenumberoftheaffectedconsumersbeforewas19.04con- sumers.However,thesehavebeenrespectivelyreducedto6.34and16.5,afterthesystem hasbeenTheThiscasescenariohasshowntheeffectoftakingthefailureratesofall systemcomponentsintotheoptimizationproblem. AsanexampleoftheeffectivenessoftheproposedmethodinminimizingtheEIU,forthecase ofnominalload,theEIUhasbeenreducedfrom0.00263beforeto0.00228after 117 thesystemhasbeen 5.4.1.3CaseScenarioIII Inrealisticdistributionsystems,thenumberofsectionalizingswitchesaresometimeslimiteddue tocertainoperationalandeconomicreasons.Therefore,thiscasescenarioissimilartocasesce- narioIIexceptthatthenumberofsectionalizingswitcheshavebeenlimitedandtheirlocations areassigned.Forthiscasescenario,thenumberofsectionalizingswitcheshasbeenassumed11 switches,inadditiontothe5tie-switches.Thepositionsofthesesectionalizingswitchesareas- sumedtobe f S7,S8,S9,S11,S12,S14,S17,S28,S29,S32,S24 g .Theresultsofthiscase scenarioaresummarizedbelowinTable5.5. Table5.5showsthesetofswitchesthathadtobeopenedtominimizethetotalloadcurtail- ment.Thissetinclude f S9,S14,S17,S28,S33 g .AscanbeseenfromTable5.5,theEUEhas beenreducedfrom85692 : 2kWh/yearfortheinitialto76107 : 7kWh/yearafterre- Thisisabout11 : 2%reductionintheexpectedunservedenergyd.Moreover,inCase ScenarioIII,fornominalloadingconditions,theaveragenumberoftheaffectedhouseholdsbefore isestimatedtobe7.32households.Thenumberoftheaffectedconsumersbefore was19.04consumers.However,thesehavebeenrespectivelyreducedto 6.5and16.91,afterthesystemhasbeenTheThiscasescenariohasshowntheeffect oftakingthefailureratesofallsystemcomponentsintotheoptimizationproblem.Asexpected, theamountofpowercurtailedisgoingtoincreaseifthenumberofsectionalizingswitchesis constrained. AsanexampleoftheeffectivenessoftheproposedmethodinminimizingtheEIU,forthecase ofnominalload,theEIUhasbeenreducedfrom0.00263beforeto0.00234after thesystemhasbeen 118 5.4.1.4CaseScenarioIV ThiscasescenarioissimilartocasescenarioIIIexceptthatthefailureratesofdistributionfeeders areincreasedby20%andthefailureratesofsectionalizingswitchesareincreasedby10%.Again, thepositionsofthesesectionalizingswitchesareassumedtobe f S7,S8,S9,S11,S12,S14,S17, S28,S29,S32,S24 g .TheresultsofthiscasescenarioaresummarizedbelowinTable5.6. Table5.6showsthesetofswitchesthathadtobeopenedtominimizethetotalloadcurtail- ment.Thissetinclude f S9,S14,S17,S28,S33 g .AscanbeseenfromTable5.6,theEUEhas beenreducedfrom91778 : 3kWh/yearfortheinitialto80972 : 4kWh/yearafterre- Thisisabout11 : 77%reductionintheexpectedunservedenergy.Moreover,inCase ScenarioIV,fornominalloadingconditions,theaveragenumberoftheaffectedhouseholdsbefore isestimatedtobe7.84households.Thenumberoftheaffectedconsumersbefore was20.4consumers.However,thesehavebeenrespectivelyreducedto 6.92and18,afterthesystemhasbeen AsanexampleoftheeffectivenessoftheproposedmethodinminimizingtheEIU,forthecase ofnominalload,theEIUhasbeenreducedfrom0.00282beforeto0.00248after thesystemhasbeen 5.4.269BusDistributionSystemTestCase The69bussystemisa12 : 66kVdistributionsystemwithtotalrealandreactivepowerloadsof 3802 : 19kWand2694 : 06kVar,respectively[33].Thesingle-linediagramofthe69bussystemis depictedinFig.5.6.Fortheinitialthenormallyopenswitchesare f S69,S70,S71, S72,S73 g . Theproposedreliabilitybaseddistributionsystemmethodisappliedonthe 