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Distributed Coordination of Smart Loads to Provide Ancillary Service to the Power Grid

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Title:
Distributed Coordination of Smart Loads to Provide Ancillary Service to the Power Grid
Creator:
Brooks, Jonathan T
Place of Publication:
[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (99 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering
Mechanical and Aerospace Engineering
Committee Chair:
BAROOAH,PRABIR
Committee Co-Chair:
HALE,MATTHEW
Committee Members:
MEYN,SEAN PETER
BRETAS,ARTURO SUMAN

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Subjects / Keywords:
control -- coordination -- distributed -- grid -- mpc -- optimization -- power -- smart
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Mechanical Engineering thesis, Ph.D.

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Abstract:
Maintaining a balance between generation and consumption in the power grid is paramount to its stable and reliable operation. Renewable energy sources are being increasingly used in the power grid, and this renewable generation brings with it volatility that acts as a large disturbance to the grid. Traditionally, generators track changes in the net load (generation minus consumption). Because of large variability of renewable generation (e.g., solar and wind), this requires the generators to ramp quickly, which may increase emission rates of greenhouse gases---counteracting the original benefit of the renewable generation. Another possibility is to manipulate the consumption of loads in order to counteract the variability in renewable generation, i.e., demand-side ancillary services. This is a low-cost and low-emission solution to the problem caused by variability of renewable energies. However, varying consumption may have a negative impact on consumers (e.g., higher air temperatures in a building). Avoiding or minimizing such negative impacts is a necessary hurdle to ensure widespread consumer participation to help the grid. Another key technical obstacle to demand-side ancillary services is the need for distributed coordination among many loads. Since each load can provide only a small amount demand variation, a large number of them must be used in a coordinated fashion. The topic of this dissertation is developing algorithms to make use of loads to provide ancillary services. We focus on two services, among many, that flexible loads can provide---contingency reserves and net-load-following. We propose two control algorithms for distributed coordination of loads to provide ancillary services, while minimizing or avoiding loads' loss in quality of service. The first, the Distributed Gradient Projection (DGP) algorithm, is designed for loads to act as contingency reserves in the event of a sudden and large change in generation. Loads eliminate the net load while minimizing the disutility associated with varying consumption. Each load uses local measurements of grid frequency to infer the global imbalance between generation and consumption. Loads use limited communication to share information with neighbors. Each load combines their neighbors' information and the local frequency measurements to determine if it should increase or decrease consumption. We provide proofs of convergence, showing the loads converge to an global optimal set that is globally asymptotically stable. Simulation results show that the DGP algorithm works as well or better than a previously proposed algorithm for the same problem. The second algorithm we propose, the Bandwidth-Limited, Disturbance-Rejecting, Decentralized Model-Predictive Control (BaLDuR-DMPC) algorithm, is designed to provide net-load-following in a slower timescale. It does not use models of economic disutility; instead it ensures the consumption changes obey predetermined constraints that ensure consumers' quality of service. In particular, the BaLDuR-DMPC algorithm minimizes predicted grid-frequency deviations while enforcing bandwidth constraints on loads' control actions. This allows loads to limit actions in frequency ranges that may negatively affect consumers; this provides bounds on the effects of the net-load-following service seen by consumers. Convergence is established under idealized conditions, and simulation results indicate that loads are successfully able to reduce frequency deviations in the grid while ensuring quality of service for consumers. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2017.
Local:
Adviser: BAROOAH,PRABIR.
Local:
Co-adviser: HALE,MATTHEW.
Statement of Responsibility:
by Jonathan T Brooks.

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DISTRIBUTEDCOORDINATIONOFSMARTLOADSTOPROVIDEANCILLARYSERVICETOTHEPOWERGRIDByJONATHANBROOKSADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2017

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c2017JonathanBrooks

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ACKNOWLEDGMENTSFirstandforemost,ImustthankEmilyandmyfamilyforsupportingmethroughoutthisendeavor.Icouldnothavesurvivedthegauntletofgraduateschoolwithouttheirencouragementandunderstanding.Iwouldalsoliketothankthemultitudeofexcellentteachersandpositiveinuencesthroughoutmylifethathelpedmereachthispoint.Ithankmyadvisor,PrabirBarooah,forrsthiringmeasanundergraduateresearcherandforcontinuallypushingmetoexpandmylimits.IalsoowealargethankstoRodrigoTrevizanandArturoBretasfortheirassistanceindesigningandimplementingadvancedpower-gridsimulations,whicharecriticalcomponentsofthisdissertation.IthankSeanMeynaswellformanyhelpfuldiscussionsabouttheSkorokhodproblem,gridmodeling,andmathematicsingeneral.Ialsothankmymanylabmatesfortheirhelpandcompanyduringthelonghoursofgraduatelife.IthankChanghongZhaoandStevenLowfortheirassistanceinreproducingtheresultsof[ 1 ],VivekBorkarforseveralusefulcomments|includingpointingustotheworkbyNagurneyonprojecteddynamicalsystems,andJamesRawlingsforhelpfulcommentsregardingMPCandbandwidthconstraints.TheresearchreportedherewaspartiallysupportedbytheNationalScienceFoundationthroughgrants1463316and1646229andbytheDOEGMLCaward(VirtualBatteries). 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 3 LISTOFTABLES ..................................... 6 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 11 1.1Overview .................................... 11 1.2Contribution ................................... 15 2PROVIDINGCONTINGENCYRESERVESBYMINIMIZINGDISUTILITY 17 2.1IntroductionandMotivation .......................... 17 2.2ProblemFormulation .............................. 19 2.3DistributedGradientProjection(DGP)Algorithm ............. 20 2.4ConvergenceAnalysis .............................. 22 2.4.1MainResults ............................... 22 2.4.2ProofofTheorem2.1 .......................... 23 2.5AuxiliaryAnalyticalResults .......................... 29 2.5.1AsymptoticStability .......................... 29 2.5.2ConstantStepSize ........................... 29 2.5.3AsymptoticBehaviorforQuadraticCost ............... 30 2.5.4CommentonStrictFeasibility ..................... 31 2.6NumericalResultsforLinearGridModel ................... 32 2.6.1SimulationSetup ............................ 32 2.6.2ResultswithNon-StrictlyConvexDisutility .............. 35 2.6.3ComparisonwithDualAlgorithm:ResultswithStrictlyConvexDisutility ................................. 35 2.7NumericalResultsforNonlinearGridModel ................. 38 2.7.1EstimatingGlobalPowerImbalance .................. 38 2.7.2SimulationSetup ............................ 40 2.7.3ComparisonwithTheoreticalPredictions ............... 40 2.7.4EvaluationofDGPAlgorithmunderPracticalLimitations ..... 41 2.7.5EectofTimeDelayandModelMismatch .............. 43 2.7.6EectofMeasurementNoise ...................... 46 2.7.7SpeedofActuation ........................... 47 2.7.8LoadQuantization(NumberofBins) ................. 48 2.7.9StochasticLoads ............................. 49 2.7.10GraphTopology ............................. 50 4

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2.8Summary .................................... 51 3BANDWIDTH-LIMITEDDEMANDDISPATCHFORNET-LOAD-FOLLOWINGUSINGVIRTUALENERGYSTORAGE ...................... 54 3.1IntroductionandMotivation .......................... 54 3.2EnvisionedGridArchitecture ......................... 56 3.3ProblemFormulation .............................. 57 3.4ProposedMethod ................................ 59 3.4.1GridModelUsedbyLoads ....................... 59 3.4.2MPCFormulationforLoadi ...................... 60 3.4.3AvoidingHigh-GainFeedback ..................... 61 3.5Feasibility .................................... 63 3.6Convergence ................................... 64 3.7NumericalResultsforLinearGridModel ................... 66 3.7.1SimulationSetup ............................ 66 3.7.2OverviewofResults ........................... 67 3.7.3EectofBandwidthConstraints .................... 70 3.8NumericalResultsforNonlinearGridModel ................. 71 3.8.1SimulationSetup ............................ 71 3.8.2ChoosingBandwidthConstraints ................... 73 3.8.3NumericalResults ............................ 76 3.9Summary .................................... 80 4CONCLUSION .................................... 82 APPENDIX:PROOFSFORDGPCONVERGENCERESULTS ........... 84 REFERENCES ....................................... 92 BIOGRAPHICALSKETCH ................................ 99 5

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LISTOFTABLES Table page 2-1Standardvaluesforparametricstudies ....................... 43 3-1Objectivevaluefordierentscenarios ........................ 68 3-2Loadtypesandcorrespondingfrequencybands .................. 75 3-3Loadcomposition ................................... 75 6

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LISTOFFIGURES Figure page 2-1Alternatemodelsofconsumerdisutilityvs.consumptionchange. ......... 18 2-2Systemarchitectureforsimulations.Inter-loadcommunicationisnotshown. .. 33 2-3Thecommunicationgraph.Toavoidclutter,notalledgesareshown. ...... 35 2-4Lineargridmodel:PerformanceoftheDGPalgorithmwithconsumerdisutilitythatisnotstrictlyconvex.Stepchangesingenerationoccurat20and50seconds. 36 2-5Lineargridmodel:PerformanceoftheDGPanddualalgorithmswithquadraticconsumerdisutilitywithprojection.Stepchangesingenerationoccurat20and50seconds. ...................................... 37 2-6SchematicoftheIEEE39-bustestsystem. ..................... 39 2-7StepresponseofidentiedLTImodelandIEEE39-bustestsystem. ....... 40 2-839-bussystem:Theconvergenceofxiindicatesthepredictionsofthetheoreticalresultsareaccurateevenwhentheassumptionsusedintheproofsofthoseresultsarenotsatised. ................................... 42 2-939-bussystem:Eectsofchangeintopologyandcommunicationdelay:generator5disconnection. ..................................... 44 2-1039-bussystem:Eectsofchangeintopologyandcommunicationdelay:150-MWloaddisturbanceatbus27. ............................. 45 2-1139-bussystem:Eectofnoiseinfrequencymeasurement:150-MWloaddisturbanceatbus27. ....................................... 46 2-1239-bussystem:Eectofcutofrequencyofloads:150-MWloaddisturbanceatbus27. ......................................... 48 2-1339-bussystem:EectofdierentsamplingperiodsontheperformanceofDGPundera150-MWloadincreaseatbus27. ...................... 49 2-1439-bussystem:Eectofloadquantization(numberofbins)ontheperformanceofDGPundera150-MWloadincreaseatbus27. ................. 50 2-1539-bussystem:EectofstochasticloaductuationsontheperformanceofDGPundera150-MWloadincreaseatbus27. ...................... 51 3-1Illustrationofthegrid'sregulationneeds. ...................... 57 3-2Gridmodel.irepresentsmeasurementnoiseforloadi.(ni)denotes\allloadsexcepti." ....................................... 58 7

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3-3BodeplotoflinearizedgridmodelG(s)from[2]. ................. 60 3-4Lineargridmodel.Noloadcontrol:Pk)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(![k]2=135. .............. 68 3-5Lineargridmodel.10loads:Pk)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(![k]2=84:3(37:6$reductionfrombaseline). 69 3-6Lineargridmodel.100loads:Pk)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(![k]2=32:6(75:6%reductioncomparedtobaseline). ....................................... 70 3-7Lineargridmodel.100loads:Pk)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(![k]2=55:6(58:8%reductioncomparedtobaseline);stricterbandwidthconstraintsthaninFigure3-6. ........... 72 3-8DiagramofthemagnitudeoftheDFTofthelteredrandomprocesswithexiblecapacity1. ....................................... 74 3-939-bussystem:Resultswithauxiliarysecondarycontrolonly:nointelligentloads.Pk)]TJ /F3 11.955 Tf 5.47 -9.69 Td[(![k]2=0:1218Hz2. .............................. 76 3-1039-bussystem.BaLDuR-DMPCalgorithm:nobandwidthconstraints.19loads:Pk)]TJ /F3 11.955 Tf 5.47 -9.69 Td[(![k]2=0:0318Hz2(74%objective-valuereductioncomparedtobaseline.) 77 3-1139-bussystem.BaLDuR-DMPCalgorithm.19loads:Pk)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(![k]2=0:0451Hz2(63%reductioncomparedtobaseline). ....................... 78 3-12Frequencycontentofloadsatbuses18(top)and31(bottom). .......... 79 3-1339-bussystem.BaLDuR-DMPCalgorithm:inaccuratefrequencymeasurements.19loads:Pk)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(![k]2=0:0878Hz2(28%reductioncomparedtobaseline).i[k]!0forallloadsduetomeasurementnoise. ...................... 80 A-1Anillustrationoftherelevantsetsfora2-Dcase.Theequalityconstraintissatisedonthelinesegment,whichisthesetU.Thethicksub-segmentistheoptimalsolutionset,X. .............................. 84 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyDISTRIBUTEDCOORDINATIONOFSMARTLOADSTOPROVIDEANCILLARYSERVICETOTHEPOWERGRIDByJonathanBrooksDecember2017Chair:PrabirBarooahMajor:MechanicalEngineering Maintainingabalancebetweengenerationandconsumptioninthepowergridisparamounttoitsstableandreliableoperation.Renewableenergysourcesarebeingincreasinglyusedinthepowergrid,andthisrenewablegenerationbringswithitvolatilitythatactsasalargedisturbancetothegrid.Traditionally,generatorstrackchangesinthenetload(generationminusconsumption).Becauseoflargevariabilityofrenewablegeneration(e.g.,solarandwind),thisrequiresthegeneratorstorampquickly,whichmayincreaseemissionratesofgreenhousegases|counteractingtheoriginalbenetoftherenewablegeneration.Anotherpossibilityistomanipulatetheconsumptionofloadsinordertocounteractthevariabilityinrenewablegeneration,i.e.,demand-sideancillaryservices.Thisisalow-costandlow-emissionsolutiontotheproblemcausedbyvariabilityofrenewableenergies.However,varyingconsumptionmayhaveanegativeimpactonconsumers(e.g.,higherairtemperaturesinabuilding).Avoidingorminimizingsuchnegativeimpactsisanecessaryhurdletoensurewidespreadconsumerparticipationtohelpthegrid.Anotherkeytechnicalobstacletodemand-sideancillaryservicesistheneedfordistributedcoordinationamongmanyloads.Sinceeachloadcanprovideonlyasmallamountdemandvariation,alargenumberofthemmustbeusedinacoordinatedfashion. Thetopicofthisdissertationisdevelopingalgorithmstomakeuseofloadstoprovideancillaryservices.Wefocusontwoservices,amongmany,thatexibleloadscanprovide|contingencyreservesandnet-load-following.Weproposetwocontrolalgorithms 9

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fordistributedcoordinationofloadstoprovideancillaryservices,whileminimizingoravoidingloads'lossinqualityofservice. Therst,theDistributedGradientProjection(DGP)algorithm,isdesignedforloadstoactascontingencyreservesintheeventofasuddenandlargechangeingeneration.Loadseliminatethenetloadwhileminimizingthedisutilityassociatedwithvaryingconsumption.Eachloaduseslocalmeasurementsofgridfrequencytoinfertheglobalimbalancebetweengenerationandconsumption.Loadsuselimitedcommunicationtoshareinformationwithneighbors.Eachloadcombinestheirneighbors'informationandthelocalfrequencymeasurementstodetermineifitshouldincreaseordecreaseconsumption.Weprovideproofsofconvergence,showingtheloadsconvergetoanglobaloptimalsetthatisgloballyasymptoticallystable.SimulationresultsshowthattheDGPalgorithmworksaswellorbetterthanapreviouslyproposedalgorithmforthesameproblem. Thesecondalgorithmwepropose,theBandwidth-Limited,Disturbance-Rejecting,DecentralizedModel-PredictiveControl(BaLDuR-DMPC)algorithm,isdesignedtoprovidenet-load-followinginaslowertimescale.Itdoesnotusemodelsofeconomicdisutility;insteaditensurestheconsumptionchangesobeypredeterminedconstraintsthatensureconsumers'qualityofservice.Inparticular,theBaLDuR-DMPCalgorithmminimizespredictedgrid-frequencydeviationswhileenforcingbandwidthconstraintsonloads'controlactions.Thisallowsloadstolimitactionsinfrequencyrangesthatmaynegativelyaectconsumers;thisprovidesboundsontheeectsofthenet-load-followingserviceseenbyconsumers.Convergenceisestablishedunderidealizedconditions,andsimulationresultsindicatethatloadsaresuccessfullyabletoreducefrequencydeviationsinthegridwhileensuringqualityofserviceforconsumers. 10

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CHAPTER1INTRODUCTION 1.1Overview Itisimportantforthestabilityandreliabilityofthepowergridthatgenerationmatchesconsumptionatalltimescales[ 3 ].Otherwise,blackoutsmayensue.Withtheincreasingamountofgenerationsuppliedbyrenewableenergysources,suchaswindandsolar,thisissueisincreasinglyimportantbecauserenewableenergiesareoftenhighlyvolatile.Toaccountforthisvolatility,fossil-fuelgeneratorsmaybepart-loadedorkeptonstandbytoallowexibilityingeneration-rampingintheeventoflargechangesinrenewablegeneration[ 4 ].Optimalloadingofresourcesinthepowergridisawellknownproblem[ 5 ].However,generatorsmaybelessecientwhenrapidlyrampingandwhenoperatingatpart-load,whichresultsinincreasedemissionrates[ 6 ].Part-loadingrequiresadditionalgeneratorstosupplytheneedsofthegridaswell,andbuildingadditionalfossil-fuelgeneratorstomitigaterenewablevolatilitywillreducetheenvironmentalbenetsoftherenewableenergies.Thus,newresourcesareneededthatcanbalanceouttheintermittencyofrenewablegenerationwithoutosettingthebenetsoftherenewablegenerationitself.Batteriesareonesuchresource,butthosemaybeprohibitivelyexpensive.Loadsareanotherresourcethatmayberecruitedtofulllthisneed;developingcontrolalgorithmsforthispurposeisthefocusofthisdissertation. Thepotentialforloadstosupportthegridiswellknown;e.g.,see[ 7 8 9 10 ]andthereferencestherein.Inparticular,loadscantrackchangesinrenewablegeneration;e.g.,increasingconsumptionwhenthesunisshining,andreducingconsumptionwhentheskiesareovercast.Thisconsumption-rampingmimicsfossil-fuel-generationramping,butitdoesnotcarrywithittheadverseeectsoffossil-fuelgeneration-ramping,suchasincreasedgreenhouse-gasemissions[ 10 11 ].Iftheseloadsareharnessedforthepurposeofprovidingwhatisknownasdemand-sideancillaryservicetothepowergrid,thenthenumberof 11

