ComponentModelingandThree-PhasePower-FlowAnalysisforActiveDistributionSystemsbyMohamedZakariaKamhAthesissubmittedinconformitywiththerequirementsforthedegreeofDoctorofPhilosophyGraduateDepartmentofElectricalandComputerEngineeringUniversityofTorontoCopyright 2011byMohamedZakariaKamhAbstractComponentModelingandThree-PhasePower-FlowAnalysisforActiveDistributionSystemsMohamedZakariaKamhDoctorofPhilosophyGraduateDepartmentofElectricalandComputerEngineeringUniversityofToronto2011This thesis presents a novel, fast, and accurate 3 steady-state power-ow analysis (PFA)tool for the real-time operation of the active distribution systems, also known as the activedistributionnetworks (ADN), inthegrid-tiedandislandedoperatingmodes. Three-phase power-ow models of loads, transformers, and multi-phase power lines and lateralsare provided. This thesis also presents novel steady-state, fundamental-frequency, power-ow models of voltage-sourced converter (VSC)-based distributed energy resource (DER)units. TheproposedmodelsaddressawidearrayofDERunits, i.e., (i)variable-speedwind-drivendoubly-fedasynchronousgenerator-basedand(ii)single/three-phaseVSC-coupledDERunits. Inaddition, acomputationally-ecienttechniqueisproposedandimplementedtoimposetheoperatingconstraintsof theVSCandthehost DERunitwithinthe context of the developedPFAtool. Novel closedforms for updatingthecorresponding VSC power and voltage reference set-points are proposed to guarantee thatthepower-owsolutionfullycomplieswiththeVSCconstraints. AlltheproposedDERmodelsrepresent(i)thesalientVSCcontrol strategiesandobjectivesunderbalancedandunbalancedpower-owscenariosand(ii)alltheoperatinglimitsandconstraintsoftheVSCanditshostDERunit.Also, theslackbusconceptisrevisited, associatedwiththePFA, wherea3dis-tributed slack bus (DSB) model is proposed for the PFA and operation of islanded ADNs.Distributingthereal andreactiveslackpoweramongseveral DERunitsisessential toprovidearealisticpower-owapproachforADNsintheabsenceoftheutilitybus. TheproposedDSBmodelisintegratedwiththedeveloped3PFAtooltoformacompleteADNPFApackage.The new PFA tool, including the proposed DER and DSB models, is tested using sev-eralbenchmarknetworksofdierentsizes,topologies,andparameters. Manycasestud-ies, encompassingawidespectrumofDERcontrolspecicationsandoperatingmodes,iiareconductedtodemonstrate(i)thenumericalaccuracyoftheproposedmodelsoftheDERunitsandtheiroperatingconstraints, (ii)theeectivenessof theproposedDSBmodel for the islanded ADN PFA, and (iii) the computational eciency of the integratedPFAsoftwaretoolirrespectiveofthenetworktopologyandparameters.iiiDedicationTomyparents,mywife,andmymotherlandEgypt..ivAcknowledgementsFirst, I wouldliketothankmyCreator, forgivingmethewisdomandfoundationtoaccomplishthisthesis.Iamdeeplyindebtedtomysupervisor, Prof. RezaIravani, forhiscontinuousaca-demic advice, constant encouragement, endless patience, and priceless guidance and sup-portthroughoutthiswork. Iamreallygratefultohim,notonlyforcontributingmanyvaluablesuggestionsandimprovementstothisthesis,butmostimportantlyfortrainingme to be an independent researcher, with the ability to identify interesting and importantresearch problems and to generate frameworks to solve them. Moreover, I am grateful tohim for exposing me to collaboration with dierent industrial and organizational institu-tions. Ifeelhonoredtohavebeengiventheopportunitytoworkunderhissupervision.IwouldalsoliketoacknowledgemyPh.D.externalexaminer,Prof. LiuchenChang,andtheesteemedinternal committeemembers: Prof. Peter Lehn, Prof. AleksandarProdic, andProf. ZebTate, forthevaluableinputtheyhavegivenintomythesis. Myappreciation also goes to my colleagues, particularly Ali Mehrizi-Sani and Amir Etemadi,andtheFacultymembersintheEnergySystemsgroup,forthefriendlyacademicatmo-sphere, theuseful discussions, andtheirencouragement. IamparticularlyindebtedtoDr. Milan Graovac and Mr. Xiaoling Wang, from the Center for Applied Power Electron-ics(CAPE)attheUniversityofToronto, fortheirpricelesshelpandvaluablefeedbackandcommentsthroughoutmyPh.D.project.Next,Iwouldliketopaymyhumblerespectandthedeepestthanksfrommyhearttomymother, Prof. SanaaKamh, myfather, General ZakariaKamh, andmylovelysisters,MaiandYasmin,fortheirendlesssupport,love,patience,andencouragement. Iam extremely grateful to my parents for giving me the best education, the warmest care,therighteousupbringingandthebestineverything.My nal warmest thanks from my heart go to my precious wife, Angie Eldamak, whoreallychangedmylifetothebestineverything,andsupportedmethroughoutthislongPh.D.path. Shealwayslightenedmybaddaysandperfectedmygoodones. Nowordscanexpressmygratitudeandappreciationtoher,andmyin-laws,forthekindsupportand tremendous care. Angie will always be my source of inspiration, the twin of my soul,andtheloveofmylife.MohamedZ.KamhJuly2011vContents1 Introduction 11.1 ActiveDistributionNetworks . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 EnablingTechnologies . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 ADNOperatingModes. . . . . . . . . . . . . . . . . . . . . . . . 31.1.4 SmartEnergyManagementSystem. . . . . . . . . . . . . . . . . 41.2 StatementoftheProblemandThesisMotivations . . . . . . . . . . . . . 41.2.1 LackofTools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 LackofDERModels . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 ThesisObjectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4.1 ModelingMethodology. . . . . . . . . . . . . . . . . . . . . . . . 71.4.2 ValidationMethodology . . . . . . . . . . . . . . . . . . . . . . . 81.5 ThesisStructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5.1 Chapter 2: AThree-Phase Sequence-Frame Power-FlowSolver(SFPS)Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5.2 Chapter 3: Steady-State Models of Three-Phase VSC-Coupled DERUnits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5.3 Chapters 4and5: Steady-State Models of Type-3WTG-BasedDERUnits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5.4 Chapter 6: Steady-State Models of Single-Phase VSC-CoupledDERUnits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5.5 Chapter 7: Three-Phase Distributed Real- and Reactive-Slack BusModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5.6 Chapter8: Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 112 Sequence-FramePower-FlowSolver(SFPS) 122.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12vi2.2 Three-PhasePower-FlowAnalysis: CriticalReview . . . . . . . . . . . . 122.2.1 Three-PhasePower-FlowAlgorithmsforRadialNetworkTopology 132.2.1.1 Backward-ForwardSweepAlgorithm(BFSA) . . . . . . 132.2.1.2 Compensation-BasedAlgorithms . . . . . . . . . . . . . 132.2.2 Three-Phase Power-Flow Algorithms for General Network Topologies 142.2.2.1 Newton-RapshonMethod . . . . . . . . . . . . . . . . . 142.2.2.2 Gauss-SeidelMethods . . . . . . . . . . . . . . . . . . . 152.3 Sequence-FrameVersusPhase-FrameinThree-PhasePower-FlowAnalysis 162.4 Sequence-FrameModelsofBasicADNComponents . . . . . . . . . . . . 172.4.1 DistributedGeneration. . . . . . . . . . . . . . . . . . . . . . . . 172.4.2 UnbalancedThree-phaseDistributionLine . . . . . . . . . . . . . 192.4.3 Three-phasePowerTransformers . . . . . . . . . . . . . . . . . . 222.4.4 Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.4.1 ConstantPowerandCurrentLoads. . . . . . . . . . . . 222.4.4.2 ConstantImpedanceLoads . . . . . . . . . . . . . . . . 232.5 TheSFPSAlgorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.6 SummaryandDiscussion. . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Power-FlowModelof3-VSC-CoupledDERUnits 323.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 ModelScopeandAssumptions. . . . . . . . . . . . . . . . . . . . . . . . 333.3 ProposedSequence-FrameModelof3VSC-CoupledDERUnits . . . . 343.3.1 InterfaceVSCPositive-SequenceModel . . . . . . . . . . . . . . . 353.3.2 InterfaceVSCNegative-andZero-SequenceModels . . . . . . . . 363.3.2.1 Three-WireVSCConguration . . . . . . . . . . . . . . 363.3.2.2 Four-WireVSCConguration. . . . . . . . . . . . . . . 383.4 ImplementationoftheVSCUniedModelintheSFPS . . . . . . . . . . 383.4.1 AccommodatingtheVSC-CoupledDERModelintheSFPS . . . 383.4.2 CalculatingtheInternalParametersoftheInterfaceVSC. . . . . 393.4.3 MitigatingtheVSCOperatingLimitsViolation . . . . . . . . . . 413.4.3.1 MitigatingthePhaseCurrentLimitViolation . . . . . . 413.4.3.2 MitigatingtheReactivePowerLimitViolation . . . . . 413.4.3.3 MitigatingthePhaseModulationIndexLimitViolation 423.4.3.4 UpdatingtheInterfaceVSCReferenceSet-Points . . . . 423.5 Sequential-SFPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.6 3VSCModelValidation . . . . . . . . . . . . . . . . . . . . . . . . . . 44vii3.6.0.5 Case-1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.6.0.6 Case-2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.6.0.7 Case-3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.6.1 Case-4: Sequential-SFPSValidation. . . . . . . . . . . . . . . . . 493.7 ComputationEciencyoftheProposedSequential-SFPS. . . . . . . . . 493.8 SummaryandDiscussion. . . . . . . . . . . . . . . . . . . . . . . . . . . 524 Power-FlowModelofType-3WTGDERUnits 534.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 ClassicationofWindTurbineGenerators . . . . . . . . . . . . . . . . . 544.2.1 Type-1WTGU . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.2.2 Type-2WTGU . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.2.3 Type-3WTGU . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.2.4 Type-4WTGU . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3 Power-FlowModelsofType-3DERUnits: Review . . . . . . . . . . . . . 564.4 TheScopeoftheChapter . . . . . . . . . . . . . . . . . . . . . . . . . . 574.5 ProposedSequence-FrameModelofAType-3DERUnit . . . . . . . . . 584.5.1 ModelAssumptions. . . . . . . . . . . . . . . . . . . . . . . . . . 584.5.2 Positive-SequenceModelofType-3DERUnit . . . . . . . . . . . 584.5.2.1 PVmodeofoperation . . . . . . . . . . . . . . . . . . . 594.5.2.2 PQmodeofoperation . . . . . . . . . . . . . . . . . . . 614.5.3 Negative-SequenceModelofType-3DERUnit . . . . . . . . . . . 614.5.4 EquivalentNegative-SequenceModelofType-3DERUnit . . . . 614.6 EvaluatingType-3Negative-SequenceModelParameters . . . . . . . . . 624.6.1 CaseA:IdleSecondaryControlofType-3Unit . . . . . . . . . . 634.6.2 CaseB:BalancingRotorCurrents . . . . . . . . . . . . . . . . . . 634.6.3 CaseC:BalancingStatorCurrents . . . . . . . . . . . . . . . . . 634.6.3.1 CaseC-1: BalancingStatorCurrentsViaRSC . . . . . . 634.6.3.2 CaseC-2: BalancingStatorCurrentsViaGSC . . . . . 644.6.4 Case D: Mitigating Stator Real Power Double-Frequency Component 654.6.5 Case E: Mitigating Double-Frequency Electromagnetic Torque (Power)Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.6.6 CaseF:CoordinatedControlofGSCandRSC . . . . . . . . . . . 664.7 Type-3DERInternalParametersCalculation . . . . . . . . . . . . . . . 674.7.1 CalculatingPositive-SequenceInternalParameters . . . . . . . . . 684.7.1.1 GSCPositive-SequenceModulationIndex . . . . . . . . 68viii4.7.1.2 GSCPositive-SequenceCurrent . . . . . . . . . . . . . . 694.7.1.3 RSCPositive-SequenceModulationIndex . . . . . . . . 694.7.1.4 StatorPositive-SequenceCurrentCalculation . . . . . . 704.7.2 EvaluatingNegative-SequenceInternalParameters . . . . . . . . 704.7.2.1 GSCNegative-SequenceModulationIndex. . . . . . . . 704.7.2.2 GSCNegative-SequenceCurrent . . . . . . . . . . . . . 704.7.2.3 RSCNegative-SequenceModulationIndex. . . . . . . . 714.7.2.4 StatorNegative-SequenceCurrent . . . . . . . . . . . . . 714.8 ImplementingType-3Power-FlowModel . . . . . . . . . . . . . . . . . . 714.8.1 AccommodatingtheType-3DERModelintheSFPS. . . . . . . 714.8.2 EstimatingType-3MaximumOperatingLimits . . . . . . . . . . 714.8.2.1 EstimatingtheMaximumRSCModulationIndex. . . . 724.8.2.2 EstimatingMaximumStatorCurrents . . . . . . . . . . 724.8.2.3 EstimatingMaximumGSCCurrents . . . . . . . . . . . 724.8.3 DEROperationalLimits . . . . . . . . . . . . . . . . . . . . . . . 734.8.3.1 FulllingGSCPositive-SequenceModulationIndexLimit 734.8.3.2 FulllingGSCPositive-SequenceCurrentLimit . . . . . 744.8.3.3 FulllingRSCPositive-SequenceModulationIndexLimit 744.8.3.4 FulllingStatorPositive-SequenceCurrentLimit . . . . 744.8.3.5 UpdatingPositive-SequenceReferenceSet-Points . . . . 744.8.3.6 FulllingNegative-SequenceOperatingLimits . . . . . . 754.8.4 SequentialSPFS . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.9 SummaryandDiscussion. . . . . . . . . . . . . . . . . . . . . . . . . . . 775 ApplicationsandValidationofType-3WTGDERModel 785.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.2 Type-3DERModelValidation . . . . . . . . . . . . . . . . . . . . . . . . 785.2.1 Case-1: InactiveNegative-SequenceControllers . . . . . . . . . . 805.2.2 Case-2: BalancingRotorCurrentViaRSC. . . . . . . . . . . . . 