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brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by VBN ControlStructuresforSmartGridBalancingMortenJuelsgaard,LuminitaC.Totu,S.EhsanShaei,RafaelWisniewskiandJakobStoustrupAutomationandControl,AalborgUniversity,DenmarkAbstract—Thisworkaddressestheproblemofmaintainingthebalancebetweenconsumptionandproductionintheelectricitygridwhenvolatileresources,suchaswindandsun,accountforalargepercentageofthepowergeneration.WepresentcontrolstructuresforSmartGridbalancingservicesonthreedifferentlevels:portfolio,largerscaleindividualpowerunits,andaggregationsofsmallpowerunits.Ourfocusisonillustratingtheconnectionbetweencoordinationandcontrolalgorithms.IndexTerms—SmartGrids,Energymanagement,Optimalscheduling,DistributedcontrolI.INTRODUCTIONDuringrecentyears,therehasbeenagrowingdesiretoincreasetheamountofrenewableenergyfromwindandsunintheoverallelectricitygeneration.Thispresentstheissueofhowtomaintainthebalancebetweenconsumptionandproductionwhenalargepartoftheelectricitygenerationcomesfromresourceswithpartiallyuncontrollablecharac-teristics.Animportantconceptthatcomesintoplayisthatofdemandmanagement,namelyusingexistingexibilitiestoshiftconsumptionfromtimeswhenrenewableresourcesarescarce,toperiodswhenresourcesareample[1],[2].Aportfoliocontainingbothtraditionalpowerplantsandcontrollableconsumersisconsideredandthebalancingprob-lemisaddressedatthedifferentlevels.Werstconsiderthehigh-leveltaskofschedulingandcoordinatingproductionandconsumptionasanoff-lineplanningprocess.Consumersareactivelyincludedinthisplanning,onlyiftheypresentsomelevelofexibility,inthesensethatsometemporalshiftsofthepowerconsumptionincurnorelevantdrawbackscomparedtonormaloperation.Wethenlookintothereal-timeoperationandcontrolforfollowingthepowerreferencesfromtheplanningprocess.Theunitsoftheportfoliocanbeindividualentities,forinstancelargeproducersaspowerplants,aswellaslargeandmediumscaleconsumers.Inparticular,commercialrefrig-erationsystems(e.g.supermarkets,coldstorages)havebeenshowntohaveahighpotentialfordemandresponseimple-mentations[3].Forthese,thereexiststwoseparatedemandresponseschemes:optimizationofthecostofoperationordirectmanagementofthepowerconsumptionforbalancingservices[4].Inthisworkwecombinethetwoschemesandperformdemandmanagementforcommercialrefrigerationsystemssuchthatobjectivesofbothrefrigerationsystemandhigherlevelserviceproviderareaccommodated.Aportfoliounitmayalsorefertoanaggregationofsmallerindividualunits,suchaswindturbinescollectedinafarm,or Contactfmju,lct,ses,raf,jakobg@es.aau.dk.ThisworkissupportedbytheSouthernDenmarkGrowthForumandtheEuropeanRegionalDevelopmentFundundertheproject”Smart&Cool”.anumberofaggregatedsmallscaleconsumersthatcancol-lectivelybeconsideredasasinglemediumscaleconsumer.Inparticular,wefocushereonthepowerexibilityofresidentialthermostaticloads.Thesehaveagoodpotentialforautomaticdemandresponseandviablecontrolstrategieshavereceivedincreasedscienticinterest.Forexample,collectivethermostatset-pointmanipulationwasproposedby[5],while[6]analyzedtheeffectivenessoftwotypesofcontrolwithpreplannedactivationwindows.Inthiswork,weuseacollectiveon-offmanipulationwithrandomizationasproposedin[7]and[8].ThehierarchicallayersofthebalancingareillustratedinFig.1,showingtheseparationbetweenoff-lineplanningandreal-timeoperation.Intheplanningprocesstheportfolioowner(PO)schedulesthepowerconsumptionand/orpro-ductionoftheunitsasaniterativeprocess,accordingtotheexternalbalanceobjectiveandthelocaloptimizationmodelofeachunit.Thelocaloptimizationencompassesspecicoperationalcostsandconstraintsofeachunit.Attheendoftheplanningprocess,eachunithasacquiredapowerscheduleinformofareferencetobetrackedduringreal-timeoperation. Portfoliodemand,Portfoliosize,etc. Coordination(CentralorDecentral) Iterations