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fossil-fuelgeneratorsrequiredforstableandreliableoperationofthepowergridmaybereduced. Thepapers[ 12 13 14 15 16 ]detailmethodsforloadcontrolforon-oloads.In[ 12 ],on-otemperaturesetpointsforthermostaticallycontrolledloadsareadjustedtoprovidedemand-sideancillaryservice.Similarlyin[ 13 ],refrigeratorsadjuston-otemperaturesetpointsbasedonlocalfrequencymeasurements;thecontrollawisdesignedtoensurerefrigeratorsinthemostneedofswitchingdosorst.In[ 14 ],loadsswitchonandobasedonfrequencymeasurements,usingmandatoryon/otimesfollowingaswitchandrandomon/otimesprecedingaswitchtoavoidhigh-frequencyswitchingandsynchronization,respectively.Aprobabilitysignalisbroadcasttoloadsin[ 15 16 ];loadsthenturnonorobasedonthebroadcastprobability,whichdiscouragessynchronizationamongloads. Inthisdissertation,weconsiderloadsthatcanvarytheirconsumptioncontinuously.Ithasbeendemonstratedthatcontinuouslyvariableloadscanprovideservicetothepowergrid[ 17 18 ].Howtocoordinateconsumptionchangesformanycontinuouslyvariableloadsremainsanopenproblem.InmanypowersystemsintheUnitedStates,abalancingauthoritysendsareferencesignalcalledtheAreaControlError(ACE)togridresourcesprovidingancillaryservices[ 19 ].Thisarchitecture(inwhichacentralauthoritybroadcastsareferencesignal)isutilizedintheliterature[ 11 20 21 17 22 ].In[ 17 23 ],scaledreferencesignalsareusedbyloadstodetermineconsumption.However,referencesignalsliketheseeitherdonotaccountforindividualloads'disutilities(costforchangingconsumption)orrequirethebalancingauthoritytohaveallproblemdata|includingsensitiveinformationsuchasconsumer'spowerdemandandtheircostofvaryingthatdemand|violatinguserprivacy. Distributedcoordinationispossiblebyusingpricesignals,whichallowloadstodeterminetheirownchangesinconsumption[ 11 24 25 ].However,consumers'nancialrewardsarethenbasedonvolatilereal-timeprices[ 26 ],whichmaylimitconsumer 12

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participation[ 27 ].Distributedcoordinationisalsopossiblebyutilizingthecyber-physicalnatureoftheelectricgridwhereby\informationcanbetransmittedthroughactuationandsensing"[ 28 ].Thefrequencyofthevoltageinthegridisafunctionofthemismatchbetweengenerationandconsumption|whenthereisexcessgeneration,thefrequencyincreases,andviceversa.Furthermore,eachloadmaymeasurethelocalfrequencyofthepowerbeingsuppliedtoit.Fromthislocalinformation,loadsmayindividuallyinferthegeneration-consumptionmismatchinthegrid.Usinglocalfrequencymeasurements,loadsmayprovidedemand-sideancillaryservices[ 29 30 31 32 33 ].Inparticular,loadsmayparticipateindemanddispatch:providingcontinuousandautomaticservicetothegridwithoutdirectinterferencefromconsumers[ 2 ]. Thereisanimportantcaveatthatmustbeaddressedregardingdemand-sideancillaryservice:itisparamountthatqualityofservicetoconsumersismaintained;i.e.,theeectoftheresultingchangesinconsumptiononconsumersshouldbeminimized.Onepossibilityistheuseofmaximizingafunctionthatmodelstheconsumer'seconomicutilityasafunctionofchangesinconsumptionor,equivalently,minimizingadisutilityfunction.Suchaformulationmayresultinanoptimizationproblemwithalargefeasibleset.Distributedoptimizationforgeneralproblemshasbeenstudiedintheliterature(e.g.,[ 34 35 36 ]),butthesolutionsproposedinthesereferencesdonotexploitthespecialstructureofourproblemduetothecyber-physicalnatureofthepowergrid.Forpower-systemapplications,frequency-basedcontrolisapopularsubjectintheliterature.Inadditiontothepapersalreadymentioned,[ 37 ]usesproportionalcontrolonmeasuredfrequencytoadjustpowerconsumption.Thepapers[ 38 39 40 1 ]proposeseveralfrequency-basedloadcontrolalgorithmsusingeconomicdisutilityfunctions.Inparticular,Zhaoetal.[ 1 ]designaloadcontrolalgorithmforfrequencyregulationduringacontingency;Zhaoetal.posetheproblemasanoptimizationproblemtominimizeeconomicdisutilityofconsumers. 13

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Quantifyingconsumers'disutilityinresponsetoconsumptionchangesischallenging,andworkinthisareaislimited.Apopularchoiceisaquadraticmodel[ 41 42 ];in[ 43 ],aquadraticdisutilitywasderivedfromananemarginal-utilitymodel.In[ 44 ],anexponentialfunctionisusedtomodeldisutility,while[ 45 ]proposesadynamicdisutilitymodel.Astudyofanindustrialaluminumsmeltingplantprovidingancillaryservicessuggeststhattheremaybenodisutilityforseveralhourswhenchangingconsumptionwithinsomethresholdofanominalvalue,butthereissignicantdisutilityifconsumptionisvariedtoomuchorfortoolong[ 46 ].Likewise,[ 17 23 ]showedthatconsumptionincommercialair-conditioningloadscanbevariedtoprovideancillaryserviceswithoutanydisutility(adverseeectonindoorclimate)aslongasthechangesinconsumptionaresmallinamplitudeandband-limited.Basedonthesestudies,wehypothesizethatanappropriatemodelofdisutilityformanyconsumersisnotstrictlyconvex;i.e.,thedisutilityiszeroforsmallchangesinconsumption,butthereisnon-zerodisutilityforlargerchangesinconsumption. Anotherpossibilityistoforgomodelingconsumers'disutilityaltogetherbyensuringchangesinconsumptionarezero-mean,whichcanprovidestrictguaranteesonQoS.Thestudiesin[ 17 23 46 ]showedthataslongasconsumptionchangesarelimitedtocertainfrequencybandswithsucientlysmallamplitude,itmaybepossibleforloadstoprovideservicetothegridwithoutincurringanydisutilityatall.Suchbehaviormaybeachievedbyenforcingbandwidthconstraintsontheconsumption'sFouriertransform,includingaDCconstraint.Thiscanprovideastrict,predeterminedconstraintonlossofQoS|eliminatinguncertaintythatmayprohibitconsumersfromparticipatingindemand-sideprograms. Intheworkpresentedhere,weproposetwonon-centralizedload-controlalgorithmsfordemanddispatch.Therstisadistributedalgorithmwithapplicationtoloadsusedasspinningreservesduringcontingencyevents.Thesecondisadecentralizedschemewithapplicationtoloadsusedasresourcesfornet-loadfollowing.Theproposedalgorithmsare 14

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non-centralizedinthatallchangesinconsumptionarecalculatedbyeachloadindividuallywithsomeornocommunicationamongloads.Eachalgorithmuseslocalfrequencymeasurementstodetermineglobalinformationaboutthegrid. 1.2Contribution InChapter 2 ,weproposetheDistributedGradientProjection(DGP)algorithmforcontingencyreserves.Loadsuselocalfrequencymeasurementsandamodelofthegridtoprovideservicetothepowergridwhilesimultaneouslyminimizingtheircollectivedisutilityusinginter-loadcommunication.Thisoptimizationproblemwasexaminedin[ 1 ].ThemaincontributionoftheDGPalgorithmoverthatof[ 1 ]isthattheDGPalgorithmhereisapplicabletodisutilityfunctionsthatarenotstrictlyconvex,whereasthealgorithmproposedbyZhaoetal.isnotimplementableforsuchfunctions.Asdiscussedintheprevioussection,non-strictlyconvexdisutilityfunctionsaremore-realisticmodelsofconsumerbehavior. ThealgorithmproposedinChapter 3 forgoesmodelingconsumers'disutilitiesaltogether.WecallitBandwidth-Limited,Disturbance-Rejecting,DecentralizedModel-PredictiveControl(BaLDuR-DMPC).TheBaLDuR-DMPCalgorithmensuresstrictboundsonconsumers'lossesinqualityofserviceusinganMPCscheme,inwhichloadsuselocallyobtainedfrequencymeasurementsanddisturbancepredictionsthatarebroadcastbyagridoperatortoallloads.ThecontributionoftheBaLDuR-DMPCalgorithmistheuseofbandwidthconstraints:constraintsareplacedonthemagnitudeoftheFouriertransformofthechangesinconsumption.Stricterconstraintsmaybeplacedonsomefrequencies,whilelooserconstraintsmaybeplacedonothers.Thisallowsloadstoensurequalityofserviceismaintainedwithinadesiredlevel.TheBaLDuR-DMPCalgorithmisalsodecentralizedinthesensethatthereisnointer-loadcommunication,andallcontrolactionsarecomputedlocallybyeachload.TheBaLDuR-DMPCalgorithmwasdesignedfornet-load-following,whichpertainstoaslowertimescalethandocontingencyreserves.Becauseofthis,theloadsareabletouseaverysimplemodelofthepowergrid 15

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(aconstantgain),whichreducescomputationtimes|asignicantfactorforreal-timeMPC. 16

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CHAPTER2PROVIDINGCONTINGENCYRESERVESBYMINIMIZINGDISUTILITY 2.1IntroductionandMotivation Inthischapter,weconsiderdemand-sidecontingencyreserves|loadsthatrespondquicklywhenacontingencyoccurs;e.g.,whenageneratorsuddenlygoeso-lineoratransmissionlinetrips.Wedevelopademand-dispatchalgorithm,whichwecallDistributedGradientProjection(DGP),basedonminimizationofconsumers'economicdisutilityduetotheresultingdemandvariations.Aproofofconvergenceundercertainassumptionsisprovided,andsimulationresultsarepresentedforalineargridmodel. TheDGPalgorithmuseslocalfrequencymeasurementsandlimitedload-to-loadcommunicationtocalculatechangesinconsumption.Loadsareassumedtoknowtheirowneconomicdisutilitiesasfunctionsofchangesinconsumptionfromnominal.Eachloadmeasureslocalfrequencyandsharesdisutility-gradientinformationwithitsneighborstodeterminewhetheritshouldincreaseordecreaseconsumption.Thisallowsconsumptionchangestobecalculatedinadistributedmannerbyeachload. Zhaoetal.[ 1 ]haveproposedasolutiontothisproblemalready.Theirsolutionsolvesthedualproblem,whiletheDGPalgorithmiseectivelyaprimal-dualmethod.ThemainadvantageoftheDGPalgorithmovertheoneproposedbyZhaoetal.isthattheDGPalgorithmdoesnotrequiredisutilitymodelsthatarestrictlyconvexinconsumptionchange|whichisrequiredbyZhaoetal..SincethealgorithmbyZhaoetal.usestheinversefunctionofthegradient,whichdoesnotexistforfunctionsthatarenotstrictlyconvex,theiralgorithmcannotbeusedwithsuchmodels.Figure 2-1 showstwopossiblechoicesfordisutilityfunctions.WhiletheDGPalgorithmmaybeimplementedinloadswitheitherofthedisutilityfunctionsinthegure,thealgorithmproposedbyZhaoetal.isnotimplementableinloadswithdisutilityfunctionsoftheform f1,whichmaymodelconsumerbehaviorthatisquitecommon[ 17 46 23 ]. 17

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Figure2-1. Alternatemodelsofconsumerdisutilityvs.consumptionchange. Ataglance,theDGPalgorithmappearssimilartothatproposedin[ 47 ],whichproposedadistributedalgorithmtosolvetheunit-commitmentproblemforgenerators:determinesetpointsforgeneratorstomatchconsumptionwhilerespectingindividualgeneratorconstraintsandminimizingthetotalgenerationcost.ThemainadvantageoftheDGPalgorithmoverthatof[ 47 ]isthatthealgorithmof[ 47 ]requiresgeneratorstoknowpartofthetotalloadsuchthatthetotalloadisfullyknownamongtheentiregenerationnetwork(evenifnosinglegeneratorknowsthetotalload).Incontrast,theDGPalgorithmrequiresnoloadstoknowthetotalmismatch;ratherthemismatchisestimatedbyeachloadindependentlyvialocalfrequencymeasurements. Therestofthischapterisorganizedasfollows.Section 2.2 formallyformulatestheproblemtobesolvedbytheDGPalgorithm.TheDGPalgorithmisintroducedinSection 2.3 .ConvergenceanalysisoftheDGPalgorithmispresentedinSection 2.4 .ResultsfromnumericalsimulationsareshowninSection 2.6 .Finally,Section 2.8 concludesthechapter. 18

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2.2ProblemFormulation Consideranelectricgridwithasinglefrequencythroughoutthegrid,whosenominalvalueisdenotedby!.Thisisthecasewhenelectricaldistancesarenegligible|suchasinamicrogrid.Therearencontrollableloads.Thedeviationofloadi'sconsumptionfromitsnominalvalueisdenotedbyxi.Loadiincursadisutilityfi(xi)asaresultofaconsumptionchangexi,withtotalconsumptionf(x),Pni=1fi(xi).Thedeviationmustlieini,[xi;xi],wherexi0xiarespecieda-priori. Letgbethegenerationdeviationfromthenominalvalue.Theproblemisfortheloadstodecidehowmuchtochangetheirownconsumptionsothattheconsumption-generationmismatchiseliminatedwhiletheresultingtotaldisutilityoftheloadsisminimized:minxi;i=1;:::;nnXi=1fi(xi);s:t:nXi=1xi=g;xi2i: (2{1) Loadicanobtainanoisymeasurementofthegridfrequencyandcanuseittomakeadecisiononxi.Inaddition,thecomputationofthedecisionvariablesximustbedistributedinthefollowingsense.ThereisaconnectedcommunicationgraphG=(V;E),wherethenodesetV=f1;2;:::;ngissimplytheloadsandtheedgesetEVV,specieda-priori,determineswhichpairsofloadscanexchangeinformation.ThesetofneighborsNiofloadi,withwhichitcanexchangeinformation,isdenedbyNi=fjj(i;j)2Eg.Thefrequencymeasurementsareessentialsinceeveryloadcanusethemtoestimatetheequalityconstraintviolationu,g)]TJ /F5 11.955 Tf 12.2 8.97 Td[(Pni=1xi.HowthisisdoneisdescribedinSection 2.2 AlthoughProblem( 2{1 )doesnotincludetime,theproposedalgorithmisaniterativeapproach,sotimedoesplayarole.Thereasonforthisistwofold:i)duetothelimitedinformationavailabletotheloads,theyareunabletondthesolutioninjustonestep,sotheymove\toward"thesolutionandthenre-evaluateatthenewlocation;ii)duetogeneratordynamics,thefrequency,!,isdependentontheloads,soastheloadschangeintime,sodoesthefrequency,whichtheloadscanthenuseforfeedbackcontrol.Time 19

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ismeasuredbyadiscreteiterationcounter: k=0;1;:::.Thegenerationattimekisdenotedbyg[k]sothatthegenerationchangefromnominalisg[k],g[k])]TJ /F3 11.955 Tf 12.02 0 Td[(g,wheregisthenominalgeneration.Weassumethat,atk=0,totalloadandtotalgenerationareequal,andwelimitourselvestostepchangesingeneration.Thatis,g[k]=0fork
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1. Obtain^ui[k]fromthemeasurement,~!i[k],usingastateestimator,whichisdescribedinSection 2.2 2. Computegradientd dxifi(xi[k]),transmitgradientvaluetoneighbors,receiveneighbors'gradientvalues,andcomputexi[k]:=)]TJ /F3 11.955 Tf 9.29 0 Td[(nirfi(xi[k])+Xj2Nirfj(xj[k]): (2{3) 3. Updateloadchangeasxi[k+1]=Pixi[k]+[k]xi[k]+[k]^ui[k]; (2{4)where Pi[]denotesthestandardprojectionoperatorand[k];[k]'sarestepsizes. Notethatloadsonlyexchangegradientinformation,soeachload'sdisutilityandconsumptioninformationremainprivateandarenotsharedwithotherloads. Weborrowtheestimationmethodproposedin[ 1 ]forestimatingu,althoughitispossibletouseanyestimatorintheDGPalgorithm.Thepowergridismodeledasadiscrete-timeLTIsystemwithconsumption-generationmismatch, u[k],astheinputandfrequencydeviationfromnominal,![k],astheoutput.Ateachtimek,loadiobtainsthenoisymeasurement,~!i[k],toestimatethestateoftheplantbyusingtheestimatorin[ 48 ],whichwasdevelopedforestimatingthestateofasystemwithanunknowninput.Oncethestateestimateisobtained,eachloadestimatestheunknowninputbyeectivelyassumingthatthemostrecentoutputiserror-freeandthensolvingforthepreviousinputfromthestateequations. Wedenotetheestimationerrorattimekby[k],^u[k])]TJ /F3 11.955 Tf 11.96 0 Td[(u[k]1; (2{5) 21