805.2.3 Case-3: BalancingStatorCurrentViaGSC . . . . . . . . . . . . . 825.2.4 Case-4: MitigatingDouble-FrequencyStatorPowerOscillations . 825.2.5 Case-5: Mitigating Double-Frequency Torque Oscillations and Bal-ancingGSCCurrents . . . . . . . . . . . . . . . . . . . . . . . . . 835.3 Case6: ValidatingtheSequential-SFPSFeasibility . . . . . . . . . . . . 845.4 ApplicationoftheSequential-SFPStoBenchmarkDistributionSystems. 855.4.1 Case7: CIGREMVDistributionNetwork . . . . . . . . . . . . . 86ix5.4.2 Case8: IEEE34-BusTestSystem . . . . . . . . . . . . . . . . . . 875.5 ConvergenceAnalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.6 SummaryandDiscussion. . . . . . . . . . . . . . . . . . . . . . . . . . . 886 Power-FlowModelof1VSC-CoupledDERUnits 906.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.3 TopologiesandControlObjectivesofSingle-PhaseInterfaceVSC . . . . 916.4 Steady-StateModelofDual-StageSingle-PhaseInterfaceVSC . . . . . . 926.4.1 ModelAssumptions. . . . . . . . . . . . . . . . . . . . . . . . . . 926.4.2 Incorporating Single-Phase VSC-Coupled DER Units in Power FlowAlgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.4.2.1 VSCModelTailoredfortheSingle-PhaseBFSA. . . . . 936.4.2.2 VSCModelTailoredfortheSFPS . . . . . . . . . . . . 946.5 EvaluatingtheInterfaceVSCInternalParameters . . . . . . . . . . . . . 956.5.1 CalculatingtheVSCPhaseCurrent. . . . . . . . . . . . . . . . . 956.5.2 CalculatingtheVSCModulationIndex. . . . . . . . . . . . . . . 966.6 ImposingtheVSCOperationalLimits . . . . . . . . . . . . . . . . . . . 976.6.1 PhaseCurrentLimit . . . . . . . . . . . . . . . . . . . . . . . . . 986.6.2 ModulationIndexLimit . . . . . . . . . . . . . . . . . . . . . . . 986.6.3 PCBusVoltageLimit . . . . . . . . . . . . . . . . . . . . . . . . 996.6.4 UpdatingtheVSCReferencePowerSet-Points. . . . . . . . . . . 1006.6.5 SequentialPower-FlowAlgorithm. . . . . . . . . . . . . . . . . . 1006.7 1VSCModelValidation . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.7.1 ValidatingtheSingle-PhaseSequentialBFSA . . . . . . . . . . . 1006.7.1.1 Case-1: MaximumPhaseCurrentLimitViolation. . . . 1026.7.1.2 Case-2: MaximumModulationIndexLimitViolation . . 1036.7.2 ValidatingtheThree-PhaseSequential-SFPS. . . . . . . . . . . . 1046.7.2.1 Case-3: ViolatingtheMaximumPCCVoltageLimit . . 1056.7.2.2 Case-4: NoDERUnitsConnectedtotheViolatingBus . 1086.7.3 ComputationalEciencyoftheSequentialAlgorithmsaccommo-datingthe1VSC-CoupledDERModel . . . . . . . . . . . . . . 1096.8 SummaryandDiscussion. . . . . . . . . . . . . . . . . . . . . . . . . . . 1107 PowerFlowAnalysisofIslandedADNs 1117.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112x7.3 TheScopeoftheChapter . . . . . . . . . . . . . . . . . . . . . . . . . . 1137.4 ProposedDistributedSlackBus(DSB)Model . . . . . . . . . . . . . . . 1137.4.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137.4.2 ParticipationFactors . . . . . . . . . . . . . . . . . . . . . . . . . 1147.4.3 DistributedSlackBusModel . . . . . . . . . . . . . . . . . . . . . 1157.4.3.1 RealPowerBalanceEquation . . . . . . . . . . . . . . . 1157.4.3.2 ReactivePowerBalanceEquation. . . . . . . . . . . . . 1157.4.3.3 Super-PQ-BusEquation . . . . . . . . . . . . . . . . . . 1157.4.4 TheUpdatingEquation . . . . . . . . . . . . . . . . . . . . . . . 1167.4.5 DERGenerationLimits . . . . . . . . . . . . . . . . . . . . . . . 1177.5 ValidationsandCaseStudies . . . . . . . . . . . . . . . . . . . . . . . . . 1177.5.1 AssumptionValidation . . . . . . . . . . . . . . . . . . . . . . . . 1177.5.2 ImpactsofDeployingtheProposedDSBModel . . . . . . . . . . 1217.5.2.1 ReferenceBusPowerOutput . . . . . . . . . . . . . . . 1227.5.2.2 RealPowerLosses . . . . . . . . . . . . . . . . . . . . . 1237.5.2.3 VoltageProle . . . . . . . . . . . . . . . . . . . . . . . 1247.5.3 ImposingtheDERPowerCapacityConstraint. . . . . . . . . . . 1247.6 SummaryandDiscussion. . . . . . . . . . . . . . . . . . . . . . . . . . . 1258 Conclusions 1278.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1278.2 GeneralConclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1278.3 QuantiableConclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1288.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1298.4.1 MajorContributions . . . . . . . . . . . . . . . . . . . . . . . . . 1298.4.2 OtherContributions . . . . . . . . . . . . . . . . . . . . . . . . . 1298.5 DirectionsforFutureResearch. . . . . . . . . . . . . . . . . . . . . . . . 129Appendices 127ADataforTestSystems 131A.1 Six-BusTestSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131A.2 Three-PhaseCIGREMVTestSystem . . . . . . . . . . . . . . . . . . . 131A.3 CIGREMVSingle-PhaseRadialFeeder . . . . . . . . . . . . . . . . . . 133A.4 IEEE34-BusTestSystem . . . . . . . . . . . . . . . . . . . . . . . . . . 133A.5 ModiedIEEE34-BusTestSystem. . . . . . . . . . . . . . . . . . . . . 137xiBSteady-StatePower-FlowModelsofDistributionPowerLines 140B.1 Three-PhasePowerLines. . . . . . . . . . . . . . . . . . . . . . . . . . . 141B.2 Two-phasePowerLines. . . . . . . . . . . . . . . . . . . . . . . . . . . . 142CSteady-StatePower-FlowModelsofElectricalLoads 144C.1 ConstantPowerLoads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144C.1.1 Four-Wire/Three-WireWye-ConnectedLoads . . . . . . . . . . . 144C.1.2 Three-WireDelta-ConnectedLoads . . . . . . . . . . . . . . . . . 145C.2 ConstantCurrentLoads . . . . . . . . . . . . . . . . . . . . . . . . . . . 146C.2.1 Four-Wire/Three-WireWye-ConnectedLoads . . . . . . . . . . . 146C.2.2 Three-WireDelta-ConnectedLoads . . . . . . . . . . . . . . . . . 147C.3 ConstantImpendenceLoads . . . . . . . . . . . . . . . . . . . . . . . . . 148C.3.1 Four-Wire/Three-WireWye-ConnectedLoads . . . . . . . . . . . 148C.3.2 Three-WireDelta-ConnectedLoads . . . . . . . . . . . . . . . . . 149DSchematicDiagramsofthePSCAD/EMTDCModels 150Bibliography 153xiiListofTables2.1 Sequence-Frame, Fundamental-Frequency, Steady-State Model of A Three-PhasePowerTransformer . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1 VSCConstant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2 ParametersofG1andG2ofFig. 3.6(Sbase=1000kVA) . . . . . . . . . 453.3 Power-FlowResultsofCase-1 . . . . . . . . . . . . . . . . . . . . . . . . 473.4 Power-FlowResultsofCase-2 . . . . . . . . . . . . . . . . . . . . . . . . 473.5 Power-FlowResultsofCase-3 . . . . . . . . . . . . . . . . . . . . . . . . 483.6 Power-FlowResultsofCase-4 . . . . . . . . . . . . . . . . . . . . . . . . 493.7 ComparisonBetweentheSequential andtheNon-Sequential Power-FlowAlgorithms inTerms of theTotal Number of JacobeanMatrixEvalua-tions/Inversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.1 ParametersoftheType-3Negative-SequenceModelFortheSixCasesofSection4.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.1 ParametersofG2ofFig. 5.1(Sbase=1000kVA) . . . . . . . . . . . . . . 795.2 Power-FlowResultsofCase-1 . . . . . . . . . . . . . . . . . . . . . . . . 805.3 Power-FlowResultsofCase-2 . . . . . . . . . . . . . . . . . . . . . . . . 815.4 Power-FlowResultsofCase-3 . . . . . . . . . . . . . . . . . . . . . . . . 825.5 Power-FlowResultsofCase-4 . . . . . . . . . . . . . . . . . . . . . . . . 835.6 Power-FlowResultsofCase-5 . . . . . . . . . . . . . . . . . . . . . . . . 845.7 LoadproleusedinSection5.3(Vbase=13.8kV,Sbase=1MVA) . . . . . 845.8 Power-FlowResultsofCase-6: OperationalLimitsofG2areDiscarded . 855.9 Power-FlowResultsofCase-6: OperationalLimitsofG2areImposed . . 866.1 Parameters of G1, G2, and G3 of Fig. 6.5 (Vbase= 7.2 kV and Sbase= 100kVA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.2 Power-FlowResultsofCase-1: MaximumPhaseCurrentLimitisDiscarded102xiii6.3 Case-1: UpdatedReal andReactivePowerSetPointsof thethreeDERunits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.4 Power-FlowResultsofCase-1: MaximumPhaseCurrentLimitisImposed 1036.5 Power-FlowResultsofCase-2: MaximumModulationIndexLimitisDis-carded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.6 Case-2: UpdatedReal andReactivePowerSetPointsof thethreeDERunits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.7 Power-FlowResultsofCase-2: MaximumModulationIndexLimitisIm-posed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.8 Case-3: DERPhaseDistribution* . . . . . . . . . . . . . . . . . . . . . . 1066.9 Case-3: Three-PhaseVoltageProleWithoutConsideringtheMaximumBusVoltageConstraint. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.10 Case-3: UpdatedRealandReactivePowerSetPointsoftheDERUnits.Sbase= 1MVA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.11 Case-3: Three-PhaseVoltageProleAfterAdjustingtheDERPowerSetPointsUsing(6.22) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.12 Case-4: Three-PhaseVoltageProleWithoutConsideringtheMaximumBusVoltageConstraint. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.13 Case-4: Three-PhaseVoltageProleAfterAdjustingtheDERPowerSetPointsUsing(6.22) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.14 Case-4: UpdatedReal andReactivePowerSetPointsoftheDERUnitsattheViolatingBusses. Sbase= 1MVA. . . . . . . . . . . . . . . . . . . 109A.1 Power Lines Phase-Frame Parameters (in pu) of the Study System of Fig.A.1(Vbase=13.8kV,Sbase=1000kVA) . . . . . . . . . . . . . . . . . . . 132A.2 LoadsoftheStudySystemofFig. A.1(Vbase=13.8kV,Sbase=1000kVA) 132A.3 PowerLinesPhase-FrameParameters(inohms)of theStudySystemofFig. A.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134A.4 Loads(inkVA)oftheStudySystemofFig. A.2 . . . . . . . . . . . . . 134A.5 Loads(inkVA)oftheStudySystemofFig. A.3 . . . . . . . . . . . . . 135A.6 PowerLinesPhase-FrameParametersoftheStudySystemofFig. A.4 . 136A.7 Loads(inkVA)oftheStudySystemofFig. A.4 . . . . . . . . . . . . . 138xivListofFigures1.1 SchematicdiagramofanActiveDistributionNetwork. . . . . . . . . . . 21.2 SchematicdiagramofanADNoperatingasavirtualpowerplant . . . . 41.3 DERModelValidationMethodology . . . . . . . . . . . . . . . . . . . . 81.4 Schematicdiagramofthethesislayout . . . . . . . . . . . . . . . . . . . 92.1 Classicationofthree-phasePFAmethods . . . . . . . . . . . . . . . . . 132.2 Sequence-frame, fundamental-frequency, steady-state model of a three-phase directlyconnectedsynchronous generator for the SFPS: (a) thepositive-sequencemodel, (b) thenegative-sequencemodel, (c) thezero-sequencemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Equivalent-modelofathree-phasedistributionline . . . . . . . . . . . 192.4 Decoupledsequence-frame,fundamental-frequency,steady-statemodelofathree-phasedistributionline: (a)thepositive-sequencemodel, (b)thenegative-sequencemodel,(c)thezero-sequencemodel . . . . . . . . . . . 212.5 Schematicdiagramofaconstantpower/currentloadconnectedtoBus-k 232.6 Decoupledsequence-frame,fundamental-frequency,steady-statemodelofaconstantpower/currentload . . . . . . . . . . . . . . . . . . . . . . . . 232.7 ModelofaconstantimpedanceloadconnectedtoBus-k: (a)phase-framemodel,(b)sequence-framemodel . . . . . . . . . . . . . . . . . . . . . . 242.8 Decoupledsequence-framemodelofaconstantpower/currentload. . . . 252.9 FlowchartoftheSFPSalgorithm. . . . . . . . . . . . . . . . . . . . . . 262.10 Zero-sequence blocking scenario: (a) phase-frame network, (b) zero-sequenceimpedancediagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1 Schematicdiagramofthree-phaseVSC-coupledDERunit . . . . . . . . 333.2 Proposedsequence-frame, fundamental-frequency, steady-statemodel ofa3interfaceVSC, (a) positive-sequencemodel, (b) negative-sequencemodel,(c)zero-sequencemodel . . . . . . . . . . . . . . . . . . . . . . . 343.3 AlternativeVSCoutputltercongurations . . . . . . . . . . . . . . . . 37xv3.4 Equivalentsequence-framecircuitsofaVSC-coupledDERunit . . . . . . 393.5 FlowchartoftheproposedSequential-SFPS . . . . . . . . . . . . . . . . 433.6 Singlelinediagramofthesix-bustestsystem . . . . . . . . . . . . . . . 453.7 Comparison betweenthesequentialandthe non-sequentialpower-owal-gorithmsintermsoftheconvergencespeed. . . . . . . . . . . . . . . . . 514.1 Schematic diagrams of the four types of wind-turbine generating systems:(a)Type-1,(b)Type-2,(c)Type-3,and(d)Type-4 . . . . . . . . . . . . 554.2 DetailedschematicdiagramofaType-3wind-drivengenerationsystem . 594.3 Proposedsequence-frame, fundamental-frequency, steady-statemodel ofType-3 DER unit, (a) positive-sequence model, (b) negative-sequence model,(d)equivalentnegative-sequencemodel. . . . . . . . . . . . . . . . . . . . 604.4 Type-3sequence-framecircuit usedtocalculatetheconverters internalparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.5 Flow chart of the proposed Sequential-SFPS algorithm including the Type-3 DER constraints evaluation and the proposed reference-set point updat-ingstrategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.1 Singlelinediagramofthesix-bustestsystem . . . . . . . . . . . . . . . 