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Energy Optimization and Control for Data Centers and Smart Grids

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Title:
Energy Optimization and Control for Data Centers and Smart Grids
Creator:
Guo, Yuanxiong
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 (170 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Electrical and Computer Engineering
Committee Chair:
FANG,YUGUANG
Committee Co-Chair:
KHARGONEKAR,PRAMOD P
Committee Members:
CHEN,SHIGANG
GUAN,YONGPEI
Graduation Date:
8/9/2014

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Subjects / Keywords:
Algorithms ( jstor )
Batteries ( jstor )
Elasticity of demand ( jstor )
Electricity ( jstor )
Energy ( jstor )
Energy storage ( jstor )
Market prices ( jstor )
Prices ( jstor )
Renewable energy ( jstor )
Workloads ( jstor )
Electrical and Computer Engineering -- Dissertations, Academic -- UF
control -- energy -- optimization
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Electrical and Computer Engineering thesis, Ph.D.

Notes

Abstract:
Today's energy systems are in the midst of a transformation driven by the global climate change and increasing demand of energy. The majority of today's energy supply comes from the combustion of fossil fuels, such as coal, petroleum, and natural gas, which not only have limited supply but also cause a lot of greenhouse gas emissions, driving climate changes. Moreover, worldwide energy demand is still growing very fast for enabling economic growth and high quality of life. Therefore, there is an urgent imperative to build a sustainable energy future. Utilizing sustainable energy sources (e.g., wind, solar, and geothermal), deploying energy-efficient technologies, and reducing energy consumption are exemplary ways to achieve such a sustainable future. The theme of this dissertation is to apply a computational approach to tackle the current energy sustainability challenges in the context of data centers and smart grids. In particular, advanced modeling, optimization, and control techniques are developed in this dissertation to manage these systems so as to improve their energy efficiency, lower their energy cost, and reduce their carbon footprint. The main contributions of this dissertation are the following. First, several novel control algorithms are developed for data center operators to effectively manage their energy usage. These control algorithms take into account important yet largely unexplored issues in practice, such as uncertain workload arrivals and electricity prices. Experimental results based on real-world traces show that one can effectively conserve the energy consumption, lower the energy cost, and increase the renewable energy utilization in these data centers without violating the service-level agreement by leveraging the proposed control algorithms. Second, two frameworks, ``Smart Home'' and ``Smart Neighborhood'', for managing distributed energy resources in residential households are proposed. These frameworks integrate all essential components such as smart appliances, storage devices, and on-site renewable generators in the future smart grid. Results in this work reveal that, by utilizing the real-time detailed data about residential customers's behaviors and power system conditions, one can greatly improve the efficiency and sustainability of current electric power systems. Third, an novel market operation strategy for a virtual power plant consisting of multiple distributed energy resources is developed to minimize the imbalance cost when participating into a wholesale electricity market. ( 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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Description based on online resource; title from PDF title page.
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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, 2014.
Local:
Adviser: FANG,YUGUANG.
Local:
Co-adviser: KHARGONEKAR,PRAMOD P.
Statement of Responsibility:
by Yuanxiong Guo.

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Copyright Guo, Yuanxiong. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
969976917 ( OCLC )
Classification:
LD1780 2014 ( lcc )

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ENERGYOPTIMIZATIONANDCONTROLFORDATACENTERSANDSMARTGRIDSByYUANXIONGGUOADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2014

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c2014YuanxiongGuo

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Tomyparentsandancee

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ACKNOWLEDGMENTS IwouldliketothankallofthosewhohavehelpedmethroughthechallengingbutrewardingPhDjourney.Firstandforemost,Iwouldliketoexpressmysinceregratitudetomytwoadvisors,Dr.YuguangFangandDr.PramodP.Khargonekar,fortheirguidanceandsupportthroughoutmygraduatestudies.Dr.Fangtaughtmehowtodoresearchandmoreimportantly,howtobeagoodperson.Dr.KhargonekarledmeintotheeldofsmartgridsandhasalwaysoeredhishelpunreservedlywheneverIneedit.Thisdissertationwouldnotbepossiblewithouttheirenthusiasmandencouragement.Theyaremyrolemodels,andbeingaprofessorlikethemisoneofthemainreasonsformetopursueanacademiccareer.Iamdeeplyindebtedtothem.IalsowouldliketothankDr.ShigangChenandDr.YongpeiGuanforservingonmysupervisorycommittee.Iamespeciallygratefultothemfortheirgreatsupportinmyjobhunting.IamalsothankfulforthefriendshipofmanycurrentandformergraduatestudentsinWirelessNetworkLaboratoryandattheUniversityofFlorida.TheseincludeMiaoPan,ZongruiDing,PanLi,ChiZhang,YangSong,JinyuanSun,RongshengHuang,XilinCheng,YiWang,KaiheXu,RuigangFang,HaoYue,LinkeGuo,HuangLin,Huai-LeiFu,GuoliangYao,KaipingXue,YueZhao,YanLong,YingCai,JieWang,LiliWang,JinlinPeng,andYanpingLi.IwouldliketoespeciallythankMiaoforhismanyhelps,bothprofessionalandpersonal,overthepastfouryears.Finally,Iwouldliketothankmyfamilyfortheirunconditionalsupportandloveinmylife.Iamluckytohavethetwomostimportantwomeninmylife:mymotherandmyancee.Mymotherhasalwaysbelievedinmeandhelpedmeinsofarassheisable.Myancee,YanminGong,hassacricedalotandchosentotravelovertenthousandmilesawayfromhometostandbymysideatUniversityofFlorida.Theirlovemotivatesmetokeepmovingforwardnomatterwhatobstacleshavecomemyway. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTION .................................. 13 1.1Motivation .................................... 13 1.2RelatedLiterature ............................... 16 1.2.1DataCenterEnergyEciency ..................... 16 1.2.2DERManagementinSmartGrids ................... 18 1.3OverviewofthisDissertation ......................... 19 2ELECTRICITYCOSTREDUCTIONINDATACENTERS ........... 25 2.1ModelingandFormulation ........................... 27 2.1.1TheWorkloadModel .......................... 27 2.1.2TheBatteryModel ........................... 28 2.1.3TheQoSModel ............................. 30 2.1.4ThePowerConsumptionModel .................... 30 2.1.5TheElectricityPriceModel ...................... 31 2.1.6TheCostMinimizationProblemWithEnergyStorage ........ 32 2.2ProposedSolution ................................ 32 2.2.1RelaxedProblem ............................ 33 2.2.2OurProposedAlgorithm ........................ 34 2.2.3SolutiontoP3 .............................. 37 2.3AlgorithmicPerformanceAnalysis ....................... 40 2.4CaseStudies ................................... 44 2.4.1ExperimentalSetup ........................... 44 2.4.2ExperimentalResults .......................... 46 2.5Summary .................................... 49 3WORKLOADMANAGEMENTFORSUSTAINABLEDATACENTERS ... 51 3.1ModelingandFormulation ........................... 53 3.1.1TheWorkloadModel .......................... 55 3.1.2TheRenewableGenerationModel ................... 57 3.1.3TheThermalStorageModel ...................... 58 3.1.4TheCostModel ............................. 60 3.1.5ProblemFormulation .......................... 62 5

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3.2AlgorithmDesign ................................ 63 3.2.1RelaxedProblem ............................ 64 3.2.2TheStochasticCostMinimizationAlgorithm(SCMA) ........ 66 3.2.3InterpretationofSCMA ......................... 71 3.3PerformanceAnalysis .............................. 72 3.4CaseStudies ................................... 75 3.4.1ExperimentalSetup ........................... 77 3.4.2ExperimentalResults .......................... 79 3.4.2.1CostSavings ......................... 79 3.4.2.2Trade-obetweenCostandDelay .............. 81 3.5Summary .................................... 82 4SMARTHOME .................................... 83 4.1ModelingandFormulation ........................... 88 4.1.1RenewableGeneration ......................... 88 4.1.2EnergyStorage ............................. 88 4.1.3ElectricityMarket ............................ 90 4.1.4ControlObjective ............................ 90 4.2InelasticEnergyDemand ............................ 90 4.2.1RelaxedProblem ............................ 93 4.2.2OurProposedAlgorithm ........................ 95 4.2.3AlgorithmicProperties ......................... 96 4.3ElasticEnergyDemand ............................ 100 4.3.1RelaxedProblem ............................ 102 4.3.2Delay-AwareVirtualQueue ...................... 103 4.3.3OurProposedAlgorithm ........................ 105 4.3.4AlgorithmicProperties ......................... 106 4.4CaseStudies ................................... 113 4.4.1ExperimentalSetup ........................... 113 4.4.2ExperimentalResults .......................... 114 4.5Summary .................................... 116 5SMARTNEIGHBORHOOD ............................. 117 5.1ModelingandFormulation ........................... 119 5.1.1LoadServingEntity ........................... 120 5.1.2EnergyLoad ............................... 120 5.1.3EnergyStorage ............................. 122 5.1.4RenewableDistributedGeneration ................... 123 5.1.5ProblemFormulation .......................... 123 5.2OnlineDistributedAlgorithm ......................... 124 5.2.1Delay-AwareVirtualQueue ...................... 125 5.2.2TheLyapunov-basedApproach ..................... 126 5.2.3DistributedAlgorithmtoP3 ...................... 129 5.3PerformanceAnalysis .............................. 130 6

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5.4CaseStudies ................................... 133 5.4.1ExperimentalSetup ........................... 134 5.4.2ExperimentalResults .......................... 135 5.5Summary .................................... 137 6IMBALANCECOSTMINIMIZATIONFORVIRTUALPOWERPLANTS .. 138 6.1ProblemFormulation .............................. 140 6.1.1RenewableDistributedGeneratorModel ............... 140 6.1.2DemandModel ............................. 141 6.1.3MarketModel .............................. 143 6.1.4VPPOperationModel ......................... 144 6.2ProposedAlgorithm .............................. 146 6.3PerformanceAnalysis .............................. 151 6.4CaseStudies ................................... 154 6.4.1PerformanceResultsUnderTypeIPrices ............... 156 6.4.2PerformanceResultsUnderTypeIIPrices .............. 157 6.4.3Worst-caseDelayPerformance ..................... 158 6.5Summary .................................... 159 7CONCLUSIONSANDFUTUREWORK ...................... 160 REFERENCES ....................................... 162 BIOGRAPHICALSKETCH ................................ 170 7

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LISTOFTABLES Table page 2-1Servercongurationsindatacenters ........................ 44 8