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where ^u[k]isthecolumnvectorof^ui[k]'s.Denethe-algebra,F[k)]TJ /F1 11.955 Tf 10.91 0 Td[(1]:=(i[`)]TJ /F1 11.955 Tf 10.9 0 Td[(1]ji2V;1`k).Itwasshownin[ 48 ]thatEi[k]jF[k)]TJ /F1 11.955 Tf 11.96 0 Td[(1]=0: (2{6)Thefollowingpropositionisreproducedfrom[ 1 ]. Proposition2.1([ 1 ]) LetA,B,andCdenotetheprocess,input,andoutputmatricesofthestate-spacemodelofthepowergridusedinthestateestimatordescribedin[ 48 ],andletIAbeanidentitymatrixthesizeofA.Ifeveryeigenvalueof(IA)]TJ /F3 11.955 Tf 12.3 0 Td[(B(CB))]TJ /F6 7.97 Tf 6.59 0 Td[(1C)Alieswithintheunitcircle,thenlimk!1E(i[k])2jF[k)]TJ /F1 11.955 Tf 11.96 0 Td[(1]exists. Thefollowingcorollaryisastraightforwardconsequenceof( 2{6 )andProposition 2.1 basedonthedenitionofamartingale-dierencesequence. Corollary1 IftheconditionforProposition 2.1 holds,then( 2{6 )andProposition 2.1 implythattheestimationerrorsequence,[k],isamartingale-dierencesequence. 2.4ConvergenceAnalysis 2.4.1MainResults Wemakethefollowingassumptionsforouranalysis. Assumption2.1 (Assumptionsondisutility). 1. fi(xi)isconvexforeachiwitha(notnecessarilyunique)minimumatxi=0. 2. fi(xi)iscontinuouslydierentiableforeachi. 3. rfi(xi)isLipschitzforeachi. Assumption2.2 (Geometricassumptions). 1. Thedomain,,iscompact. 2. Thecommunicationgraph,G,isconnected. 3. Thedisturbanceisaconstant:g[k]gforallk0. Assumption2.3 (Technicalassumptions). 1. [k]=c[k]forsomepositiveconstantc. 2. Thefunction,[k]!0,satisesP1k=0[k]=1andP1k=0([k])2<1. 22

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3. Theestimationerrorsequence,[k],isamartingale-dierencesequence. Assumption 2.1 isreadilysatisedbecausethedisutilityfunctionsareamodelingchoice.Assumption 2.2 (1)isalwaysmetinpracticesincealoadcannotvarydemandoutsideitsmaximumratedpowerand0.Assumption 2.2 (3)meansthatweonlyconsiderastep-changeingeneration,whichisareasonableapproximationofacontingencyevent(e.g.,ifageneratortrips[ 1 ])andwhichwemakeforeaseofanalysis.Assumptions 2.3 (1-2)aredesignchoicesforthealgorithm,sotheymaybesatisedinpracticebychoosing[k]and[k]appropriately.Assumptions 2.3 (2-3)arestandardintheeldofstochasticapproximation[ 49 ].Assumption 2.3 (3)holdsinthisworkduetotheestimatorusedin[ 1 ]andProposition 2.1 [ 50 ]. Themainconvergenceresultisthefollowing. Theorem2.1 IfAssumptions 2.1 2.2 ,and 2.3 holdandallsolutionstoProblem( 2{1 )arestrictlyfeasible,x[k]convergestoasolutiontoProblem( 2{1 ),almostsurely(a.s.). 2.4.2ProofofTheorem 2.1 TheproofofTheorem 2.1 reliesonthesocalledo.d.e.methodofstochasticapproximation,whichestablishesarigorousconnectionbetweennoisydiscreteiterationsandacontinuous-timeo.d.e.[ 49 51 ],statednext. Proposition2.2(Theorem2.1(Chapter5)in[ 51 ]) Considerthesequencefy[k]ggeneratedbytheiterationy[k+1]=Phy[k]+[k])]TJ /F3 11.955 Tf 5.48 -9.68 Td[(h(y[k])+[k]i;where Pistheprojectionoperatoronto,h(y):Rn!RnisLipschitz,andf[k]gisamartingale-dierencesequence.If[k]satisesAssumption 2.3 (2)theny[k]convergesa.s.tosomelimitsetoftheo.d.e.,_y(t)=)]TJ /F6 7.97 Tf 27.61 -1.86 Td[(;y(t)[h(y(t))]; 23

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where )]TJ /F6 7.97 Tf 7.31 -1.85 Td[(;y(t)[]denotesthecontinuous-timeprojectionoperatorofy(t)2Rnonto.Thatis,theithcomponentof)]TJ /F6 7.97 Tf 7.32 -1.8 Td[(;y[z]is )]TJ /F1 11.955 Tf 5.47 -9.69 Td[()]TJ /F6 7.97 Tf 7.32 -1.79 Td[(;y[z]i=8>>>><>>>>:0;yi=mini;zi<00;yi=maxi;zi>0zi;o:w: Byusing( 2{5 )andAssumption 2.3 ,theupdatelawoftheDGPalgorithmcanbewrittenasxi[k+1]=Pixi[k]+[k]cXj2Ni)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(rfj(xj[k]))-222(rfi(xi[k])+u[k]+i[k]; (2{7)orcompactly,x[k+1]=Px[k]+[k]()]TJ /F3 11.955 Tf 9.3 0 Td[(cLrf(x[k])T+u[k]1+[k]); (2{8)where rf(x)isthegradientoff(x):=Pni=1fi(xi)andL2RnnisthegraphLaplacianofG[ 52 ]:Lii=ni(numberofneighborsofnodei),andforj6=i,Lij=)]TJ /F1 11.955 Tf 9.3 0 Td[(1ifj2Ni,Lij=0ifj=2Ni.ByProposition 2.2 ,theiterates,x[k],convergea.s.toalimitsetoftheo.d.e.,_x(t)=)]TJ /F6 7.97 Tf 27.61 -1.86 Td[(;x(t)[)]TJ /F3 11.955 Tf 9.29 0 Td[(cLrf(x(t))T+u(t)1]; (2{9)u(t):=g)]TJ /F8 11.955 Tf 11.95 0 Td[(1Tx(t) (2{10)Thetaskistoprovethatthetrajectoriesofo.d.e.( 2{9 )convergetothesetofsolutionstoProblem( 2{1 ),whichwedenoteby X.Becauseoftheconvexity,x2Xifandonlyifitsatisestherst-ordernecessaryconditionsofoptimality[ 53 ].Wedenotetheboundaryofby@,andwedenotetheinteriorofbyo.BecauseofthehypothesisaboutstrictfeasibilityinTheorem 2.1 ,onlytheequalityconstraintisactive,whichisalsoregular,and 24

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thusitisstraightforwardtoverifythatX=fx2oj1Tx=g;rfi(xi)=rfj(xj);8i;j2Vg: (2{11)Becauseeach fiisconvex,eachgradientrfi(xi)isnondecreasing,andtheoptimalset,X,isconnected.Itfollowsthatthegradient,rf(x),atanyoptimalpointx2Xisunique,whichcanbeseenviacontradiction.Supposex1;x22Xwithkrf(x1)k
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dierentialinclusionisdenedplaysacrucialroleinbothexistenceanduniqueness.Thiscanfurthercomplicatetheanalysis. Theo.d.e.( 2{9 )iscloselyrelatedtothesocalledSkorokhodproblem,whichdealswithstochasticdierentialequationswithboundaries[ 56 ].TheconnectionwiththeSkorokhodproblemandoursisuseful;knownresultsontheSkorokhodproblemprovideguaranteesofnotonlyexistencebutalsouniquenessofCaratheodorysolutionstoo.d.e.( 2{9 )[ 54 ].Forcompleteness,therelevantresultfrom[ 54 ]ispresentedbelow. Proposition2.3(Theorem2.5in[ 54 ]) Leth()beLipschitzcontinuousand_x(t)=)]TJ /F6 7.97 Tf 27.61 -1.86 Td[(;x(t)h(x(t)); (2{13)where )]TJ /F9 11.955 Tf 11.49 0 Td[(istheprojectiondenedinProposition 2.2 andiscompact.Foranyx(0)2,thereexistsauniqueCaratheodorysolution,x(t),to( 2{13 )startingfromx(0).Furthermore,x(t)iscontinuouswithrespecttotheinitialcondition,x(0). Thesecondchallengeisanalyzinglimitingbehavioroftheo.d.e.Thetrajectorymay\evolve[]alonga`section'of[@]....Atalatertimethesolutionmayre-enter[o],oritmayenteralower[-]dimensionalpartof[@]."[ 54 ].Thetrajectorymaygoondoingsowithouteverconvergingtoalimitsetwithtractablestructure.Thisisaseverehurdleinanalyzingdynamicalsystemsevolvinginaboundedregion,wheretheboundednessisenforcedthroughaprojectionoperator.In[ 54 ],thishurdlewassidesteppedbyassumingthatthe!-limitsetiscontainedinthesetofxedpoints.Thatisequivalenttoassumingthatcomplicatedlimitsetssuchaslimitcyclesdonotarise.Thereference[ 49 ]alsomentionsthat,inpresenceoftheprojectionoperatorontodomainswithnon-smoothboundaries(suchasinourcase),existenceofano.d.e.limitofthediscrete-timealgorithmmaybeanon-trivialissue.Theclassicreference[ 51 ]onstochasticapproximationando.d.e.methodsmentionsthatbasicresultsonconvergencetoalimitset(suchasProposition 2.2 )israrelyusefulsincethelimitsetcanbethewholeoftheboundeddomaininwhichtheo.d.e.evolves.Thiswasalsothemainargumentin[ 54 ] 26

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thatismentionedabove.Onlyinthespecialcasewhentheright-handsideoftheo.d.e.isagradient-descentsystemcanconvergencetoasetofxedpointsbeestablished[ 51 ].However,( 2{9 )isnotagradient-descentsystem. Ourmainconvergenceresult,Theorem 2.1 ,isobtainedwithoutmakingthestrongassumptionsthataretypical,suchasassumingthatnocomplicatedlimitsetsexist.ThestructureoftheLaplacianmatrixplaysanimportantroleintheproofsofthemaintheoremandthetechnicalresultsneededforthetheoremproof.Akeytechnicalresultisthattheequalityconstraintiseventuallysatised,whichisstatedinthenextlemma. Lemma1 IfAssumptions 2.1 2.2 ,and 2.3 holdandallsolutionstoProblem( 2{1 )arestrictlyfeasible,u(t)!0,whereu(t)isdenedin( 2{10 )andx(t)|inthedenitionofu(t)|isgovernedbyo.d.e.( 2{9 ). Asimilarresultwasobtainedin[ 57 ].However,theanalysisinthatworkusesapenaltymethodinplaceofprojection.Consequently,theresultofLemma 1 isobtainedimmediately.Inourcase,whereweretaintheprojectionoperation,theproofofLemma 1 isnotasstraightforward.TheproofofLemma 1 isprovidedintheAppendix.WearenowreadytoproveTheorem 2.1 ProofofTheorem 2.1 :RecallthediscussionafterProposition 2.2 :wemustprovethatthetrajectoriesofo.d.e.( 2{9 )convergetotheset X.ByProposition 2.3 ,thesolutiontoo.d.e.( 2{9 )iscontinuouswithrespecttotheinitialcondition,so!Lisaninvariantset[ 58 ].Ourgoalistousethisinvariancepropertytocharacterize!Landshowthat!LX.Forconvenience,wewritethe ithelementof( 2{9 )below:_xi(t)=)]TJ /F6 7.97 Tf 27.61 -1.86 Td[(i;xi(t)hcXj2Nirfj(xj(t)))-222(rfi(xi(t))+u(t)i: (2{14)ByLemma 1 x(t)!fx2j1Tx=gg=:U;thisimpliesthatalllimitsetsarecontainedwithinU.Becausethe!-limitsetistheunionofalllimitpoints,wehave 27

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!LU.Let rf (t),minirfi(xi(t)): (2{15)Imagineatrajectorystartingin !L:x(0)2!LU,whichimpliesx(t)2Uforalltbyinvarianceof!L.Sinceu(t)0forthistrajectory,o.d.e.( 2{9 )reducesto_x(t)=)]TJ /F6 7.97 Tf 32.11 -1.86 Td[(;x(t))]TJ /F3 11.955 Tf 12.85 0 Td[(cLrf(x(t))T.Thatis,foreveryi2V,_xi(t)istheaverageofitsneighbors'gradientsatt,whichshowsthat_xr(t)0,wherer(t):=argminirfi(xi(t)).Therefore, rf (t)=rfr(t)(xr(t)(t))isnondecreasingbyconvexity.(Notethatthisdoesnotrequireuniquenessofr(t),sowhentwogradientsarebothminimal,botharenondecreasing.)Because rf (t)isboundedaswell,whichcomesfromboundednessofthedomain,, rf (t)converges.DenotethelimitbyFsothattrajectoriesstartingin!LconvergetoF:=fx2jminirfi(xi)=F;1Tx=gg!LU.Therefore,foranyx(0)2!L,x(t)!F. Wenowshowthatx(t)!Fforanyx(0)2.Supposewehavesometrajectory,x(t)withx(0)=xo2.Fromthedenitionofthe!-limitset,thereexistsapoint,y2!L,andasequence,t0;t1;:::withtn!1asn!1,suchthatx(tn)!yasn!1.Letxn(t):=x(tn+t)foreverytsothatxn(0)=x(tn).Letx!Lbeatrajectorythatstartsfromy:x!L(0)=y2!L.Byconstructionlimn!1xn(0)=y=x!L(0).Therefore,foranyT2[0;1),xn(T)!x!L(T)asn!1bycontinuityofx(t)withrespecttox(0)(seeProposition 2.3 ).Now,wehave limn!1x(tn+T)=limn!1xn(T)=x!L(T).ThisistrueforanyT0,sowemaytakethelimitasT!1toobtainx(t)!Fbecausex!L(T)!FasT!1.Therefore,x(t)!Fforanyx(0)2,so!LF.SinceF!Lalso,wehaveF=!L.Therefore,Fisinvariantbyinvarianceof!L. Next,weshowthatFisinvariantonlyifF=rf.ConsideragainatrajectorystartinginF:x(t0)2Fforsomet0.Ifrfi(xi(t0))=Fforeveryi2V,thentherst-ordernecessaryconditionsforoptimalityaresatised,soF=rf,andF=X,andtheproofiscomplete.Supposethereexistsan`2Vsuchthatrf`(x`(t0))>F.Let 28

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q:=argminirfi(xi(t0));then_xq(t0)0withequalityifandonlyifrfj(xj(t0))=Fforallj2Nq,inwhichcaseqisnotunique.Denotethedierentchoicesforqbyqi;i2V.Because rf`(x`(t0))>FandthegraphGisconnected,_xqj(t0)>forsomeqjandsome>0.Itfollowsthatthereexistssomet1>t0suchthatrfqj(xqj(t1))>F.Notethatu(t)=0impliesrfqj(xqj(t))>Fforalltt1;thisisbecause_xqj(t)!R0asrfqj(xqj(t))!F(see( 2{14 )).Hence,_xqi(t1)>forsomeqi2Nqjandsome>0.Bycontinuingthisargument,weseethateventuallythereexistssomet2>t0suchthat rf (t2)>F;thatis,whenthegradientofoneqjincreases,thatcausesthegradientofallqi2Nqjtoeventuallyincrease,andsoon.However, rf (t2)>Fimpliesx(t2)=2F,soFisnotinvariant|acontradiction.Therefore,ifx(t0)2F,thereexistsno`2Vsuchthatrf`(x`(t0))>F.Hence,rfi(xi(t0))=Fforalli2V,whichsatisestheoptimalityconditions,soF=rf,and!L=F=X,whichcompletestheproof. 2.5AuxiliaryAnalyticalResults 2.5.1AsymptoticStability Thefollowinglemmashowsthattheoptimalsolutionset,X,isnotonlygloballyattractive,butalsogloballyasymptoticallystable.Thenotionofstabilityofaninvariantsetusedhereistheonefrom[ 58 ].TheproofofthislemmaisprovidedintheAppendix. Lemma2 LettheconditionsofTheorem 2.1 hold.Thentheoptimalset,X,isgloballyasymptoticallystable. 2.5.2ConstantStepSize Theorem 2.1 requiresAssumption 2.3 (2)(decayingstepsize).ThefollowingtheoremcharacterizesthelimitingbehavioroftheDGPalgorithmwhenthestepsizeisconstant. Theorem2.2 LetAssumptions 2.1 2.2 ,and 2.3 (1,3)hold.If [k]=isaconstant,thenforany>0,thefractionoftimethatx[k]in( 2{8 )spendsinthe-neighborhoodofXon[0;T]goesto1inprobabilityas!0andT!1. Thegeneralresultforaconstantstepsizeiswellknownintheeldofstochasticapproximation[ 51 ],butitsspecicapplicationtoourproblemrequirestheresultof 29

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Theorem 2.1 .Thatis,theresultholdsifthecontinuous-timeo.d.e.analogueconvergestoX[ 51 ],andTheorem 2.1 showsthatitdoesindeedconvergetoX. AlthoughTheorem 2.2 |withconstantstepsize|isnotasstrongasthealmost-sureconvergenceresultofTheorem 2.1 |withdecayingstepsize|itindicatesthattheDGPalgorithmstillworkswellwithaconstantstepsize,provideditissmall.ThenumericalresultspresentedinSection 2.7 areconsistentwiththisprediction. 2.5.3AsymptoticBehaviorforQuadraticCost HerewecharacterizetheasymptoticbehavioroftheDGPalgorithmforaconstantstepsize.Thenonlinearityinthedynamics( 2{8 )duetotheprojectionoperatormakesanalysischallenging[ 49 ].However,basedonTheorem 2.2 wecanarguethatforasymptoticanalysis,theprojectionoperatorcanbeignored,asfollows.Theiteratesspendalmostalloftheirtimeclosetotheoptimalsolutionofproblem( 2{1 )aslongasthestepsizeissmall.Sincethesolutionsetofproblem( 2{1 )lieswithinthestrictinteriorof,thenforasmall,theiteratesoftheDGPalgorithmspendmostoftheirtimewithinthestrictinteriorof.Hence,asymptoticallytheprojectionislikelytobemostlyinactive.Therefore,weconsidertheunprojecteddynamicsobtainedbyremovingtheprojectionoperatorinthesubsequentanalysisinthissection.Wefurtherlimitourselvestoaquadraticmodelofconsumerdisutility.Althoughtheseareratherstrongassumptions,thenumericalresultsinthenextsectionshowtheanalyticalresultswiththeseassumptionsstillaccuratelypredictthebehavioroftheDGPalgorithmwhentheassumptionsareviolated.TheproofofthefollowingresultisintheAppendix. Proposition2.4 Leti=Randfi(xi)=qix2iforalli2V,whereqi>0foreachi.LetbetheeigenvalueofA=I)]TJ /F3 11.955 Tf 12.16 0 Td[((cLQ+11T)withmaximumrealpart,whereIistheidentitymatrixofsizen,ListhegraphLaplacianofthecommunicationgraphG[ 52 ],andQ=diag(q1;q2;:::;qn).Thenthedynamicsoftheloads'changesindemandarestableifandonlyif<2=jj.Furthermore,themean, [k],andcovariance,[k],oftheloads' 30