795.2 Voltageproleof theCIGREdistributionnetworkwith2Type-3basedDERunits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.3 Voltage prole of the IEEE 34-bus distribution feeder with 3 Type-3 basedDERunits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.4 Convergence patternof the Sequential-SFPSalgorithmaccommodatingtheType-3DER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.1 Schematicdiagramofdual-stagesingle-phaseVSC-coupledDERunit . . 926.2 Steady-state,fundamental frequencyBFSA modelof asingle-phase VSC-coupledDERunit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.3 Steady-state,fundamentalfrequencySFPSmodelofasingle-phaseVSC-coupledDERunit, (a)positive-sequencemodel, (b)zero- andnegative-sequencemodels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.4 Equivalentcircuitofthesingle-phaseVSC-coupledDERunitusedtocal-culatetheVSCinternalparameters . . . . . . . . . . . . . . . . . . . . . 966.5 Singlelinediagramoftheradialtestfeeder. . . . . . . . . . . . . . . . . 1016.6 Singlelinediagramofmodiedthree-phaseIEEE-34busradialfeeder . . 1057.1 SinglelinediagramoftheCIGREdistributionbenchmarknetwork . . . . 119xvi7.2 SinglelinediagramofthemodiedIEEE34-busfeeder . . . . . . . . . . 1197.3 ValidatingtheassumptionstatedinSection7.4.1 . . . . . . . . . . . . . 1207.4 Eectof theproposedDSBmodel ontheapparentpoweroutputof thereferencebusDERunit. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1227.5 Eect of the proposed DSB model on the real-power losses of the IEEE-34bustestfeeder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.6 EectoftheproposedDSBmodel onthevoltageproleoftheIEEE-34bustestfeeder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247.7 Eect of imposing the DER power capacity constraint on the PQ-controlledDERoutput . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125A.1 Singlelinediagramofthesix-bustestsystemusedinChapters3and5 . 132A.2 Singlelinediagramof thethree-phaseCIGREMVdistributionnetworkusedinChapters3,5,and7 . . . . . . . . . . . . . . . . . . . . . . . . . 133A.3 SinglelinediagramoftheCIGREMVsingle-phaseradial feederusedinChapter 6. All power lines areidentical. Theseries impedanceof anypowerline=0.219+0.14 . . . . . . . . . . . . . . . . . . . . . . . . . 135A.4 Single line diagram of the IEEE 34-bus distribution system used in Chap-ters3and5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136A.5 Single line diagram of the Modied IEEE 34-bus distribution system usedinChapters6and7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137B.1 Phase-framemodelofthree-phasepowerline . . . . . . . . . . . . . . . 140C.1 Phase-framemodelofthree-phasefour-wire,constantpowerload . . . . 144C.2 Phase-framemodelofthree-phasedelta-connectedconstantpowerload . 145C.3 Phase-framemodelofthree-phasefour-wire,constantcurrentload . . . . 147C.4 Phase-framemodelofthree-phasedelta-connectedconstantcurrentload 148C.5 Phase-framemodelofthree-phasefour-wire,constantimpedanceload . . 148C.6 Phase-framemodelofthree-phasedelta-connectedconstantimpedanceload149D.1 PSCADmodelofthetestsysteminFig. A.1. . . . . . . . . . . . . . . . 150D.2 PSCADmodelofathree-phaseVSC-coupledDERunit. . . . . . . . . . 151D.3 PSCADmodelofathree-phaseType-3basedDERunit . . . . . . . . . . 152xviiNomenclatureAcronymsADN ActiveDistributionNetworkADS ActiveDistributionSystemAVR AutomaticVoltageRegulatorBESS BatteryEnergyStorageSystemBFSA Backward-ForwardSweepAlgorithmDER DistributedEnergyResourceDFAG DoublyFedAsynchronousGeneratorDG DistributedGeneratorDNO DistributionNetworkOperatorDS DistributedStorageDSB DistributedSlackBusFC FuelCellsFIT Feed-in-TarrifG-S Gauss-SeidelMethodICT InformationandCommunicationTechnologyMPPT MaximumPowerPointTrackingMT MicroturbineN-R Newton-RaphsonMethodPC PointofConnectionPEV PluggedElectricVehiclePFA Power-FlowAnalysisxviiiPHEV PluggedHybridElectricVehiclePLL Phase-lockedloopPQ Power-controlledPV Real-powerandvoltage-controlledSEMS SmartEnergyManagementSystemSFPS Sequence-FramePower-owSolverSM SmartMeterSPV SolarPhotovoltaicSPWM SinusoidalPulse-WidthModulationSSB SingleSlackBusSVM SpaceVectorModulationTDSG Three-phaseDirectly-connectedSynchronousGeneratorV2G Vehicle-to-GridVPP VirtualPowerPlantVSC Voltage-sourcedconverterWTG WindturbinegeneratorWTGU WindturbinegeneratingunitSymbolsInthisthesis,italicsymbolsindicatedphasorquantities,e.g.,I,whilemulti-dimensionalmatricesareshowninbolduppercase,e.g.,I.xixChapter1Introduction1.1 ActiveDistributionNetworks1.1.1 DenitionsDrivenbytechnical advancement, political readiness, social awareness, andeconomicalincentives,theanticipatedproliferationofdistributedenergyresources(DER)units,in-cludingdistributedgeneration(DG), distributedstorage(DS), andcontrollableloads,into the current distribution grids close to the load sites, is emerging as a complementaryinfrastructure to the traditional central power plants [1, 2]. Unless optimally coordinatedandecientlyintegrated,thehigh-depthofDERpenetrationwillbringmanytechnicalchallenges interms of planningandoperationof thedistributionnetworks including,amongothers, protectionmal-coordination, voltagelevel violations, powerqualitycon-cerns, and increasing line losses. The need for ecient and safe DER integration schemesbringsabouttheconceptoftheactivedistributionsystems.The active distributionsystem(ADS), alsoknownas active distributionnetwork(ADN),isthenewgenerationoftodaysdistributionnetworkswheretheDERunitsareoptimallyoperatedandecientlyintegrated. TheCIGREC6.11workinggroupdenesthe ADN as a distribution network whose operator can remotely and automatically con-trol the DER units and network topology to eciently manage and optimally utilize thenetworkassets [3]. IntheADN, thepower owbetweenbusses is bidirectional, andvariablesaremeasured(orestimated)andcontrolledbasedonacentralizedandintelli-gentsystem. Thecentral control systemiscapableof makingdecisionsandoperatingtheADNbasedonmonitoringthenetworkconditions. Ingeneral, theultimategoal oftheADNis to(i) enhancetheDERobservabilityandcontrollability, (ii) deliver costecientintegrationoftheDERunitsintothedistributionnetwork, and(iii)maximize1Chapter1. Introduction 2Figure1.1: SchematicdiagramofanActiveDistributionNetworkthetechnical andeconomical benetsachievedbyboththeDERownersandthehostgrid[46].1.1.2 EnablingTechnologiesAschematicdiagramofanADNisdepictedinFig. 1.1. AsrepresentedinFig. 1.1,theenablingtechnologiestorealizetheADNinclude: Informationandcommunicationtechnology(ICT)infrastructuretoestablishfastandreliabletwo-waycommunicationbetweentheDERunits,circuitbreakers,in-terconnectionswitches, smartmeters, andthelocal energymanagementsystem.IntheADN,thebi-directionalcommunicationisessentialformonitoring,control,Chapter1. Introduction 3and protection. The most deployed communication standards are the IEC61850 [7]andtheIEEE802.15(ZigBEE)[8]standards. Smartmeters(SM)installedattheloadpremisestomonitorandsendreal-timedata of the load consumption,voltage prole,and total harmonic distortion to thedistributionnetworkoperator(DNO). Inaddition, proliferationof theSMallowsthe DNO to control the load prole by utilizing the demand response functions [9]. Storage devices, e.g., batteries, to maximize the utilization of the renewable energyresources,e.g.,photovoltaicandwindenergy. Integratingthestoragedeviceswithrenewable-basedDERunits overcometheintermittent natureof theseresourcesandenhancetheirdispatchability[10, 11]. Power electronicconverters tointerfacetheDERunits tothegrid. Inparticu-lar, AC-DCandDC-ACvoltage-sourcedconverters (VSC) arethemost widely-adoptedinterface mediumfor DERunits, e.g., batteryenergystorage systems,ywheelenergystoragesystems,fuelcells,solarphotovoltaicunits,variable-speedwind turbine generators with full-scale and partial power-electronic converters, andmicro-turbinesystems[12]. Real-time centralized and decentralized control techniques to optimally control thelargeeetofDERunits[13, 14].1.1.3 ADNOperatingModesTheADNcanoperateintwodistinctmodes,i.e., Utility-ConnectedMode: Inthegrid-tiedscenarios, theSEMSoptimallycontrolstheADNcomponents tomaximizethetechnical andeconomical benets of theexisting DER eet. One way of achieving this is to coordinate the ADN apparatustoprovideapre-speciedperformanceproleat thepoint of commoncoupling(PCC),i.e.,theADNoperatesasavirtualpowerplant(VPP)thatiscomparabletoaconventional plant[1517]. Aconceptual representationoftheVPPisgiveninFig. 1.2. IslandedMode: Whentheinterconnectionswitchof Fig. 1.1opens, eitherinten-tionally or in response to an accidental disturbance, the entire ADN, or pre-speciedpartsofit,shouldbeabletooperatesafelyandreliablyintheautonomousmode,i.e.,formanislandedmicrogrid(grid)[18, 19].Chapter1. Introduction 4Figure1.2: SchematicdiagramofanADNoperatingasavirtualpowerplant1.1.4 SmartEnergyManagementSystemThebrainof theADNis theSmart EnergyManagement System(SEMS), Fig. 1.1.SEMS is a local energy and power control and management center that collects real-timedataabout (i) thestatus of theADNcomponents, e.g., DERpower output, on-loadtapchangers,circuitbreakers,theinterconnectionswitch,(ii)loadandvoltageproles,and(iii)lineows. Several intelligentdistributionautomationfunctionsareintegratedwiththeSEMS, e.g., faultdetectionandisolation, networkreconguration, congestionmanagement, blackout and brownout management, optimal asset utilization, and optimaldispatchandcontrol[2023].1.2 Statement of the ProblemandThesis Motiva-tionsFortheSEMStoconductalltheaforementioneddistributionautomationfunctions,ac-curatepower-owanalysis(PFA)resultsmustbeavailable. ThePFAisthekernel oftheSEMStoguaranteeasatisfactoryandreliableoperationof theADNbycalculat-ingtheappropriatevoltageandpowerreferencesetpointsforall theDERunits[24].Inaddition, duringthe planningphase, PFAis requiredto(i) determine the eectsof adding/removingdierent apparatus, i.e., loads, interconnections, generation, volt-ageregulators,andVArdevicesontheADNperformanceand(ii)evaluatetheoptimalsizingof dierentpowersystemcomponents. Moreover, thePFAisusedtoprovideaChapter1. Introduction 5preciseandecient initializationapproachfor eigenanalysis, transient stability, andelectro-magnetictransientsprograms[25, 26].1.2.1 LackofToolsThePFAof largepower systemsisamaturesubject andawidearrayof productiongradepower-owsoftwaretoolsthatrepresentvariouspower-systemcomponents, e.g.,HVDCconvertersandFACTScontrollers, arewidelyavailable[2729]. However, suchtoolsareneithertailoredfornoradequatelyaddressthePFArequirementsoftheADN.Themainreasonsare: theinherentlargedegreeof imbalanceof distributionnetworksdueto(i)single-phaseandtwo-phaseloads, (ii)single-phaseDERunits, (iii)untransposedlines,and(iv) single-phase laterals [30]. As such, the distributionlevel PFAshouldsimultaneouslyaddressthethreephases, the presence of three-phase VSC-based DER units which can include (i) three-wirecongurations, (ii) four-wire congurations, and (iii) various control strategies thatcanrespondtoselectedsequence-framevoltageandcurrentcomponents[3137].Some productiongrade software tools provide three-phase power-owanalysis fordistribution networks accommodating DER units [3840]. However, these tools have thefollowingshortcomings: Thecommercially-availablepower-owtoolsaredevelopedforradial distributionnetworks. However,theADNmaybemesh-connectedtomaximizetheDERben-ets. Thus, the available tools cannot handle multi-directional power-owandgeneralnetworktopologies[34, 41]. Theavailablethree-phasePFAalgorithmsadoptthephase-frameratherthanthesequence-frame. As will be detailed in Chapter 2, sequence-frame based PFA algo-rithmsareeasiertoimplement, computationallymoreecient, andprovideexi-bilitytomodelVSC-coupledDERunitsthantheirphase-framecounterparts. The available three-phase power-ow engines are not suitable for the islanded gridoperatingmode. Thereasonisthatthesetoolslackanappropriatethree-phasedistributedreal- andreactive-slackbus model toconduct the PFA. As will bediscussedinChapter 7of this thesis, distributedslackbus(DSB)-basedPFAisvital to conduct the power-ow analysis of the islanded grid operating mode sinceChapter1. Introduction 6itguaranteesthattheDERunitconnectedtothereferencebusneednottobeaninnitepowersource[42].1.2.2 LackofDERModelsToobtainunerringthree-phasePFAresults, detailedandaccuratethree-phase, steady-state, fundamental-frequency DER models are required. These models should adequatelyaddress (i) dierent DERtypes, i.e., rotatingmachine-basedDERunits, single-phaseVSC-coupled, three-phase three-wire VSC-coupled, and three-phase four-wire VSC-coupledDERunits, (ii)dierentcontrol objectivesunderbalancedandunbalancedgridcondi-tions,i.e.,voltage/frequencyandreal/reactive-powercontrol,speciccontrolactionstomitigate the voltage and/or current unbalance at the point of connection, and (iii) the op-erating limits of the interface VSC and the host DER unit, e.g., the converter