119 Figure5.6:Single-linediagramofthe69busdistributionsystem 69bussystemandtheEUEreliabilityindexiscalculatedforeveryfeasibleWe considerthenominalloadingconditionsinthiscasescenario.Itisappropriatetomentionherethat variouscasestudieswereconductedonthe69bussystem,however,owingtospaceconstraints, theresultsofonecasescenario,inwhichthefailureratesofalldistributionfeedersareconsidered, arepresentedinTable5.7.AscanbeseenfromTable5.7,theamountoftheannualunserved energyisreducedfrom31498 : 4kWh/yearfortheinitialnetworktopologyto23124 : 5kWh/year afterThisisequivalentto26 : 58%reductionintheannualunservedenergy. Fornominalloadingconditions,theaveragenumberoftheaffectedhouseholdsbeforerecon- isestimatedtobe2.96households.Thenumberoftheaffectedconsumersbeforerecon- was7consumers.However,thesehavebeenrespectivelyreducedto1.98and 5.14,afterthesystemhasbeenMoreover,theEIUhasbeenreducedfrom0.00095 beforeto0.00069afterthesystemhasbeen 5.4.3118BusDistributionSystemTestCase Theproposedframeworkhasbeenappliedtoamorecomplicatedandrealisticdistributionsystem tovalidateitsfeasibilityinsuchconditions.Thesystemunderconsiderationisan11kV,118- 120 nodeslarge-scaleradialdistributionsystem[121].Thesingle-linediagramofthe118bussystem isdepictedinFig.5.7.Thissystemconsistsof117branchesandhas15tie-lines.Theresults ofonecasescenarioperformedonthissystemarepresented.Thefailureratesofdistribution feedersareconsideredinthiscase.FortheinitialtheEUEhasbeencalculatedusing theproposedalgorithmsandwasfound111 : 29MW/year.TheEUEhasbeenreducedto108 : 75 MW/yearafterthenetworkbeing Fornominalloadingconditions,theaveragenumberoftheaffectedhouseholdsbeforerecon- isestimatedtobe9.51households.Thenumberoftheaffectedconsumersbeforerecon- was24.73consumers.However,thesehavebeenrespectivelyreducedto9.29 and24.16,afterthesystemhasbeen 5.4.4VoltagePrImprovement Wediscusstheeffectoftheonimprovingtheoverallvoltageofthedistribu- tionsysteminthissection.Weconsiderthe33bus,69bus,and118bussysteminthiscasestudy. However,weprovidetheresultsofthebasiccasescenarioinwhichcircuitbreakers,switches,and thedistributiontransformerareassumedtobeperfectlyreliableexceptdistributionfeeders.Fur- ther,weassumethateverydistributionlinehasasectionalizingswitch.AscanbeseenfromTable 5.9,allbusvoltageshavebecomewithinthelimit,whichisassumedtobe5%inthis work.Thisisakeyadvantageoftheproposedmethodofreliabilityenhancementsincekeeping busvoltageswithinthepredeterminedlimitsreliveloads,reducethepossibilityofloadshedding; andtherebyminimizeserviceinterruptions. 121 Figure5.7:Single-linediagramofthe118buslarge-scaledistributionsystem 122 5.5Summary Thischapterhasproposedaframeworkfordistributionsystemwithanobjective ofreliabilitymaximization.Themainfeaturesoftheworkdescribedaboveinclude: 1. Theframeworkisdevelopedbasedonanovellinearizedpowerwmodel,inwhichthecou- plingbetweenactivepowerandvoltagemagnitudeaswellasthecouplingbetweenreactive powerandvoltageangleismaintained. 2. Voltagemagnitudes,reactivepowerws,andallshuntelementshavebeentakenintocon- siderationintheproposedmodel. 3. Thecurrentlimitsconstraintscanbeapproximatedusingpiecewiselinearmodel.Therefore, thedegreeofapproximationcanbeimprovedtothedesiredlevelbyincreasingthenumber oflinesegmentsused,withoutsubstantialaffectonthemainroutine. 