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LISTOFFIGURES Figure page 2-1Geographicallydistributeddatacenters ....................... 28 2-210-minaveragereal-timeelectricitypricesin3hoursatfourlocations ...... 46 2-310-minaverageworkloadin3days ......................... 46 2-4Performancecomparisons .............................. 48 3-1TypicalcloudnetworkarchitectureofaCSP .................... 54 3-2Sustainabledatacenter ................................ 54 3-310-minaverageworkloadarrivalsforoneday .................... 76 3-4Hourlyelectricitypricesinday-aheadmarketsforonedayatfourlocations ... 76 3-510-minaveragesolarandwindenergygenerationforoneweek .......... 77 3-6Averageenergycostcomparison ........................... 80 3-7Averageoperatingcostcomparison ......................... 81 3-8Cost-Delaytradeo .................................. 82 4-1Residentialrenewableenergysystem ........................ 86 4-2Residentialenergymanagementsystemwithoutelasticdemands ......... 91 4-3Residentialenergymanagementsystemwithelasticdemands ........... 101 4-4AveragehourlyelectricitypriceforoneweekatLA ................ 113 4-5AveragehourlysolarpowerforoneweekatLA .................. 113 4-6Cost-Storagetradeowithoutelasticdemands ................... 114 4-7Costcomparisonamongthreeschemes ....................... 115 4-8Impactofdelaysensitivityofelasticdemands ................... 115 5-1Neighborhoodenergymanagementsystem ..................... 119 5-2FlowchartfortheAlgorithm 5 ........................... 130 5-3Performanceoftheproposedalgorithm ....................... 134 5-4Impactofparametersoncostsaving ........................ 135 6-1StructureofaVPP .................................. 140 9

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6-2PerformancecomparisonwithdierentV ...................... 156 6-3Performancecomparisonwithdierent ...................... 157 6-4Delayperformanceoffourtypesofelasticdemands. ................ 158 6-5Imbalancecostunderdierenti;8i. ........................ 158 10

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AbstractofdissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyENERGYOPTIMIZATIONANDCONTROLFORDATACENTERSANDSMARTGRIDSByYuanxiongGuoAugust2014Chair:YuguangFangCochair:PramodP.KhargonekarMajor:ElectricalandComputerEngineeringToday'senergysystemsareinthemidstofatransformationdrivenbytheglobalclimatechangeandincreasingdemandofenergy.Themajorityoftoday'senergysupplycomesfromthecombustionoffossilfuels,suchascoal,petroleum,andnaturalgas,whichnotonlyhavelimitedsupplybutalsocausealotofgreenhousegasemissions,drivingclimatechanges.Moreover,worldwideenergydemandisstillgrowingveryfastforenablingeconomicgrowthandhighqualityoflife.Therefore,thereisanurgentimperativetobuildasustainableenergyfuture.Utilizingsustainableenergysources(e.g.,wind,solar,andgeothermal),deployingenergy-ecienttechnologies,andreducingenergyconsumptionareexemplarywaystoachievesuchasustainablefuture.Thethemeofthisdissertationistoapplyacomputationalapproachtotacklethecurrentenergysustainabilitychallengesinthecontextofdatacentersandsmartgrids.Inparticular,advancedmodeling,optimization,andcontroltechniquesaredevelopedinthisdissertationtomanagethesesystemssoastoimprovetheirenergyeciency,lowertheirenergycost,andreducetheircarbonfootprint.Themaincontributionsofthisdissertationarethefollowing.First,severalnovelcontrolalgorithmsaredevelopedfordatacenteroperatorstoeectivelymanagetheirenergyusage.Thesecontrolalgorithmstakeintoaccountimportantyetlargelyunexploredissuesinpractice,suchasuncertainworkloadarrivalsandelectricityprices.Experimentalresultsbasedonreal-worldtracesshowthat 11

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onecaneectivelyconservetheenergyconsumption,lowertheenergycost,andincreasetherenewableenergyutilizationinthesedatacenterswithoutviolatingtheservice-levelagreementbyleveragingtheproposedcontrolalgorithms.Second,twoframeworks,\SmartHome"and\SmartNeighborhood",formanagingdistributedenergyresourcesinresidentialhouseholdsareproposed.Theseframeworksintegrateallessentialcomponentssuchassmartappliances,storagedevices,andon-siterenewablegeneratorsinthefuturesmartgrid.Resultsinthisworkrevealthat,byutilizingthereal-timedetaileddataaboutresidentialcustomers'sbehaviorsandpowersystemconditions,onecangreatlyimprovetheeciencyandsustainabilityofcurrentelectricpowersystems.Third,annovelmarketoperationstrategyforavirtualpowerplantconsistingofmultipledistributedenergyresourcesisdevelopedtominimizetheimbalancecostwhenparticipatingintoawholesaleelectricitymarket. 12

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CHAPTER1INTRODUCTION 1.1MotivationEnergyisattheheartofmostcriticaleconomic,environmentalanddevelopmentalissuesfacingtheworldtoday.Clean,ecient,aordable,andreliableenergyservicesareindispensablefortheprosperityofourcountry.AsreportedbytheU.S.EnergyInformationAdministration,theworldenergyconsumptionisexpectedtogrowby56percentbetween2010and2014[ 10 ].Electricitysuppliesanincreasingshareoftheworld'stotalenergydemandandistheworld'sfastest-growingformofdeliveredenergy.Itisexpectedthattheworldnetelectricitygenerationwillgrowby93percentbetween2010and2040[ 10 ].Inadditiontothechallengesbroughtbytheincreasedenergyconsumption,theenvironmentalimpactofenergyconsumptionisgainingworldwideattention.Globalincreasesingreenhousegas(GHG)emissionsaredueprimarilytofossilfueluse;energy-relatedgreenhousegasemissionsaccountedformorethan80percentoftotalU.S.greenhousegasemissionsin2007.Theincreasedgreenhousegasemissionsresultinglobalclimatechangesandcallformoreutilizationofrenewableenergysourceswithoutcarbonemissions.SomecountrieshavecommittedtoagreementssuchasKyotoProtocol.IntheUnitedStates,severalpolicyinitiativesatbothstateandfederallevelshavebeensetup.Forinstance,Californiahassettheambitiousrequirementthatrenewablesotherthansmallhydroplantswillberequiredtoaccountfor33percentofelectricitysupplyby2020.However,unlikeconventionalenergysourcessuchascoal,naturalgas,andoil,thepoweroutputfrommostrenewableenergysourcessuchaswindorsolaris\variable",i.e.,intermittent,uncertain,andunpredictable.Thisvariabilitywillhaveatremendousimpactonourelectricityservice,andnewtechnologiesneedtobedevelopedinordertotackleitinacost-eectivewaywhilemaintainingthereliabilityofourelectricitysupply. 13

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Inviewoftheabovechallenges,thereisanimperativeneedfordeployingenergy-ecienttechnologies,utilizingsustainableenergysources,andreducingenergyconsumptionforawiderangeofapplicationsinoursociety.Inthisdissertation,wefocusonenergyoptimizationandcontrolmethodsfordatacentersandsmartgrids.Ourgoalistondecientwaystomanagethesesystemssothattheirenergyutilizationcanbemadesustainable.DatacentersprovidethesupportinginfrastructureforawiderangeofITservicesandconsumeahugeamountofelectricity.AccordingtotheU.S.EPAReporttotheCongressonServersandDataCenterEnergyEciencyin2007,USdatacentersconsumes61billionkWhin2006(1.5%oftotalUSelectricity).Moreimportantly,itisgrowingexponentiallyatanannualrateof15%(morethan12timesthegrowthrateofthetotalUSelectricityusage).Actually,theenergycosthasgrowntoexceedtheservercostindatacenters.Therefore,itisimperativetoimprovingtheenergyeciencyandreducingtheenergycostindatacentersfordatacenteroperators.Notethatenergyconsumptionindatacenterscomesfromtwoparts:IT-relatedandnon-ITrelated.TheIT-relatedpartincludespowerconsumptionfromservers,datastorage,andnetworkequipments.Thenon-ITrelatedpartincludespowerconsumptionfromthecoolingandprovisioninginfrastructure.Tocapturethisconsumption,powerusageeectiveness(PUE)measurestheratiooftotalbuildingpowertoITpower.TheaveragevalueofPUEindatacenterindustryis2whilethemostenergy-ecientdatacenterstodaycanachieve1.2.Previousstudiesshowthatsavingenergyandimprovingperformanceareusuallyinconictofeachother.Hence,moreresearchisneededfordesigningecientalgorithmstosolveenergyeciencyandenergycostproblemsindatacenterswhileensuringperformanceconstraints.Moreover,withrisingpublicawarenessofclimatechange,theenvironmentalimpactofelectricityconsumptionindatacentersisreceivingintensiveattention[ 27 ].Thecarbonintensityhasbecomeanimportantmetricformeasuringtheenergyeciencyofadatacenterinindustry.SeveralleadingITserviceproviderssuchasAppleandGooglehave 14

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installedon-siterenewablegeneratorssuchassolarandwindfarmsneartheirdatacenterssothatlessenergyisusedfromtheutilitygrid.Then,howtoeectivelyutilizethisrandomrenewableenergyisabigchallengefordatacenteroperators.Besidesdatacenters,wealsostudyresidentialdistributedenergyresources(DER)insmartgrids.Motivatedbyenergysecurityandenvironmentalconcerns,manystatesinU.S.havesetaggressiverenewableportfoliostandards(RPSs)topromotetheuseofrenewableenergyresources.However,thevariabilityofthepoweroutputfromtheserenewablesisthemostimportantobstacletohighrenewableenergypenetration.Incurrentoperatingparadigm,operatingreserveisusedtoabsorbthisvariability.Althoughthisapproachworksatmodestpenetrationlevels,itfailswhenwehavetomeetaggressivetargets.Astudy[ 47 ]conductedinCaliforniaestimatesthatinordertomeetits33%renewableportfoliostandard,themaximumregulation-upcapacityrequirementincreasesfrom277MWto1135MW,andthemaximumregulation-downcapacityincreasesfrom382MWto1097MW.Theselargeincreasesinreservesareeconomicallyuntenableandosettheenvironmentalbenetsofrenewables.Duetothelackoftransmissioncapacityincurrentpowergrids,muchofthisrenewablegenerationisbeingdeployedatthedistributionlevelintheformofrooftopPVpanelsandsmall-scalewindfarms.Distributedgenerators,distributedstorage,controllableloads,andadvancedpowerelectronicsareexamplesofdistributedenergyresources(DER),whichrefertosmall-scalepowersourcesthatarelocatedonusers'sites(distributionlevel)wheretheenergytheygenerateisused.ItiswidelyanticipatedthatDERwillplayasignicantroleinfutureelectricitysupply[ 54 ].TherearearangeofbenetsfordeployingDERunitssuchas(1)reducingcarbonemissions,(2)increasingenergyeciencythroughcombinedheatandpower(CHP)generation,(3)improvingpowerqualityandreliability,and(4)reducinglinelossesanddeferringgridexpansionduetothepresenceofgenerationclosetodemand[ 45 ].Enabledbytheinformationandcommunicationinfrastructureavailableinsmartgrids,theseDERunitscanbeaggregated 15