modulationindexlimit,phasecurrentlimit,powercapacitylimit,andterminalvoltagelimit.ModelingofelectronicconvertersforinterfacingDERunitsisoneofthemostchal-lengingproblems, andis becominganareaof active researchrelevant topower owstudies[43]. Developingdetailedsingle-andthree-phaseVSC-coupledDERmodelsfortheADNpower-owanalysisapplicationshasnotbeensystematicallyaddressedinthetechnical literature. Thereasons are(i) thecurrent relativelylimitedproliferationoftheelectronically-coupledDERunitsinthedistributiongridsand(ii)mostofthecur-rentlyavailable DERunits are basedonconstant speeddirectly-coupledthree-phasesynchronousgenerators. Theavailableelectronically-coupledDERmodels, bothinthetechnical literatureandtheproduction-gradesoftwaretools, (i) assumeonlypositive-sequenceDERrepresentation, (ii)donotrepresentthesingle-phaseVSC-coupledDERunits, (iii) do not address most of the aforementioned control objectives, and (iv) neglecttheinterfaceconverteroperatinglimits[31, 32, 34, 4454].1.3 ThesisObjectivesBasedonthediscussionofSection1.2,thethesisobjectivesare:1. Develop detailed and accurate steady-state, fundamental-frequency models of VSC-basedDERunitsforthree-phasePFA.TheDERunitsunderstudyinclude: DER units interfaced via a three-phase front-end three-wire or four-wire VSC, Variable-speedwind-turbine-generator(WTG)-basedDERunitsdeployingaChapter1. Introduction 7three-phase doubly-fed asynchronous generator (DFAG), also known as Type-3WTG, Single-phaseVSC-coupledDERunits,The developed DER models should address (i) balanced and unbalanced power-owscenarios,(ii)variousVSCcontrolstrategiesunderbalancedandunbalancedgridconditions, and(iii)theoperatinglimitsandconstraintsoftheVSCanditshostDERunit.2. Developathree-phasePFAtool for PFAof analysis andreal-timeoperationofADNs. Thedevelopedprogrammust: befastandaccurateforthereal-timemanagementandcontroloftheADN, incorporatethedevelopedsingle-phaseandthree-phaseDERmodels,includ-ing the interface VSC operating constraints in a computationally-ecient man-ner, becapableof analyzingdierentnetworktopologies(radial, weaklymeshedandmeshednetworks),includinghighdegreesofunbalance, accommodatemodelsofmulti-phasepowerlinesandsingle-phaselaterals, contain models of single, two, and three-phase loads with dierent connections,includingconstantimpedance,current,andpower(ZIP)loadmodels, accommodate models of three-phase distributiontransformers withvariousconnections, including the phase shift introduced by dierent transformer con-nections.3. Developathree-phasedistributedslackbus (DSB) model. ThedevelopedDSBmodel will be integrated with the aforementioned tool to conduct three-phase PFAofislandedADNs.1.4 Methodology1.4.1 ModelingMethodologyThe developed DER models and the three-phase PFA tool are developed in the sequence-componentsframe. Themeritsofthesequence-framecomparedtothephase-frameforthethree-phasePFAaredetailedinChapter2ofthisthesis.Chapter1. Introduction 8Figure1.3: DERModelValidationMethodology1.4.2 ValidationMethodologyThedevelopedthree-phasePFAtool, includingdierent DERandADNcomponentsmodels, isimplementedintheMATLABplatform. Toverifythenumerical accuracyof the developed three-phase VSC-coupled and Type-3 WTG-based DER models, a rela-tively small three-phase unbalanced test system, with unbalanced loads and untransposedlines, is used where a DER unit is connected to one of the system busses. The reason forselectingthisrelativelysmall systemistobeabletosimulatethesystem, includingallthe required details, in time-domain in the PSCAD/EMTDC platform, and consequently,validatethenumericalaccuracyoftheDERmodelsandtheproposedthree-phasePFAtool. The detailed time-domain model of the study system, including the DER convertersandcontrollers, isdevelopedinthePSCAD/EMTDCtime-domainsoftwaretoserveasthebenchmarkforvalidatingtheproposedDERmodels. ThePSCAD/EMTDCsim-ulationenvironmentisselectedsinceitcontainsdetailedandwidelyusedtime-domainmodels of power system components and allows representation of various controls for theVSC-interfacedDERunits. Thesingle-linediagramandparametersofthetestsystemaregiveninAppendixA.1. Aconceptual representationof theDERmodel validationmethodisdepictedinFig. 1.3.1.5 ThesisStructureAschematicdiagramofthethesislayoutisshowninFig. 1.4. Thisthesisisstructuredasfollows.Chapter1. Introduction 9Figure1.4: Schematicdiagramofthethesislayout1.5.1 Chapter2: AThree-PhaseSequence-FramePower-FlowSolver(SFPS)ToolInthis chapter, afast, accurate, androbust three-phasepower-owanalysis softwaretool is developedfor ADNapplications, i.e., the SFPStool. Three-phase, sequenceframe-based,fundamental-frequency,steady-statemathematicalmodelsoftransformers,loads, andpower-linesaredescribedandaccommodatedintheSFPS.TheSFPSisthehubintowhichdierenttypesof DERunits, includingtheirinterfacingmedia, controlcapabilities, andoperatinglimits, areaccommodatedandtested. Thiscontributionisdetailed in Chapter 2, and published in an IEEE Power Delivery Transactions paper [55]andareviewedconferencepaper[56].1.5.2 Chapter 3: Steady-State Models of Three-Phase VSC-CoupledDERUnitsInthis chapter, a uniedsequence frame-based, fundamental-frequency, steady-statemathematical model of aDERunitinterfacedtothegridviaathree-phasefront-endVSCisproposedanddeveloped. AsdetailedinSection1.1.2,thedevelopedmodelrep-resentsawidespectrumof DERtypes, e.g., batteryenergystoragesystems, fuel cells,solar photovoltaicunits, variable-speedwindturbinegenerators withfull-scalepower-Chapter1. Introduction 10electronic converters (Type-4 WTG), and micro-turbine systems. The proposed model isgeneric since it accurately addresses (i) three-wire and four-wire VSC congurations, (ii)balancedandunbalancedpower-owscenarios,(iii)variousVSCcontrolstrategies,and(iv)theoperatinglimitsandconstraintsof theinterfaceVSCandthehostDERunit.ThedevelopedmodelisincorporatedwiththeSFPStoolofChapter2.Inaddition, anenhancementtothebasicSFPSalgorithm, presentedinChapter2,isproposedtomodel andimposetheVSCoperatingconstraintsinacomputationally-ecient way. The enhanced algorithm is called the Sequential-SFPS. The computationaleciencyof theSequential-SFPSis veriedbycomparingtheconvergencepatternofthe proposed algorithm against other reported methods. This contribution is detailed inChapter3andispublishedinanIEEEPowerDeliveryTransactionspaper[57].1.5.3 Chapters4and5: Steady-StateModelsofType-3WTG-BasedDERUnitsIn Chapter 4, I applied the VSC model of Chapter 3 to develop a detailed sequence frame-based, fundamental-frequency, steady-state mathematical model of a Type-3 WTG-basedDERunitsubjectedtounbalancedvoltageandcurrentconditions. TheproposedmodelisincorporatedwiththeSFPS, andtheSequential-SFPSof Chapter3isextendedtoaccommodate all the operating limits of the WTG unit and its associated VSCs. A widearrayof casestudiesareconductedinChapter5toverifythenumerical accuracyandthe computational eciency of the Sequential-SFPS accommodating the proposed Type-3WTG-basedDERmodel. Atwo-partIEEESustainableEnergyTransactionspaper,describing the details of the proposed DER model and its applications, has been acceptedforpublication[58, 59].1.5.4 Chapter 6: Steady-State Models of Single-Phase VSC-CoupledDERUnitsIn this chapter, I developed fundamental-frequency, steady-state models of a single-phaseVSC-coupled DER unit for the PFA of single-phase laterals and three-phase distributionfeeders. TheproposedmodelsrepresentdierentVSCoperatingmodesandconstraints.Thesequential approachof Chapter3isdeployedtoimposetheVSCconstraintsintothepower-owalgorithm. Theproposedmodelsareintegratedwith(i)theSFPSforthePFAof three-phasenetworks and(ii) asingle-phasePFAalgorithmtostudytheradial single-phaselaterals. Several casestudiesareconductedtoevaluateandverifyChapter1. Introduction 11theaccuracyoftheproposedmodel. ThiscontributionisdetailedinChapter6,andanIEEEPowerDeliveryTransactionspaper,describingthedetailsandapplicationsoftheproposedmodels,hasbeensubmittedandisinreview[60].1.5.5 Chapter7: Three-PhaseDistributedReal-andReactive-SlackBusModelInthischapter, anovel three-phasesequenceframe-basedDSBmodel isdescribedandaugmented with the SFPS of Chapter 2 to conduct three-phase PFA for islanded ADNs.Unlike the existing DSB models, the proposed formulation (i) simultaneously distributesthe real andreactive power slackand(ii) involves DERunits withdierent controlstrategies in slack compensation. A wide array of case studies is conducted to investigatethe impacts of distributing the real and reactive power slack using the three-phase DSB-SFPS tool. This contribution is detailed in Chapter 7. In addition, an IEEE Smart GridsTransactionspaper[61]issubmittedtoreportthiscontribution,andisinreview.1.5.6 Chapter8: ConclusionsThemainconclusions of thethesis andthesuggestions for futureresearchtopics arelistedinChapter8.Chapter2Sequence-FramePower-FlowSolver(SFPS)12.1 IntroductionThis chapter lays the foundation of a three-phase power-ow algorithm, in the sequence-components frame, for ADNapplications. The algorithmis calledSequence-FramePower-ow Solver (SFPS). Basic ADN components (power-lines, transformers, and loads)aremodeledinthesequence-frameandaccommodatedintheSFPStoobtainanaccu-rate three-phase steady-state solution. The SFPS is the hub to which models of dierenttypesofelectronically-coupledDERunits,includingtheirinterfacingmedia,controlca-pabilities, and operating limits under balanced and unbalanced power-ow scenarios, areaccommodated to construct the integrated 3 power-ow analysis (PFA) tool, as will bedetailedinthenextfourchapters.2.2 Three-PhasePower-FlowAnalysis: Critical Re-viewTheconceptof three-phasePFAhasbeenextensivelyaddressedintheliterature. AsshowninFig. 2.1, the relatedalgorithms are classiedaccordingto(i) the network1The work presented in this chapter has been published and appears in M.Z. Kamh and R. Iravani,UnbalancedModel andPower-FlowAnalysisofMicrogridsandActiveDistributionSystems,IEEETrans. Power Delivery, vol.25, no.4, pp.2851-2858, Oct. 2010. An earlier version of this work has beenpresentedandappearsinM.Z. KamhandR. Iravani, Three-PhaseModel andPower-FlowAnalysisof Microgrids and Virtual Power Plants, Proc. oftheFourthCanadianCIGREConferenceonPowerSystems, Toronto, Ontario, Canada, October 2009.12Chapter2. Sequence-FramePower-FlowSolver(SFPS) 13structure, i.e., radial [48, 6268] andgeneral networktopologies[25, 30, 6975] and(ii)theadoptedreferenceframetomodelthepowersystemapparatus,i.e.,thephase-frame[25, 48, 6271, 75],thesequence-frame[72, 73],andahybridofbothframes[74].Figure2.1: Classicationofthree-phasePFAmethods2.2.1 Three-PhasePower-FlowAlgorithmsforRadialNetworkTopology2.2.1.1 Backward-ForwardSweepAlgorithm(BFSA)BFSA, alsoknownas themodiedladder networkalgorithm[62], exploits theradialtopology and unidirectional power ow of distribution feeders to evaluate the power-owsolutionbysuccessivelyapplyingKirchoscircuitlaws. Bothunbalancedthree-phaseandbalancedsingle-phasepower-owproblemsaredoable. Balancedandunbalancedloads as well as shunt elements aremodeledas anequivalent current injection. Thebackward sweep calculates the currents through each line segment. Using these currents,thevoltageofeachnodeiscalculatedintheforwardsweep. Theprocesscontinuesuntilconvergenceisachieved. Thetechniqueasdescribedin[62] canaccommodateneithervoltage-controlled(PV)bussesnorweaklymeshednetworksinthealgorithm.2.2.1.2 Compensation-BasedAlgorithmsCurrent compensation PFA algorithm is developed in [63] to address weakly-meshed dis-tributionnetworkswithfewnumberof PVbuses. Thealgorithmisbasedonbreakingtheloopsatanumberofbreakpointstoconvertthenetworktoaradialequivalent. TheChapter2. Sequence-FramePower-FlowSolver(SFPS) 14BFSA is then used to conduct the PFA of the equivalent radial network, where the break-pointscurrentinjectionsarecalculatedusingthemulti-portcompensationmethod. ThePV busses are modeled as PQ busses with negative real power consumption and variablereactivecurrentinjection. Subsequenttoeachiteration, thereactivecurrentinjectionis updatedusingthesecant methodtoregulatethevoltagemagnitudeof PVnodes.However,thisalgorithmdivergesasthenumberofPVbusesand/orloopsincreases.AnimprovedPV-busmodel isintroducedin[64] and[65]. ThenewPVmodel isalinearizedapproximationof the automatic voltage regulator (AVR) of synchronousgenerators. ThePVbus is initializedas PQbus withreactivepower injectionset totheminimum. Thealgorithmconsistsof threeindependentiterativesubroutines: theBFSA, thebreakpointvoltagecompensation, andthePVnodevoltagecompensation.Powercompensation,insteadofcurrentcompensation,isproposedin[48]toadjustthePV bus voltage. The convergence rate of this algorithm is very sensitive to the degree ofunbalance[65]. Inaddition, thecompensation-basedPFAalgorithmwasnottestedonreal-sizedistributionnetworks.Dierentloadconnectionsandshuntcapacitorsaremodeledin[66, 67] usingtheirequivalent current injections, and are incorporated in the compensation-based PFA method.Power-controlled (PQ) DER units are modeled