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changesindemandaregivenby[k+1]=A[k]+Bg1;[k+1]=A[k]AT+BW[k]BT;respectively,where B=Iand W[k]=E[k][k]T. Theorem 2.2 doesnotprovideanestimateofhowsmallshouldbe.Byassumingquadraticdisutilities,Proposition 2.4 doesprovidesuchanestimate.Althoughtheloaddynamicsaremadestabletriviallybytheprojectionoperator,theresultofthepropositionimpliestheDGPalgorithmmaybepoorlybehavediftheconstantstepsizeistoolarge,despitethetheoreticalguaranteesofTheorem 2.2 .Althoughaquadraticdisutilitywasassumed,resultsinSection 2.7 indicatetherobustnessofthisresulttothatassumption. 2.5.4CommentonStrictFeasibility TheassumptionofstrictfeasibilityofallsolutionstoProblem 2{1 isnecessaryfortheconvergenceoftheDGPalgorithmtohold.IfXisnotstrictlyfeasible,thentheDGPalgorithmisnotguaranteedtoconvergetoX;infactitmayconvergetonon-optimalpoints.Thismaybeseenthroughthefollowing2Dcounterexample.Letx2R2,g=1,f1(x1)=(x1)2,f2(x2)=(x2)2,=[0;1=4][0;1],andc=1.Gconsistsofthetwonodesandoneedgeconnectingthem.Theconstraint,u=0,issatisedonthelinex1+x2=1.Itisstraightforwardtoverifyx=[1=4;3=4]TistheuniquesolutiontoProblem( 2{1 ),butxisnotstrictlyfeasiblesinceitliesontheboundaryof.Now,from( 2{9 ),_x2(t)jx=)]TJ /F6 7.97 Tf 19.73 -1.86 Td[(2;x2(t)[)]TJ /F1 11.955 Tf 9.3 0 Td[(2x2(t)+2x1(t)+u(t)]x=)]TJ /F6 7.97 Tf 19.73 -1.8 Td[(2;0:75[)]TJ /F1 11.955 Tf 9.3 0 Td[(20:75+20:25+0]=)]TJ /F1 11.955 Tf 9.3 0 Td[(1;whichshowsthat [1=4;3=4]Tisnotanequilibriumpointofo.d.e.( 2{9 ).Therefore,theDGPalgorithmisnotguaranteedtoconvergetoX.Itcanalsobeshownthatforthis 31

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example[1=4;5=12]Tisanequilibriumpointofo.d.e.( 2{9 )andthatitisattractive,but[1=4;5=12]TisnotasolutiontoProblem( 2{1 ).Therefore,fromProposition 2.2 ,theiterateswillconvergealmostsurelytoanon-optimalpoint. 2.6NumericalResultsforLinearGridModel Inthissection,wepresentsimulationresultsoftheDGPalgorithm,andwecompare,whenapplicable,withthoseofthealgorithmdescribedin[ 1 ].Werefertothealgorithmproposedin[ 1 ]asthe\dualalgorithm". 2.6.1SimulationSetup Figure 2-2 showsthesystemarchitectureusedfordesignandsimulation.ThegeneratordynamicsblockshowninFigure 2-2 alsoincludeslocalcontrolsthatareusuallypresentingenerators.Theinputtothesystemmodelistheload-generationmismatch, u,anditsoutputisthefrequencydeviation,!,fromthenominalfrequency,!=60Hz.ThelineargridmodelisG(s)=N(s) D(s); (2{16)whereN(s)=)]TJ /F1 11.955 Tf 9.29 0 Td[(0:00979s11+0:06216s10)]TJ /F1 11.955 Tf 11.95 0 Td[(0:1707s9+0:2644s8)]TJ /F1 11.955 Tf 11.95 0 Td[(0:2526s7+0:1526s6)]TJ /F1 11.955 Tf 11.96 0 Td[(0:05744s5+0:01277s4)]TJ /F1 11.955 Tf 11.96 0 Td[(0:00152s3+8:93210)]TJ /F6 7.97 Tf 6.59 0 Td[(5s2)]TJ /F1 11.955 Tf 11.96 0 Td[(2:31210)]TJ /F6 7.97 Tf 6.59 0 Td[(6s+2:4510)]TJ /F6 7.97 Tf 6.59 0 Td[(8;D(s)=s12)]TJ /F1 11.955 Tf 11.96 0 Td[(7:282s11+23:35s10)]TJ /F1 11.955 Tf 11.96 0 Td[(43:25s9+50:94s8)]TJ /F1 11.955 Tf 11.96 0 Td[(39:6s7+20:38s6)]TJ /F1 11.955 Tf 11.95 0 Td[(6:782s5+1:389s4)]TJ /F1 11.955 Tf 11.95 0 Td[(0:1638s3+0:01072s2)]TJ /F1 11.955 Tf 11.96 0 Td[(0:0003201s+1:46210)]TJ /F6 7.97 Tf 6.59 0 Td[(6 (2{17)Thelossofgenerationismodeledasanexogenousdisturbance, g,inFigure 2-2 .TheestimatorinthegureistheonedescribedinSection 2.2 toestimatetheconsumption-generationmismatchfromlocal,noisyfrequencymeasurements.Theprocessdisturbance,,andmeasurementnoise,i,ateachloadaremodeledaswide-sensestationarywhitenoise.The 32

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processdisturbance,,canbeinterpretedassmall,high-frequencychangesingenerationinadditiontothenominaldisturbance,g. Figure2-2. Systemarchitectureforsimulations.Inter-loadcommunicationisnotshown. TheestimatordescribedinSection 2.2 isusedtoestimatetheunknowninputu.Eachloaduses( 2{16 ){( 2{17 )foritsgridmodel.Theprocessdisturbance,,andmeasurementnoise,i,ateachloadaremodeledaswhitenoise,andisalsoindependentacrossloads.Attimek,thecovariancematrixoftherandomvector[k]isB(0:002MW)2BT,whereBistheinputmatrixofthestate-spacemodel,andthevariancematrixforloadiofi[k]is10)]TJ /F6 7.97 Tf 6.58 0 Td[(6Hz2. Evenwithouttheuseofintelligentloads,thelocalgeneratorcontrolwillchangethegeneratorsetpointinresponsetofrequencydeviationtomatchconsumption,whichwillrestorethefrequencytoitsnominalvalueonitsown.Intelligentloadsaresupposedtohelpthegeneratorinreactingtofrequencydeviationsfastersothatlargeexcursionsofsystemfrequencyareavoided. Foreachloadi,wechoosei=[)]TJ /F1 11.955 Tf 10.02 0 Td[(xi;xi],wherexiischosenfromauniformdistributionandthennormalizedsothatPni=1xi=60MW(asin[ 1 ]). 33

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WetesttheperformanceoftheDGPalgorithmwithtwodistinctdisutilityfunctions.Therstisaconvexbutnotstrictlyconvexfunction:fi(xi)=8>>>>>><>>>>>>:0;jxij>>><>>>>: 200MW;0skT<20s 190MW;20skT<50s 170MW;50skT;where T=0:1secondsisthediscretizationinterval.Thatis,a10-MWgeneratortripsat20seconds,andanother20-MWgeneratortripsat50seconds. Thesimulationsareconductedwitha1D-gridcommunicationnetwork,whereeachloadicommunicateswithloadsfrommaxf1,i)]TJ /F3 11.955 Tf 11.99 0 Td[(n0gtominfn,i+n0g,wheren0n.Forthesimulationresultsreportedhere,weusen=1000andn0=1.ThenetworkisshowninFigure 2-3 Additionally,weusec=10=3and[k]=[0]=(k0:8)fork>0,with[0]=1:5q =n,whereq ,miniqi. 34

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Figure2-3. Thecommunicationgraph.Toavoidclutter,notalledgesareshown. Withalloftheseparameterchoices,Assumptions 2.1 2.2 ,and 2.3 aresatised.NotethatAssumption 2.3 (3)issatisedfromthediscussioninSection 2.2 2.6.2ResultswithNon-StrictlyConvexDisutility Figure 2-4 showssimulationresultsfortheDGPalgorithmwithconsumerdisutilityfunction( 2{18 ).Thedualalgorithm(DA)from[ 1 ]isnotapplicablebecausetheinverseofrf(x)mustexistintoimplementDA,whichisnotthecasewhenjxijai. Thesystemfrequencywithoutsmartloads(i.e.,withgenerator-onlycontrol)isshowninred.UsingDGP,theloadsareabletoassistthegeneratorinavoidinglargefrequencydeviationsfromthenominalwheneachcontingencyoccurs. 2.6.3ComparisonwithDualAlgorithm:ResultswithStrictlyConvexDisu-tility Figure 2-5 showstheresultsofDGPandDAwithquadraticdisutilities( 2{19 ).DGPresultsinasignicantlysmallerfrequencydropcomparedtobothgenerator-onlycontrolandDA.AlthoughDAreturnsthefrequencytothenominalvaluefasterthangenerator-onlycontrol,itdoesnotreducetheinitialfrequencydropasmuchasDGP. However,theconsumerdisutilityislowerforDAthanforDGP.ThisisbecauseDAisrespondingmoreslowlythanDGP,sotheequalityconstraintisnotbeingsatised|resultinginalowercost(andalargerfrequencydeviation).TheslowerresponseofDAisduetotheinversionofthederivativeofeachload'sdisutilityfunction.Becausethederivativeofeachdisutilityfunctionisrathersteep,theinverseisquiteat,solargechangesinitsargumentstillresultinsmallchangesinitsvalue|leadingtosmallchangesinconsumption.Conversely,DGPaggressivelymeetstheequalityconstraintbecause 35

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Figure2-4. Lineargridmodel:PerformanceoftheDGPalgorithmwithconsumerdisutilitythatisnotstrictlyconvex.Stepchangesingenerationoccurat20and50seconds. 36

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Figure2-5. Lineargridmodel:PerformanceoftheDGPanddualalgorithmswithquadraticconsumerdisutilitywithprojection.Stepchangesingenerationoccurat20and50seconds. 37

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ofthegeneration-matchingstep.Thisresultsinalowerfrequencydeviationbutmoredisutility. Althoughthedisutilitiesappeartoreachsteadystate,whereasthatinFigure 2-4 doesnot,neitheralgorithmactuallyreachessteadystateineithergureduetothefundamentalslownessofstochasticapproximation[ 51 ].TherateofconvergenceofthedisutilitycanbeincreasedfortheDGPalgorithmbyincreasingthegain,c,butatthecostofslowerconvergenceoftheconsumption-generationmismatch. Simulationswithvaryingnumberofloads( n=10;100)andvaryingamountofcommunication(n0=10;100;1000)showedsimilartrendsasinthen=1000,n0=1case.Itwasobservedin[ 1 ]thatthedualalgorithmshowedsimilarbehavior.Weconcludethatperformanceofbothalgorithmsislargelyunaectedasthenumberofloadsanddegreeofcommunicationincrease.Thereasonforthisinsensitivitytonetworksizeandstructureislikelytheuseoflocalfrequencymeasurementswhichprovidesglobalinformation,eectivelycreatingavirtualcommunicationlinkbetweenactitiouscentralnodeandallnodesofthenetwork. 2.7NumericalResultsforNonlinearGridModel ThesimulationresultsinSection 2.6 wereobtainedusingalinearmodelofthepowergrid.Inthissection,theDGPalgorithmisimplementedintheIEEE39-bustestsystem[ 59 ],whereeachagentintheDGPalgorithmisanaggregationofloadsateachloadbus.Forsimplicity,wewillcontinuetorefertotheseaggregateagentsas\loads."Theimplementationofthe39-bustestsystemhasthemulti-bandPowerSystemStabilizer(PSS)activatedforsystemstabilizationasimplementedin[ 60 ].Figure 2-6 showsaschematicofthe39-bussystem. 2.7.1EstimatingGlobalPowerImbalance AsdescribedinSection 2.2 ,eachDGPagentrequiresadiscrete-timeLTImodelofthepowergridtoinferpowerimbalancethroughfrequencymeasurements.Inpractice,the 38

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Figure2-6. SchematicoftheIEEE39-bustestsystem. balancingauthorityhasadetailedmodelofthetransmissiongrid,fromwhichtheycanextractsimpliedmodels.Thesemodelsinturncanbeprovidedtoloads. Forthissection,weusetheleast-squaresidenticationmethod[ 61 ],inwhichunknownmodelparametersareestimatedusinginput-outputdatafromthe39-bussystem.Apseudorandom-binary-sequencedisturbancewasappliedtothe39-bussystem,andasecond-orderLTItransferfunctionwasidentiedrelatingglobalpowerimbalancetothemeanfrequencyofthegenerators.Figure 2-7 showsthestepresponseoftheidentiedmodelandthe39-bussystem.Itcanbeseenthattheidentiedmodelishighlyinaccurateduetothecomplexityofthe39-bussystemandthesimplicityofasecond-orderlinearmodel. 39

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Figure2-7. StepresponseofidentiedLTImodelandIEEE39-bustestsystem. 2.7.2SimulationSetup TheIEEE39-bustestsystemhas19loads,andeachloadcanmodulateitsconsumptionby5%;thatis,ifthenominalconsumptionofloadiisxoi,theni=[)]TJ /F1 11.955 Tf 9.3 0 Td[(0:05xoi;0:05xoi]. AsinSection 2.6 ,wetesttheperformanceoftheDGPalgorithmwithaconvex(butnotstrictlyconvex)disutilityfunction( 2{18 ).Thereisacommunicationdelayforeachload,i.e.,eachloadonlyhasaccesstopastvaluesofitsneighbors'gradients.Forfrequencymeasurements,eachloadhasaccesstothespeed(frequency)oftheclosestgenerator,andthosemeasurementsarecorruptedbyzero-meanGaussiannoisewithastandarddeviationof0:01%ofthevalueofsynchronousfrequency(60Hz),unlessotherwisenoted.Theloadsusetheidentiedmodeltoestimatepowerimbalanceusingthefrequencymeasurements.Theconsumptionofloadsvariesovertime.Thisuctuationismodeledasi.i.d.Gaussianadditivenoiseinthenominalpowerconsumptionofeachloadwithzero-meanandstandarddeviationequalto0:01%ofthenominalpowerconsumptionoftheload(exceptforcasestudy5).ThegainsusedintheDGPalgorithmarec=10=3and[k]=0:06=(19maxifig)forallk. 2.7.3ComparisonwithTheoreticalPredictions First,theDGPalgorithmwastestedinanidealscenario,i.e.,assumingloadscancontinuouslyvarydemandwithnoconstraintsintermsofsaturation,speedof 40

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actuation,ordelayincommunications.Theonlynon-idealaspectconsideredisthenoiseinfrequencymeasurements,modeledaszero-meanGaussiannoisewithastandarddeviation(f)of0:01%ofthevalueofthesynchronousfrequency(60Hz).Inthissimulation,qi(see( 2{18 ))ischosenequalto5forallloads. Thegoalofthistestistoobserveifthesystemwillbehaveaspredictedbytheorywhenadisturbanceintheformofa150MWloadincreaseinbus27isappliedatt=10seconds.Twotestswererun:i)ai=0MWandii)forai=2MW,whichisroughlyonequarterof7:89MW(theexpectedchangeindemandofall19loadsiftheyequallysharethecontroleort).Theresultinggridfrequencydeviation,!,isshowninFigure 2-8 (a),andvaluesofxiforthecasesofai=0MWandai=2MWaredepictedinFigure 2-8 (b)and 2-8 (c),respectively.Thespeeddeviation,!,shownisthemeandeviationofall10generatorsfromthenominalfrequency(60Hz). InFigure 2-8 (a),wenoticethattheDGPalgorithmquicklyreducesthefrequencydeviationofthesystemtozeroinbothcases,inspiteofthehighlynoisyfrequencymeasurements.Figure 2-8 (b)and 2-8 (c)showthatthedemandchangesremainwithin3standarddeviationsofthemean([k]3[k]),where[k]wascalculatedfromtheformulainProposition 2.4 |numericallyconrmingthetheoreticalresultspredictedbytheproposition.Thatis,despitethestrongassumptionsusedinitsproof,theresultofProposition 2.4 stillaccuratelypredictstheasymptoticbehavioroftheloads.Therefore,theresultofProposition 2.4 maybeusedtochooseanappropriatestepsize. Itisimportanttostressthatevenwhenanon-strictlyconvexandnon-quadraticdisutilityfunctionwasused(ai6=0),whichviolatestheassumptionProposition 2.4 ,theresultsarenumericallyidentical.Therefore,weusethenon-strictlyconvexcostfunctiondenedby( 2{18 )intheremainderofthechapter. 2.7.4EvaluationofDGPAlgorithmunderPracticalLimitations Real-lifeimplementationsoftheDGPalgorithmwouldfaceanumberoflimitationsthatarenotincludedintheformulationoftheproposedsolution.Wepresentcase 41

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Figure2-8. 39-bussystem:Theconvergenceofxiindicatesthepredictionsofthetheoreticalresultsareaccurateevenwhentheassumptionsusedintheproofsofthoseresultsarenotsatised. 42

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studiesthatassesstheeectofthefollowingpracticallimitations:i)modelmismatch,ii)communicationdelaybetweenloadaggregators,iii)communicationtopology,iv)frequencymeasurementnoise,v)presenceofuncertain,uncontrollablerenewablesandstochasticloads,vi)samplingperiodofdiscretecontrollerandmeasurements,vii)quantizationofcontrolleroutput,andviii)bandwidthoftheactuator.Theparametersrelatedtoeachofthesepracticalconsiderations,alongwiththeirstandardvaluesusedinthesimulations,arepresentedinTable 2-1 ,andthedenitionofallpracticalconsiderationsareprovidedalongwiththepresentationofthecorrespondingparametricstudy.Whenoneparameterisvariedtostudyitseect,othersareheldconstantattheirnominalvalues. Table2-1. Standardvaluesforparametricstudies ParameterNominal dDelayincommunicationbetweenaggregators100ms TsSamplingperiodofcontroller2=60s jijNumberofbins(discretestates)ofloads51 fStandarddeviationoffrequencymeasurements0:01% lStandarddeviationofnoiseinloaddemand0:01% fcCutofrequencyofloaddynamics0:32Hz Wehavestudiedtwotypesofdisturbance.Therstisadisturbanceintheformofa150-MWincreaseinloadatbus27appliedtothesystembetween10secondsand50seconds.Thesecondtypeofdisturbanceisthelossofgenerator 5att=5seconds,whichalsorepresentstheintroductionofamodelerrorbecausethemodelusedbythestateestimator(Figure 2-7 )thatconsiderstheoriginal10generatorsdoesnotmatchthemodelofthe9-generatorsystemunderdisturbance.Allparametrictestswereperformedforbothdisturbancetypes.Becausemostofthemshowedanalogousresults,wewillshowtheresultsforthe150-MWloaddisturbance,exceptwhenmodelmismatchisconcerned. 2.7.5EectofTimeDelayandModelMismatch Toevaluatemodelmismatchandtheeectofcommunicationtimedelay,wedisconnectgenerator5(508MW)fromthesystemat5seconds;generator5accountsfornearly10%ofthetotalgenerationinthesystem.Thereisacommunicationdelay(d) 43

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foreachload,i.e.,eachloadonlyhasaccesstopastvaluesofitsneighbors'gradients.Toassesstheeectsofcommunicationdelay,weusethreedierentvaluesforeachscenario:d=0ms(nodelay),d=100ms,andd=1secondbetweenadjacentnodes.Figure 2-9 showstheresultsofthesimulationwheregenerator5isdisconnectedfromthegrid.Resultsfornominaloperation(withoutsmartloads)areshownforcomparison.Thefrequencydeviationcausedbythegenerator'sdisconnectionishalvedcomparedtothescenariowithoutsmartloads.TheDGPalgorithmachievesthiswhileusingtheoriginalmodel,whichisnolongeraccurateduetochangescausedbythegeneratordisconnection. Figure2-9. 39-bussystem:Eectsofchangeintopologyandcommunicationdelay:generator5disconnection. Forthistypeofdisturbance,theperformanceofthemethodisonlymarginallyaectedbythetimedelayincommunications,asshowninFigure 2-9 .Forthe150-MWload-increasedisturbance(Figure 2-10 ),alargertimedelayincommunicationshasamorenoticeabledegradationintheresponsetimeoftheDGPalgorithm.Inparticular,thelongerdelayresultsinalongertimeforthedisutilitytoreachsteadystate.Thishappensbecausethedelayaectsthegradient-descentstepinthealgorithm,whichcorrespondstominimizingdisutility.Thisdierencebetweentheresponseofbothtypesofdisturbance 44

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happensbecausethelossofthegeneratorcreatesaverylargemismatchbetweenloadandgeneration,thusresultinginaveryhighvaluefor ^ui[k]in( 2{4 ).Therefore,inthecaseofalargepowerimbalance,suchasthecasewhereageneratorisdisconnected,theresponseoftheDGPalgorithmisdominatedbythegeneration-matchingterm( [k]^ui[k]),whichisnotaectedbycommunicationdelaybecauseitonlydependsonlocalfrequencymeasurements,whilethegradient-descentterm(c[k]xi[k])hassmallervalues.Thesameeectisnotreproducedinthecaseofloadincrease,wherethegeneration-matchingtermisnotasdominant. Figure2-10. 39-bussystem:Eectsofchangeintopologyandcommunicationdelay:150-MWloaddisturbanceatbus27. 45

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2.7.6EectofMeasurementNoise Fromthissectionon,wewillonlyconsiderthecasewherea150-MWincreaseinloadoccursatbus27between10and50seconds.Toevaluatetheeectoffrequencymeasurementnoise,wehavesetthestandarddeviationofthefrequencynoisemeasurementstof=0:01%,0:1%,and1%. TheresultsshowninFigure 2-11 demonstratethataverynoisymeasurementcanseverelyharmtheperformanceoftheDGPalgorithmintermsofreducingfrequencydeviation.Whencomparedtotheperformanceofthesystemwithoutsmartloads,theresultsaresuperiorforthecaseswherethestandarddeviationofnoiseis0:01%and0:1%,andtheyaresimilarwhenitisequalto1%.Weconcludethatthequalityoffrequencymeasurementsiscriticalforthisapplication.Itisimportanttonotethatthecontrolsystemsofthesynchronousmachinesthatkeepthesystemstabledonotrelyonnoisymeasurementsinthesesimulations. Figure2-11. 39-bussystem:Eectofnoiseinfrequencymeasurement:150-MWloaddisturbanceatbus27. 46

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IfwecomparethisresulttoFigure 2-8 ,wenoticethatthepracticallimitationsintroducedfromSection 2.7.4 onwardshavegreatlydegradedtheperformanceoftheDGPmethod,whichisnotcapableofreturningthefrequencyofthesystemtoitsnominalvalue. 2.7.7SpeedofActuation Weconsiderthatthereisadelaybetweenthecontrolactionandthechangeindemandduetothedynamicsofloadsanddelayincommunications(betweenaggregatorsandtheirsubordinateloads).Becauseloadscannotrespondinstantaneously,theirdynamicsweremodeledasalow-passlterwithgain1inthepassband.Weexaminetheperformancewithvariouscutofrequencies(fc),including0:032Hzand0:32Hz,whichwereinspiredbythecutofrequenciesexperimentallyobtainedforcommercial-buildingHVACfans[ 17 ]andvariable-speedheatpumps[ 21 ],respectively.Wealsotestedforcutofrequenciesof 0:01Hzand3:2Hztoemulateloadswithslowerandfasterresponses,respectively,asshowninFigure 2-12 Theresultsshowthatgoodperformancecanbeobtainedformostcases,whencomparedtothecasewithoutsmartloads.Testsforfcof0:01Hzandbelow,however,haveshownthattheperformanceofsmartloadswhoseresponseisveryslow(canonlyrespondintensofseconds)islesseectiveorevenineectiveforthetimerangestudiedinthissimulation. Inpractice,thecontrolsystemandmeasurementswouldbedigital(thereforecalculatedindiscretetime).Unlessotherwisenoted,thevaluesforsamplingperiod(Ts)ofcontrollerandmeasurementsis2=60seconds.Testswereperformedforsamplingperiodsof Ts=2=60,5=60,10=60,and0:5seconds.TheresultsinFigure 2-13 showthattheperformanceoftheDGPalgorithmisharmedbyincreasingthesamplingperiodbeyond2cycles,andforTs>10=60seconds,largeexcursionsofallsignals|especially 47

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Figure2-12. 39-bussystem:Eectofcutofrequencyofloads:150-MWloaddisturbanceatbus27. totaldisutility|appear.ForTs=0:5seconds,themethodisnolongermoreecientthanthebaselinecasewithoutsmartloads. 2.7.8LoadQuantization(NumberofBins) Eachagentimaychangeitsdemandbydiscreteincrements;thatis,changesindemandareplacedinto\bins."BecausetheDGPalgorithmwasdesignedforcontinuouschangesindemandratherthandiscrete,forimplementation,eachloadroundsitscalculateddemandchangetoitsclosestbin.Theeectofchangingthenumberofbinswasstudiedbyconsideringjij=5;9;and51.TheresultsshowninFigure 2-14 demonstratethattheDGPalgorithmcanfunctionadequatelyforallthreecases.However,thefrequencydeviationwaslargerforasmallernumberofbins. 48

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Figure2-13. 39-bussystem:EectofdierentsamplingperiodsontheperformanceofDGPundera150-MWloadincreaseatbus27. 2.7.9StochasticLoads Itisconsideredthatthenetdemandofloadsvariesovertimeduetothestochasticnatureofloadsandrenewableenergygeneration.Theeectofwidespreadintermittent,uncertain,anduncontrollablerenewableenergygenerationwasconsideredasanincreaseintheuctuationofthedemandfromloads.Thosevariationscanbedecomposedintosmall-andlarge-scaledeviations.Largevariationscanbemodeledasastepinconsumptionofagivenload,suchasthe 150MWincreasedisturbanceappliedtobus27describedinpreviousresults.Therandomsmall-scaleuctuationsofbothrenewablegenerationandloadaremodeledasGaussianadditivenoiseindemandofeachloadwithzero-meanandstandarddeviation(l)equalto0:01%ofthedemandoftheload.Wehavetestedstandarddeviationofeachloadfroml=0:01%to1%.Figure 2-15 showsthattheDGPalgorithmcancompensateforthe10-foldincreaseintheuncertaintyofload, 49

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Figure2-14. 39-bussystem:Eectofloadquantization(numberofbins)ontheperformanceofDGPundera150-MWloadincreaseatbus27. evenwhenl=1%.Incomparisontothecasewithoutsmartloads,therearegainswithreducedspeeddeviationfromthenominalvalueforallcases. 2.7.10GraphTopology Thesetupofdisturbancebyloadincreasewasreproducedusingadierentcommunication-graphtopology,havingaconnectedgraphwithsmallergraphdiameter.Theresults(omittedforbrevity)aresimilartothosetothegraphwiththeoriginaltopology. 50

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Figure2-15. 39-bussystem:EectofstochasticloaductuationsontheperformanceofDGPundera150-MWloadincreaseatbus27. 2.8Summary TheDGPalgorithmsolvesaconstrainedoptimizationprobleminadistributedmannertoaidapowergridinmaintainingsystemfrequencynearitsnominalvalueduringcontingencyeventswhileminimizingconsumers'disutility.ThemainadvantageoftheDGPalgorithmoverthedualalgorithmof[ 1 ]isthatitisapplicabletodisutilityfunctionsthatareconvexbutnotnecessarilystrictlyconvex,whilethedualalgorithmcanbeusedonlyforstrictlyconvexfunctions.Disutilityfunctionsthatarenotstrictlyconvexmodelmorerealisticconsumerbehaviorthatisinsensitivetosmallchangesinconsumption. 51

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SimulationsalsoshowthattheDGPalgorithmperformedeitherbetterthanorsimilarlytothedualalgorithmfrom[ 1 ]inmaintainingfrequency. TheconvergenceoftheDGPalgorithmwasestablishediftheoptimalpointslieinthestrictinteriorofthedomain.Whencapacityoftheloadsissmall,theoptimalsolutionislikelytolieontheboundary.Anopenproblemisthedesignofadistributedprimalalgorithm,ifsuchanalgorithmexists,thatisguaranteedtoconvergetoanoptimalsolutionlyingontheboundary.Additionally,theanalysisinthischapterassumedatime-invariantcommunicationgraph.Incontrast,thedualalgorithmin[ 1 ]wasprovedtoconvergeevenwithatime-varyingcommunicationgraph.ConvergenceanalysisoftheDGPalgorithmforthetime-varyingcaseisanareaforfuturework. Anotherissuethatwasignoredintheanalysisinthischapter|aswellasthatin[ 1 ]|istheeectoffeedbackinterconnectionbetweenthegeneratorcontrolsystemandtheloadcontrolalgorithm.However,analysistoruleoutpossibleinstabilitiesislacking.Arelatedissueisactuatordynamics.Itisassumedthatloadscanreactasfastasaskedbytheloadcontrolalgorithm.Thephaselagduetoloads'inertiacanreduceperformanceorevencauseinstability.However,bothlinearandnonlinearsimulationsindicatethatthetheoreticalresultsarerobusttothevariousassumptionsusedintheanalysis. TheDGPalgorithmwasshowntosuccessfullyarrestfrequencydeviationsfromthenominalvalueinmostscenariosconsidered.Inparticular,throughsimulationsweexaminedtheeectsofmodelmismatch,communicationdelay,communicationtopology,frequency-measurementnoise,presenceofuncertain,uncontrollable,stochasticrenewablegeneration,samplingperiodfordiscrete-timecontrolandmeasurements,quantizationofcontrolleroutput,andspeedofactuatorresponse.ThesimulationsrevealedthattheDGPalgorithmisrobusttomostofthesefactors,butitissensitivetonoiseinfrequencymeasurementsandsamplingfrequencyofimplementation.AlthoughtheDGPalgorithmimprovedperformance(comparedtothescenariowithoutsmartloads)forallactuatorbandwidthsconsidered,acutofrequencyof 0:01Hzsignicantlydegraded 52

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performancecomparedtohighercutofrequencies.ThisimpliesthatonlyveryfastloadscaneectivelyprovidecontingencyserviceusingtheDGPalgorithm,asexpected.However,acutofrequencyof0:03Hz,whichcorrespondstothespeedofacommercialHVACfan,stillsignicantlyimprovedperformancecomparedtothescenariowithoutsmartloads.ThisresultindicatesthatcommercialHVACfansarefastenoughtoprovidemeaningfulcontingencyservicetothegrid. 53

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CHAPTER3BANDWIDTH-LIMITEDDEMANDDISPATCHFORNET-LOAD-FOLLOWINGUSINGVIRTUALENERGYSTORAGE 3.1IntroductionandMotivation Inthepreviouschapter,weproposedanalgorithmthatmaybeusedforprovidingdemanddispatchforcontingencyevents,whosedynamicsareonascaleofseconds.However,thegridfrequencymustbemaintainedonlongertimescalesaswell.Inthischapter,weproposeademand-dispatchalgorithmforlongertimescales,i.e.,load-following.Inparticular,weareinterestedinnet-load-following:trackingchangesinthenetload,whichincludesrenewablegeneration. Mostworksondemanddispatchinvolveuseofreal-timeelectricitypricestoencourageloadstoshiftorreducedemand[ 11 24 ].Anadvantageofaprice-basedschemeisthatitenablesdecentralizedcoordinationwheremarketclearingprice(computedandbroadcastbythebalancingauthority(BA))providesafeedbacksignalallloadscanuse[ 62 ].Aprice-basedschemesuersfromseveraldiculties,too.First,itassumesthatconsumersarewillingtoenduresomelossofQoSinreturnforapayment[ 63 24 64 ].TherehasbeensomeworkonminimizingtheQoSlossexperiencedbyconsumerswhendecidingdemandchanges[ 1 44 45 65 ].However,modelingthelossofQoSasafunctionofdemandvariationandcomputingthedollarvalueofthatlossischallenging.Second,consumersfacelargeuncertaintyinhowmuchnancialrewardtheymayreceivesincetherewarddependsonreal-timepricesthathistoricallyshowspikybehavior[ 26 ].Infact,asurveyofdemandresponsefoundpricevolatilitytobeasignicantlimitingfactorinconsumerparticipation[ 27 ].Thethirdproblemisuncertaintyfacedbythegridoperator.Currently,generatingunitsarecommittedthroughabiddingprocessinday-aheadandhour-aheadmarkets,whoseclearingrequiressolvinganoptimizationproblem.Theinformationneededtosolvethisprobleminvolvescapacity,ramprate,andmarginalcostofallgeneratingunits,alongwithforecastofdemand-supplyimbalancein 54

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thegrid.Includingalargenumberofconsumerloads|withanuncertaincostofdemandvariation|intothisoptimizationisimpractical. Inthischapter,weproposeanalternativeapproachforprovidingdemanddispatchtoreducedemand-supplyimbalanceinthegrid.Theproposedapproachdoesnotinvolvereal-timeprices;itreducesuncertaintytobothconsumersandgridoperators.Itconsistsoftwomainideas:(1)constrainingthebandwidthofdemandvariation(i.e.,magnitudeoftheFouriertransformofdemandvariation)toensurestrict,predeterminedboundsontheQoS,and(2)decentralizedcoordinationbasedonlocalgrid-frequencymeasurements. Constrainingbandwidthisinspiredbypriorworkthatarguesthatthechangeinaconsumer'sQoSisafunctionofthefrequencycharacteristicsofthedemandvariation[ 66 ].E.g.,variationsintheairowrateofanHVACsysteminalargebuildingwillhaveanegligibleeectontheindoorclimateifthevariationisofsucientlyhighfrequency[ 67 17 ].DemandvariationatlowerfrequencieswillleadtolossofQoS;e.g., 2-Fvariationinindoortemperature[ 18 ].Similarly,poolpumpscanprovideservicetothegridwithoutaectingpoolowners'QoSaslongasthedemandvariationsarelimitedtocertaintimescales(frequencies)[ 68 69 ],andthismayapplytoaluminum-smeltingplantsaswell[ 46 ].Thismotivatesanoptimizationproblemwithbandwidthconstraintsplacedonthedecisionvariables. SomeworksusestrictoutputconstraintstoensureQoSwhileoptimizingamarket-basedobjectivefunction[ 70 71 72 73 74 ].Thisapproach,however,requiresanaccuratemodelofeachload'sdynamics.Incontrast,theproposedapproachrequiresanimprecisecharacterizationofaload:thelocalcontrolleronlyneedsaconservativeestimateoftheregioninthemagnitude-frequencyplotthatthedemandvariationneedstostaywithintosatisfytheQoSrequirementsoftheconsumer.Suchacharacterizationcanbedoneatthefactoryforconsumerloadslikewaterheaters. Intheproposedapproach,eachloademploysalocalmodel-predictive-control(MPC)scheme[ 75 76 ]tocomputethecontrol(demandvariation)usingprediction 55

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ofdemand-supplyimbalancefromthegridoperator.ItisassumedthatthebalancingauthoritybroadcaststhesepredictionsovertheInternet,whichitobtainsfromforecastsofdemandandrenewablegeneration.Thecontrollersolvesanoptimizationproblemthatdoesthebestitcantorejectthedisturbancefacedbythegridsubjecttolocalbandwidthconstraints.Coordinationisperformedbyusingmeasurementsofthegridfrequency,whichcanbemeasuredbyeachloadlocally[ 30 77 ]andisrelatedtothegrid-leveldemand-supplyimbalance[ 19 38 78 ].Thus,everyloadisabletoinferglobalinformationfromlocalmeasurementswithoutrequiringinter-loadcommunication.Wecalltheproposedcontrolschemethebandwidth-limited,disturbance-rejecting,decentralizedmodel-predictivecontrol(BaLDuR-DMPC). Thischapterisorganizedasfollows.WedescribetheenvisionedgridarchitectureinSection 3.2 .Section 3.3 describestheproblemtobesolved.TheproposedcontrollerformulationisdescribedinSection 3.4 .ConvergenceanalysisispresentedinSection 3.6 .WepresenttheresultsofseveralsimulationscenariosinSection 3.7 .Finally,Section 3.9 concludesthechapter. 3.2EnvisionedGridArchitecture ItisenvisionedthateachloadsignsacontractwiththeBAthatspeciesthebandwidthofdemandvariationtowhichtheconsumeragrees.ThatguaranteestheQoSlosstheconsumermayexperienceforalltime.Forexample,ifaconsumeriswillingtotolerateamaximumtemperaturedeviation(fromthenominalvalue)of2F,thenthebandwidthconstraintwillbetighter,andmonthlypaymentwillbesmallercomparedtothoseforaconsumerwhoiswillingtotoleratea4-Fchange.Thiscontract-basedmechanismmakesiteasyforthegridoperatortoestimatehowmuchVEScapacityitcancountonateveryinstant.Bythesamereason,theconsumerknowsaheadoftimehowmuchmoneyshewillmakeandmaximumdiscomfortshemayexperience. Oneclassofloadswillnothavesucientexibilitytomeetalltheneedsofthegrid.Itisarguedin[ 66 ]thataspectraldecompositionofthedemand-supplyimbalancecanbe 56

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usedtoassigntheappropriatepartthereoftotheappropriateresource.Forinstance,theslowlyvaryingpartofthenetdemand(demandminusrenewablegeneration)ofafuturegridcanbehandledbytraditionalgenerators.A\mid-pass"componentcanbeprovidedbypoolandirrigationpumping,anda\high-pass"componentcanbeprovidedbyHVACsystemsinbuildings,etc.ThisisillustratedinFigure 3-1 .SincetheBAsignslong-termcontractswithvariousexibleloads,itknowsaheadoftimewhetherithasadequatecapacityatallfrequencies(timescales).Eachclassofresourcesthereforeonlyfocusesonthetaskofrejectingthepartoftheimbalancethatitsbandwidthconstraintsallow.Thisarchitecturewasexploredinsimulationsin[ 79 ],whichshowedheterogeneousresourcestrackingreferencesignalsindierentfrequenciescanprovidevaluableservicetothegrid. Figure3-1. Illustrationofthegrid'sregulationneeds. 3.3ProblemFormulation Lettimebemeasuredbythediscreteiterationcounter: k=0;1;:::,andletTbethediscretizationinterval.Thecontrolactionatloadiattimekisthedemandvariationfromthenominalvalue,denotedbyvi[k]. Thegridismodeledasaplantwithinputequaltotherealpowerinjectedandoutputequaltothegridfrequency.Weonlyconsiderthelinearizedgriddynamicsaround 57

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Figure3-2. Gridmodel.irepresentsmeasurementnoiseforloadi.(ni)denotes\allloadsexcepti." thenominalsystemfrequency,60Hz,withthenominalinputbeingthenetrealpower(generationminusnominaldemand)injectedtomaintainthenominalfrequency.Thelinearizedplant,discretizedwithsamplingperiodT,isdenotedbyG(z).TheinputtoG(z)isthesumofdisturbance,d[k],andthecontrolactionsoftheloads;denotethisinputby[k].Let#[k]bethetotalcontrolactionbytheloads: #[k]=Pni=1vi[k].Then,[k]:=d[k]+#[k].Theoutputof G(z)isthedeviationofthesystemfrequencyfrom60Hzandisdenotedby![k].IntheZ-transformdomain,!(z)=G(z)(z).SeeFigure 3-2 foraschematicrepresentationofthesevariables. Thegoaloftheloadsistokeepfrequencydeviationsfrom60HzsmallwhileensuringlossinQoSforconsumersismaintainedwithinpredeterminedbounds.WeuseMPCbecauseitisreadilyabletoenforceconstraints.Let NbethepredictionhorizonfortheMPCateveryload. Thecontrolatloadi|itsdemandvariation|mustsatisfythebandwidthconstraintsofthatload.ThesearespeciedintermsoftheDFTofthecontrolsignal.TheL-point 58

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DFTofthecontrolinputs,vi[k],atloadiisVi[m]:=LXk=1vi[k]e)]TJ /F4 7.97 Tf 6.59 0 Td[(j2 Nmk;m=0;:::;L)]TJ /F1 11.955 Tf 11.95 0 Td[(1Bandwidthconstraintsonthecontrolactionsareenforcedbyspecifyingupperbounds, i[m],onthemagnitudeofthecontrolsignal'sL-pointDFT:jVi[m]ji[m],m=0;:::;L)]TJ /F1 11.955 Tf 12.16 0 Td[(1,foranappropriatelychosenL.Thecontrolactionsmustalsosatisfyupperandlowerboundconstraints,whicharedenotedbyuiandu i,respectively,forloadi. Weassumethatapredictionofdisturbancesaectingthegridareavailabletoallloads(throughperiodiccommunicationfromtheBA).Let^d[k](BA):=^d(BA)[kjk];:::;^d(BA)[k+N)]TJ /F1 11.955 Tf 11.17 0 Td[(1jk]Tbethepredictionofthedisturbance,d[k];:::;d[k+N)]TJ /F1 11.955 Tf 11.16 0 Td[(1]T,aectingthegridthatisavailabletoallloadsattimek. 3.4ProposedMethod 3.4.1GridModelUsedbyLoads SincetheproposedcontrolschemeisMPC-based,eachloadneedsamodeloftheplant,G(z).Intheproposedmethod,loadsuseanextremelysimplemodelofG(z)forcontrolcomputations:aconstantgain,whichwecallg.Therationaleforchoosingsuchasimplemodelistwofold.First,asimplermodelaidsinkeepingcontrolcomputationssimple.Second,sincethefocusofthisworkisdemand-supplybalanceattheslowertimescaleofafewminutestoafewhours,i.e.,inthefrequencyrange f2[1 5min1 120min],thelinearizedmodelofapowergridatthesefrequencies,G(ejf),isanearlyconstantgain.Forexample,Figure 3-3 showstheBodeplotofalinearizedmodeloftheERCOTgrid[ 2 ].Thisleadstothefollowingmodelusedbyeveryload:![k]=g[k]=g(d[k]+#[k])=g(d[k]+nXi=1vi[k]): (3{1)Notethatload ionlyknowsvi[k]andanoisyversionof![k];itdoesnotknowd[k]orvj[k];j6=i. 59

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Figure3-3. BodeplotoflinearizedgridmodelG(s)from[ 2 ]. 3.4.2MPCFormulationforLoadi Letui[k]denotethedecisionvariablesofloadiattimek:ui[k],ui[kjk];:::;ui[k+N)]TJ /F1 11.955 Tf 11.96 0 Td[(1jk]T:Therstentryoftheoptimalvalue, ui[k],whichisdenotedby ui[kjk],isimplementedasthecontrol,i.e.,vi[k]=ui[kjk].Theoptimalvalueui[k]isobtainedbysolvingthefollowingoptimizationproblemattimekbyloadi:minui[k]k+N)]TJ /F6 7.97 Tf 6.59 0 Td[(1X`=k)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(![`]2 (3{2)subjecttothefollowingconstraintsoverthehorizon k`k+N)]TJ /F1 11.955 Tf 11.95 0 Td[(1:![`]=g(ui[`jk]+^di[`jk]); (3{3)u iui[`jk]ui; (3{4)Vi[mjk]i[m];0m2N)]TJ /F1 11.955 Tf 11.96 0 Td[(1; (3{5) 60

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where ^di[`jk]isadisturbancepredictionusedbyloadiandVi[mjk]:=k)]TJ /F6 7.97 Tf 6.58 0 Td[(1X`=k)]TJ /F4 7.97 Tf 6.59 0 Td[(Nvi[`]e)]TJ /F4 7.97 Tf 6.59 0 Td[(j2 2Nm`+k+N)]TJ /F6 7.97 Tf 6.58 0 Td[(1X`=kui[`jk]e)]TJ /F4 7.97 Tf 6.59 0 Td[(j2 2Nm` (3{6)isthe 2N-pointDFTofthearray, vi[k)]TJ /F3 11.955 Tf 11.42 0 Td[(N];:::;vi[k)]TJ /F1 11.955 Tf 11.42 0 Td[(1];ui[kjk];:::;ui[k+N)]TJ /F1 11.955 Tf 11.42 0 Td[(1jk];recallthat vi[`]isthecontrolactionpreviouslyimplementedbyloadiattime`. Thereasonforincludingpastcontrolinputsinthebandwidthconstraint( 3{6 )isthefollowing.Ifonlypredictedvaluesareused,evenifthesolutiontotheoptimizationproblemsatisesthebandwidthconstraint,becauseMPConlyimplementstherstentryofthesolution,theclosed-loopcontrolactionmaynotsatisfythebandwidthconstraint.ThisissueisfacedinMPCschemesthatlimittherateofchangeincontrol,wherepastdataareusedtoenforcetheconstraintintheclosedloop[ 72 ].TheuseofpastdataincomputingtheDFTin( 3{6 )wasfoundtohelpmaintainthebandwidthconstraintintheclosedloop.Next,wedescribehowloadsobtain^di[k]=^di[`jk]j`=kk+N)]TJ /F6 7.97 Tf 6.59 0 Td[(1. 3.4.3AvoidingHigh-GainFeedback Notethatthesystem-modelconstraint( 3{3 )indicatesthateveryloadthinksitistheonlyloadinthegrid.TheMPCschemeencourageseachloadtorejectthepredicteddisturbancethroughitslocalcontrol.Thecombinedactionsofallloadsmayleadtohigh-gainfeedbackandinstability.Toavoidthis,load idoesnotusethegrid-leveldisturbanceprediction, ^d(BA)[k],for^di[k]insolvingproblem( 3{2 )-( 3{5 ).Rather,thedisturbancepredictionisscaledaccordingtothefollowing: ^di[k]:=i[k]^d(BA)[k],where i[k]>0isatime-varyinggainthatisupdatedviai[k]=ri[k]i[k)]TJ /F1 11.955 Tf 11.96 0 Td[(1]; (3{7)theratio, ri[k],iscomputedfromri[k]=min(max^d(BA)[k] ^#i[k)]TJ /F1 11.955 Tf 7.08 -3.45 Td[(];r;r); (3{8) 61

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where 0
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Theinputstothealgorithmforloadiattimekare:the Npast(executed)inputsvi[`]( `=k)]TJ /F3 11.955 Tf 12.09 0 Td[(N;:::;k)]TJ /F1 11.955 Tf 12.1 0 Td[(1),thenoisyfrequencymeasurement,~!i[k)]TJ /F1 11.955 Tf 7.08 -4.34 Td[(],andthegrid-leveldisturbanceprediction^d(BA)[k].Theoutputisthecontrolinput,vi[k].Nointer-loadcommunicationisneeded. Obtainingnecessaryconditionsforfeasibilityoftheoptimizationproblem( 3{2 )-( 3{5 )isleftforfuturework.However,if( 3{2 )-( 3{5 )isfeasible,thenitcanbeshownthattheproblemisconvex. 3.5Feasibility Wenowusetheconstraints( 3{4 )-( 3{5 )todescribethefeasiblesetofproblem( 3{2 )-( 3{5 )attimek. DeneCm;`:=cos(2 2Nm`)and Sm;`:=)]TJ /F1 11.955 Tf 11.29 0 Td[(sin(2 2Nm`),andlet[Cm;`]k+N)]TJ /F6 7.97 Tf 6.58 0 Td[(1`=k)]TJ /F4 7.97 Tf 6.59 0 Td[(N=[Cm;k)]TJ /F4 7.97 Tf 6.59 0 Td[(N;Cm;k)]TJ /F4 7.97 Tf 6.59 0 Td[(N+1;:::;Cm;k+N)]TJ /F6 7.97 Tf 6.59 0 Td[(1]Tand[Sm;`]k+N)]TJ /F6 7.97 Tf 6.59 0 Td[(1`=k)]TJ /F4 7.97 Tf 6.59 0 Td[(N=[Sm;k)]TJ /F4 7.97 Tf 6.59 0 Td[(N;Sm;k)]TJ /F4 7.97 Tf 6.59 0 Td[(N+1;:::;Sm;k+N)]TJ /F6 7.97 Tf 6.59 0 Td[(1]T.Now,foragivenk,denetheouterproduct,Qm;k,anditsfourNNsubmatrices:Qm;k=[Cm;`]k+N)]TJ /F6 7.97 Tf 6.59 0 Td[(1`=k)]TJ /F4 7.97 Tf 6.59 0 Td[(N[Cm;`]k+N)]TJ /F6 7.97 Tf 6.59 0 Td[(1`=k)]TJ /F4 7.97 Tf 6.58 0 Td[(NT+[Sm;`]k+N)]TJ /F6 7.97 Tf 6.58 0 Td[(1`=k)]TJ /F4 7.97 Tf 6.59 0 Td[(N[Sm;`]k+N)]TJ /F6 7.97 Tf 6.58 0 Td[(1`=k)]TJ /F4 7.97 Tf 6.59 0 Td[(NT;=:264Q1;m;kQ2;m;kQ3;m;kQ4;m;k375:Attime k,forsomei,letvi[k]:=vi[k)]TJ /F3 11.955 Tf 11.96 0 Td[(N];:::;vi[k)]TJ /F1 11.955 Tf 11.95 0 Td[(1]T.Now,denethesetUm;k:=(uuTQ4;m;ku+vTi[k](Q2;m;k+QT3;m;k)u)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(i[m]2)]TJ /F10 11.955 Tf 11.96 0 Td[(vTi[k]Q1;m;kvi[k]):Finally,let [u i;ui]`R`denotethe`-dimensionalproductoftheinterval[u i;ui].Then F[k]:=)]TJ /F2 11.955 Tf 8.71 -9.68 Td[(\2N)]TJ /F6 7.97 Tf 6.59 0 Td[(1m=0Um;kT[u i;ui]Nisthefeasiblesetattimek.Therefore,F[k]6=;isasucientconditionforfeasibilityattimek. 63

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3.6Convergence Wenowshowthattheproposedalgorithmdrivestheoutput,![k],to0undersomeassumptions. Assumption3.1 1. (Perfectdisturbanceprediction)^d(BA)[k]=d[k]forallk. 2. (No\plant-modelmismatch")G(z)=g. 3. (Perfectfrequencymeasurements)^![k)]TJ /F1 11.955 Tf 7.08 -4.34 Td[(]=![k)]TJ /F1 11.955 Tf 7.08 -4.34 Td[(]forallk. 4. (Loosebandwidthconstraints)[m]issucientlylargesothatVi[mjk]<[m]foralli,m,andk. 5. (Sucientactuation)Pni=1uimaxkjd[k]j,andPni=1ju ijmaxkjd[k]j. Assumptions 3.1 (2),(3)imply^#i[k)]TJ /F1 11.955 Tf 7.09 -4.34 Td[(]=#[k)]TJ /F1 11.955 Tf 7.08 -4.34 Td[(]foralli,k;Assumptions 3.1 (4),(5)implystrictfeasibilityofproblem( 3{2 )-( 3{5 ). Since!isthedeviationfromthenominalsystemfrequency,60Hz,thedesiredequilibriumoftheclosed-loopcorrespondsto ![k]=0withPivi[k]=)]TJ /F3 11.955 Tf 9.3 0 Td[(d[k]. Theorem3.1 Supposethedisturbanceaectingthegridisconstant:d[k]=forall kandsomexed.IfAssumption 3.1 holds,r =0,andr=+1,then![k]!0ask!1. Itisstraightforwardtoseefrom![k]=g(d[k]+#[k])that#=)]TJ /F3 11.955 Tf 9.3 0 Td[(and!=0isanequilibrium:oncethecontrolreachesthispointatsomek,ri[k+1]=1from( 3{8 ),andi[k+1]=i[k]by( 3{7 ),meaningeachloadwillusethesamedisturbancepredictionatk+1asatk,whichwillleadtothesamecontrolactionat k+1asatk;hence,#k+1=)]TJ /F3 11.955 Tf 9.29 0 Td[(and![k+1]=0.Theassumptionsofanaccurategridmodelandperfectfrequencymeasurementsanddisturbancepredictionsareutilizedinthisargument.Thetheoremstatesthattheclosed-looptrajectoriesconvergetothisequilibrium. ProofofTheorem 3.1 Withoutlossofgenerality,suppose>0;symmetricargumentsapplyfor<0.Thesolutiontoproblem( 3{2 )-( 3{5 )istheonethatexactlycancelsoutthescaledpredicteddisturbanceifactuatorconstraintsallow.Thatis,ignoringactuator 64