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andcontrolledtogethertoabsorbthevariabilityassociatedwithrenewablegenerationandmitigatetheincreasedreservecostofhighrenewableenergypenetration.However,newschedulingandcontrolmethodsneedtobedevelopedinordertohandleapotentiallylargenumberofDERunitsaswellastheirrandomness.Sinceresidentialsectoraccountsfor37%ofallU.S.electricityuse,wefocusontheresidentialDERmanagementinsmartgrids.Withtheactivedeploymentofadvancedmeteringinfrastructures(AMIs)andreal-timepricingprogramsinresidentialhouseholdsnationwide,residentialcustomershavebothmechanismsandincentivestoactivelymanagetheirenergyusage.HowtoecientlymanagingDERunitsinresidentialhouseholdstohelprealizethevisionofsmartgridisalargelyunsolvedproblem. 1.2RelatedLiteratureInthissection,wereviewtheliteraturerelatedtoenergymanagementfordatacentersanddistributedenergyresourcesinsmartgrids. 1.2.1DataCenterEnergyEciencyThehugeenergyconsumptionindatacentershasmotivatedalotofresearchtoreducetheelectricitycostindatacenters.Thesestudiescanberoughlydividedintotwocategories:oneisfromtheperspectiveofhardwaredesignandengineering;theotherisfromtheperspectiveofalgorithmicdesign.Ourresearchfocusesonthealgorithmicperspective.Intherstcategory,manyenergy-ecienthardwaredesignssuchasenergy-ecientchipsandmulti-coreservershavebeendevelopedtoreducethepowerconsumptionofIT-relatedpart[ 56 ].Fornon-ITrelatedpart,lowerPUEisusuallyachievedbyusingadvancedengineeringtechniquessuchashighertemperaturesetting,DCpowerdistribution,andbetterbuildingdesign(See[ 12 ]forasurveyontheseissues).Inthesecondcategory,theresearchcanbedividedintothefollowingthreedierentlevels.Therstoneistheserverlevel,whereonlythepowerconsumptionofasingleserverisconsidered.Awidely-usedtechniqueisthedynamicvoltageandfrequencyscaling 16

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(DVFS),wheretheoperatingvoltageandfrequencyoftheserver'sCPUcanbeadjustedaccordingtotheintensityoftheworkloadontheserver.SincetherstanalyticalstudyofDVFSbyYaoetal.[ 92 ],theschedulingandspeedscalingalgorithmstominimizethetotalenergyusedinordertomeetjobdeadlineshavebeenaddressedin[ 75 ].Theobjectiveofminimizingtheaverageresponsetimegivenanenergybudgetisaddressedin[ 25 ],whiletheobjectiveofminimizingaweightedcombinationofexpectedresponsetimeandenergyusageperjobisconsideredin[ 16 ].Thesecondlevelisthelocaldatacenterlevel.Thegoalofresearchatthislevelistodesigndatacentersthatare\power-proportional",i.e.,usingpoweronlyinproportiontotheload.However,today'sdatacentersarefarfromthisgoal-theyconsumealmosthalfoftheirpeakpowerwhennearlyidle.Dynamiccapacityprovisioning(DCP)hasbeenproposedtodynamicallyadjustthenumberofactiveserverstomatchthecurrentworkloadsinadatacenter.Linetal.proposeanovelonlinealgorithmforcostreductiontodynamicallyright-sizeadatacenter,whichisproventobe3-competitive,whiletakingintoaccounttheswitchingcostduringturningon/oservers[ 57 ].Electricenergystoragedevices,typicallyUPSunits,indatacentersisexploitedin[ 86 ]toreducetheelectricitycostinadatacenter.ThesystemimplementationissueofusingUPSunitstohelpreduceelectricitycostisanalyzedin[ 36 ].However,batteriessuchasUPSunitsarequiteexpensiveandcannotbeoverusedsincefrequentlycharginganddischargingseverelyimpactstheirlifetimes.Ontheotherhand,thermalstorageismuchcheaperandcanbeutilizedtoreducethecoolingcostindatacentersasshownin[ 91 ].Thethirdlevelistheglobaldatacenterlevel,whichisbasedonthefactthatITserviceprovidersoftenhavemultipledistributeddatacenterstoprovidebetterQoSforusers,andelectricitypricesatthesedatacentersexhibitbothspatialandtemporaldiversity.Qureshietal.arethersttodiscusstheopportunityofutilizingsuchelectricitypricevariationtoreducetotalelectricitycostbydistributingmoretractodatacenterswithlowelectricityprice[ 77 ].Raoetal.investigatetheproblemofminimizingthetotal 17

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electricitycostfordatacentersinamulti-electricity-marketenvironmentsubjecttoQoSguaranteeandproposealinearprogrammingformulationtoapproximatelysolveit[ 79 ].Thesestudiesfocusondirectlyreducingthetotalelectricitycostbyexploitingthespatialvariationofelectricityprices.Thefocusofourworkisonthisglobaldatacenterlevel.Besidesresearchonenergycostreductionindatacenters,renewable-powereddatacentersarereceivingmoreandmoreattentionbothinindustry[ 4 , 5 ]andinacademia[ 34 , 35 , 59 { 61 , 81 ].Previousstudies[ 60 , 61 ]explorethefeasibilityandbenetsofusinggeographicalloadbalancingfordelay-sensitiveinteractiveworkloadstofacilitatetheintegrationofrenewablesourcesintodatacenters.Schedulingofdelay-tolerantbatchworkloadandenergystoragetohelpintegraterenewablesourcesintoadatacenterwithon-siterenewablegenerationisdiscussedin[ 59 , 81 ].Systemimplementationissueswithrenewableenergy-awarebatchworkloadschedulerarediscussedin[ 34 , 35 ]andprototypesarebuilttoshowtheeectivenessofthesejobschedulers.However,alltheaforementionedpaperseitherconsiderasingledatacenter,asingleclassofapplication,noenergy/thermalstoragefacility,onlydelay-sensitiveinteractiveworkloads,orassumeperfectfutureinformation. 1.2.2DERManagementinSmartGridsTherehavebeenmanypreviousstudiesonenergyconsumptionschedulinginhouseholds,renewableenergyintegration,anddemandresponseschemes.Ontheresidentialenergyconsumptionschedulingside,Mohsenian-Radetal.[ 64 ]formulatetheoptimalcontrolofmultipleexibleappliancesasalinearprogramtoachieveadesiredtrade-obetweentheelectricitypaymentandthewaitingtimefortheoperationofeachapplianceinahousehold,wherecustomersaresubjecttoareal-timepricingtaricombinedwithincliningblockrates.Agametheorybasedapproachisproposedin[ 63 ]tohandlethecaseofmultiplehouseholds.Kimetal.[ 53 ]usedynamicprogrammingtosolvetheproblemofschedulingpowerconsumptionofasingleapplianceinordertominimizetheexpectedcost.Onthesideofrenewableenergyintegration,theproblem 18

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ofsupplyingrenewableenergytodemand-exiblecustomersisinvestigatedin[ 69 , 73 ].Specically,Papavasiliouetal.[ 73 ]addresstheoptimalallocationforrenewablesourcestodemand-exiblecustomersinareal-timepricingenvironmentusingdynamicprogramming,whilein[ 69 ],theauthorsdevelopaLyapunovoptimizationbasedmethod.Bitaretal.[ 22 ]consideroptimalsellingstrategiesforuncertainandvariablewindproductionintothecurrentelectricitymarket.Onthedemandresponseside,Lietal.[ 55 ]considertheproblemofoptimaldemandresponseasaconvexoptimizationproblemandstudytheroleofdynamicpricing.Jiangetal.[ 51 ]proposeamodelthatintegratestwo-periodelectricitymarkets,uncertaintyinrenewablegeneration,andreal-timedynamicdemandresponse,andderivetheoptimalcontroldecisionstooptimizethesocialwelfare.However,mostofthesepreviousstudieseitherassumeperfectfutureinformation,ordonotconsideron-sitedistributedgenerationandenergystorage. 1.3OverviewofthisDissertationThisdissertationisorganizedasfollows.InChapter 2 wefocusontheproblemofreducingtheelectricitycostindatacentersbyusingenergystorage.InChapter 3 westudytheworkloadmanagementproblemforrenewable-powereddatacenterswiththermalstorage.Then,weinvestigatetheresidentialDERmanagementproblemsinsmartgridsundertheframeworksof\SmartHome"and\SmartNeighborhood"inChapter 4 andChapter 5 ,respectively.Afterthat,westudytheoptimalmarketoperationstrategyforavirtualpowerplantconsistingofmultipleDERunitswhenparticipatinginawholesaleelectricitymarketinChapter 6 .Finally,inChapter 7 weconcludethisdissertationanddiscussfuturework.Chapter 2 :ElectricityCostReductioninDataCenters Traditionaltechniquessuchasdynamicvoltage/frequencyscaling,dynamiccapacityprovisioning,andenergy-awareloadbalancingoftenresultinatrade-obetweencostandperformance.However,duetotherecentderegulationofelectricitymarkets,electricitypricesarenolongerxedbuttime-varying.Inthisscenario,energystoragecanbe 19

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leveragedtoreducetheenergycostofdatacenters.Comparedtoprevioustechniques,themostattractingpointofusingenergystorageisthatitwouldnotcauseanyperformancedegradationfordatacenterworkloads.Ithasbeenobservedthatdatacentersusuallyinstallmanyuninterruptiblepowersupply(UPS)unitstoprovidetransitionpowerforthemtoswitchfromtheutilitypowersupplytotheirback-updieselgenerators.Whereasthistransitiontakesonly10-20seconds,UPSunitshaveenoughbatterycapacitytokeeptheentiredatacenterpoweredatitsmaximumpowerneedsforanywherebetween5-30minutes.Thisexcessenergystoragecapacityoerstheopportunitytoreducetheenergycostbystoringsomeenergyduringperiodsoflowpriceswhiledischargingenergytoreducethepowerdrawnfromtheutilitygridduringperiodsofhighprices.Althoughtheintuitionissimple,severalchallengesexist.First,bothelectricitypricesanddatacenterworkloadsareuncertain,makingithardtodeterminetheoptimalcharge/dischargedecisionsofenergystorage.Second,frequentcharge/dischargewouldseverelyimpacttheUPSbatterylifetime,andweneedtoconsiderthecostofusingbatteries.Toaddresstheseissues,werstformulateastochasticprogramtomodelthefactorsappropriately.Then,weproposeandevaluateanovelcontrolmethodwhichcanadaptivelylearntheopportunitiestocharge/dischargetheenergystoragewhileminimizingtheimpactonthebatterylifetime.Weshowthatourcontrolmethodissimplebutecientinreducingtheenergycostofdatacentersasveriedbyreal-worldtrace-basedsimulations.Inaddition,weshowthatcombingexistingtechniqueswithenergystoragecanfurtherreducetheenergycostofdatacenterstoalargeextent.Insummary,weareamongthersttoidentifytheopportunityofUPSunitsindatacentersforenergycostreduction,andarethersttoaddresstheintegrationofprevioustechniqueswithenergystorage.OurworkmakesanimportantsteptowardsbestutilizingUPSunitsindatacenters.Moreover,thecontrolmethodologywedevelopedinthisworkcanbeeasilyextendedtostudyotherproblemsinvolvedwithstoragedevices. 20