as balanced negative constant power loads.High-resolution one-minute time-series data is used to model the loads in [68]. However,allthelinesareassumedtobeperfectlytransposedandbalanced.2.2.2 Three-Phase Power-Flow Algorithms for General NetworkTopologiesToenableecientandeconomicDERintegration, radial topologyislesslikelytosuittheADNrequirements. Assuch,Newton-Raphson(N-R)[25, 6974]andGaussand/orGauss-Seidel (G-S) [30, 75] methods aremoresuitablefor general networktopologies.ApplyingthesemethodsforbalancedPFAiswellestablishedintheliterature[76]. Theformerisknowntohavegoodconvergencecharacteristics,butthecontinuousupdateoftheJacobianmatrixmakesthisapproachlessattractive. Ontheother, G-Smethodisknowntohaveoscillatorynatureandmorevulnerabletodivergence.2.2.2.1 Newton-RapshonMethodIn[25], ageneralizedthree-phasepower-owsolverisdevelopedusingtheN-Rmethodintherectangularco-ordinates. Phase-framemodelsof powerlines, transformers, andloads aredeveloped. TheonlyDERtypeconsideredinthat workis thethree-phaseChapter2. Sequence-FramePower-FlowSolver(SFPS) 15directly-connectedsynchronous generator, whichis modeledas abalanced3voltagesourcebehindanimpedance. However, theN-RPFAintherectangularcoordinatesisnot common since the size of the Jacobean matrix increases as the number of PV bussesincreases.A modied N-R method is developed in [69]. The Jacobian matrix is decomposed intoaconstantuppertriangularmatrixandadiagonal matrixwhoseelementsareupdatedprior to each iteration. However, the results indicate that this method suers from weakconvergence patterns. In [70], new variables are dened to formulate the N-R power-owequationsasaset of 3Nequations(2Nlinear plusNnon-linear equations), Nbeingthenumberofbusses. Thedevelopedapproachshowsgoodconvergencepatternswhenappliedtobalancednetworks.In[71],optimalstep-sizemultipliersareaugmentedwiththeN-RPFAalgorithmtoimprove its convergence when used to study networks with high R/X ratio. However, theapplicabilityofthealgorithmforADNisdoubtful sinceitisnottestedondistributionnetworkswithPVbusesandloops.Sequence-components frame is exploitedin[72] and[73] todevelopathree-phasepower-owalgorithm. Thethree-phaseunbalancedpower-owproblemisdecomposedintothreesub-problemswithweakmutuality. However,onlythree-phasefour-wirecon-stant power loads are considered in the analysis. In addition, only the voltage-controlledsynchronousgenerator-basedDERunitisconsideredandmodeledasanidealbalancedvoltagesource.Phase- andsequence-frames are combinedtosolve the 3power-owproblemofdistributionnetworkswithsingle-phaselaterals[74]. Thenetworkisdecomposedintotwoparts,thethree-phasetrunkandthesingle-phaselateral. Thealgorithmof[72]andtheBFSAareiterativelyinterleavedleadingtodeterioratingtheconvergencespeedofthecombinedalgorithm.2.2.2.2 Gauss-SeidelMethodsIn[30], optimalorderingandtriangularfactorizationareusedtodevelopathree-phasedistributionpower-owmethodusingimplicitbusimpedance(Z-BUS)Gaussmethod.Detailed phase-frame models of lines, loads and transformers are presented. The conver-gencebehaviorofthemethodishighlydependentonthenumberofthePVbusesandloops,whichhinderstheapplicabilityofthismethodtoADNapplications.Phase-decoupledformulationisdevelopedin[75]tosolvethethree-phasepower-owproblem. Afterdecouplingthethreephases, theimplicitZ-BUSGaussmethodisusedtosolvethepower-owequations. TheformulationcannotaccommodatePVbusses.Chapter2. Sequence-FramePower-FlowSolver(SFPS) 162.3 Sequence-FrameVersusPhase-FrameinThree-PhasePower-FlowAnalysisAsindicatedinSection2.2, boththephase- andthesequence-framesareexploitedindevelopingthree-phasepower-owalgorithms. Intheformermethod,thepowersystemcomponents are modeled using their phase frame data. This method is very accurate sinceit includes the coupling between the un-transposed lines as well as the phase shifts due todierenttransformers connections. Themainreporteddrawbackofphase-frame-basedthree-phasepower-owalgorithms is theunacceptedcomputational timeandrequire-ments. ForanN-buspowersystem,thisisequivalenttosolvingagroupof6Nstronglycouplednonlinear simultaneousequations: real andimaginarypartsof 3Nequations,oneforeachphase.However,usingthesequence-framehasthefollowingmerits: Using the sequence-components frame in the power-ow analysis eectively reducestheproblemsizeandthecomputational burdenascomparedtothephase-frameapproach. This is true since solving the power-ow equations in the sequence-frameis equivalent to solving three sets of weakly coupled equations: 2Nnonlinear simul-taneousequationsforthepositive-sequencenetwork, andtwosetsof Ncomplexlinear simultaneous equations for thenegative- andzero-sequencenetworks [73].AswillbediscussedinSections2.4and2.5,thecouplingbetweenthethree-phasepower-ow equations is due to the non-zero o-diagonal elements of the 3x3 admit-tance matrices that represent untransposed power lines and unbalanced multi-phaseloads. However, the sequence-frame o-diagonal elements are smaller in magnitudethantheirphase-framecounterparts[62]. The three sets of sequence-frame power-ow equations are weakly coupled comparedtotheirphase-framecounterparts. Assuch, thesequence-framePFAalgorithmsare less vulnerable to divergence and faster to convergence compared to the phase-framealgorithms[72]. Inaddition, thesequence-framealgorithmscaneasilybeimplementedusingtheparallelprogrammingtechniques. Parallelimplementationof thesequence-framepower-owanalysis algorithmis beyondthescopeof thisthesis. Undertheunbalancedgridconditions, theVSCcontrollersarerealizedbasedontwosynchronouslyandoppositelyrotatingreferenceframes[3537, 7783]. Thus,itiseasiertodevelopthree-phasemodelsof theVSC-coupledDERunitsintheChapter2. Sequence-FramePower-FlowSolver(SFPS) 17sequence-componentsframecomparedtothephase-frame, aswill bedetailedinthenexttwochapters.Motivatedbytheaforementionedmeritsofthesequence-framePFAalgorithms,thischapterdevelopsanintegratedSequence-FramePower-owSolver(SFPS)toolfortheADNplanningandoperation. Thedevelopedtoolincorporatesmodelsofelectronically-coupledDERunitsrepresentingtheircontrolcharacteristicsandoperatinglimitsunderbalancedandunbalancedgridconditions. Inaddition, unlike the algorithms of [72]and[73], theproposedSFPSaccommodatesdierentclassesof loadsincludingsingle-phase,two-phase,andthree-phase,three-wire(WyeandDeltaconnected)andfour-wireloadswithconstantpower/current/impedancemodels.2.4 Sequence-FrameModelsof BasicADNCompo-nentsThis sectionbrieydescribes the sequence-frame models of some basic ADNcompo-nents, i.e., synchronousgenerators, powerdistributionlines, three-phasetransformers,and loads, tailored for the SFPS. More details of the sequence-frame-based, fundamental-frequency,steady-statemodelsofmulti-phasepower-linesandloadsaregiveninappen-dicesBandC,respectively.2.4.1 DistributedGenerationDespitethefactthatconsiderableattentionhasbeenpaidtoconverter-baseddistribu-tiongenerationtechnologies, e.g. fuel cells, photovoltaicarrays, andvariablefrequencymicroturbines,somedistributedgenerationsitesintodaysdistributiongridstillemploysynchronousgeneratorsforpowerandheatco-generation[84]. ThebasicDGmodelde-scribedinthissectiononlyencompassesTDSGinPVandPQoperatingmodes. Thenext four chapters are dedicated to develop detailed sequence-frame-based, fundamental-frequency,steady-statemodelsofthree-phaseandsingle-phaseVSC-coupledDERunitsinthecontextoftheSFPS.Figure 2.2 shows the sequence-frame-based, fundamental-frequency, steady-state modelofaTDSGconnectedtoBus-k. Thepositive-sequencemodel,Fig. 2.2(a),representsitscontrolstrategy. IftheunitoperatesinthePVmode,itismodeledasanidealvoltagesource behindBus-k. Under this mode of operation, the magnitude of the positive-sequence component of the TDSG terminal voltage, |V1k|, and the corresponding injectedChapter2. Sequence-FramePower-FlowSolver(SFPS) 18positive-sequencerealpower,P1DER,bothinper-unit,aregivenby|V1k| = Vsp, (2.1)P1DER=PspDER3, (2.2)where Vspand PspDERare the specied per-unit positive-sequence terminal voltage andthetotalthree-phaseinjectedrealpoweroftheTDSGunitrespectively.(a) (b) (c)Figure 2.2: Sequence-frame, fundamental-frequency, steady-state model of a three-phasedirectly connected synchronous generator for the SFPS: (a) the positive-sequence model,(b)thenegative-sequencemodel,(c)thezero-sequencemodelIftheTDSGunitoperatesinthePQmode,itspositive-sequencerepresentationisaconstant power source (or negative constant power load) [85]. The positive-sequence realpowerP1DERisgivenby(2.2). Thepositive-sequencereactivepower(Q1DER)injectedbytheTDSGunit,inper-unit,isQ1DER=QspDER3, (2.3)whereQspDERisthetotalthree-phasereactivepowerinjectedbytheTDSGunit. Thefactor1/3in(2.2)and(2.3)isusedtocalculatethesequence-framepowercomponentsfromtheir phase-frame counterparts, assumingthe same base-power is usedinbothreferenceframes.Thenegative-andzero-sequencemodelsoftheTDSGunitareshowninFig. 2.2(b)and2.2(c),respectively. Thenegative-andzero-sequenceadmittances(y2,0DER)are[86]y2,0DER=1_R2,0SG +X2,0SG_, (2.4)Chapter2. Sequence-FramePower-FlowSolver(SFPS) 19whereR2SG= RaSG,R0SG= RaSG + 3RnSG, (2.5)andX2SG=_Xdunsat +Xqunsat_/2,X0SG=X2SG4+ 3XnSG, (2.6)where Xdunsat (Xqunsat) is the direct (quadrature) unsaturated sub-transient reactance,XnSG(RnSG) is the TDSG neutral grounding reactance (resistance), and RaSGis thearmatureresistanceoftheTDSG.2.4.2 UnbalancedThree-phaseDistributionLineAn(un)balancedthree-phasedistributionlineconnectingBus-kandBus-m,ismodeledasasingleequivalentpi-section, Fig. 2.3. Thismodel adequatelyaddressestwo-phaseandthree-phasethree-wireandfour-wiremulti-groundeddistributionlines, whicharepredominanttheNorthAmericansdistributionsystems[8789].Figure2.3: Equivalent-modelofathree-phasedistributionlineInFig. 2.3, seriesandshuntbranchesaregivenbytwo33admittancematrices,Y012seriesandY012shunt, respectively. Also, I012kmandV012kare 31vectors, andrepresentthesequence-framefundamental-frequency, steady-statecurrentowingfromBus-ktoBus-mandvoltagecomponentsofBus-k,respectively. I012kmisgivenbyI012km=Y012shunt2V012k+Y012series_V012kV012m_, (2.7)Chapter2. Sequence-FramePower-FlowSolver(SFPS) 20whichisequivalentto__I0kmI1kmI2km__=12__y00shunty01shunty02shunty10shunty11shunty12shunty20shunty21shunty22shunt____V0kV1kV2k__+__y00seriesy01seriesy02seriesy10seriesy11seriesy12seriesy20seriesy21seriesy22series____V0kV0mV1kV1mV2kV2m__.(2.8)Equation (2.8) is the coupledsequence-frame model of the distribution line of Fig. 2.3.The coupling between the sequence networks is due to the o-diagonal elements in Y012seriesandY012shunt. Thesetermsarezeroforaperfectlytransposedpowerline, anuncommoncaseintheNorth-Americandistributionnetworks. Themagnitudesof theo-diagonalelementsof Y012seriesandY012shuntarelessthantheirdiagonal counterparts[62, 73]. Thus,(2.8)canbewrittenas__I0kmI1kmI2km__=12__y00shunt0 00 y11shunt00 0 y22shunt____V0kV1kV2k__+__y00series0 00 y11series00 0 y22series____V0kV0mV1kV1mV2kV2m____I0shuntkmI1shuntkmI2shuntkm____I0serieskmI1serieskmI2serieskm__, (2.9)where__I0shuntkmI1shuntkmI2shuntkm__= 12__y01shuntV1k +y02shuntV2ky10shuntV0k +y12shuntV2ky20shuntV0k +y21shuntV1k__, (2.10)__I0serieskmI1serieskmI2serieskm__= __y01series_V1kV1m_+y02series_V2kV2m_y10series_V0kV0m_+y12series_V2kV2m_y20series_V0kV0m_+y21series_V1kV1m___, (2.11)and__I0kmI1kmI2km__=__I0shuntkmI1shuntkmI2shuntkm__+__I0serieskmI1serieskmI2serieskm__. (2.12)Chapter2. Sequence-FramePower-FlowSolver(SFPS) 21Equation(2.12)representstheequivalentcompensationcurrentsthatshouldbein-jectedintoBus-ktodecouple the three sequence networks. The equivalent positive-sequencecompensationpower(S1km)injectedatBus-kisgivenbyS1km= P1km +Q1km= V1k_I1km_. (2.13)The decoupledsequence-frame model of the three-phase distribution line is given by (2.9)-(2.13), andisdepictedinFig2.4. Implementingthemodel of Fig. 2.4intheSFPSisdiscussedinSection2.5(a)(b)(c)Figure2.4: Decoupledsequence-frame, fundamental-frequency, steady-statemodel of athree-phase distribution line: (a) the positive-sequence model, (b) the negative-sequencemodel,(c)thezero-sequencemodelChapter2. Sequence-FramePower-FlowSolver(SFPS) 222.4.3 Three-phasePowerTransformersSeveral three-phasetransformer connections exist indistributionnetworks. Theycanbecategorizedinto: /Yg, Yg/Yg, /Y, Yg/Y ,and/. Thesetransformerconnectionsintroducedierent phaseshifts betweentheir primaryandsecondarysides. Accuratetransformermodelsmustconsiderthesephaseshiftsinthesequence-framePFA.IntheSFPS, athree-phasetransformer that connects Bus-ktoBus-mandhas ashort circuit admittance of ySC, is modeledusingthree decoupled2 2admittancematrices [73]. The entries of these three matrices depend on the transformer connection,asshowninTable2.1.Table2.1: Sequence-Frame, Fundamental-Frequency, Steady-StateModel of AThree-PhasePowerTransformerTransformerConnectionBus-k Bus-m Bus-k Bus-m Bus-k Bus-m Bus-k Bus-m Bus-k Bus-mYgYgYg Y YgY Positive-SequenceModel y1trkkySCySCySCySCySC__y1trkky1trkmy1trmky1trmm__y1trmmySCySCySCySCySCy1trkmySCySC