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constraints,ui[k]=argminui[k]k+N)]TJ /F6 7.97 Tf 6.58 0 Td[(1X`=k)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(g(ui[`jk]+^di[`jk])2=)]TJ /F1 11.955 Tf 10.02 3.16 Td[(^di[k]:Therefore,ui[kjk]=)]TJ /F3 11.955 Tf 9.3 0 Td[(i[k]i[k]; (3{10)wherei[k],minj u ij i[k];1: (3{11)Now,wehave#[k]=)]TJ /F3 11.955 Tf 9.29 0 Td[(nXi=1i[k]i[k]: (3{12)Theproofwillproceedbyshowing i[k]converges,whichimpliesj#[k]j!by( 3{7 )and( 3{8 ).Since#andareoppositesignsby( 3{10 ),#[k]!)]TJ /F3 11.955 Tf 24.89 0 Td[(,whichimplies![k]!0by( 3{1 ). From( 3{8 ),using( 3{7 ),( 3{12 ),and ^#i[k)]TJ /F1 11.955 Tf 7.08 -4.34 Td[(]=#[k)]TJ /F1 11.955 Tf 7.09 -4.34 Td[(],weobtaini[k+1]= Pnj=1j[k]j[k]i[k]=i[k] Pnj=1j[k]j[k]: (3{13)Now,suppose j#[k0]j>forsomek0;symmetricargumentsmaybemadeifj#[k0]j<.Thenthereistoomuchtotaldemandvariationatthatinstant.From( 3{12 ),wethenhave Pni=1i[k0]i[k0]>1.By( 3{13 ),wethenhave i[k0+1]
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i[k0+1]i[k0]by( 3{11 ).Now,byiterating( 3{13 ),wemayobservei[k0+2]=i[k0] nP`=1`[k0]`[k0] nPj=1j[k0] nP`=1`[k0]`[k0]j[k0+1]i[k0] nPj=1j[k0]j[k0]=i[k0+1]: Now,i[k0+2]i[k0+1]impliesri[k0+1]1by( 3{7 ).Therefore,#[k0+1]by( 3{8 ).Hence, #[k0]>impliesi[k0+2]i[k0+1],whichimplies#[k0+1].Byinduction,itfollowsthat #[k]forallkk0.Therefore,i[k+1]i[k]foralliandallkk0.Hence,if#[k]>foranyk,i[k]convergesbecauseitisboundedbelowby0.Thishappensforeveryi. Now,supposej#[k]j<.Then,wehavei[k0]forlargekbyAssumption 3.1 (5)|acontradiction.Therefore, i[k]converges(foralli),andtheproofiscomplete. 3.7NumericalResultsforLinearGridModel 3.7.1SimulationSetup Weconsiderthelinearizedgridmodel,G(s)=0:644s+0:147 s2+0:4797s+0:147; (3{14)whichwasbasedonastudyoftheTexasgridin[ 80 ].Forsimulations,weuseadiscretizationintervalof1second.Eachloadusesthegridmodel g=1fortheirconstant-gainmodel,withinputinGWandoutputinHz.Controlactionsarediscretizedinto5-minute 66

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intervals;i.e., T=300seconds.Loadsusea2-hourpredictionhorizon,so N=24;thismeansloadsalsousea2-hourpasthorizonforbandwidthconstraints. Thecontinuous-timedisturbancetothegridisd(t)=0:05sin2 3600t+0:1sin2 23600t+(t);where tisinsecondsand(t)ischosenfromN(0;0:052).ThedisturbancepredictionsuppliedtotheloadsfromtheBA, ^d(BA),isthediscreteformof^d(BA)(t)=d(t))]TJ /F3 11.955 Tf 11.37 0 Td[((t)withasamplingintervalofT=300seconds. Themeasurementnoise(forgrid-frequencymeasurements)ismodeledasazero-meannormalrandomvariablewithvarianceof10)]TJ /F6 7.97 Tf 6.59 0 Td[(4Hz2.Foreach i,weuseui=)]TJ /F3 11.955 Tf 9.3 0 Td[(ui=1sothattheeectsofbandwidthconstraintsareclearlyseen(ratherthanthoseofsaturation).Finally,weuse r=1=r=0:95andi[0]=0:01foralli. Tosolveproblem( 3{2 )-( 3{5 ),eachloadusesasequential-quadratic-programming(SQP)formulation.Anymethodmaybeused,butSQPisusefulinthepresenceofnonlinearconstraints,suchas( 3{5 ),andhasrobusttheoreticalfoundations[ 81 ]. Inthesequel,wepresentsimulationresultsoftheBaLDuR-DMPCalgorithmforn=10loadsandn=100loadswithdierentchoicesfor[m],m=0;:::;2N)]TJ /F1 11.955 Tf 11.95 0 Td[(1. 3.7.2OverviewofResults Table 3-1 showstheobjectivevalue,Pk)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(![k]2,forvaryingnumberofloads.Therstrowservesasabaseline.Increasingthenumberofloadssignicantlyimprovedperformancebecause,withmoreloads,eachloadisresponsibleforasmallerportionofthedisturbance, d.Asaresult,theenergyofeachload'scontrolactionsinthefrequenciesofinterestisreduced. 67

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Table3-1. Objectivevaluefordierentscenarios Numberofloads Pk)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(![k]2 0(baseline)1351084.310032.6100(strictconstraints)55.6 Figure 3-4 showsthedisturbanceandsystemfrequencywhenthereisnodemandvariation.Figure 3-5 showsthedisturbance,d,totaldemandvariation,#,andgridfrequency,!,forn=10loads.ThebottomplotshowsthemagnitudeoftheDFToftheoverallcontrolactionsforload1aswellasthevaluesof[m]foreachm(i.e.,eachfrequency).With 10loads,BaLDuR-DMPCreducedtheobjectivevalueby37:6%comparedtowhenthereisnoloadcontrol(compareFigure 3-6 withFigure 3-4 ). Figure3-4. Lineargridmodel.Noloadcontrol:Pk)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(![k]2=135. NotefromFigure 3-5 thatthebandwidthconstraintsarenotcompletelysatisedatfrequency1=2hours.SincethebandwidthconstraintisposedintermsoftheDFTofanite-lengthsignal,theclosed-loopcontrolsignalmaynotsatisfythebandwidth 68

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Figure3-5. Lineargridmodel.10loads:Pk)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(![k]2=84:3(37:6$reductionfrombaseline). constraintevenifthesolutiontoproblem( 3{2 )-( 3{5 )ateveryksatisesit.ThisisawellknownissuewithMPCwheretheconstraintsdependonpastdata.However,theloadsdidnotviolatethebandwidthconstraintsatanyindividualinstantintime;thatis,theloadswerealwaysabletosuccessfullylocateafeasiblesolutiontoproblem( 3{2 )with 2-hourpastandpredictionhorizons. 69

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3.7.3EectofBandwidthConstraints Figure 3-6 showsthedisturbance,d,totaldemandvariation,#,andgridfrequency,!,forn=100loads.ThebottomplotshowsthemagnitudeoftheDFToftheoverallcontrolactionsforload1aswellasbandwidthconstraints,[m],foreachfrequency.Therewasa 75:6%reductionintheobjectivevaluecomparedtothecasewithoutintelligentloads. Figure3-6. Lineargridmodel.100loads:Pk)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(![k]2=32:6(75:6%reductioncomparedtobaseline). 70

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InFigure 3-7 ,n=100,andthevaluesof[m],m=0;1;2weretentimessmallerthanthoseinFigure 3-6 ,while[m]wasthesameasinthatgureforallotherm;i.e.,thelowestthreefrequenciesintheDFTweresubjectedtostricterconstraints.TheresultofstricterconstraintswasahigherobjectivevaluerelativetothatforFigure 3-6 .Evenso,Figure 3-7 stillshowsa58:8%reductioninobjectivevaluecomparedtothebaselinewithnoloadcontrol.Additionally,theenergyofthecontrolactionsinthemorestrictlyconstrainedfrequencywassignicantlyattenuatedcomparedtothatinFigure 3-6 ,whiletheenergyinthelessconstrainedfrequencyremainednearlythesame|eectivelyshapingthefrequencycontentoftheloads'actionstomaintainQoS. 3.8NumericalResultsforNonlinearGridModel Inthissection,wepresenttheresultsoftheBaLDuR-DMPCalgorithmfromsimulationsoftheIEEE39-bustestsysteminSimscape[ 60 ],justasinChapter 2 (seeFigure 2-6 ). 3.8.1SimulationSetup AsinSection 3.7 ,controlactionsarediscretizedinto5-minuteintervals.Thereare19loads(oneateachloadbus),andeachloadusesg=0:0003foritsconstant-gaingridmodel,withinputinMWandoutputinHz.Loadsusea2-hourpredictionhorizon,aswellasa2-hourpasthorizonforbandwidthconstraints. Thedisturbancetothegridisnet-loaddatafromtheBonnevillePowerAuthorityonOctober10,2017,scaleddowntomatchtheactuationavailabletotheloadsandinterpolatedinto30-secondincrements(from5-minutedata).Loadshaveaccesstothe5-minutedataaspredictionsanddonotknowwhichbusissubjectedtothedisturbance.Aprocessdisturbanceisappliedtothenet-loaddataintheformofband-limited,Gaussianwhitenoisewithavarianceof5=6MW2.Becausethedisturbancetothegridisnotzero-meanandtheloadsareconstrainedinlowerfrequencies,theycannotrejectthelower-frequencycomponentsofthedisturbance.Becausetheloadswillsaturateandbeunabletorejectpartofthedisturbance,[k]!1from( 3{7 )and( 3{8 ). 71

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Figure3-7. Lineargridmodel.100loads:Pk)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(![k]2=55:6(58:8%reductioncomparedtobaseline);stricterbandwidthconstraintsthaninFigure 3-6 Apossiblesolutiontothisistosupplyloadswithahigh-pass-ltereddisturbanceprediction.However,thiswillresultinatheDCcomponentofthedisturbancetoaectthegrid|resultinginanonzerofrequencydeviation.TheloadswillinterpretthisDCosetasalargemagnitudeof#[k](seeSection 3.4.3 ),whichwillresultin[k]!0from( 3{7 )and( 3{8 ).Therefore,additional,auxiliarysecondarycontrolisappliedatbus39in5-minuteincrementstorejectthelow-frequencycomponentsofthedisturbance.Theloadsarethensuppliedwiththehigh-pass-ltereddisturbanceprediction.The 72

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high-pass-ltertransferfunctionusedis:0:9116z2)]TJ /F1 11.955 Tf 11.95 0 Td[(1:8232z+0:9116 z2)]TJ /F1 11.955 Tf 11.95 0 Td[(1:8153z+0:8310: Themeasurementnoise(forgrid-frequencymeasurements)ismodeledasband-limitedwhitenoisewithavarianceof10)]TJ /F6 7.97 Tf 6.59 0 Td[(7Hz2.Thiscorrespondstonoisemagnitudesintheorderof10)]TJ /F6 7.97 Tf 6.59 0 Td[(4Hz.Foreachload i,weuseui=)]TJ /F3 11.955 Tf 9.29 0 Td[(ui=5%uo,whereuoisthenominalvalueoftheload.Finally,weuse r=1=r=0:95andi[0]=1=19foralli. 3.8.2ChoosingBandwidthConstraints Appropriatebandwidthconstraintsi[m]mustbechosenforeachfrequencymandeachloadi.Todothis,werequireinformationofthefrequencycontentoftheactionsoftheloads,whicharetime-varyingsignals.Wemodeltheseactionsasarandomprocess,Y[k].Wesayaload's\exiblecapacity"iscifP(jY[k]jc)<,whereisa(consumer-dened)smallnumberandY[k]istherandomprocessmodelingtheload'scontrolactions.Toensurethisprocesshasenergyinthecorrectfrequencies,wepassGaussianwhitenoisewithpowerspectraldensityof2throughabandpasslterwithpassband[1=6hours,1=1minute].Foranidealbandpasslter,thevarianceofthelteredoutputisapproximately0:033.ByChebyshev'sinequality,P(jY[k]j1)0:033,whereY[k]istheoutputofthelter.Therefore,wecansaythatthislteredoutputisarandomprocessthatmodelsthecontrolactionsofaloadwhoseexiblecapacityis1.WemaythentaketheDFTofthissignaltodetermineappropriatevaluesof[m](seeFigure 3-8 ).From 10;000randomprocesses(withlengthof4hoursand5-minutesampleperiod),2standarddeviationsabovethemeanvalueofthemagnitudeoftheDFToftheload'scontrolactionswasapproximately5.Therefore,themagnitudeoftheDFTofaload'scontrolactionsisexpectedtobeapproximately5timestheexiblecapacityoftheloadmostofthetime.Hence,[m]shouldbeabout5timestheexiblecapacityoftheloadineachload'srespectivefrequencyband.Insummary,i)whitenoisewithappropriatepowerspectraldensityislteredtoobtainaband-limitedrandomprocessrepresentinga 73

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load'scontrolactionswithexiblecapacity1;ii)twostandarddeviationsabovethemeanmagnitudeoftheDFTofthelteredrandomprocessformanysamplepathsgivesanappropriatebandwidthconstraintforaloadwithexiblecapacity1. Figure3-8. DiagramofthemagnitudeoftheDFTofthelteredrandomprocesswithexiblecapacity1. Becauseloadsinthe39-bussystemrepresentlarge,aggregateloads,eachloadbusisassumedtobecomposedofdierenttypesofloadsthatcanprovidezero-meanband-limitedservicetothegrid,summarizedinTable 3-2 .ThecompositionofeachloadissummarizedinTable 3-3 .Animportantquestionishowtoconstructcompositebandwidthconstraintsforeachloadcomposedofmultipleloadtypes.Toconstructthebandwidthconstraintsforload i,rst,5%oftheload'sexiblecapacity(ui)isaddedinallfrequenciestoensurestrictfeasibility.Then,thebandwidthconstraintsforeachloadtypearecombinedusingthe1-normaccordingtotheirexiblecapacities.Thisprovidesanupperboundonthebandwidthoftheservicethatloadicanprovide.Forexample,ifloadiiscomposedsolelyofpoolpumpsandsmeltingloads,thenthe 74

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compositebandwidthconstraintforload iandfrequencymisi[m],pool[m]+smelt[m]+0:05uiwhere pool[m]andsmelt[m]are5timestheload'sexiblecapacityscaledbythepercentageoftheloadthatiscomposedofpoolpumpsandsmeltingloads,respectively,inthecorrespondingfrequenciesinTable 3-2 Table3-2. Loadtypesandcorrespondingfrequencybands LoadtypeFrequencyband Refrigerationloads[ 1=30minutes,1=5minutes]HVACloads[ 1=1hour,1=5minutes]Industrialsmeltingloads[ 1=2hours,1=1hour]Poolpumps[ 1=6hours,1=1hour] Table3-3. Loadcomposition BusRefrigeration%HVAC%Smelting%Poolpump% 355405044050100610107010840202020121090001520104030166020101018101008020525304021152520402302080024551080253020401026202040202730501010282525252529405055319010003920302030 75

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3.8.3NumericalResults Figure 3-9 showsthemeanfrequencydeviationamongthegeneratorswhenonlytheauxiliarysecondarycontrolisactive(i.e.,whenthereisnointelligentloadcontrol).ThismaybeusedasareferencetoevaluatetheperformanceoftheBaLDuR-DMPCalgorithm.Theobjectivevalueoftheoptimizationproblem( 3{2 )forthisscenariowas0.1218Hz2. Figure3-9. 39-bussystem:Resultswithauxiliarysecondarycontrolonly:nointelligentloads.Pk)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(![k]2=0:1218Hz2. Figure 3-10 showstheresultsofapplyingtheBaLDuR-DMPCalgorithmwithoutbandwidthconstraints.Thismaybeusedasabenchmarkforwhatispossiblewithasamplingtimeof 300seconds.TheBaLDuR-DMPCalgorithmreducedtheobjectivevalueof( 3{2 )by74%. 76

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Figure3-10. 39-bussystem.BaLDuR-DMPCalgorithm:nobandwidthconstraints.19loads:Pk)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(![k]2=0:0318Hz2(74%objective-valuereductioncomparedtobaseline.) Figure 3-11 showstheresultsoftheresultsoftheBaLDuR-DMPCalgorithmwithbandwidthconstraints.Theobjectivevalueoftheproblem( 3{2 )was0:0451Hz2|a63%decreasecomparedtothebaselinescenariowithoutintelligentloads. Figure 3-12 showsthemagnitudeoftheDFTofthechangesinconsumptionfortheloadsatbuses18(top)and31(bottom)overthe6-hoursimulationperiod.FromTable 3-3 ,wemayseethattheloadatbus18iscomposedprimarilyofpoolpumps|resultinginlooserbandwidthconstraintsinlowerfrequencies.Conversely,theloadatbus31iscomposedofmostlyrefrigerationloads|resultinginlooserbandwidthconstraintsinhigherfrequencies.TheBaLDuR-DMPCalgorithmrestrictsthefrequencycontentofeachload'scontrolactionstotheirrespectivefrequencybands;themagnitudesoftheDFTfortheloadatbus31aremuchsmallerthanthosefortheloadatbus18becauseitscapacityismorethan10timessmaller.Theloadsdidnotviolatethe 77

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bandwidthconstraintsatanyindividualinstantintime;thatis,theloadswerealwaysabletosuccessfullylocateafeasiblesolutiontoproblem( 3{2 )with 2-hourpastandpredictionhorizons. Figure3-11. 39-bussystem.BaLDuR-DMPCalgorithm.19loads:Pk)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(![k]2=0:0451Hz2(63%reductioncomparedtobaseline). Figure 3-13 showsresultsforthescenariowherethevarianceofthemeasurementnoiseisincreasedto10)]TJ /F6 7.97 Tf 6.58 0 Td[(3Hz2.Becausethefrequencymeasurementsaresoinaccurate,theloadsinterprettheselargemeasuredfrequencydeviationsasalargepowerimbalanceinthegridcausedbytheactionsofotherloads.Asaresult,i!0from( 3{8 )and( 3{7 )foralli.Thismeansthattheloadsasymptoticallyloseactuationbecausethepredicteddisturbanceusedbytheloadsgoesto0.However,thisshowsthatinaccuratefrequencymeasurementsdonotcausetheBaLDuR-DMPCalgorithmtoharmthegrid.Rather,theloadsslowlystopprovidingsupporttothegrid.Infact,theobjectivevalueforthis 78

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Figure3-12. Frequencycontentofloadsatbuses18(top)and31(bottom). 79