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Theworkpresentedinthischapterisbasedonthepublications[ 37 , 38 ]Chapter 3 :WorkloadManagementforSustainableDataCenters ReducingthecarbonfootprintofdatacentersisbecomingaprimarygoaloflargeITcompanies.However,unlikethepowerdrawnfromthepowergrid,thelocallygeneratedrenewableenergyisintermittent,uncontrollable,andunpredictable,dependingontheambientenvironment.Themismatchbetweenintermittentrenewableenergysupplyandcontinuousdatacenterenergydemandwouldcauserenewableenergywaste,leadingtolowrenewableenergyutilizationindatacenters.Tosolvethischallenge,weobservethefollowingtwoaspectsfrompracticalsystems:(1)Datacenterssupportawiderangeofworkloads,includingbothdelay-sensitive,interactiveworkloadsanddelay-tolerant,batchworkloads.Whilethedelay-sensitiveworkloadsmustbeservedimmediatelywithminimaldelay,delay-tolerantworkloadscanbeusedtoincreasetherenewableenergyutilizationbydelayingtheirservicestoperiodswhenrenewablegenerationisabundantwithoutexceedingtheirexecutiondeadlines.(2)Datacentersspendalmostasmuchpoweroncoolingastheydoonservers.Unliketheexpensiveelectrochemicalstoragesuchasbatteries,thermalstorageismuchcheaperandcanbeleveragedtostoreexcessrenewableenergyandthenprovidecoolingpower.Wearethersttoidentifythesetwoaspectstofacilitatetherenewableenergyintegrationindatacenters.Wethendesignanewcontrolstrategytoschedulethedelay-tolerantworkloadsandmanagethethermalstoragefacilitiessuchthatboththerenewableenergyutilizationismaximizedandtheenergycostofpowerdrawnfromtheutilitygridisminimized.Moreover,byconsideringthemechanismofloadbalancing,weproposeaninnovativedistributedcontrolstrategytofacilitaterenewableenergyintegrationingeographicallydistributeddatacenters,whichcanbereadilyimplementedincurrentcloudsystems. 21

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Insummary,wearethersttoproposetheuseofthermalstorageanddelay-tolerantworkloadstoincreaserenewableenergyutilizationindatacentersandourcontrolstrategiesholdgreatpotentialforpracticaldeploymentinthenearfuture.Theworkpresentedinthischapterisbasedonthepublications[ 39 , 40 ]Chapter 4 :SmartHome Weconsiderasmarthomewithsmartappliances,energystoragedevices,anddistributedrenewablegeneratorssuchasrooftopPVpanels.Ratherthanatelectricityrate,thesmarthomewouldreceiveviaAMIatime-varyingelectricityrate,whichtheutilitycompanycanadjustinrealtimebasedonthepriceinthewholesaleelectricitymarketorthegridcongestioncondition.Boththelocalrenewablegenerationandthereal-timeelectricitypricevaryovertimeandarehardtopredict.Therefore,homeenergymanagementsystemsforresidentialhouseholdsinthisnewcontextneedtobedevelopedinordertoachievevariousobjectssuchasmaximizingtherenewableenergyutilizationorminimizingtheelectricitybill.Inthischapter,toexploretheopportunitiesinresidentialhomeenergymanagement,wenovellydividetheenergyloadsinresidentialhouseholdsintotwocategories:elasticloadsandinelasticloads.Elasticloadssuchasrefrigerators,waterheaters,andairconditionerscanbecontrolledtoadjusttheiroperatingtimeandpowerlevelswithoutaectingend-usersatisfaction.RecentmeasurementsshowthatinatypicalU.S.household,elasticenergyloadsaccountfor59%oftheaverageenergyconsumption.Therefore,thereisgreathiddenpotentialinexploitingtheinherentexibilityofsuchelasticloadsforvariousimportantindividualandsystemlevelobjectives.Moreover,energystoragedevicessuchaselectricvehiclebatteriescanalsobeleveragedtohelpmatchtheenergydemandwithlocalrenewablesupplyorreducetheelectricitybill.Wearethersttoproposeanintegratedframeworktomodeltheinteractionsamongthesecomponentsinaresidentialhouseholdforsmartgrids.Furthermore,insteadofunrealisticallyassumingthatrandomeventssuchasrenewablegeneration,real-time 22

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priceandenergyloadsareknownaprioriasinpreviousstudies,weemployastochasticapproachanddesignanovelcontrolalgorithmwithoutrequiringknowledgeoffutureinformation.Weshowthatbyutilizingourcontrolalgorithmconsideringbothelasticloadsandstoragedevices,theelectricitybillofasmarthomecanbereducedgreatlyinsuchastochasticenvironment.Theworkpresentedinthischapterisbasedonthepublication[ 41 ]Chapter 5 :SmartNeighborhood WhilethesmarthomeframeworkdiscussedinChapter 4 iseectiveinmanagingtheenergyusageinasinglehousehold,thefollowingquestionremainsopen:whatistheaggregateeectifthesesmarthomesoperateindividuallyaccordingtothestrategywedevelopedbefore?Forexample,whenaverylowelectricitypriceisbroadcastedtoallsmarthomes,itmayincuranewpeakdemand,whichisnotdesirablefromtheperspectiveofpowersystemoperator.Mostpreviousstudieshaveignoredtheimpactsfromasystemperspectivewhenmanagingtheenergyusageinresidentialhouseholds.Consideringthisaspect,weproposeaframeworkcalled\SmartNeighborhood",inwhichmultiplesmarthomesinaneighborhoodareservedbythesameloadservingentity(LSE).Inthesmartneighborhood,smarthomeswouldcollaboratewitheachotherviaAMItocoordinatetheirenergyutilizationsoastoreducetheaggregatepeakdemandandtominimizethetotalcostofsupplyingenergytotheneighborhood.ThetraditionalapproachistolettheLSEcontroltheenergyusageinallsmarthomestoachievethesesystemicobjectives.However,thisapproachishardtoimplementinpractice.Moreover,itcouldviolateuserprivacybycollectingdetailedenergyusageprolesfromcustomers.Toaddresstheseissues,wedevelopanovelpricingstrategyfortheLSEsuchthatateachtimeinstant,theLSEwouldonlybroadcastapricinginformationtoallsmarthomes,andeachsmarthomewouldautonomouslymanageitsenergyusageaccordingtothereceivedpricinginformation.Weshowthatbyproperlydesigningthepricingstrategy, 23

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boththeenergycostandtheaggregatepeakdemandcanbereducedsimultaneouslyinaprivacy-preserving,distributedway.Theworkpresentedinthischapterisbasedonthepublications[ 43 , 44 ]Chapter 6 :ImbalanceCostMinimizationofVirtualPowerPlants Manygovernmentsaroundtheworldhavesetuprenewableportfoliostandards(RPSs)topromotetheproductionofpowerfromrenewableenergysources.However,sellingrenewableenergy,especiallywindandsolar,intoelectricitymarketsishighlyriskybecauseofthepossibleimbalancebetweenday-aheadscheduledgenerationandreal-timedeliveredamount.Inelectricitymarketswithhighpenetrationofrenewables,theimbalancewouldincuralargepenaltywhichisdetrimentaltorenewablepowerproducers.Weanalyzedtheeconomicissuesofintegratingrenewableenergyintocurrentelectricitymarketsandproposedcost-eectivewaysofintegratingdistributedrenewableenergyforutilitycompaniesandrenewablepowerproducers.Specically,weconsideravirtualpowerplantconsistingofvariousdistributedenergyresourcesandinvestigateitsreal-timeoperationstrategytoreducetheimbalancecost.Amajornoveltyofourmodelisthatitcanensureacertainportionofrenewableenergygenerationtobeutilized.Ourstudiesshowthatbyintelligentlycontrollingthesystemoperationandaggregatingotherdistributedenergysources(e.g.,distributedstorageandcontrollableloads),itispossibletoachieveaelectricenergysystemwithhighrenewableenergypenetrationinacost-eectiveway.Theworkpresentedinthischapterisbasedonthepublication[ 39 ]. 24

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CHAPTER2ELECTRICITYCOSTREDUCTIONINDATACENTERSWiththepopularityofcloudcomputing[ 14 ],moreandmoredatacentersareanticipatedtobebuiltinthefutureinordertomeetthegrowingdemandforlarge-scalecomputingresources.Itiscommonforacloudserviceprovidertohavemultipledatacenterseachhavinghundredsofthousandsofservers.Thosedatacentersaregeographicallydistributedforreliabilityaswellasperformanceimprovement.Acriticalissueintheoperationsofthosedatacentersistheenergyconsumption,includingbothserversandairconditioners.Accordingtotheestimationfrom[ 77 ],largecompaniessuchasGoogleandMicrosoftpaytensofmillionsofdollarsforjustelectricityusageeveryyear,and30%-50%percentageofoperationalexpensesindatacenterscomefromelectricity.Further,fromaserviceoperator'sviewpoint,theelectricitycosthasgrowntoexceedtheservercostsindatacenters.Therefore,minimizingtheelectricitycostisreceivingmoreandmoreattention.However,savingelectricitycostandimprovingperformanceareusuallyinconictwitheachother,thusjointoptimizationisneeded.Anaturalwaytoreduceelectricitycostistoconserveenergyconsumptionortoimprovetheenergyeciencysuchthatthesameamountofworkloadcanbeservedwithlessenergy.Notethatinlarge-scaledatacenters,computingequipmentsgenerallyexhibithighpowerintensitywithallofitsconsumedelectricpowerconvertedtoheat.Inordertoensurethereliableoperationofdatacenters,airconditioningisrequiredtoextracttheheatdissipatedbytheITcomputingdevices.Thus,additionalpowerisrequiredtooperatethecoolingsystem.Ontheotherhand,moreandmoreelectricitymarketsareundergoingderegulationwheretheelectricitymarketoperatorsoerdynamicelectricityratestolargeindustrialandcommercialcustomersinsteadoftraditionalatratesattheretaillevel.Therefore,minimizationofelectricityconsumptiondoesnotnecessarilytranslateintothatoftheelectricitycostsincethecostshouldbethepricetimestheenergyamount.AlthoughtechniquessuchasDVFS,DCP,GLBareeectiveinpractice, 25