Units LocalOptimizer(Unit1) LocalOptimizer(Unitn) Off-LinePlanning LocalController(Unit1) ProcessDynamics(Unit1) LocalcontrolAggregatedunits LocalController(Unitn)

2 ProcessDynamics(Unitn) LocalcontrolStan
ProcessDynamics(Unitn) LocalcontrolStand-aloneunits PowerReference PowerReference Real-TimeOperation Fig.1.Hierachicalillustrationoftheoff-lineplanningandcoordinationofportfoliounits,andreal-timeoperationofeachunitgovernedbylocalcontrollers.ThecoordinationmechanismandplanningprocessoftheportfolioarepresentedinSectionII.SectionIIIpresentsthecontrolstructureforasupermarketrefrigerationsystemconsideredasanindividualunitintheportfolio.SectionIVpresentsthecontrolstructureforaportfoliounitthatisanaggregationofresidentialrefrigeratorsdeviceswithsmallindividualconsumption.NumericalexperimentsarepresentedinSectionVandSectionVIprovidesclosingremarks.II.DECENTRALIZEDPOWERPORTFOLIOPLANNINGWeconsideraparticipantintheenergymarket,whoman-agesaportfoliocomposedofdifferentunits,asoutlinedinFig.2(Left).Specically,theportfolioincludesconsumptionunitswithexibleconsumption.WeassumethatthePOhas committedtoanoverallpowerschedule,sothattheportfoliomustbeoperatedtocomplywiththisschedule.Sincetheportfoliocontainsbothproductionandconsumptionunits,thetotaloutputoftheportfoliocanbemanagedbyadjustingeithertheproductionortheexibleconsumption.OperatingtheportfolioentailsthatthePOmustcreateindividualpowerschedulesforeachunitintheportfolio.Thisshouldbedoneinaneconomicalway,meaningthatthePOmustminimizeoperatingcosts,whileobeyingoperationalconstraintsofeachunit. LargescaleSmallscaleGridAggregatedConsumersTraditionalPowerPlantsConsumers x1 x2 x3 x4 x5 x6 xn Fig.2.Left:Theportfolioconsistingofbothtraditionalpowerplants,largescaleconsumersandaggregatedsmallscaleconsumers.Right:Graphstructureoftheportfolio,whereunitsshouldonlycommunicatewithafewneighboringunits.Theunitsintheportfolioaredistributedacrossgeographicalareas.Further,thenumberofunitsintheportfolio,whenincludingalargenumberofcontrollableconsumers,canmaketheportfolioverylarge.Consequently,bothcollectinganddistributingdataacrosstheportfolio,aswellasoptimizingin-dividualpowerschedulesinacentralizedmanner,maybecomecumbersome.Therefore,weproposetosolvetheprobleminadistributedfashionasillustratedinFig.2(Right).Inthegure,eachboxillustratesaunitoftheportfoliothatoperatesindependently,theblackarrowsindicatecommunicationpathsusedtocoordinateunits,andthegraylinesindicatethepowergrid.Allunitsareconnectedtothepowergrid,buteachunitisonlyallowedtocommunicatewithafewneighboringones.Thefollowingoutlineshowsatisfactoryoperationofthecollectedportfoliomaybemaintained,evenwhenenforcingsuchdistributedcomputationandcommunicationstructure.A.TechnicalapproachLetT=f1;:::;Ngdenotethediscretetime-frameforwhichtheportfoliobalancingisconducted.Lettheportfolioconsistofnunits,andleteachportfoliouniti2S=f1;:::;ngbecharacterizedbyastrictlyconvexcostfunctionfi:RN!R,andconvexlocalconstraints,denotedPi.Lettingc2RNdenotetheenergyscheduletobehonoredbythePO,theoveralloptimizationproblemisformulatedasminimizepi;i2SPi2Sfi(pi)subjecttopi2PiPi2Spi=c;(1)wherepi2RN;i2Sisthepowerconsumed(orproduced)byeachportfoliounit.Problem(1)isconvexandmaybereadilysolvedinacentralizedfashion.However,asmentioned,wedesiretoderiveadecentralizedapproach,thatdoesnotrelyoncentralizedcomputations.Amethodthathasprovedviableinseveralcontexts([9],[10])andcanalsobeshowntoworkinthiscase,isthemethodofdualdecomposition,followingtheapproach:1)Initialize:k=0;(1)2RN;�0;f(j)g;j2N2)Letk=k+1,andp?(k)i=arginfpi2Pi(fi(pi)+pTi(k))3)g(k)=Pi2Sp?(k)i�c4)(k+1)=(k)+(k)g(k)5)Ifkg(k)k,repeatfromitem2Above,kisaniterationindexand(k)areLagrangemultipli-ersforthepoweraccumulationconstraint.Thesemultipliersareupdatedateachiteration.Dualdecompositionseparatesthemainproblem(1)intosmallersubproblemsthatcanbesolvedlocallybyeachunit(Step2above),andutilizesthatg(k)canbeshowntobeagradienttothedualofProblem(1).Thusforsuitablechoicesof(k),theiterativemethodconverges.Thedrawbackishoweverthattheimplementationofthisstrategyrequiresacentralunittocollectp?(k)i;8i,andcalculateaswellasdistribute(k+1)for

3 thefollowingiteration.Theworkby[11]showe
thefollowingiteration.Theworkby[11]showedhowthiscanbeavoidedbythereformulation p(k)=1 nXi2Sp?(k)i;g(k)=n p(k)�c:ByemployingthegraphstructureoftheportfoliooutlinedinFig.2(Right),theaveragevector p(k)canbecomputedlocallybyeachunit,throughdata-exchangeonlywithneighboringunitsinthegraph[12].ThisallowsalocalupdateoftheLa-grangemultipliers.Employingaveragingeliminatestheneedforacentralunitforgathering/scatteringdata.Inaccuraciesinthelocalcalculationof p(k)foreachnode,entailthatthisapproachdoesnotconvergetotheglobaloptimum.However,itwasshownby[11]thattheseinaccuraciescanbemadearbitrarilysmall,andconvergencecanthereforebeguaranteedtoapointarbitrarilyclosetotheglobaloptimum.Thealgorithmsuppliedabovecomprisesthetop-leveloff-linecoordinationalgorithm,renderingrun-timepowerrefer-encesfortheportfoliounits.Thefollowingsectionsdescribehowportfoliounitscanparticipateinthecoordinationprocess,aswellastracktheresultingreferencesinreal-time.Asmentioned,wefocusonconsumptionforthermalprocesses,inthetwocasesoflargescaleconsumersandaggregatedsmallscaleconsumers.III.COORDINATIONOFLARGESCALECONSUMERSAtypicalsupermarketrefrigerationsystem(SRS)iscom-posedofseveralcompressorsplacedinoneortworacks(forinstanceinaboosterconguration),severaldisplaycases,andfreezerrooms.Themainpowerconsumingcomponentsofthesystemarethecompressors.Supermarketrefrigerationsystemshavesignicantpotentialforthermalenergystorage,whichisnotcurrentlyusedforgridbalancing.Inthispartofthework,wefocusonhowtomanagetheoperationofasupermarketinapowerexiblemannerandhaveitconnectedasanodetothepowerportfolioofSectionII.ForthispurposewerstderiveandvalidateadynamicalmodeloftheSRS,whichisalsocapableofpredictingthepowerconsumptionofthesystem.Secondly,weformulatetheminimizationoftheSRSoperatingcost,asastrictlyconvexoptimizationproblem,forparticipatinginthetop-levelcoordination.Finally,asupervisorycontrollerisneededtoregulatethepowerconsumptiontotheassignedreference,duringreal-timeoperation. TheSRSisalreadyequippedwithvariouscontrollersdistributedacrosstheinstallation.Thesehavetheprimarypurposesofmaintainingthefoodtemperatureswithinsafetyregulations,andnominaloperatingconditionsfortherefriger-ationequipment.Thecontrolapproachforpowerexibilityistokeepthestandardlocalcontrollers,anddesignasupervisorycontrol,responsibleforprovidingrequiredset-points,asshowninFig.3. CoordinationInterface