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scenariowas0:0878Hz2|areductionof28%comparedtothebaselinescenariowithouttheBaLDuR-DMPCalgorithm. Figure3-13. 39-bussystem.BaLDuR-DMPCalgorithm:inaccuratefrequencymeasurements.19loads:Pk)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(![k]2=0:0878Hz2(28%reductioncomparedtobaseline).i[k]!0forallloadsduetomeasurementnoise. 3.9Summary Inthischapter,weproposedtheBaLDuR-DMPCalgorithmfordemand-sidenet-load-followinginthepowergrid.ThealgorithmusesanMPCformulationwithconstraintsonthefrequencycontentoftheloads'controlactions.Theloadsuseasimpleconstant-gainmodelofthepowergrid.Loadsthenusedisturbancepredictionsthattheyreceiveasbroadcastfromagridoperatortominimizefrequencydeviations.Thereisnocommunicationamongloads.Wehavepresentedproof-of-conceptanalyticalresultsthatshowthealgorithmconvergesundersomeidealizedassumptions.Simulationresultsforbothalinearizedmicrogridandthe39-bustestsystemshowthatwhentheloads 80

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havesucientactuation,thealgorithmperformswellinrejectingthedisturbanceandmaintainingsystemfrequency|despitetheoverlysimplemodelusedbytheloads. Forthe39-bussystem,auxiliarysecondarycontrolwasusedtorejectthelow-frequencycomponentsofthedisturbance.Itwasfoundthatthealgorithmisineectivewhenfrequencymeasurementsarehighlyinaccurate.However,ratherthantheloadscausingadditionalproblemsforthegrid,theiractuationdecays|eectivelyshuttingothealgorithm.Thisshowsthatevenwhenthealgorithmbreaksdown,itdoesnotnegativelyimpactthegridcomparedtothescenariowithoutintelligentloads. 81

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CHAPTER4CONCLUSION Inthisthesis,twonon-centralizedalgorithmsforcoordinatingloadstoprovidedemanddispatchinthepowergridwereproposed,analyzed,andtested.Bothalgorithmsachievenon-centralizationbyusinggridfrequencymeasurementsthatcanbelocallyobtainedtoinfertheglobaldierencebetweentotalconsumptionandtotalgenerationinthegrid. InChapter 2 ,theDGPalgorithmwasproposedforloadstoprovidecontingencyservicetoapowergridatfasttimescales.Eachloadhasadisutilityfunctionthatmapschangesinconsumptiontolossofqualityofservice.OneadvantageoftheDGPalgorithmoverotherproposedalgorithmsisthatitisimplementablefordisutilityfunctionsthatarenotstrictlyconvex.Loadssharegradientinformationwitheachother,andusethisinformationtominimizethetotaldisutilityamongtheloads.Simultaneously,localfrequencymeasurementsareusedtorestorethebalancebetweentotalconsumptionandtotalgenerationinthegrid.ConvergenceoftheDGPalgorithmwasestablishedundersomeidealizedassumptions,butnumericalsimulationsintheIEEE39-bustestsystemshowedthealgorithm'srobustnesstothoseassumptions.TheDGPalgorithmwasfoundtobesensitivetosensornoiseinthefrequencymeasurementsusedbytheloads,anditwasdeterminedthatperformanceofthealgorithmdegradesassamplingtimeincreases. InChapter 3 ,theBaLDuR-DMPCalgorithmwasproposedfornet-load-followingatslowertimescales.Nointer-loadcommunicationisused.Instead,loadsonlyreceivedisturbanceforecastsfromacentralizedauthority,suchasabalancingauthority.Loadsthenusemodel-predictivecontroltorejecttheforecastdisturbance.BecausetheBaLDur-DMPCalgorithmwasdesignedforslowertimescales,loadsareabletouseanextremelysimplegridmodel|aconstantgain.Morecomplexmodelsmaybeused,butthiswillrequiremorecomputation.Loadsuselocalfrequencymeasurementstoestimatethetotalimbalancebetweengenerationandconsumptioninthegrid,andtheyusethose 82

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estimatestoscalethedisturbanceforecastsfromthebalancingauthority;thishelpstheloadsavoidhigh-gainfeedbackthatcouldharmthegrid.Closed-loopstabilityoftheBaLDuR-DMPCalgorithmwasestablishedundersomeidealizedassumptions.NumericalsimulationsintheIEEE39-bustestsystemshowedthealgorithm'srobustnesstothoseassumptions. AlthoughnumericalsimulationsindicatedtheDGPalgorithmisrobusttotheassumptionsusedintheanalysisinChapter 2 ,analyzingtheconvergenceoftheDGPalgorithmwithoutsomeofthelesspracticalassumptions(e.g.,communicationdelay)isanavenueforfuturework.Likewise,theanalysisinChapter 3 usedseveralstrongassumptions,suchasnomeasurementnoise.EventhoughthenumericalresultsalsoindicaterobustnessoftheBaLDuR-DMPCtothoseassumptions,removingsomeofthestrongeronesisstillasubjectforfuturework.Additionally,itmaybepossibletoadjustthesub-algorithmforavoidinghigh-gainfeedbacktoscalethedisturbanceforecastsinindividualfrequencies.E.g.,ifloadsareadequatelyrejectingthedisturbanceinonefrequencybutnotanother,loadsmaybeabletoscalethedisturbanceforecasttoemphasizethis.ThedevelopmentandsubsequentanalysisofsuchamodicationtotheBaLDuR-DMPCalgorithmisaninterestingpathforfuturework. 83

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APPENDIXPROOFSFORDGPCONVERGENCERESULTS ProofofLemma 1 :First,weintroduceafewdenitions.Denetheprojection-lessderivativeattime tforloadi,_pi(t),suchthat_xi(t)=)]TJ /F6 7.97 Tf 27.61 -1.86 Td[(i;xi(t)[_pi(t)].Thatis,_pi(t),cXj2Ni)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(rfj(xj(t)))-222(rfi(xi(t))+u(t): (A{1)LetXi:=fm2Rj9x2Xs.t.xi=mg: (A{2)Figure A-1 showsanillustrationfora2-Dcase.Notethat xi2Xiforallidoesnot FigureA-1. Anillustrationoftherelevantsetsfora2-Dcase.Theequalityconstraintissatisedonthelinesegment,whichisthesetU.Thethicksub-segmentistheoptimalsolutionset,X. implyx2Xbecauseitmaynotsatisfytheequalityconstraint,1Tx=g.However,thesets,Xi,satisfyapropertythatisusefulintheproofs:ifx2X1X2:::Xn,thenrfi(xi)=rfforeveryi;whererfistheoptimalgradient;see( 2{12 ).Itshouldbenoted,however,thattheconverseistrueonlyiffiisstrictlyconvex.Forthisreason,we 84

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shallcallXithecriticalgradientsetofloadi.Letusdenethecollectionsofloads,A(t):=fi2Vjxi(t)>Xig;B(t):=fi2Vjxi(t)0=_xi(t)gS(t);SB(t):=fi2Bj_pi(t)<0=_xi(t)gS(t): (A{4)Thatis, S(t)isthesetofloadsforwhichtheprojectionoperationiscurrentlyactive,SA(t)isthesetofloadsinA(t)currentlyconstrainedattheirupperboundsduetoprojection,andSB(t)isthesetofloadsinB(t)constrainedattheirlowerboundsduetoprojection.Recallthatoistheinteriorof,andnotethatXioibecauseXo;therefore,projectiondoesnotaectloadsinM(t).ItfollowsthatS(t)=SA(t)[SB(t).Likewise,wehaveSA(t)A(t)andSB(t)B(t).Now,wemayobserve_u(t)=)]TJ /F8 11.955 Tf 9.3 0 Td[(1T_x(t)=)]TJ /F8 11.955 Tf 9.3 0 Td[(1T_p(t)+Xi2S(t)_pi(t)=)]TJ /F3 11.955 Tf 9.3 0 Td[(nu(t)+Xi2SA(t)_pi(t)+Xi2SB(t)_pi(t); (A{5)wherethenalequalityfollowsfrom S(t)=SA(t)[SB(t). 85

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Letthefunction,yi:R!R0,i=1;:::;n,andthefunction,y:Rn!R0,bedenedas yi(m),infrfjm)]TJ /F3 11.955 Tf 11.95 0 Td[(rj:r2Xig;i2V;y(x),Xi2Vyi(xi):(A{6)Attime t,yi(xi(t))istheEuclideandistancebetweenthecomponent,xi(t),andthecriticalgradientset,Xi,andy(t)isthetotaldistanceofallloadsfromtheirrespectivecriticalgradientsets.Forthesakeofcompactness,wewillwriteyi(t)andy(t)inplaceofyi(xi(t))andy(x(t))inthesequel.Next,wemorecloselyexaminetheterm,)]TJ /F3 11.955 Tf 9.3 0 Td[(Lrf(x(t)),in( 2{9 ).Withoutlossofgenerality,letrf1(x1(t))rf2(x2(t)):::rfn(xn(t)).CB(t),cXi2B(t)Xj2Ni)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(rfj(xj(t)))-222(rfi(xi(t))=cb(t)Xi=1Xj2Ni)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(rfj(xj(t)))-222(rfi(xi(t)): (A{7) Fori=1,everytermin( A{7 )isnonnegative.Suppose1=2N2;thenfori=2,everytermin( A{7 )isnonnegative.Nowsuppose12N2;then22N1,andtherespectivetermsin( A{7 )cancelfori=1;j=2andi=2;j=1.Therefore,thetermsof( A{7 )fori=1;2arenonnegativeregardlessofwhether1and2areneighbors.Thisargumentmaybecontinuedforalli2B(t).ItfollowsthatCB(t)0. WemaydeneasimilarsumoverA(t):CA(t),cXi2A(t)Xj2Ni)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(rfj(xj(t)))-222(rfi(xi(t)): (A{8)Bythesameargumentasabove,wemayshowthat CA(t)0. Combining( A{1 )and( A{7 ),wecanseethatthetotalchangeinconsumptionforloadsinB(t)withoutprojectionisXi2B(t)_pi(t)=CB(t)+b(t)u(t): 86

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Similarlyfrom( A{1 )and( A{8 ),thetotalchangeinconsumptionforloadsin A(t)withoutprojectionisXi2A(t)_pi(t)=CA(t)+a(t)u(t): NotethatXi2B(t)_x(t)=Xi2B(t))]TJ /F1 11.955 Tf 7.77 -9.69 Td[(_pi(t))]TJ /F5 11.955 Tf 18.13 11.36 Td[(Xi2SB(t))]TJ /F1 11.955 Tf 7.77 -9.69 Td[(_pi(t)=CB(t)+b(t)u(t))]TJ /F5 11.955 Tf 18.13 11.36 Td[(Xi2SB(t)_pi(t)andXi2A(t)_x(t)=Xi2A(t))]TJ /F1 11.955 Tf 7.77 -9.68 Td[(_pi(t))]TJ /F5 11.955 Tf 17.99 11.36 Td[(Xi2SA(t))]TJ /F1 11.955 Tf 7.76 -9.68 Td[(_pi(t)=CA(t)+a(t)u(t))]TJ /F5 11.955 Tf 17.98 11.35 Td[(Xi2SA(t)_pi(t):Because x(t)isaCaratheodorysolution(seeProposition 2.3 ),itisabsolutelycontinuousanddierentiablealmosteverywhere.Itfollowsthaty(t)isabsolutelycontinuousanddierentiablealmosteverywhere.Thenwehave_y(t)=Xi2A(t))]TJ /F1 11.955 Tf 7.5 -9.68 Td[(_xi(t))]TJ /F5 11.955 Tf 15.28 11.35 Td[(Xi2B(t))]TJ /F1 11.955 Tf 7.5 -9.68 Td[(_xi(t)=CA(t)+a(t)u(t))]TJ /F5 11.955 Tf 17.99 11.36 Td[(Xi2SA(t)_pi(t))]TJ /F5 11.955 Tf 11.95 13.27 Td[(CB(t)+b(t)u(t))]TJ /F5 11.955 Tf 18.14 11.36 Td[(Xi2SB(t)_pi(t)=CA(t))]TJ /F3 11.955 Tf 11.95 0 Td[(CB(t)+)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(a(t))]TJ /F3 11.955 Tf 11.95 0 Td[(b(t)u(t))]TJ /F5 11.955 Tf 17.98 11.36 Td[(Xi2SA(t))]TJ /F1 11.955 Tf 7.76 -9.68 Td[(_pi(t)+Xi2SB(t))]TJ /F1 11.955 Tf 7.77 -9.68 Td[(_pi(t): (A{9) 87

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Observethat,for u(t)0,wehave_y(t)+_u(t)=CA(t))]TJ /F3 11.955 Tf 11.95 0 Td[(CB(t)+)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(a(t))]TJ /F3 11.955 Tf 11.95 0 Td[(b(t))]TJ /F3 11.955 Tf 11.95 0 Td[(nu(t)+2Xi2SB(t)_pi(t) (A{10))]TJ /F3 11.955 Tf 21.91 0 Td[(u(t)=ju(t)j0;wherewehaveused a(t)n)]TJ /F1 11.955 Tf 11.98 0 Td[(1ifu(t)0(i.e.,ifu(t)0,atleastoneloadimustbeatorbelowitcriticalgradientset,Xi,bydenitionofu(t)).Similarly,for u(t)0,wehave_y(t))]TJ /F1 11.955 Tf 13.98 0 Td[(_u(t)=CA(t))]TJ /F3 11.955 Tf 11.95 0 Td[(CB(t)+)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(a(t))]TJ /F3 11.955 Tf 11.96 0 Td[(b(t)+nu(t))]TJ /F1 11.955 Tf 11.96 0 Td[(2Xi2SA(t)_pi(t) (A{11)u(t)=ju(t)j0:Considerthefunctionz:Rn!R0:z(x)=y(x)+jg)]TJ /F8 11.955 Tf 11.95 0 Td[(1Txj (A{12))z(t)=y(t)+ju(t)j:Justas y(t)isabsolutelycontinuousanddierentiablealmosteverywhere,soisju(t)j,andtherefore,z(t).Then,wemaycombine( A{10 )and( A{11 )toobtain_z(t)ju(t)j0: (A{13)Hence, z(t)isnon-increasingalmosteverywhere. Now,supposeu(t)doesnotconvergeto0.Forsomet1fort10.From( A{13 )andtherstfundamentaltheoremofcalculus,itthenfollowsthatz(t2))]TJ /F3 11.955 Tf 11.96 0 Td[(z(t1))]TJ /F4 7.97 Tf 28.94 18.7 Td[(t2Zt1dt: 88

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Because u(t)iscontinuousandu(t)90,thereexistsasequence,t1;t2;:::;withtk!1ask!1suchthatju(t)j>forallt1
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theorem[ 82 ]to(cLQ)Tandutilizingthefactthatamatrixanditstransposehavethesameeigenvalues,weconcludethatcLQhaseigenvalueswithnonnegativerealparts.BecauseLandQarereal,symmetricmatricesandQispositivedenite,cLQhasrealeigenvalues 1 Nowsuppose(;v)isaneigenpairofcLQ+11T.Thatis, (cLQ+ 11T)v=v: (A{15)Weconsidertwopossibilities: 1Tv6=0and1Tv=0.If1Tv6=0,thenpre-multiplying( A{15 )by1Tyieldsn1Tv=1Tv=)=n>0.If1Tv=0,then( A{15 )reducestocLQv=v,so(;v)isalsoaneigenpairofcLQ.Thusisrealandnonnegative.For=0,wendacontradiction.Qhasatrivialnullspace,whichimpliesthatQvisaneigenvectorofLassociatedwithazeroeigenvalue.BecauseGisconnected,thereisonlyonesucheigenvalue,andanycorrespondingeigenvectorisparalleltothe 1vector,buteachnonzeroentryof Qispositive,soeachelementofQ)]TJ /F6 7.97 Tf 6.58 0 Td[(11isalsopositive.Therefore1Tv6=0.Thuswehaveacontradiction,so6=0.Therefore 11T+cLQhasstrictlypositiveeigenvalues. Now,letibeaneigenvalueofcLQ+11T.ThentheeigenvaluesofAare1)]TJ /F3 11.955 Tf 12.44 0 Td[(i.Thus,if<2=,whereisthemaximumeigenvalueofcLQ+11T,thenj1)]TJ /F3 11.955 Tf 11.97 0 Td[(ij<1foralli.Therefore,Aisstable. That[k+1]=A[k]+Bg1followsfrom( A{14 )andthefactthattheestimatorin[ 1 48 ]isunbiased.Todeterminethesteady-statecovariance,let ~x[k],x[k])]TJ /F10 11.955 Tf 12.49 0 Td[([k],andnotethattheexpressionforW[k]=E[k][k]isgivenin[ 1 ].Thecovarianceattime 1 Thisfactmustbewellknown,butwewereunabletondareference.Itcanbeprovenasfollows.Let(;v)beaneigenpairofcLQ,socLQv=v.ThencvQLQv=vQv.wherevdenotestheconjugatetransposeofv.BecauseLandQarereal,symmetricmatrices,takingtheconjugatetransposeyieldscvQLQv=vQv.Fromthisandthepreviousequation,itfollowsthat0=()]TJ /F3 11.955 Tf 14.47 0 Td[()vQvSinceQispositivedenite,vQv6=0,whichimplies=.Thereforeisreal. 90

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k+1is[k+1]=E(x[k+1])]TJ /F10 11.955 Tf 11.96 0 Td[([k+1])(x[k+1])]TJ /F10 11.955 Tf 11.96 0 Td[([k+1])T=E(A~x[k]+B[k])(A~x[k]+B[k])T=A[k]AT+BW[k]BT;wherewehaveusedthat x[k]and[k]areuncorrelated. 91

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BIOGRAPHICALSKETCHJonathanBrookswasborninOrlando,Floridain1990oneminutebeforehissister,Amber.Initially,Jonathanhadplannedtopursueacareerinanimationbuteventuallychoseengineeringsothathecouldhaveamore-physicalpositiveeectontheworld.Anatural-bornGator,Jonathanreceivedhisbachelor's(2012),master's(2013),anddoctoral(2017)degreesinmechanicalengineeringfromtheUniversityofFlorida.Jonathanbeganhisresearchcareerasanundergraduateresearchassistantin2011,workingonfaultdetectionincommercialheating,ventilation,andair-conditioningsystemsundertheadvisementPrabirBarooah.Asagraduatestudent,hetransitionedtoresearchingalgorithmsforenergy-ecientcontrolofHVACsystemsandthentocontrolalgorithmsfordemand-sideancillaryservicetothepowergrid. 99