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alargelyignoredfactorislargelyignoredfactoristheexistenceofenergystoragefacilitieswithindatacenters,whichcanprovidefurthercostsavingifutilizedintelligentlyincombinationoftheprevioustechniques.Comparedwithexistingtechniquesforelectricitycostreduction,themethodofenergystoragehasnoperformancedegradation.Datacentershaveuninterruptedpowersupply(UPS)unitstokeepthempoweredusingstoredenergyincaseofelectricutilityfailure,whichistheirprimarypowersource,beforethebackupdieselgenerationcanstartupandprovidepowerassecondarypowersource.Usually,thetransitiontousedieselgenerationtakesonly10-20secondswhileUPSunitshaveenoughcapacitytopowerthedatacenteratitsmaximumpowerneedbetween5-30minutes.Thisexcessenergystoragecapacitycanbeusedtosavetheelectricitycostbythesimpleintuitionofchargingwhentheelectricitypriceislowwhiledischargingwhentheelectricitypriceishighintheutilitygrid.Inthischapter,weinvestigatetheproblemofexploitingtheUPSunitswithindatacenterstominimizethecloudserviceprovider'selectricitycost.Weproposeajointloadbalancing,serverconguration,andbatterymanagementschemeformultipledistributeddatacenters.Sincethetracarrivalsandelectricitypricesarebothrandomprocesseswithpossiblyunknownstatistics,theproblemisformulatedasastochasticprogramandthen,anecientonlinealgorithmbasedontheLyapunovoptimizationtechniqueisproposedtosolveit.Insummary,thischaptermakesthefollowingcontributions: Weinvestigatetheproblemofminimizingthetotalelectricitycostofmultipledatacentersforacloudserviceproviderunderwholesaleelectricitymarketsbytakingintoaccountthebatterieswithinthesedatacenters. Weformulatetheproblemasastochasticprogram,whichcapturesthecenter-levelloadbalancing,theserver-levelconguration,andthebatterymanagementwhileatthesametimeguaranteeingthequality-of-service(QoS)experiencebyendusers. WeproposeanecientonlinealgorithmbasedontheLyapunovoptimizationtechniquetoobtaintheoptimaljointloadbalancing,servercongurationandbatterymanagementschemeforthetotalelectricitycostminimization.Moreover, 26

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ouralgorithmoersanexplicittradeobetweenthecostsavingandthebatterycapacity. Weevaluateouralgorithmbasedonreal-worlddatasetsandtheresultsshowthatourapproachcanachievesignicantelectricitycostsaving.Therestofthischapterisorganizedasfollows.Section 2.1 presentsthemodelsofelectricitycostinmultipledatacenters,whichisformulatedasastochasticprogramtominimizethetime-averageexpectedelectricitycost.Section 2.2 solvestheoptimizationproblembyrstconsideringarelaxedproblemandthen,usingthetheLyapunovoptimizationtechniquetodesignacontrolalgorithmtoapproximatelysolvetheoriginalproblem.Section 2.3 givesthealgorithmicperformanceanalysisandSection 2.4 givesthenumericalevaluationresultsbasedonreal-worlddatasets.ThischapterendswithasummaryinSection 2.5 . 2.1ModelingandFormulationWenowdescribethemodelsweuseinthispapertominimizethetime-averageexpectedelectricitycostindatacenters.Assumethesystemisdiscrete-timewithtimeperiodmatchingthetimescaleatwhichtracdistribution,serverconguration,andcharging/dischargingdecisionscanbeupdated(e.g.,10min).WeconsideracloudserviceproviderhavingNgeographicallydistributeddatacenters,denotedbyD=fD1;:::;DNgandKfront-endproxyservers,denotedbyS=fS1;:::;SKg.EachdatacenterDihasatotalnumberofMihomogeneousservers.Thesystemoperatesinslottedperiod,i.e.,t=f0;1;:::g.TheblockdiagramofoursystemmodelisshowninFigure 2-1 ,whichisdescribedindetailasfollows. 2.1.1TheWorkloadModelIneveryperiodt,customerrequestsarriveateachfront-endproxyserver.WedenotetheaveragearrivalrateofworkloadatSjbyAj(t),j2f1;:::;Kg,whereA(t)=(A1(t);:::;AK(t))denotesthetracarrivalvector.Theworkloadarrivalratedistributedfromthefront-endproxyserverSjtothedatacenterDiisdenotedasji(t).ThiscanbedonebydynamicallygeneratedDNSresponses,HTTPredirections,orusing 27

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Figure2-1. Blockdiagramforthesystemmodel. persistentHTTPproxiestotunnelrequests.Weassumethatthereexistsaproxy/DNSserverco-locatedwitheachrequestsource.Therefore,wehave NXi=1ji(t)=Aj(t);8j=1;:::;K(2{1) ji(t)0:(2{2)DenethetotalarrivalratedistributedtodatacenterDiasi(t)andthedistributedworkloadvectoras(t)=(1(t);:::;N(t)).Then,wehave i(t)=KXj=1ji(t);8i=1;:::;N:(2{3) 2.1.2TheBatteryModelWeassumethateachdatacenterpossessessomekindofbattery.ForeachdatacenterDi,wedenotebyEi;maxthebatterycapacity,byEi(t)theenergylevelofthebatteryatperiodt,andbyPi(t)thepower(energyperperiod)chargedto(whenPi(t)>0)ordischargedfrom(whenPi(t)<0)thebatteryduringperiodt.Assumethatthebatteryenergyleakageisnegligibleandbatteriesatdatacentersoperateindependentlyofeach 28

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other.Thenwemodelthedynamicsofthebatteryenergylevelby Ei(t+1)=Ei(t)+Pi(t):(2{4)ForeachdatacenterDi,thebatteryusuallyhasanupperboundonthechargerate,denotedbyPi;max,andanupperboundonthedischargerate,denotedbyPi;min.Pi;maxandPi;minarepositiveconstantsdependingonthephysicalpropertyofthebattery.Therefore,wehavethefollowingconstraintonPi(t): )]TJ /F4 11.955 Tf 11.96 0 Td[(Pi;minPi(t)Pi;max:(2{5)Thebatteryenergylevelshouldbealwaysnonnegativeandcannotexceedthebatterycapacity.Therefore,ineachtimeperiodt,weneedtoensurethatforeachdatacenterDi, 0Ei(t)Ei;max:(2{6)Fromconstraints( 2{4 ),( 2{5 ),and( 2{6 ),wegetthefollowingequivalentconstraintsineachperiodtfordatacenterDi: Pi(t))]TJ /F1 11.955 Tf 23.91 0 Td[(minfPi;min;Ei(t)g;(2{7) Pi(t)minfPi;max;Ei;max)]TJ /F4 11.955 Tf 11.95 0 Td[(Ei(t)g:(2{8)However,thecostofusingbatterycannotbeignored.Inpractice,therearelimitedtimesofcharging/discharingcyclesforeachbattery.Besides,conversionlossoccursbothincharginganddischargingprocesses.Storedenergyisalsosubjecttoleakagewithtime.Allthesefactorsdependonhowfast/much/oftenitischargedanddischarged.Insteadofmodelingthesefactorsexactly,weuseaamortizedcostCb(inunitofdollars)tomodeltheimpactofperchargingordischargingoperationonthebatteryduringoneperiod.Therefore,duringonetimeperiod,anoperatingcostofCbisincurredwheneverthebatteryischarging(Pi(t)>0)ordischarging(Pi(t)<0). 29

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2.1.3TheQoSModelInpractice,accordingtotheservicelevelagreement(SLA)betweentheserviceproviderandcustomers,customerrequestsshouldhavesomekindofQoSrequirements.Inthispaper,weusetheaverageresponsetimeastheQoSmetric.Asin[ 79 ],weuseanM=M=nqueuingmodeltoanalyzetheaverageresponsetimeindatacenterDiwhenthetracarrivalrateisi(t)andtherearemi(t)activeservers,eachwithserviceratei(t).Notethatmi(t)isanintegralvariableandhasthemaximumvalueMiateachdatacenterDi.Also,thereexiststhemaximumserviceratei;maxforeachserverindatacenterDi.WhenthereistracdistributedintoDi,usingtheresultsfromqueuingtheory[ 21 ],theaverageresponsetimeWi(t)isPQ=(mi(t)i(t))]TJ /F4 11.955 Tf 12.59 0 Td[(i(t))wherePQisthequeuingprobability.Withoutlossofgeneralityinadatacenter,weassumetheserversarealwaysbusyifturnedon.Hence,PQ=1andWi(t)=1=(mi(t)i(t))]TJ /F4 11.955 Tf 12.14 0 Td[(i(t)).TomeettheQoSrequirementofcustomers,themaximumaverageresponsetimeWi;maxisimposedoneachdatacenterDi.Therefore,wehavethefollowingQoSconstraints: mi(t)i(t))]TJ /F4 11.955 Tf 11.96 0 Td[(i(t)1 Wi;max;8i=1;:::;N(2{9) 0mi(t)Mi;mi2N;8i=1;:::;N(2{10) 0i(t)i;max;8i=1;:::;N:(2{11) 2.1.4ThePowerConsumptionModelIneachtimeperiodt,thenormalpowerconsumptionHi(t),includingthecoolingenergyconsumptionateachdatacenterDi,byrunningmi(t)serversatratei(t)canbeapproximatedbythefollowingformula[ 30 ]: Hi(t)=mi(t)(iii(t)+i)PUEi;(2{12)wherei,i,i,andPUEiareconstantsdeterminedbythedatacenterDi.Specically,iistheaverageidlepowerconsumptionofaserver,andiii(t)+igivesthepowerconsumptionofoneserverrunningatratei(t)atDi.PUEiistheratioofthetotal 30