30oySC

30oySCySCy1trmkySCySC

30oySC

30oySCySCNegative-SequenceModel y2trkkySCySCySCySCySC__y2trkky2trkmy2trmky2trmm__y2trmmySCySCySCySCySCy2trkmySCySC

30oySC

30oySCySCy2trmkySCySC

30oySC

30oySCySCZero-SequenceModel y0trkk((ySC)1+ 3zn)1((ySC)1+ 3zn)10 ((ySC)1+ 3zn)10__y0trkky0trkmy0trmky0trmm__y0trmm((ySC)1+ 3zn)10 0 0 0y0trkm((ySC)1+ 3zn)10 0 0 0y0trmk((ySC)1+ 3zn)10 0 0 02.4.4 LoadsThesequence-frame, fundamental-frequency, steady-statemodelsoffour-wire, constantpower loads onlyare consideredin[72] and[73]. However, electric loads inthe dis-tributionnetworksincludesingle-phase, two-phase, orthree-phasethree-wireconstantpower/current/impedanceloads[62]. Thissectionbrieydescribesthesequence-framemodelsof3constantpower/current/impedanceloadsinthecontextoftheSFPS.ThedetaileddevelopmentofdierentloadmodelsiscoveredinAppendixC.2.4.4.1 ConstantPowerandCurrentLoadsFigure 2.5 shows a schematic diagram of a generic constant power/current load connectedto Bus-k. A unied sequence-frame model for these kinds of loads is depicted in Fig. 2.6.Chapter2. Sequence-FramePower-FlowSolver(SFPS) 23Figure2.5: Schematicdiagramofaconstantpower/currentloadconnectedtoBus-k(a) (b) (c)Figure2.6: Decoupledsequence-frame, fundamental-frequency, steady-statemodel of aconstantpower/currentloadThepositive-sequenceloadmodelisapowersource,Fig. 2.6(a),whosevalueS1loadistheproductofthepositive-sequencevoltageatBus-kV1kandthecomplexconjugateofthepositive-sequenceloadcurrentinjectedintoBus-kI1load. Subsequenttoeachpower-owiteration,V1k,I1load,andconsequentlyS1loadareupdated.As showninFig. 2.6(b) and2.6(c), thenegative- andzero-sequenceloadmodelsaretwocurrentsourceswhosevaluesareequal tothenegative-andzero-sequenceloadcurrentsinjectedintoBus-krespectively(I2,0load). Three-wireloadconnection(WyeandDelta)doesnotprovideapathforthezero-sequencecurrent. Assuch,I0loadissetequaltozeroforsuchloads.2.4.4.2 ConstantImpedanceLoadsRegardless of the load connection, the phase-frame representation of a constant impedanceloadisa3 3diagonaladmittancematrix,Fig. 2.7(a). Thevaluesofthethreeadmit-tances,yaaload,ybbload,andyccload,dependontheloadsconnection,ratedvoltage,andratedpower[62].Chapter2. Sequence-FramePower-FlowSolver(SFPS) 24(a) (b)Figure2.7: Model of aconstantimpedanceloadconnectedtoBus-k: (a)phase-framemodel,(b)sequence-framemodelInthesequence-frame,theequivalentsequence-frameloadadmittancematrixis__y00loady01loady02loady10loady11loady12loady20loady21loady22load__= T1__yaaload0 00 ybbload00 0 yccload__T, (2.14)whereT =__1 1 11 a2a1 a a2__,a = 1

120. (2.15)Theo-diagonal elementsof thesequence-framematrixarezeroonlyforperfectlybalancedloads,i.e.,yaaload= ybbload= yccload.Thesequence-frameloadcurrents(I012load)andvoltages(V012k)ofFig. 2.7(b)are__I0loadI1loadI2load__= __y00loady01loady02loady10loady11loady12loady20loady21loady22load____V0kV1kV2k__. (2.16)Eq. (2.16)canbewrittenas__I0loadI1loadI2load__= __y00load0 00 y11load00 0 y22load____V0kV1kV2k__+__I0loadI1loadI2load__, (2.17)Chapter2. Sequence-FramePower-FlowSolver(SFPS) 25where__I0loadI1loadI2load__= __y01loadV1k +y02loadV2ky10loadV0k +y12loadV2ky20loadV0k +y21loadV1k__. (2.18)Equation(2.18) represents the sequence-frame compensationcurrents requiredtodecouplethethreesequencenetworks. Theequivalentpositive-sequencecompensationpower(S1load)injectedintoBus-kisgivenbyS1load= P1load +Q1load= V1k_I1load_. (2.19)Thesequence-frameconstantimpedanceloadmodel incorporatedintheSFPSisgivenby(2.17)-(2.19),andisdepictedinFig. 2.8.(a) (b) (c)Figure2.8: Decoupledsequence-framemodelofaconstantpower/currentload2.5 TheSFPSAlgorithmTheSFPSalgorithmisdepictedinFig. 2.9. ThedetailsoftheSFPSstepsfollow.Step1: ConstructtheSequence-FrameBusAdmittanceMatrices:Theentrycorrespondingtothekthrowandmthcolumnofthepositive-,negative-,andzero-sequencebusadmittancematrices(Y1,2,0BUS)isgivenbyy1BUSkm=___