4. Theframeworkpresentedinthispaperisgenericinthesensethatitcanbeusedforsolv- ingseveralpowersystemoptimizationproblemssuchastheconstrainedeconomicpower dispatchforactivedistributionsystemsandenergymarketsimulations,inwhichrepetitive solutionsarerequired. 5. Theprobabilisticreliabilityevaluationwasconductedusingtheeventtreeanalysiswith higher-ordercontingencyapproximation,hencethecomputationalburdenhasbeentremen- douslyreducedwhencomparedtosomeotherapproachesavailableintheliterature,which sometimestendtobeinfeasibleevenforoff-lineplanningstudies. 123 Table5.3:ResultsofCaseScenarioI NetworkStatus& BestSetofTie& EUE Affected Affected LoadingConditions SectionalizingSwitches kWh/year Households Consumers Nominal InitialConf. S33,S34,S35,S36,S37 17521.3 1.5 3.89 OptimalConf. S6,S10,S14,S27,S36 13536.6 1.16 3.0 Light InitialConf. S33,S34,S35,S36,S37 7008.54 0.60 1.56 OptimalConf. S6,S10,S14,S27,S36 5414.63 0.46 1.20 Heavy InitialConf. S33,S34,S35,S36,S37 24529.9 2.10 5.45 OptimalConf. S7,S10,S13,S27,S36 18951.2 1.62 4.21 Table5.4:ResultsofCaseScenarioII NetworkStatus& BestSetofTie& EUE Affected Affected LoadingConditions SectionalizingSwitches kWh/year Households Consumers Nominal InitialConf. S33,S34,S35,S36,S37 85692.2 7.32 19.04 OptimalConf. S6,S10,S13,S27,S36 74235.6 6.34 16.49 Light InitialConf. S33,S34,S35,S36,S37 34276.9 2.93 7.62 OptimalConf. S7,S10,S14,S27,S36 29694.2 2.54 6.60 Heavy InitialConf. S33,S34,S35,S36,S37 119969 10.25 26.65 OptimalConf. S7,S10,S13,S27,S36 103930 8.88 23.10 Table5.5:ResultsofCaseScenarioIII NetworkStatus& BestSetofTie& EUE Affected Affected LoadingConditions SectionalizingSwitches kWh/year Households Consumers Nominal InitialConf. S33,S34,S35,S36,S37 85692.2 7.32 19.04 OptimalConf. S9,S17,S28,S33,S34 76107.7 6.50 16.91 Light InitialConf. S33,S34,S35,S36,S37 34276.9 2.93 7.62 OptimalConf. S9,S17,S28,S33,S34 30443.1 2.60 6.77 Heavy InitialConf. S33,S34,S35,S36,S37 119969 10.25 26.65 OptimalConf. S9,S14,S17,S28,S33 106551 9.11 23.68 Table5.6:ResultsofCaseScenarioIV NetworkStatus& BestSetofTie& EUE Affected Affected LoadingConditions SectionalizingSwitches kWh/year Households Consumers Nominal InitialConf. S33,S34,S35,S36,S37 91778.3 7.84 20.40 OptimalConf. S9,S17,S28,S33,S34 80972.4 6.92 18 Light InitialConf. S33,S34,S35,S36,S37 36711.3 3.14 8.16 OptimalConf. S9,S17,S28,S33,S34 32389.0 2.77 7.20 Heavy InitialConf. S33,S34,S35,S36,S37 128490 10.98 28.55 OptimalConf. S9,S14,S17,S28,S33 113361 9.69 25.19 124 Table5.7:Resultsof69BusSystem Network BestSetofTie& EUE Affected Affected Status SectionalizingSwitches kWh/year Households Concumers InitialConf. S69,S70,S71,S72,S73 31498.4 2.96 7 FinalConf. S14,S18,S24,S56,S69 23124.5 1.98 5.14 Table5.8:Resultsof118BusSystem Network BestSetofTie& EUE Affected Affected Status SectionalizingSwitches MWh/year Households Concumers InitialConf. S33,S34,S35,S36,S37 108.75 9.51 24.73 FinalConf. S6,S10,S14,S27,S36 111.29 9.29 24.16 Table5.9:ComparisonofBusVoltagesBeforeandAfter Test Min.Voltage Max.Voltage Min.Voltage Max.Voltage System BeforeReco. BeforeReco. AfterReco. AfterReco. 