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buildingpower(includingcoolingpower)toITserverpower,whosevalueliesbetween1.3and2intoday'senergy-ecientdatacenters[ 48 ].Duetotheintroductionofenergystorage,thetotalamountofenergyGi(t)drawnfromthegridtosupplythedatacenterDiduringtimeperiodtisgivenby Gi(t)=Hi(t)+Pi(t);8i=1;:::;N:(2{13)Specically,whenPi(t)>0,someenergydrawnfromthegridisusedtochargethebatterybesidesservingthenormaldatacenteroperation.WhenPi(t)<0,someenergyisdischargedfromthebatterytosupplementtheenergydrawnfromthegridsoastomeettheenergydemandofthedatacenter. 2.1.5TheElectricityPriceModelTheelectricpowergridinU.S.isorganizedintodierentreliabilityregions,whereeachregionhasitsownregionaltransmissionorganization(RTO)orindependentsystemoperator(ISO)[ 6 ].TheRTOorISOisacentralauthoritythatdirectstheowofelectricitybetweengeneratorsandconsumersandensuresthereliabilityofthegrid.Italsooperateswholesaleelectricitymarkets,whichusuallyincludeday-aheadandreal-timeelectricitymarkets.Theelectricitypricesinthesemarketsaredeterminedbytheclearingprocessesofsupplyanddemandbidswhilesatisfyingthetransmissionconstraints.Asanalyzedin[ 77 ],theelectricitypricesinwholesaleelectricitymarketshavebothspatialandtemporalvariations.AteachdatacenterDi,weassumeatime-varyingelectricitypriceCi(t)withthemaximumvalueCi;maxandtheminimumvalueCi;min,respectively.DenoteC(t)=(C1(t);:::;CN(t))astheelectricitypricevectorandG(t)=(G1(t);:::;GN(t))asthegridenergyconsumptionvector.WefurtherassumethatC(t)andG(t)areindependent.Dierentdatacentersmayhavedierentelectricitypricesatthesametimeduetobeinglocatedindierentelectricitymarkets. 31

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2.1.6TheCostMinimizationProblemWithEnergyStorageAsdiscussedbefore,thetotalelectricitycostofNdatacentersduringtimeperiodtisgivenbythefollowing: NXi=1fGi(t)Ci(t)+1fPi(t)6=0gCbg:(2{14)Inthispaper,weareinterestedinchoosingthefollowingthreecontroldecisionstominimizethelong-termtime-averageexpectedelectricitycost:(i)theworkloaddistributedfromthefront-endwebportalstodierentdatacenters-(t);(ii)thenumberofactiveserversatdierentdatacenters-m(t)andthecorrespondingservicerates-(t);(iii)thecharge/dischargerateatdierentdatacenters-P(t).Basedonthemodelsabove,ourproblemcanbeformulatedasthefollowingstochasticprogram,calledP1:minimizelimsupT!11 TT)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0NXi=1EfGi(t)Ci(t)+1fPi(t)6=0gCbg;subjecttoconstraints( 2{1 ),( 2{2 ),( 2{6 ),( 2{7 ),( 2{8 ),( 2{9 ),( 2{10 ),( 2{11 ),and( 2{13 ),wheretheconstraintsareforeachtimeperiodtanddatacenterDi. 2.2ProposedSolutionOnechallengeofsolvingthestochasticoptimizationproblemaboveistheunawarenessoffutureworkloadarrivalsaswellastime-varyingandlocation-varyingelectricityprices.Moreover,theconstraintsonEi(t)bringthe\time-coupling"propertytothestochasticoptimizationproblemabove.Itmeansthatthecurrentcontrolactionmayimpactthefuturecontrolactions,makingitmorechallengingtosolve.Asmentionedbefore,thestatisticsofA(t)andC(t)maynotbeknownandweneedtodesignanoptimalcontrolalgorithmunderuncertainty.WeusetherecentlydevelopedtechniqueofLyapunovoptimization[ 67 ].ThealgorithmweproposecanachievetherangeofO(1=V)withintheoptimalobjectivevalue,whereVisaparameterrelatedtothebatterycapacityofeachdatacenterDi.Onesalientfeatureofouralgorithmisthatitdoesnotneedanyfutureknowledgeofthesystemandcanbeeasilyimplementedonline. 32

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2.2.1RelaxedProblemBeforegivingthesolutiontoouroriginalproblemP1,weconsiderarelaxedproblem.Denethetime-averageexpectedvalueofchargeordischargerateatdatacenterDiunderanyfeasiblecontrolpolicyofP1asfollows: Pi=limsupT!11 TT)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0EfPi(t)g:(2{15)Sincethebatteryenergylevelisevolvingaccordingto( 2{4 ),summingoverallt2f0;1;2;:::;T)]TJ /F1 11.955 Tf 11.95 0 Td[(1g,andtakingexpectationofbothsides,wehaveEfEi(T)g)]TJ /F4 11.955 Tf 20.58 0 Td[(Ei;ini=T)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0EfPi(t)g;whereEi;ini=Ei(0)istheinitialbatteryenergylevelatdatacenterDi.As0Ei(t)Ei;maxforalltimeperiodst,dividingbothsidesbyT,andtakingT!1yields Pi=0.Hencewehavethefollowingrelaxedproblem,calledP2:minimizelimsupT!11 TT)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0NXi=1EfGi(t)Ci(t)+1fPi(t)6=0gCbg;subjecttoconstraints( 2{1 ),( 2{2 ),( 2{5 ),( 2{9 ),( 2{10 ),( 2{11 ),( 2{13 ),and Pi=0;wheretheconstraintsareforeachtimeperiodtanddatacenterDi.DenotetheoptimalobjectivevalueofP1asQOPTandtheoptimalobjectivevalueofP2asQREL.Asdiscussedbefore,anyfeasiblesolutiontoP1isalsoafeasiblesolutiontoP2.Hence,QRELQOPT.NotethatP2isadecoupledcontrolproblemsincenocorrelationexistsinanyconstraint.FromtheframeworkofLyapunovoptimization[ 67 ],wehavethefollowingtheoremforthesolutiontoP2: Theorem2.1. IfC(t)andA(t)arei:i:d:overslots,thenthereexistsastationary,randomizedpolicythattakescontroldecisionsstat(t),mstat(t),stat(t),andPstat(t)everyperiodpurelyasafunction(possiblyrandomized)ofthecurrentworkloadvectorA(t)and 33

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theelectricitypricevectorC(t)whilesatisfyingtheconstraintsofP2andprovidingthefollowingguarantees:EfPstati(t)g=0;EfNXi=1[Gstati(t)Ci(t)+1fPstati(t)6=0gCb]g=QREL;wheretheexpectationsabovearewithrespecttothestationarydistributionsofA(t),C(t),andtherandomizedcontroldecisions. Proof. ItcanbeprovenusingCaratheodory'stheoremin[ 67 ]andissimilartothatin[ 86 ].Itisomittedhereforbrevity. Inordertoderivesuchapolicy,weneedtoknowthestatisticaldistributionsofallcombinationsofC(t)andA(t),whichusuallyhastheproblemof\curseofdimensionality"[ 20 ]ifsolvedbydynamicprogramming.Moreover,thiscontrolpolicymaynotbeafeasiblesolutiontoP1.Instead,weusetheexistenceofsuchapolicytohelpusdesignourcontrolpolicythatmeetsallconstraintsofP1andderivethealgorithmicperformanceofouralgorithmasillustratedintheproofofouralgorithmicpropertieslater. 2.2.2OurProposedAlgorithmTheideaofouralgorithmistoconstructaLyapunovschedulingalgorithmwithperturbedweightsfordeterminingthetracdistribution,datacentersizing,servicerate,andcharging/dischargingdecisions.Bycarefullyperturbingtheweights,wecanensurethatwheneverwechargeordischargethebattery,theenergylevelinthebatteryalwaysliesinthefeasibleregion.First,wedeneamodiedLyapunovfunctionasfollows: L(t),1 2NXi=1[Ei(t))]TJ /F4 11.955 Tf 11.96 0 Td[(VCi;max)]TJ /F4 11.955 Tf 11.96 0 Td[(Pi;min]2:(2{16)Tosimplifythenotation,wedeneavariableSi(t)foreachdatacenterDi,i2f1;:::;Ngasfollows: Si(t)=Ei(t))]TJ /F4 11.955 Tf 11.95 0 Td[(VCi;max)]TJ /F4 11.955 Tf 11.95 0 Td[(Pi;min:(2{17) 34

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LetS(t)=(S1(t);:::;SN(t)).ItisobviousthatSi(t)isjustashiftedversionofEi(t)andhasthesamedynamicsasEi(t)withthefollowingequation: Si(t+1)=Si(t)+Pi(t):(2{18)Then,theLyapunovfunctioncanberewrittenasfollows:L(t),1 2NXi=1S2i(t):Nowdenetheone-periodconditionalLyapunovdriftasfollows: 4(t)=EfL(t+1))]TJ /F4 11.955 Tf 11.96 0 Td[(L(t)jS(t)g:(2{19)Heretheexpectationistakenovertherandomnessofelectricitypricesandworkloadarrivals,aswellastherandomnessinchoosingthecontrolactions.Then,followingtheLyapunovoptimizationframework,weaddafunctionoftheexpectedelectricitycostoveroneperiod(i.e.,thepenaltyfunction)to( 2{19 )toobtainthefollowingdrift-plus-penaltyterm: 4V(t),4(t)+VE(NXi=1Gi(t)Ci(t)+1fPi(t)6=0gCbjS(t)):(2{20)Wehavethefollowinglemmaregardingthedrift-plus-penaltyterm: Lemma1. Underanyfeasibleactionthatcanbeimplementedatperiodt,wehave4V(t)B+NXi=1EfSi(t)Pi(t)jS(t)g+VNXi=1EGi(t)Ci(t)+1fPi(t)6=0gCbjS(t); (2{21)whereB,NPi=1maxfP2i;max;P2i;ming 2. Proof. From( 2{18 ),squaringbothsides,wehaveforeachdatacenterDi, S2i(t+1))]TJ /F4 11.955 Tf 11.95 0 Td[(S2i(t) 2=P2i(t) 2+Si(t)Pi(t):(2{22) 35