SupervisoryControl DistributedControllers RefrigerationSystem Iterations set-points controlsignals Localfeedbacks Set-pointsandpowerfeedbacks Innercontrolloop Outercontrolloop-Realtimeoperation Fig.3.Supermarketcontrolsystemstructure[13],alongwithinterfacetohigh-levelcoordination.A.DynamicModelAmodular,nonlinear,continuoustimemodel,suitableforsupervisorycontroldesigninthesmartgrid,isdevelopedby[14].Themodelingisbasedonarstprincipleapproach,wherethecompletesystemisseparatedintomodules,eachaddressedandvalidatedseparately.Thismodularityleavesopenthepossibilityofmodelingvariouscongurationsfordifferentsupermarkets.Eachmodule,aswellastheoverallintegratedsystem,isvalidatedagainstmeasureddata,collectedfromasupermarketinDenmark.Foranelaborationonthemodelingandvalidation,thereaderisreferredto[14].B.CostOptimizationTheobjectiveofthelocaloptimizationproblemtheSRSistokeepoperationclosetoidealconditions.Toelaborateonthis,weletp(t);(t)2Rdenotethetotalpowercon-sumptionandcommoncoefcientofperformance(COP)ofthecompressors,andletu(t)=(u1(t);


;um(t))2Rm;t2Tdenotethecoolingcapacityappliedbyeachofmcoolingunits.Thecoherencebetweentheelectricalpowerconsump-tionandcoolingcapacityisp(t)=mXi=1ui(t)=(t);8t2T;(2)andweletp=(p(1);:::;p(N))2RN.Ingeneral,bothu(t)and(t)arenonlinearfunctionsofothermanipulatedvariablesoftherefrigerationsystem.However,[15]and[16]explainhowu(t)canbedealtwithasactitiousmanipulatedvariable,andfurtherhow(t)canbeobtainedbyalinearestimationprocess.Itisfurtherpossibletoset-upalineardynamicalmodelthatusesu(t)asinputandactsasahigh-levelap

4 proximationofthesupermarketoperatingcond
proximationofthesupermarketoperatingconditions:x(t+1)=Ax(t)+Bu(t)+Bdd(t);whered(t)encapsulatesdisturbances,e.g.,buildingindoorthermalconditions.Thestatevectorx(t)containsfoodandairtemperaturesforthedisplaycasesintheSRS.Thedetailsofthemodelareelaboratedin[15].Wedenotebyx(t)thedesiredoperatingconditionsasset-pointvaluesforthestates.Givenestimatesofthemodelparametersforallt2T,anoptimizationproblemthatgivesthepowerconsumptionreferencefortheSRSisexpressedasminimizeu;pPt2Tjjx(t)�x(t)jj2+Tpsubjecttox(t+1)=Ax(t)+Bu(t)+Bdd(t)xminx(t)xmax0u(t)umax
uminu(t)�u(t�1)
umaxt2T;(3)wheretheconnectionbetweenpanduisstillgovernedby(2).Abovexmin;xmax;umaxdenoteupperandlowerboundsonstatesandactuators.Similarly,
uminand
umaxdenoteupperandlowerboundsontherateofchangeofcoolingcapacityu(t)betweentime-steps.Thisisbothtoavoidlargeoscillationsinthecoldstoragetemperatures,andfurthertoproducesmootherpowerconsumptiontrajectoriesthataremoreappropriatetotrackduringreal-timeoperation.ThenaltermTpinthecost,isonaccountofthehigh-levelcoordinationofSectionII.Problem(3)correspondstoasinglesubproblemintheoverallbalancingdescribedby(1).LetSsrsSdenotethepartoftheportfolioconsistingofSRS's.Thengiventhecorrespondencebetweencoolingcapacityandpowerin(2),thesubproblem(3)denesfiandPiforalli2Ssrs.C.PowermanagementTheSRSoptimizationdescribedin(3)isthelocaloptimizerthatcoordinatestheSRStotherestoftheportfolioasillustratedinFig.1.Afterconvergenceofthecoordination,theSRSshouldfollowthepowerconsumptionproleagreedwiththehigherlevelserviceprovider.Forthis,amodelpredictivecontrol(MPC)schemewitha5minutesamplingtimeiscon-sidered.Asamplingtimelargerthanfourminutesensuresthatthemassowdynamicsinsidetheevaporatorsareinsteadystate,andtheirdynamicsmaythusbeneglected.Further,assumingconstantpressureset-pointsforthesuctionmanifoldgivesalinearizedrelationbetweenthecoolingcapacityandtheopeningdegreeofexpansionvalves(asmanipulatedvariables).Thedetailsofthecontrollerareelaboratedin[14].IV.COORDINATIONOFSMALLSCALECONSUMERSResidentialpowerconsumptioniscurrentlyinexibleinpractice,asitisnotdirectlyaffectedbypricesfromtheenergymarketsandisnotcontrolledbyothermechanismsexceptcompleteloadsheddinginexceptionalcases.However,asforSRS's,thereexistsanobviouspotentialforexibility.Wefocushereonthermostaticappliances,inparticularcoolingdevicessuchasresidentialrefrigeratorsandfreezers.Thesecanberegardedasmanysmallandleakythermalenergystor-ageseachconsumingpowerinanon/offpatterndeterminedbythethermostatcontrol.Weconsideranaggregatorthatisresponsibleforthepowerconsumptionofalargepopulationofsuchthermostatically controlledappliances.Theobjectiveistoincludetheaggrega-torasasingleexibleconsumptionnodeinthepowerportfoliopresentedinSectionII.A.DynamicmodelLetFdenotethesetofallrefrigeratorsinthepopulation,withcardinalityjFj=F.Also,letpi(t)denotethepowerconsumptionforFi2F;i2f1;:::;Fg,andletp(t)=XFi2Fpi(t)denotethetotalpopulationconsumptionfort2T.Strictlyspeaking,p(t)isaquantizedvariable,however,undertheassumptionoflargepopulationsizeF,wewilltakep(t)tobecontinuous.EachunitFi2Fismodeledwithrst-orderdynamicsandequippedwithalocallogicalcontrollerKithathasthemainfunctionofathermostat.Inaddition,thelogicalcontrollercanswitchtheon/offstateoftherefrigeratorasaresponsetoabroadcastsignalfromtheaggregator,seealsoSectionIV-C,butonlyifdoingsodoesnotcontravenewiththesafetyofthelocaloperation.Forthepurposeofoff-lineplanning,wewillmodeltheaggregatedpopulationasasingle,leakingenergystorage.LetE(t)denotethevirtualstoredenergy,E(t+1)=aE(t)+Tsp(t);wherea2(0;1)isaleakageparameter.Wedenetheexibilityofthepopulationasanintervalofvirtualenergylevels,EminE(t)Emax.Areferencep(t)isdenedtobevalidaslongas,startingfromaninitialE0,itmaintainsthevirtualenergylevelwithinthisinterval.Avalidreferencemaybesuccessfullytrackedbythepowermanagementstructureofthepopulation.Numericalsimulationsshowthat,whilesi

5 mple,thismodelandtheassociatedconstraint
mple,thismodelandtheassociatedconstraintsprovideanefcientwayofchar-acterizingthepowerconsumptionexibilityoftherefrigeratorpopulationunderourchosenexibilitycontrol.Theparametersoftheleakystoragemodel(a,Emin,Emax,E0)havetobeestimatedusingMonteCarlosimulationsofthethermostatloadpopulationundersteppowerreferences.B.CostoptimizationAspreviously,letp=(p(1);:::;p(N))2RN.Further,letpbas2RNdenotethepreferredbaselineoftheaggregatedconsumptionforthepopulationofrefrigeratorsacrossthehori-zonT.Giventhemodelingdescribedabove,theexibilityoftheconsumerscanbemanagedthroughthelocaloptimizationminimizepkp�pbask2+TpsubjecttoE(t+1)=aE(t)+Tsp(t)EminE(t)Emax0p(t)pmax;(4)wherepmaxisthehardupperlimitforthepopulationconsump-tion.ThelasttermTpisthecoordinationcontributionfromthetop-levelcoordinationalgorithm.SimilarlytothepreviouscaseoftheSRS,problem(4)makesupthelocaloptimizationforoneoftheportfoliounitsinthemainproblem(1).C.PowermanagementTheoverallapproachforenablingthepowerexibilityoftherefrigeratorpopulationconsistsofatwo-levelcontrolstructure;asupervisorandthelocalcontrollersKi,seeFig.4.Thesupervisorycontrollermeasuresthetotalpoweroutputp(t)ofthepopulation,predictstheconsumptiononestep-aheadintimeandcomparesthepredictionwiththepredeter-minedreferencetrajectory.Thedifferencesarebalancedoutbybroadcastingan-signaltorefrigeratorpopulation.The-signalinuencestheon/offswitchingofthelocalcontrollersKiandrepresentsarandomizeddispatchstrategy.Detailsareincludedin[7],andwenotethattechniqueissimilartothatproposedin[8]. CoordinationInterface SupervisoryControl Iterations K2 F2 K1 F1 KF FF   p1(t) p2(t) pF(t) p(t) Realtimeoperation Fig.4.Overviewofthecontrolapproach,alongwithinterfacetothehigh-levelcoordination.V.NUMERICALEXPERIMENTSThefollowingpresentsacombinednumericalexperimentillustratingthedifferenthierarchicallevelsofthebalancingstrategyoutlinedintheprevioussections:A.toplevelportfoliobalancingandcoordinationB.referencetrackingbylargescaleandaggregatedsmallscaleconsumersA.PortfoliobalancingWeconsiderasmallportfolioconsistingofasupermarket,anaggregatedpopulationof1000householdrefrigerators,andapowerplant.Thecoordinationandlocaloptimizationfromthesupermarketandrefrigeratorpopulation,areperformedasdescribedintheprevioussections.Thepowerplantismodeledsimilartotheworkof[17],asathirdordersystemGpp(s)=1=(spp+1)3,describingtheclosedlooptransferfunctionfrompowerreferencetoactualpowerproduction.Thepowerplantissubjectamaximumcapacityconstraintof20kW,aswellasarampingconstraintsontheinput,i.e.,thepowerreference.Thepowerplantoptimizationattemptstominimizeaquadraticfuelcostpertainingtothepowerproduction.Thetimehorizonoftheexamplespans24hours,witha5minutesampletime.Thepowerschedulec(t)thatthePOhastocomplywith,isshowninFig.5(Top)alongwiththeaccumulationoftheportfolioreferencesobtainedfromthedistributedcoordinationprocessdescribedinSectionII.ItcanbeseenthatthecommittedPOscheduleiscloselyfollowedbythecoordination.Forcomparison,Fig.6showsthescheduleinthecasewhenthepowerplantisactingalonewithoutcontributionsfromtheexibledemandunits.Inparticular,duetothecapacityconstraint,thepowerplantalonecannotmeettheschedule. Fig.5.Top:Thepowerdemand(solid,black)andtheaccumulatedportfolioreferences(dotted,red),aftercoordination.Bottom:Thecoordinatedreferencesforthepowerplant(dotted,magenta),thepopulation(asterisk,blue),andSRS(circles,red),aswellasbaselineconsumptions(dashed,black). Fig.6.Thedemand(solid,black),andthebestobtainabledemandtrackingfromthepowerplantalone(circles,red).B.ReferencetrackingThepowerreferencestobetrackedinrealtimebythesu-permarketandtherefrigerationpopulationrespectively,canbeseeninFig.5(Bottom).FollowingtheMPCschemeoutlinedpreviouslyfortheSRS,resultsinthereferencetrackingpre-sentedinFig.7(Top).Similarly,bytheaggregatedcontrolleroutlinedinSectionIV,thedemandtrackingoftherefrigeratorpopulationispresentedinFig.7(Bottom). Fig.7.Top:Thecoordinatedreference(circl

6 ed,red),andtheruntimepowerconsumption(do