33BusSystem 0.9113@18 1.0@1 0.95@25 1.0474@1 69BusSystem 0.9175@64 1.0@1 0.9654@65 1.01@1 118BusSystem 0.8844@77 1.0@1 0.9528@77 1.0414@1 125 Chapter6 ConclusionsandFutureWork 6.1Conclusions Electricpowerdistributionsystemshavebeenoperatedinaverticalandcentralizedmannerforsev- eralyears.However,duetocertainenvironmental,economic,andpoliticalreasons,thisstructure hasbeenchangedandseveralrealŒtimeengineeringapplicationsinbothoperationalandplanning stageshaveemerged.Examplesoftheseapplicationsincludeoptimalsizingandplacementof distributedgenerationunits,optimalpowerwofactivedistributionsystems,andfeederrecon- forreliabilityenhancementandservicerestoration,andsoforth.Itisverywellknown thattheseapplicationsrequireapowerwstudyatthestepofthesolution.Nevertheless,the vastmajorityoftheseapplicationsrequirerepetitiveandpromptpowerwsolutions.Performing fullACpowerw,ononehand,giveshighcalculationprecisionbutrequiresaquiteexten- sivecomputationalburdenandstoragerequirements.Ontheotherhand,andmoreprominently, thelargestpartoftheaforementionedapplicationsisessentiallynonlinearcomplexcombinatorial constrainedoptimizationproblems.Theformulationofthenonlinearproblemtendstobeatedious taskandcomputationallycumbersomeintermsofexecutiontime,storagerequirements,andpro- gramming.Theseconstraintscombinedwiththelargenumberofnodes,branches,andswitches ofdistributionsystemwillincontestablyincreasethecomplexityoftheproblem.Ithastherefore becomenecessarytodevelopmorepowerfultoolsforbothplanningandoperationalstudiesnot 126 onlytoaccompanytheaforementionedapplications,butalsotohandletheothernewtasks,which arecomingintheimmediatefuture. Thepartofthisdissertationdevelopedanovelpowerwmodel,whichisequallyappro- priateforuseatbothdistributionandtransmissionlevelsandcanbeextremelyusefulwhenever fast,robust,andrepetitivepowerwsolutionsarerequired.Wedevelopedtheproposedlinearized ACpowerwmodel(LACPF)basedonlinearizationofthefullsetofconventionalpowerw equations,andthereforeincludesvoltagemagnitudesolutionsandreactivepowerws,unliketra- ditionallinearizedpowerwmethods.Themodeldevelopedinthisdissertationisnon-iterative, direct,andinvolvesnoconvergenceissuesevenwithill-conditionedsystemsandsystemswithhigh R/Xbranchratios.Theincaseofunbalanceddistributionnetworksarestraightfor- wardandlargelylieincertainelementsinthebusadmittancematrix;andthustheadvantages obtainedwithbalancedoperationarepreserved.WetestedtheproposedLACPFmodelonseveral balanced,unbalanced,andweakly-mesheddistributionsystemsandfoundtoperformwithspeed andaccuracyappropriateforrepetitivesolutions.Weprovidedandthoroughlydiscussedtheresults ofvarioustestdistributionsystems,includingalarge-scalesystemtestcase,inChapter2. Thesecondpartofthisdissertationdevelopedanefoptimizationframeworktohandle severaldistributionsystemoperationalandplanningproblems.Theproposedframeworkuseslin- earprogrammingsincelinearprogrammingbasedformulationstendtobexible,reliable,and fasterthantheirnonlinearcounterparts.Weconsideredvoltagebounds,reactivepowerlimits,and allshuntelementsintheproposedoptimizationmodel.AswasdiscussedinChapter1,distribution systemsarethemostextensivepartintheentirepowersystemduetotheirspanningtreestructure andthehighR/Xbranchratios.Therefore,weproposedanewanalyticalmethodforoptimalplace- mentandsizingofdistributedgenerationunitsondistributionsystemsinChapter3.Theobjective oftheanalyticalmethodpresentedinChapter3istominimizethedistributionsystemlosses.An- 127 