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Moreover,wehavethefollowinginequality:P2i(t) 2maxP2i;max;P2i;min 2,Bi:Takingexpectationsofbothsidesof( 2{22 )givenS(t)andsummingoveralldatacentersDi,wehave4(t)NXi=1Bi+NXi=1EfPi(t)Si(t)jS(t)g:AddingpenaltytermVPNi=1EGi(t)Ci(t)+1fPi(t)6=0gCbjS(t)tobothsidesoftheinequalityabove,wearriveattheconclusion. Wenowpresentouralgorithm.ThemaindesignprincipleofouralgorithmistochoosecontrolactionsthatminimizetheR.H.S.of( 2{21 )subjecttotheconstraintsinP2.Ouralgorithmworksasfollows: Algorithm1:CostMinimizationwithEnergyStorage foreachTimeperiodtdo 1ObservethesystemstatesA(t),C(t),andS(t). 2Choosecontroldecisions(t),m(t),(t),andP(t)astheoptimalsolutiontothefollowingoptimizationproblem,calledP3:minimizeNXi=1nSi(t)Pi(t)+VGi(t)Ci(t)+1fPi(t)6=0gCbo;subjecttoconstraints( 2{1 ),( 2{2 ),( 2{5 ),( 2{9 ),( 2{10 ),( 2{11 ),and( 2{13 )wheretheconstraintsareforeachtimeperiodtanddatacenterDi. 3UpdateEi(t)accordingtothedynamics( 2{4 ). NotethatthealgorithmaboveonlyrequirestheknowledgeoftheinstantvaluesofelectricitypricesC(t),tracarrivalratesA(t),andbatteryenergylevelsE(t).Itdoesnotrequireanyknowledgeofthestatisticsofthesestochasticprocesses.TheremainingchallengeistosolveP3,whichisdiscussedbelow. 36

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2.2.3SolutiontoP3Aswecanobserved,foreachtimeperiodt,theoptimizationproblemaboveisamixed-integernonlinearprogramming(MINLP),whichisNP-hardingeneral.Asin[ 78 ],byusingtheKKTconditions,wecanobtainthattheoptimalsolution,ifexists,mustsatisfythefollowing: i(t)=mi(t)i(t))]TJ /F1 11.955 Tf 26.99 8.09 Td[(1 Wi;max(2{23)Itcanbeobservedthatoncetheoptimalmi(t)andi(t)areobtained,wecansolvethecorrespondingij(t)bythefollowingtwoequations: NXi=1ji(t)=Aj(t);ji(t)0;(2{24) KXj=1ji(t)=mi(t)i(t))]TJ /F1 11.955 Tf 26.99 8.09 Td[(1 Wi;max:(2{25)Inthefollowingpart,wefocusonhowtoobtaintheoptimalmi(t)andi(t).WerstdeneIi(t)astheindicatorvariabletodescribethebatteryusageassociatedwitheachdatacenterDiduringtimeperiodt.Whenthebatteryisused(eitherchargingordischarging)atdatacenterDi,Ii(t)=1.Otherwise,Ii(t)=0.Then,theoptimizationproblemcanrewrittenasfollows,namedP4:minimizeNXi=1f(Si(t)+VCi(t))Pi(t)+VCbIi(t)+VCi(t)Hi(t)g;s.t.NXi=1mi(t)i(t)=KXj=1Aj(t)+NXi=11 Wi;max;0mi(t)Mi;mi(t)2N;0i(t)i;max;)]TJ /F4 11.955 Tf 11.96 0 Td[(Pi;minIi(t)Pi(t)Pi;maxIi(t);Ii(t)2f0;1g;0Pi(t)+Hi(t)Gi;max; 37

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wheretheconstraintsareforeachtimeperiodtanddatacenterDi.AgeneralMINLPisknowntobeNP-hardandnoecientsolutionsexistwhentheproblemsizeislargebecausethesearchspacewouldincreaseexponentially.However,insomepracticalsituations,theMINLPproblemoftenhavesomespecialstructuresthatcanbeexploitedfordesigningeectivesolutions.Oneparticularsituation,asinourproblem,isthatbyxingthediscretevariablesrst,theremainingproblembecomesconvexforcontinuousvariables.Inthispaper,weusethetechniqueofgeneralizedBendersdecomposition[ 32 ]tosolveit.Theproposedalgorithmisexpectedtoconvergetotheoptimalsolutionwithinanitenumberofiterations.Tosimplifythenotation,weignorethetimetinthefollowingalgorithm. Denition1. LetX,f;PgandY,fm;Ig.Wedenotethetotalelectricitycostfunctionf(X;Y),theworkloadconstraintfunctiong(X;Y),andthecharging/dischargingconstraintfunctionsh1i(X;Y),h2i(X;Y),k1i(X;Y),andk2i(X;Y)asfollows:f(X;Y)=NXi=1n(Si+VCi)Pi+VCbIi+VCiHio;g(X;Y)=KXj=1Aj+NXi=11 Wi;max)]TJ /F5 7.97 Tf 16.8 14.94 Td[(NXi=1mii;h1i(X;Y)=Pi)]TJ /F4 11.955 Tf 11.95 0 Td[(Pi;maxIi;8i2f1;2;:::;Ngh2i(X;Y)=)]TJ /F4 11.955 Tf 9.29 0 Td[(Pi;minIi)]TJ /F4 11.955 Tf 11.95 0 Td[(Pi;8i2f1;2;:::;Ngk1i(X;Y)=)]TJ /F4 11.955 Tf 9.3 0 Td[(Pi)]TJ /F4 11.955 Tf 11.95 0 Td[(Hi;8i2f1;2;:::;Ngk2i(X;Y)=Pi+Hi)]TJ /F4 11.955 Tf 11.96 0 Td[(Gi;max;8i2f1;2;:::;Ng: Denition2. Let=fmi2[0;Mi];Ii2f0;1gj9i2[0;i;max];Pisuchthatg(X;Y)0;h1i(X;Y)0;h2i(X;Y)0;k1i(X;Y)0;k2i(X;Y)0;8ig.Foranyxed 38

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^Y=f^m;^Ig2,wedenethesubproblemNLP(^Y)asfollows:minimizef(X;^Y)s.t.g(X;^Y)0;h1i(X;^Y)0;8i2f1;2;:::;Ngh2i(X;^Y)0;8i2f1;2;:::;Ngk1i(X;^Y)0;8i2f1;2;:::;Ngk2i(X;^Y)0;8i2f1;2;:::;Ng: Denition3. Let=f 2R4N+1: 0and4N+1Xi=1 i=1g:DeneJ(X;Y; ), 1g(X;Y)+NXi=1n i+1h1i(X;Y)+ i+N+1h2i(X;Y)+ i+2N+1k1i(X;Y)+ i+3N+1k2i(X;Y)o:Forany^Y2,wedenethefeasibility-checkproblemNLPF(^Y)asfollows:minimizes.t.J(X;^Y; );8 2: Denition4. Let2R4N+1and0.DeneL(X;Y;),f(X;Y)+1g(X;Y)+NXi=1i+1h1i(X;Y)+i+N+1h2i(X;Y)+i+2N+1k1i(X;Y)+i+3N+1k2i(X;Y): 39

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Themasterproblemisstatedasfollows:minimizez0s.t.z0minXL(X;Y;);800minXJ(X;Y; );8 2: Denition5. TherelaxedmasterproblemRMGBD(p,q)isstatedasfollows:minimizez0s.t.z0minXL(X;Y;i);8i2f1;2;:::;pg0minXJ(X;Y; j);8j2f1;2;:::;qg:Basedonthedenitionsabove,aniterativealgorithmbasedonthegeneralizedBendersdecompositiontechniqueisdescribedasfollows: 2.3AlgorithmicPerformanceAnalysisInthissection,weanalyzethefeasibilityandalgorithmicperformanceofouralgorithm.Tostartwith,wedeneanupperboundVmaxonparameterVasfollows: Vmax=miniEi;max)]TJ /F4 11.955 Tf 11.95 0 Td[(Pi;max)]TJ /F4 11.955 Tf 11.96 0 Td[(Pi;min Ci;max)]TJ /F4 11.955 Tf 11.96 0 Td[(Ci;min:(2{26)Next,theoptimalsolutiontoP3hasthefollowingpropertiesthatareusefulforthefollowinganalysisofalgorithmicperformance: Lemma2. TheoptimalsolutiontoP3hasthefollowingproperties: 1. IfSi(t)>)]TJ /F4 11.955 Tf 9.3 0 Td[(VCi;min,theoptimalsolutionalwayschoosePi(t)0. 2. IfSi(t)<)]TJ /F4 11.955 Tf 9.3 0 Td[(VCi;max,theoptimalsolutionalwayschoosePi(t)0. Proof. ForeachdatacenterDiandtimeperiodt, 1. WhenSi(t)>)]TJ /F4 11.955 Tf 9.3 0 Td[(VCi;min,supposePi(t)>0,thenwehaveGi(t)>mi(t)Hi(t).AccordingtotheobjectiveofP3,inthiscase,thevalueoftheobjectiveshouldalwaysbelargerthanthecasethatPi(t)=0andGi(t)=mi(t)Hi(t)wheremi(t) 40

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Algorithm2:GeneralizedBendersDecompositiontoP4 Input: AninitialfeasiblesolutionY1andaconvergencetolerantparameter>0 Output: Anoptimalsolution(X;Y)toP4LB0=,UB0=+1;p=q=0,k=1;foreachiterationkdo SolvetheNLP(Yk);ifNLP(Yk)isfeasiblethen ObtainanoptimalsolutionXkandanoptimalmultipliervector;p p+1;p=;UBk=minfUBk)]TJ /F6 7.97 Tf 6.59 0 Td[(1;f(Xk;Yk)g;ifUBk=f(Xk;Yk)then (X;Y)=(Xk;Yk); else SolvetheNLPF(Yk);ObtainanoptimalsolutionXkandanoptimalmultipliervector ;q q+1; q SolvetheRMGBD(p;q);Obtainanoptimalsolution(Yk+1;z0);LBk=z0;ifLBk+UBkthen Stopandreturn(X;Y);else k k+1; andHi(t)donotchange.Thisresultsinthecontradictionbecauseouralgorithmisalwaystryingtominimizetheobjectivefunction.Hence,whenSi(t)>)]TJ /F4 11.955 Tf 9.3 0 Td[(VCmin,Pi(t)cannotbestrictlygreaterthanzero,i.e.,thebatterywouldnotcharge. 2. WhenSi(t)<)]TJ /F4 11.955 Tf 9.29 0 Td[(VCi;max,supposePi(t)<0,thenwehaveGi(t)
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Si(t)<)]TJ /F4 11.955 Tf 9.3 0 Td[(VCi;max,Pi(t)cannotbestrictlylessthanzero,i.e.,thebatterywouldnotdischarge. Then,wehavethefollowingtheoremaboutthealgorithmicperformanceofourproposedalgorithm: Theorem2.2. SupposetheinitialbatteryenergylevelEi;ini2[0;Ei;max].ImplementingtheabovealgorithmwithanyxedparameterV2(0;Vmax]forallperiods,wehavethefollowingperformanceguaranteesforeachdatacenterDi: 1. ThebatteryenergylevelEi(t)isalwaysintherange[0;Ei;max]foralltimeslotst. 2. Allcontroldecisionsarefeasible. 3. IfA(t)andC(t)arei:i:d:overslots,thenthetime-averageexpectedelectricitycostunderouralgorithmiswithinboundB=Voftheoptimalvalue:limsupT!11 TT)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0NXi=1EfGi(t)Ci(t)+1fPi(t)6=0gCbgQOPT+B=V; (2{27)whereBisaconstantgivenby B,NXi=1maxfP2i;min;P2i;maxg 2:(2{28) Proof. Inthefollowing,weproveTheorem 2.2 . 1. Toshow0Ei(t)Ei;max,accordingtothedenitionofSi(t),itisequivalenttoshowthatforeachdatacenterDi, Si(t))]TJ /F4 11.955 Tf 21.92 0 Td[(VCi;max)]TJ /F4 11.955 Tf 11.96 0 Td[(Pi;min;(2{29)and Si(t)Ei;max)]TJ /F4 11.955 Tf 11.96 0 Td[(VCi;max)]TJ /F4 11.955 Tf 11.95 0 Td[(Pi;min:(2{30)Since0Ei;iniEi;max,theaboveinequalitiesholdfort=0.Weproveinthefollowingthatthisconstraintissatisedforallperiodsbyinduction. 42