Ai=1y11seriesi +12

Ai=1y11shunti +

Bi=1y1trkmi +

Ci=1y11loadi,k = m,

`Ai=1y11seriesi

`Bi=1y1trkmi,k = m,(2.20)Chapter2. Sequence-FramePower-FlowSolver(SFPS) 26Figure2.9: FlowchartoftheSFPSalgorithmChapter2. Sequence-FramePower-FlowSolver(SFPS) 27y2,0BUSkm=___

Ai=1y22,00seriesi +12

Ai=1y22,00shunti +

Bi=1y2,0trkmi +

Ci=1y22,00loadi+

Di=1y2,0DERi,k = m,

`Ai=1y22,00seriesi

`Bi=1y2,0trkmi,k = m,(2.21)whereAisthetotal numberofpowerlinesconnectedtoBus-k, Bisthetotal numberoftransformersconnectedtoBus-k,CisthetotalnumberofconstantimpedanceloadsconnectedtoBus-k,`Aisthetotal numberof power-linesconnectingBus-ktoBus-m,`Bisthetotal numberof transformersconnectingBus-ktoBus-m, andDisthetotalnumberofTDSGunitsconnectedtoBus-k.Step2: CheckfortheZero-SequenceBlockingCondition:Inpracticalpowersystems,therewillalwaysbeapathforzerosequencecurrent. How-ever,duetothecomponentsmodelsincorporatedintheSFPS,somebusesofthezero-sequence networkmight be isolated, Fig. 2.10. The cloudedbus inFig. 2.10(a) isconnectedtoathree-wireloadandthedeltasideof athree-phasetransformer. Thecorrespondingzero-sequenceimpedancediagramisgiveninFig. 2.10(b). Inthezero-sequencenetwork,thecloudedbusbecomesisolatedfromtherestofthenetwork.(a)(b)Figure 2.10: Zero-sequence blocking scenario: (a) phase-frame network, (b) zero-sequenceimpedancediagramIn the SFPS, this condition causes some rows and columns of the Y0BUSwith all-zeroentries. This is called zero-sequence blocking[90]. If this situation occurs, Y0BUSbecomessingular. Inthiswork, oncethezero-sequenceblockingagisraised, amodiedY0BUSisconstructedbyeliminatingthecorrespondingall-zerosrowsandcolumns.Chapter2. Sequence-FramePower-FlowSolver(SFPS) 28Step3: InitializethePositive-,Negative-,andZero-SequenceBusVoltages:Thesequence-framecomponentsofthevoltagesatBus-kareinitializedaccordingtoV1k=___1.0 Bus-kisaPQbus,VspBus-kisaPVorslackbus,V0,2k= 0.0. (2.22)Ifthezero-sequenceblockingagisraised,omitthecorrespondingentriesofV0BUS.Step4: ResettheAlgorithmCounter:ct = 0. (2.23)Step5: UpdatetheNegative-andZero-SequenceSpeciedBus-CurrentIn-jectionVectors:Priortothectthiteration, theentrycorrespondingtothekthrowof thenegative-andzero-sequencespeciedbus-currentinjectionvectors(I2,0BUS,ct)isI2,0BUSk,ct=`A

i=1I2,0kmi,ct +C

i=1I2,0loadi,ct +E

i=1I2,0loadi,ct, (2.24)whereEisthetotalnumberofconstantpower/currentloadsconnectedtoBus-k. Ifthezero-sequenceblockingagisraised,omitthecorrespondingentriesofI0BUS.Step 6: Update the Positive-Sequence Specied Bus-Power Injection Vectors:Prior tothe ctthiteration, the kthentryof the positive-sequence speciedbus-powerinjectionvectors(P1BUS,ctandQ1BUS,ct)isgivenbyP1BUSk,ct=D

i=1P1DERi +_E

i=1S1loadi,ct_+P1kcomp,ct, (2.25)Q1BUSk,ct=D

i=1iQ1DERi +_E

i=1S1loadi,ct_+Q1kcomp,ct, (2.26)wherei=___1 DER-iisPQcontrolled,0 DER-iisPVcontrolled.(2.27)Chapter2. Sequence-FramePower-FlowSolver(SFPS) 29P1kcomp,ctand Q1kcomp,ctare the positive-sequence real and reactive power compensationatBus-k,andarecalculatedpriortothectthiterationusingP1k,comp= __`A

i=1S1kmi,ct +C

i=1S1loadi,ct__, (2.28)Q1k,comp= __`A

i=1S1kmi,ct +C

i=1S1loadi,ct__. (2.29)Step7: ConductthePositive-SequenceN-RPFAUpdatethepositive-sequenceJacobianmatrixJctandthepositive-sequencepowermis-matchvectorS1mismatch,ct. Solve(2.31)forthemagnitudesandanglesofthepositive-sequencebusvoltagesofthect + 1thiteration.Jct_____1BUS,ct+1V1BUS,ct+1__ __1BUS,ctV1BUS,ct_____ = S1mismatch,ct, (2.30)whereJ =__PBUS/BUSPBUS/VBUSQBUS/BUSQBUS/VBUS__ , (2.31)S1mismatch,ct=__P1BUS,ctQ1BUS,ct__ __P1calc.,ctQ1calc.,ct__ , (2.32)andP1calc.i= __V1i_N

k=1V1ky1BUSik___Q1calc.i= __V1i_N

k=1V1ky1BUSik___(2.33)Step8: ConducttheNegative-andZero-SequencePFA:Solvethetwocomplexmatrixequations(2.34)forthenegative-andzero-sequencebusvoltagesatiterationct + 1.I2,0BUS,ct= Y2,0BUSV2,0BUS,ct+1. (2.34)Chapter2. Sequence-FramePower-FlowSolver(SFPS) 30Step9: EvaluatethePhase-FrameBusVoltages:Reconstruct V0BUSbyinsertingazeroentrycorrespondingtoeachbus witharaisedzero-sequenceblockingag. Evaluatethephase-framebus-voltagesusing___VaBUS,ct+1__VbBUS,ct+1__VcBUS,ct+1___= T___V0BUS,ct+1__V1BUS,ct+1__V2BUS,ct+1___, (2.35)where(.)

isthetransposeofavector.Step10: EvaluatetheTerminationCriterion:Evaluatetheterminationcriterion(ter cri)usingter cri = max__|VaBUS,ct+1| |VaBUS,ct||VbBUS,ct+1| |VbBUS,ct||VcBUS,ct+1| |VcBUS,ct|__(2.36)Themaximumpowermismatchcouldalsobeusedasaterminationcriterion:ter cri = max_S1mismatch,ct_(2.37)Updatethealgorithmcounter(ct):ct = ct + 1. (2.38)If ter cri islargerthanapre-speciedtolerance, gotoStep5, elseprinttheSFPSresults.2.6 SummaryandDiscussionThischapterpresentsthedevelopmentofathree-phasepower-owanalysismethodforthe ADNapplications, under balancedandunbalancedconditions. The power-owalgorithm, Sequence-Frame Power-owSolver (SFPS), is developedinthe sequence-Chapter2. Sequence-FramePower-FlowSolver(SFPS) 31componentframe. Thischapteralsodescribessequence-frame, fundamental-frequency,steady-statemodelsof: athree-phasedirectly-connectedsynchronousgeneratorDGunitunderbalancedand unbalanced power-ow scenarios with dierent control characteristics, i.e., con-stantpower(PQ)andvoltage-regulated(PV)operatingmodes, (un)transposedthree-phasethree-andfour-wirepowerlines, three-phasetransformerswithdierentconnections, multi-phaseconstantpower/current/impedanceloads.TheSFPSalgorithm,includingthedetailsofincorporatingtheaforementionedcom-ponents, ispresentedanddiscussed. TheSFPSisthehubintowhichtheVSC-basedDERmodelsareincorporatedandtested,aswillbedetailedinthenextfourchapters.Chapter3Power-FlowModelof3VSC-CoupledDERUnits13.1 IntroductionThischapterpresentsauniedfundamental-frequency, steady-statemodel of athree-phaseVSC, inthesequence-componentsframe, forPFAofVSC-interfacedDERunits.Theproposedmodel is uniedsinceit encompasses awidearrayof DERunits, e.g.,batteryenergystoragesystems(BESS),fuelcells(FC),solarphotovoltaicunits(SPV),variable-speed wind turbine generators with full-scale power-electronic converters (Type-4WTG),andmicro-turbine(MT)systems. Inaddition,themodelrepresents(i)three-wireandfour-wireVSCcongurations,(ii)balancedandunbalancedpower-owscenar-ios,(iii)variousVSCcontrolstrategies,and(iv)operatinglimitsandconstraintsoftheVSCandits host DERunit. Toachievenumerical andcomputational eciency, theSFPSalgorithm, presentedinChapter2, ismodiedtoimposethetheinterface-VSCoperatinglimits. Theaccuracyofthedevelopedmodelandthecomputationaleciencyof the modied SFPS are demonstrated based on several case studies. Where applicable,thenumerical accuracyof theVSCmodel is validatedbasedoncomparisonwiththeexacttime-domainsolution,usingthePSCAD/EMTDCplatform.1The work presented in this chapter has been published and appears in M.Z. Kamh and R. Iravani,A Unied Three-Phase Power-Flow Analysis Model for Electronically-Coupled Distributed Energy Re-sources, IEEE Trans. Power Delivery, vol.26, no.2, pp.899-909, April 2011.32Chapter3. Power-FlowModelof3-VSC-CoupledDERUnits 33Figure3.1: Schematicdiagramofthree-phaseVSC-coupledDERunit3.2 ModelScopeandAssumptionsAs earlier statedinChapter 1, three-phase DC-ACVSCis the most widely-adoptedinterface mediumfor DERunits [12]. Toobtainanaccurate power-owsolutionofADN,three-phaseVSC-coupledDERunitsshouldbemodeledandincorporatedintheSFPS.Theproposedmodelencompasses: three-wireandfour-wireVSCcongurationsunderbothbalancedandunbalancedpower-owscenarios, various VSC control strategies, i.e., voltage/frequency and four-quadrant real/reactivepower controls, andspeciccontrol actionstoinject an/or respondtonegative-and/orzero-sequencecomponents, various VSC operational modes, e.g., four-quadrant real-/reactive-power exchange, the VSC operational limits and constraints, i.e., maximum phase current, maximummodulationindex,andmaximumreactivepowerlimits.Figure3.1showsaschematicdiagramofaDERunitcoupledtothehostsystematBus-k, whichrepresentsthepointof connection(PC), viaathree-wireorafour-wireVSC. TheprimarysourceisconnectedtotheVSCeither(i)directly, e.g., BESSandFC,(ii)byaDC-DCconverter,e.g.,SPV,or(iii)byanAC-DCconverter,e.g.,Type-4WTG[12, 34]. Thefollowingassumptionsaremade: The DER primary source is not directly represented in the model as the controllersof thefront-endinterface-VSCortheback-endDC-DCconverterareassumedtofullyregulatetheDC-linkvoltageundersteady-stateconditions.Chapter3. Power-FlowModelof3-VSC-CoupledDERUnits 34(a) (b) (c)Figure3.2: Proposedsequence-frame, fundamental-frequency, steady-statemodel of a3interfaceVSC, (a)positive-sequencemodel, (b)negative-sequencemodel, (c)zero-sequencemodel Only the fundamental frequency model of the interface-VSC is considered, i.e., theharmoniceectsarediscarded. The VSC synchronization to the grid is based on a phase-locked loop (PLL) systemthatalsoextractsthesequence-framecomponentsofthePCvoltage[36, 77]. The interface VSC is equipped with dedicated controllers to provide specic controlfunctions with respect to the negative- and/or zero-sequence components of the PCvariables. Double-frequency voltage and current components, present at the converter dc sideduetopossiblesystemunbalance,areassumedtobenegligible. theVSClossesareneglected. TheVSCnegative-andzero-sequencepowerexchangewiththehostgridareneg-ligiblysmall comparedtotheirpositive-sequencecounterpart. Assuch, theVSCpositive-, negative-, andzero-sequence-framemodelsareassumedtobefullyde-coupled.3.3 Proposed Sequence-Frame Model of 3 VSC-CoupledDERUnitsFigure 3.2 shows the proposed sequence-frame, fundamental-frequency, steady-state rep-resentationofa3VSCinterfacingaDERunit. Underbalanced/unbalancedgridcon-ditions, the primary control objective of the interface VSC controllers is to either achieveconstant-power regulation (PQ) [91] or voltage and frequency regulation (PV) [92] at theChapter3. Power-FlowModelof3-VSC-CoupledDERUnits 35PC. Thisisrealizedbysensingthepositive-sequencevoltageandcurrentcomponentsatthePC, andutilizingthemthroughafeedbackprocesstocontrol thecouplingVSCtogeneratetherequiredpositive-sequencevoltageatitsterminals. Thus, thepositive-sequencemodeloftheVSC-coupledDERunitreectseitherthePQorthePVcontrolmode.ShouldtheDERunitbesubjectedtounbalancedpower-ow, thenegative-and/orzero-sequence components of the PCvariables alsocanbe exploitedtoaugment theVSCswitchingprocesstoprovidespeciccontrol functions. Forexample, athree-wireVSCcanbecontrolledtoinject (i) onlybalancedthree-phasecurrents[36], or (ii) inaddition, apre-speciedamount of negative sequence current [37], tocounteract thesystemimbalanceorforactiveislandingdetection. If desired, afour-wireVSCcanbecontrolled also to compensate for the neutral (zero-sequence) current of a local unbalancedthree-phaseload[35]. Thedevelopednegative-andzero-sequenceframeVSCmodelsinthischapterareintendedtoreectany/alloftheseVSCfunctionalities.3.3.1 InterfaceVSCPositive-SequenceModelTheVSCpositive-sequencemodelissimilartothatoftheTDSGshowninFig. 2.2(a),duplicatedasFig. 3.2(a)foreaseofreference.a) WhentheDERunitoperatesinthePVmode[92],itspositive-sequencemodelwithrespecttothePCisanideal voltagesourcebehindthePCbus. Thespeciedvolt-agemagnitude, |V1k|, at thePCandthepositive-sequencereal power injected(orabsorbed),P1DER,bytheunitare|V1k| = Vsp, (3.1)P1DER=PspDER3, (3.2)where Vspand PspDERare the per-unit voltage and the three-phase real power refer-enceset-pointsoftheVSC.b) WhentheDERunitiscontrolledtooperateinthePQmode, itisrepresentedasaconstant power source. In this case, the real power injected/absorbed is given by (3.2)andtheexchangedreactivepower,Q1DER,isQ1DER=QspDER3, (3.3)whereQspDERistheper-unitthree-phasereactivepowerreferenceset-pointof theChapter3. Power-FlowModelof3-VSC-CoupledDERUnits 36VSC. In Fig. 3.2(a), I1DER is the positive-sequence current exchange between the DERunitandthesystem.3.3.2 InterfaceVSCNegative-andZero-SequenceModelsThenegative-andzero-sequencemodelsoftheinterfaceVSCareshowninFigs. 3.2(b)and 3.2(c), respectively. A parallel combination of a current source (I0,2CTRL) and ctitiousadmittance (Y0,2CTRL) can represent any control objective for both the three-wire and four-wireVSC,correspondingtonegative-andzero-sequenceframes. InFig. 3.2(b)(3.2(c)),thenet current exchangebetweentheVSCnegative- (zero-) sequencemodel andthesystemispresentedbyI2DER(I0DER).3.3.2.1 Three-WireVSCCongurationIf a three-wire interface-VSC is controlled only based on the positive-sequence dq currentcontrol method[91], theDERunitalsoexchangesnegative-sequencecurrentwiththeunbalancedsystem. Inthiscase, thenegativeandzerosequence-framecomponentsofthemodelarespeciedasI0,2CTRL= 0,Y2CTRL= 1/Zf, (3.4)Y0CTRL= 0,whereZfistheequivalentseriesimpedanceof theVSCoutputlterbetweenthePCandtheshort-circuitedVSCterminals. TheVSCoutput lter is usedtoreducetheharmoniccurrentinjectedintheutilitysystem[93]. Thesimplestoutputltertopologyisaseries-connectedinductor,Fig. 3.1,forwhichZfisgivenbyZf= Rf+Xf, (3.5)where Xf/Rfis the VSCoutput lter net reactance/resistance. Other output ltercongurationsarereportedin[94]. Thesealternativecongurationslieundereitheroneof thetwotopologiesshowninFig. 3.3. Theequivalentseriesimpedance, Zf, fortheChapter3. Power-FlowModelof3-VSC-CoupledDERUnits 37(a) (b)Figure3.3: AlternativeVSCoutputltercongurationsltercongurationsshowninFig. 3.3(a)andFig. 3.3(b),iscalculatedasZf=Z1Z2Z1 +Z2, (3.6)Zf= Z3 +Z1Z2Z1 +Z2, (3.7)respectively,whereZ1,Z2,andZ3areshowninFig. 3.3.If a three-wire VSC is equipped with both positive- and negative-sequence current con-trolschemes,andthenegative-sequencecurrentcontrolisassignedtopreventnegative-sequencecurrentexchangewiththesystem[36], thenthenegative- andzero-sequencemodelscomponentsareI0,2CTRL= 0,Y0,2CTRL= 0. (3.8)Forsomeapplications,thenegative-sequencecontrollerofthethree-wireVSCisde-signed to inject a pre-specied negative-sequence current into the system, e.g., for island-ingdetection[37]. Inthiscase, thenegativeandzero-sequencemodelsoftheinterface-VSCareI2CTRL= INSCI