alyticalmethodsarereliable,computationallyefandaresuitableforplanningstudiessuch asdistributedgenerationplanning.Furthermore,analyticalapproachescouldleadtoanoptimal ornear-optimalglobalsolution.Wedevelopedaprioritylistbasedonlosssensitivityfactorsto determinetheoptimallocationsofthecandidatedistributedgenerationunits,afteridentifyingthe penetrationlevelofthedistributedgenerationunits.Weperformedsensitivityanalysisbasedonthe realpowerinjectionofthedistributedgenerationunittoestimatetheoptimalsizeandpowerfactor ofthecandidatedistributedgenerationunits.Weconsideredvarioustypesofdistributedgenerators andalsoproposedviablesolutionstoreducetotalsystemlosses.Wevalidatedtheeffectiveness oftheproposedmethodbyapplyingitonthesamebenchmarksystemsusedbeforeinChapter2, inparticularthe33busandthe69busdistributionsystems,sincebothsystemshavebeenexten- sivelyusedasexamplesinsolvingtheplacementandsizingproblemofdistributedgenerators.In addition,weperformedexhaustivepowerwstudiestoverifythesizesobtainedbytheanalytical method.Wevalidatedtheoptimallocationsandsizesobtainedbytheproposedanalyticalmethod bycomparingthemwithsomeotheranalyticalmethodsavailableintheliterature.Weshowthat theproposedanalyticalmethodcouldleadtoanoptimalornear-optimalglobalsolution,while requiringlowercomputationaleffort. Weproposedanewmethodtosolvetheoptimaleconomicpowerdispatchproblemofactive distributionsystemsinChapter4.Nonlinearprogrammingandlinearprogrammingbasedmethods arewidelyusedintheliteraturetosolvetheoptimaleconomicdispatchproblem.Nevertheless,the vastmajorityofthelinearprogrammingbasedmethodsweredevelopedbasedontheDCPFmodel, whichhasseveraldrawbackswerediscussedearlierinSection1.5.InChapter4,inadditiontothe piecewiselinearmodelwehavedevelopedearlierinChapter2tohandlethethermalcapacities oftransmissionlines,wedevelopedpiecewiselinearmodelstodealwiththeexponentialloads, costcurvesofgeneratingunits,andtotalpowerlosses.Wetaketheeffectofdistributedgeneration 128 unitsinseveralcasescenariosbyconsideringdifferentpenetrationlevels.Wedemonstratedthe effectivenessoftheproposedmethodbyperformingnumerouscasestudies.Wewereableto showthattheresultsobtainedbytheproposedmethodcorrespondcloselywiththoseobtainedby nonlinearmeans,whilerequiringlowercomputationaleffort. Weproposedamethodtosolvethedistributionsystemproblemwithanob- jectiveofreliabilityimprovementinChapter5.Reliabilityenhancementofdistributionsystems throughfeederisnotwellstudiedintheliterature.Inthecontext,weintroduceda completeoptimizationframeworktohandlethereliabilitymaximizationproblem.Sincethetime andcomputationaleffortspentinevaluatingreliabilityindicesareofgreatconcerninbothplanning andoperationalstages,weusedaprobabilisticreliabilityassessmentmethodbasedoneventtree analysiswithhigher-ordercontingencyapproximation.Therefore,theeffectofthehigher-order contingenciesislimitedand,atthesametime,thecomputationalburdenisimproved.Weselect theexpectedunservedenergyastheenergyindexthatneedstobeminimized.However,toknow howmuchreliablethesystemis,weintroducedanotherreliabilitymeasure,whichistheenergy indexofunreliability.Frompracticalperspective,theradialtopologicalstructurehasbeentaken asanecessaryconditionduringtherealizationofthiswork.Therefore,wedevelopedanothercon- straintsbasedontheoreticgraphtopreservethespanningtreestructureofthedistributionsystem. Weproposedasearchmethodbasedonparticleswarmoptimizationtechnique.Wedemonstrate theeffectivenessoftheproposedmethodonseveraldistributionsystemsandshowthattheamount oftheannualunservedenergy,thenumberofaffectedhouseholds,andtheaffectednumberof consumerscanbetremendouslyreducedusingtheproposedmethod. 