ed,red),andtheruntimepowerconsumption(dotted,cyan),oftheSRS.Bottom:Thecoordinatedreference(asterisk,blue),andtheruntimepowerconsumption(dotted,magenta),oftheconsumerpopulation.BoththeSRSandrefrigeratorpopulationcontrollermanagetotracktheirrespectivereferencesclosely,withonlyminorde-viations.Overall,thetrackingresultsindicatethatthebalanceobtainedbytheoff-linecoordinationillustratedinFig.5willbecloselymaintainedduringreal-timeexecutionaswell.VI.CONCLUSIONThisworkhasaddressedthetaskofmaintainingthebalancebetweenconsumptionandproductionintheelectricalgrid.Wehaveconsideredthisissueondifferenthierarchicallevels,encompassingbothdistributedoff-linecoordination,aswellasthereal-timeexecutionandreferencetracking.Ourworkhasfocusedoncontrollableconsumption,andwehaveshownhowabalancedgridcanbemaintainedbyconsumptioncontrol,bothforlargescaleconsumers,aswellasaggregationsofapopulationofsmallscaleconsumers.Ournumericalexamplehasillustratedhowcontrollableconsumptioncanassistandimprovethegridbalancing,comparedtothecasewhereexclusivelytraditionalpowerplantsmaintainsbalance.REFERENCES[1]F.RahimiandA.Ipakchi,“Demandresponseasamarketresourceunderthesmartgridparadigm,”IEEETranscationsonSmartGrid,vol.1,no.1,2010.[2]P.Varaiya,F.Wu,andJ.Bialek,“Smartoperationofsmartgrid:Risk-limitingdispatch,”ProceedingsoftheIEEE,vol.99,no.1,2011.[3]S.Goli,A.McKane,andD.Olsen,“Demandresponseopportunitiesinindustrialrefrigeratedwarehousesincalifornia,”in2011ACEEESummerStudyonEnergyEfciencyinIndustry,NiagaraFalls,NY,USA,Jul.2011.[4]T.Hovgaard,L.Larsen,K.Edlund,andJ.Jørgensen,“Modelpredictivecontroltechnologiesforefcientandexiblepowerconsumptioninrefrigerationsystems,”Energy,vol.44,pp.105–116,2012.[5]D.S.Callaway,“Tappingtheenergystoragepotentialinelectricloadstodeliverloadfollowingandregulation,withapplicationtowindenergy,”EnergyConversionandManagement,vol.50,no.5,pp.1389–1400,2009.[6]M.Stadler,W.Krause,M.Sonnenschein,andU.Vogel,“Modellingandevaluationofcontrolschemesforenhancingloadshiftofelectricitydemandforcoolingdevices,”EnvironmentalModelling&Software,vol.24,no.2,pp.285–295,2009.[7]L.C.Totu,J.-J.Leth,andR.Wisniewski,“Controlforlargescaledemandresponseofthermostaticloads,”AmericanControlConferenceProceedings,inpress.[8]C.-Y.Chang,W.Zhang,J.Lian,andK.Kalsi,“Modelingandcontrolofaggregatedairconditioningloadsunderrealisticconditions,”inInnovativeSmartGridTechnologies(ISGT),2013IEEEPES.IEEE,2013,pp.1–6.[9]B.Biegel,P.Andersen,J.Stoustrup,andJ.Bendtsen,“Congestionmanagementinasmartgridviashadowprices,”inProceedingsofIFACPPPSC,sep.2012.[10]H.Terelius,U.Topcu,andR.M.Murray,“Decentralizedmulti-agentoptimizationviadualdecomposition,”Proceedingsofthe18thIFACWorldCongress,Aug.2011.[11]M.Juelsgaard,R.Wisniewski,andJ.Bendtsen,“Faulttolerantdis-tributedportfoliooptimizationinsmartgrids,”UnderreviewforIn-ternationalJournalofRobustControl,2013.[12]L.XiaoandS.Boyd,“Fastlineariterationsfordistributedaveraging,”SystemsandControlLetters,vol.53,pp.65–78,Feb.2004.[13]S.E.Shaei,H.Rasmussen,andJ.Stoustrup,“Modelingsupermarketrefrigerationsystemsforsupervisorycontrolinsmartgrid,”inproceed-ingsoftheAmericanControlConference,WashingtonDC,USA,Jun.2013.[14]——,“Modelingsupermarketrefrigerationsystemsfordemand-sidemanagement,”Energies,vol.6,no.2,pp.900–920,2013.[15]——,“Modelpredictivecontrolforathermostaticcontrolledsystem,”inSubmittedtoEuropeanControlConference,Z¨urich,Switzerland,2013.[16]S.E.Shaei,J.Stoustrup,andH.Rasmussen,“Asupervisorycontrolapproachineconomicmpcdesignforrefrigerationsystems,”inpro-ceedingsoftheEuropeanControlConference,Z¨urich,Switzerland,Jul.2013.[17]K.Edlund,J.Bendtsen,andJ.Jørgensen,“Hierarchicalmodel-basedpredictivecontrolofapowerplantportfolio,”ControlEngineeringPractice,vol.19,no.10,pp.1126–1136,201