NSCI,I0CTRL= 0, (3.9)Y0,2CTRL= 0,whereINSCIandNSCIarethemagnitudeandphaseangleofthepre-speciednegativesequence current injection. In (3.4), (3.8), and (3.9), the parameters of the zero-sequencemodel of Fig. 3.2(c) are always zero since there is no path for zero-sequence current ow.Chapter3. Power-FlowModelof3-VSC-CoupledDERUnits 383.3.2.2 Four-WireVSCCongurationThe split-capacitor and the four-leg interface-VSC congurations can enable neutral con-nectionandestablishafour-wireVSCsystem[35]. Withrespect tothesteady-statepower-ow,bothcongurationsareequivalentaslongastheassumptionsstatedinSec-tion3.2areapplicable. Inadditiontothepositive-sequencecurrentcontrol,afour-wireinterface-VSCcanexchangecontrollednegative-andzero-sequencecurrentcomponentswith the system, e.g., to counteract the imbalance due to unbalanced load at the PC [35].ThemodelsofFig. 3.2(b)andFig. 3.2(c)canrepresentthisstrategybysettingI0,2CTRL=N

i=1V0,2iy0,2BUSkiI0,2loadk,Y0,2CTRL= 0, (3.10)where I0,2loadkis the equivalent zero- andnegative-sequence loadcurrent componentsinjectedintothePC(Bus-k).If the four-wire interface-VSC only injects a controlled positive-sequence current in thesystemandpermitsthesystemtodeterminetheexchangednegative-andzero-sequencecurrentcomponents,thenthecorrespondingmodelparametersareI0,2CTRL= 0,Y2CTRL=1Rf+Xf, (3.11)Y0CTRL=1(Rf+ 3Rn) +(Xf+ 3Xn),whereRnandXnaredenedinFig. 3.1.3.4 Implementation of the VSC Unied Model in theSFPS3.4.1 Accommodating the VSC-Coupled DERModel in theSFPSConsideraVSC-interfacedDERunitcoupledtothehostdistributionnetworkatBus-k(DER-k). ToembedthemodelofFig. 3.2intheSFPSalgorithm:1. Add the terms Y0,2CTRLto the entries of the zero- and negative-sequence bus admit-Chapter3. Power-FlowModelof3-VSC-CoupledDERUnits 39Figure3.4: Equivalentsequence-framecircuitsofaVSC-coupledDERunittancematricescorrespondingtothekthrowandthekthcolumn(y0,2BUSkk), givenby(2.21),respectively.2. Addtheterms I0,2CTRLtothekthentries of thezero- andnegative-sequencebus-currentinjectionvectors(I0,2BUSk),givenby(2.24),respectively.3. AddthetermsP1DERandQ1DER, givenby(3.2)and(3.3), tothekthentryofthespeciedreal andreactivepowervectors(P1BUSkandQ1BUSk), givenby(2.25)and(2.26),respectively.3.4.2 Calculating the Internal Parameters of the Interface VSCToimposetheoperatinglimitsoftheinterface-VSCofDER-kintheSFPS,itsinternalparametersvalues, i.e., modulationindices, phasecurrents, andreactivepower(forPVunits), must be determined rst. Figure 3.4 shows the equivalent sequence-frame circuitsoftheinterfaceVSCusedtodeterminetheseparameters.Subsequenttoeachpower-owiteration, thePCterminal conditionsareevaluated.TheVSCnetsequence-framecurrentcomponentsinjectedintothePCareI0,2DER= I0,2CTRLY0,2CTRLV0,2k

0,2k, (3.12)I1DER=P1DERQ1DERV1k 1k, (3.13)where the elements of the VSC net current injection vector I012DER=_I0DERI1DERI2DER_arespeciedonFig. 3.2. Thenthethree-phasecurrentIabcDERinjectedbytheDERunitiscalculatedusingIabcDER= TI012DER, (3.14)Chapter3. Power-FlowModelof3-VSC-CoupledDERUnits 40Table3.1: VSCConstantModulationStrategy KinvSinusoidalPulse122WidthModulation(SPWM)SpaceVector16Modulation(SVM)wherethematrixTisgivenby(2.15)andIabcDER=_IaDERIbDERIcDER_. (3.15)Finally, the sequence components of the interface-VSC terminal voltage, Fig. 3.4, can bedeterminedusingV0,2t

0,2t= I0,2DER_R0,2+X0,2_+V0,2k

0,2k, (3.16)V1t 1t=_P1DERQ1DERV1k 1k__R1+X1_+V1k 1k, (3.17)whereR1,2= Rf,R0= Rf+ 3Rn, (3.18)X1,2= Xf,X0= Xf+ 3Xn.TheVSCsequence-andphase-framemodulationindicesaregivenbym012=V012tKinvVdc, (3.19)mabc= Tm012, (3.20)whereKinvisthephase-to-neutral converterconstantandisdeterminedbasedontheadoptedmodulationtechnique,asshowninTable3.1[95],andmabc=_ma

atmb

btmc

ct_, (3.21)m012=_m0

0tm1

1tm2

2t_. (3.22)Chapter3. Power-FlowModelof3-VSC-CoupledDERUnits 413.4.3 MitigatingtheVSCOperatingLimitsViolationSubsequent toevaluatingthe internal parameters of the interface-VSC, its operatinglimits are checked to achieve a power-ow solution that satises all the VSCs constraints.Ifanyviolationisdetected, thevoltageand/orpowersetpointsofeachconverter, i.e.,Vsp,PspDER,andQspDER,areupdatedbasedonthefollowingproposedstrategies.3.4.3.1 MitigatingthePhaseCurrentLimitViolationIfanyoftheDERphasecurrents(Ia,b,cDER)exceedsthemaximumphasecurrentlimit,thespecied real and reactive power components associated with each VSC unit are updatedusingP1DER,ct+1|x=13PspDER,ctIphmaxmax violate_Ia,b,cDERct_,Q1DER,ct+1|x=13QspDER,ctIphmaxmax violate_Ia,b,cDERct_, (3.23)where ct is the SFPS current iteration index and maxviolate_Ia,b,cDER,ct_is the magnitudeof the VSC largest violating phase current corresponding to iteration ct. Sux x refers tothe VSCpowerset-points satisfying theVSC maximum current constraint. It should benotedthatiftheDERunitiscontrolledtooperateataconstantpower-factor,thenthereal andreactivepowerset-pointsaresimultaneouslyupdatedusing(3.23). Otherwise,theDERreactivepowercontributionisreducedrsttoalleviatetheviolationwhilethereal-powerset-pointremainsunaltered.3.4.3.2 MitigatingtheReactivePowerLimitViolationFor PV-controlledDERunits, if thereactivepowerlimit ishit, thespeciedpositivesequencevoltageforthecorrespondingbusisupdatedusingVsp,ct+1|y=QmaxX1cct, (3.24)wherecct=V1t,ctcos_1t,ct1k,ct_Vsp,ct1. (3.25)Equation(3.24)isthesolutionof(3.17)afterreplacingQ1withQmaxandassumingthatR1ismuchsmallerthanX1. SuxyreferstotheVSCvoltageset-pointsatisfyingChapter3. Power-FlowModelof3-VSC-CoupledDERUnits 42themaximumreactivepowerconstraint.Itshouldbenotedthat, unliketheconventional N-Rpower-owalgorithmswheretheviolatingPVbusisswitchedtoPQbus[72, 96, 97], theproposedapproachof Sec-tion3.4.3.2usestheclosedformgivenby(3.24)and(3.25)toupdatethespeciedbusvoltageforthenextiteration, Vsp,ct+1, suchthatthereactivepowerremainswithintheacceptable operating limits. Consequently, the Jacobean matrix and states vector do notrequirerestructuring,correspondingtobus-typechanging. Thissimpliesthealgorithmimplementationandprogramming.3.4.3.3 Mit