129 6.2FutureWork Aswithanyresearchtopic,themodelsandmethodspresentedinthisdissertationcanbeextended inseveraldirections.Someotherresearchdirectionsaresummarizedhere. InChapter3,weaddressedtheproblemofoptimalplacementandsizingofdistributedgen- erationunitsusinganalyticaltechniques.Weproposedformulasforboththeoptimallocation andoptimalsizeofthecandidatedistributedgenerationunits.Weconsideredoneloadscenario,in whichpeakloadsareassumed.Infact,theworkpresentedinChapter3canbeextendedindifferent directions.Forinstance,theproblemcanbesolvedusingtheproposedoptimizationframeworkin additiontoamethod,whichcanbedevelopedbasedonswarmintelligencesuchasparticleswarm optimizationorgeneticalgorithms.Weanticipatethatboththeanalyticalandtheswarmintelli- gencebasedmethodwouldeventuallyleadtosimilarresults.However,thelattermethodwillbe moreappropriateincasethatdifferentloadscenariosaresought. InChapter4,weproposedamethodforsolvingtheproblemofoptimaleconomicpowerdis- patchofactivedistributionsystems.Weassumedaconstantpowerdistributedgenerationunits whileperformingthisstudy.Infact,inarelatedwork[122],weconsideredintermittentwindtur- bineinductiongeneratorsbaseddistributedgeneratorstoaccountforpenetrationofthedistributed generators.However,thestudyweconductedin[122]wascompletelydevelopedbasedonthe DCPFmodel.TheworkpresentedinChapter4canbeextendedusingintermittentdistributed generationunitssuchwindturbineinductiongeneratorsandsolargenerators.Theintermittencyof thedistributedgeneratorscanbeaccountedforusingahybridoptimizationmodelsuchasHomer [105],forinstance,whichhasbeenusedin[122].Otherconstraintscanalsobeaddedtothe optimizationframeworkinordertoperformsomeotheroptimaloperationandplanningstudies. Weproposedanewmethodtosolvethedistributionsystemproblemwithan 130 objectiveofreliabilityimprovementinChapter5.Inthecontext,weintroducedacompleteop- timizationframeworktohandlethereliabilitymaximizationproblem.Weusedaprobabilistic reliabilityassessmentmethodbasedoneventtreeanalysisandselectedtheexpectedunserveden- ergyastheenergyindexthatneedstobeminimized.Wedevelopedanotherconstraintsbased ontheoreticalgraphtopreservethespanningtreestructureofthedistributionsystem.wepro- posedasearchmethodbasedonparticleswarmoptimizationtechnique.However,inChapter5 wedidnotconcentrateonthefaultlocationandisolation.TheworkpresentedinChapter5can beextendedtosolvetheproblemofservicerestorationindistributionsystems.Inthiscontext, theobjectivefunctionhastobesothattheexpectedunservedenergyandthenumber switchingoperationshavetobeminimizedsimultaneously.Inordertoincreasetheamountofthe totalrestoredload,distributedgenerationunitscanbeaddedtothesystem.Theamountofthe totalrestoredloadcanbeincreasedbymeansofdistributedgenerationunits.Forfur- therloadrestoration,sensitivityanalysiscanbeperformedtoselecttheoptimalsizeandlocation ofthesedistributedgenerators,whichwouldresultinminimizingofthetotalloadcurtailmentof in-serviceconsumers. 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