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Supposeinequalities( 2{29 )( 2{30 )holdfortimeperiodt,weneedtoshowthatitalsoholdsfortimeperiodt+1. WerstproveSi(t+1)Ei;max)]TJ /F4 11.955 Tf 12.47 0 Td[(VCi;max)]TJ /F4 11.955 Tf 12.48 0 Td[(Pi;min:if)]TJ /F4 11.955 Tf 9.3 0 Td[(VCi;min
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DivingbothsidebyT,letT!1,andusingthefactsthatEfL(0)gareniteandEfL(t)garenonnegative,wearriveatthefollowingperformanceguarantee:limsupT!11 TT)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0NXi=1EfGi(t)Ci(t)+1fPi(t)6=0gCbgQOPT+B=V;whereQOPTistheoptimalobjectivevalue,Bisaconstant,andVisacontrolparameterwhichhasthemaximumvaluegivenby( 2{26 ). 2.4CaseStudiesInthissection,weevaluatetheperformanceofourproposedalgorithmbasedonreal-worldworkloadandelectricitypricedatasets.Toreducetheoverheadbetweenswitchingtheserverson/oacrossdierenttimeperiods,theschedulinghorizonisdividedintodiscretetimeperiodswith10-minateachperiod.Weconsiderarequest-responsetypeofcloudservice.Toaccommodateotherkindsoftypicalcloudservicessuchasbatchcomputingandsessionbasedapplication,onlyminormodicationsneedtobemadeforthetracdistributionconstrainsandapplicationQoSrequirementsinourmodel.Werstdescribethereal-worlddatasetsandsystemparametersusedinthispaper.Then,weillustratetheimprovedenergycostsavingofourschemeincomparisonwithsomebenchmarkschemes.Theperformancechangesduetodierentbatterycapacities,batterycost,andQoSrequirementsarealsoanalyzed. 2.4.1ExperimentalSetup Table2-1. Serverparametersinfourlocations LocationServeri;maxiiiMi(#ofrequest/sec)(Watt)(Watt) ChicagoAMDAthlon212.5150315000NewYorkAMDAthlon212.5150310000PaloAlto,CAINTELPentium6301.544.44100310000Houston,TXINTELPentium9502.59.6100310000 44

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Forthepurposeofillustration,weconsiderseveralsimulateddatacenterswherethemultiplefront-endproxyserversaremergedintoone.Thefront-endproxyserveractsasaloadbalancerwhichreceivestheincomingtracrequestsanddistributesworkloadtodierentdatacentersindierentlocations.Itisalsoresponsibleforsendingcontroldecisionstoback-endserverstoconguretheserversandmanagethebatterycharging/dischargingoperation.Fourdierentdatacenters,eachhavingabattery,atdierentgeographiclocationsareassumedinthisevaluation.ElectricityPriceData:Weusethe5-minlocationalmarginalprices(LMP)inreal-timeelectricitymarketsatfourdierentlocations:Chicago,NewYork,PaloAlto,CAandHouston,TXthathostGoogle'sdatacenters.Thedatasetisobtainedfromthepubliclyavailablegovernmentsources.Basedonthisrawdata,wecalculatetheaverageelectricitypriceoverdisjoint10minuteintervals.ThetimehorizonweconsiderinthispaperisfromJan.1,2011toJan.21,2011.Intotal,thisdurationincludesthreeweeksor3,02410-minperiods.Aportionoftheaverage10-minreal-timeelectricitypricesduringtherstweekofJanuary2011atthefourdierentlocationsisplottedinFigure 2-2 .Theelectricitypriceisinunitof$/MWh.WorkloadData:Thereal-worldworkloaddataweuseintheevaluationsisasetofI/Otracestakenfrom6RAIDvolumesatMSRCambridge[ 66 ].Theoriginaltracedperiodisonlyoneweekandwerepeatittogeta3-weekworkloadtraces.Figure 2-3 showstherequestnumbervariationsindierent10-minperiodsfor3days.Thepeak-to-averageratiooftheworkloadis4.5.SystemParameters:Weassumethattheserversatonelocationarehomogeneous.Notethatourmodelisquitegeneralandcanbeeasilyextendedintotheheterogenouscasewithonlyadditionalnotations.TheserverparametersateachlocationsarepresentedinTable 2.4.1 ,wheretheservicerate(inunitofrequestspersecond)isestimatedbytheaveragesizeoftheworkloadrequest(inunitofbytesperrequest)aswellasCPUandserverarchitecture.WechoosePUEi=1:3inallourevaluationstogetaconservative 45

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estimateofthecostsavings.Thedelayconstraintateachdatacenterischosentobe1ms,i.e.,Wi;max=0:001s. Figure2-2. 10-minaveragereal-timeelectricitypricesin3hoursatfourlocations[ 6 , 7 ]. Figure2-3. 10-minaverageworkloadin3days[ 66 ]. 2.4.2ExperimentalResultsInordertoanalyzetheperformanceimprovementduetoourscheme,wecompareitwiththefollowingschemesthateitherrepresentthecurrentpracticeindatacenterpowermanagementorarethestart-of-arttechniquesproposedbypreviouswork[ 77 { 79 ]:(i)Staticloadbalancing(SLB):Currentdatacentersusuallyrunsaconstantnumberofserverstoservetheworkload.Inordertosatisfythetime-varyingdemand,datacentersusuallyoverlyprovisionandkeepmorerunningserversthanwhatisneededtomeetthe 46

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peakload.Intheevaluation,weassumethatthedatacenterhascompleteworkloadinformationaheadoftimeandprovisionsexactlytosatisfythepeakload.Moreover,theamountofworkloadroutedtodierentdatacentersisproportionaltotheservicecapacityofdatacentersregardlessofelectricityprice.Weassumethatallserversareactivatedatalltimes.However,theserviceratesoftheserverscanbeadjustedineveryperiod;(ii)Price-awareloadbalancing(PLB):Theschemeissimilartotheheuristicproposedby[ 77 ]thatroutesmorejobstodatacenterswithlowerelectricityprice.Intheevaluation,weassumethattheworkloadisrstroutedtothedatacenterwiththelowestelectricityprice.Then,weroutetheremainingworkloadtothedatacenterwiththesecondlowestprice,andsoon.Again,allserversareassumedtobeactivatedatalltimesandserviceratescanbetunedineveryperiod;(iii)Price-awaredynamicprovisioning(PDP):Thisschemeisproposedby[ 78 , 79 ],whichconsiderbothtracroutinganddynamicserverprovisioningtoexploitthespatialvariationofelectricitypriceinreal-timeelectricitymarkets.However,energystoragefacilitiesindatacentersarenotconsideredinthesework.Therefore,thetemporalvariationofelectricitypricesisnotutilized.Intherstevaluation,wecompareouralgorithmwiththethreebenchmarkschemesaboveusingthereal-worldtraces.Notethattheperformanceofourschemedependsonthebatterycapacityandthebatterycost.WechooseCb=0:1$andEi;max=300kWh.ThemaximumchargeanddischargeratearesettobePi;max=Pi;min=10kWh.Theinitialbatteryenergylevelateachdatacenterischosentobezero,i.e.,Ei;ini=0.LetV=Vmax.TheresultisshowninFigure 2-4 (a).Fromthegure,wecanseethatbothourschemeandthePDPcangetmuchbetterperformancethantheSLBandthePLBbecauseofturningounnecessaryserversratherthanleavingthemidle.Also,ourschemeperformsbetterthanthePDPbecauseoftheintroductionofenergystorage,whichcanchargewhenthepriceislowwhiledischargingwhenthepriceishigh.Inthefollowing,weconsidertheimpactofindividualparametersontheperformanceofourschemecomparedtothePDP. 47

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Figure2-4. (a)Comparisonofthetotalenergycostinfourapproaches;(b)Theimpactofbatterycapacityonthecostsaving;(c)Theimpactofbatterycostonthecostsaving;(d)TheimpactofQoSrequirementsonthecostsaving. 48

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ImpactofBatteryCapacity:Inthisevaluation,wevarythebatterycapacitiesofdatacenterswithotherparametersxed.WesetEi;max=f300;400;500gkWhandV=Vmax.TheresultisillustratedinFigure 2-4 (b).Fromthegure,itisclearthatthelargerthebatteryis,themorecostsavingourproposedalgorithmcanobtain,whichcoincideswiththealgorithmicperformanceresultsofouralgorithminTheorem 2.2 .Aswehavementionedbefore,thesavingcomesfromthefactthatouralgorithmwouldchargethebatterywhentheelectricitypriceislowwhiledischargingitwhentheelectricitypriceislow.ImpactofBatteryCost:Currently,thebatteryisstillexpansive.Thechargingordischargingoperationwouldreducethelifetimeofthebattery.However,itisexpectedthatthecostofbatterywoulddecreasegreatlyinthenextdecade.Inthisevaluation,weestimatetheimpactofbatterycostonthecostsavingofouralgorithm.WesetCb=f0:1;1;10;100g$andkeepEi;max=300kWhxed.TheresultisshowninFigure 2-4 (c).Notethatwhenthebatterycostperoperationisverylarge(e.g.,100$),ouralgorithmwouldnotchargeordischargethebatteryatall,soitisthesameastheschemein[ 78 ].Asthebatterycostincreases,thetotalcostsavingcomparedwithPDPwoulddecreasesincetheopportunitytoutilizethetemporalvariationofelectricitypricesissmaller.ImpactofQoSRequirement:Inthissetting,weadjusttheQoSrequirementsofcustomerrequestswhilexingotherparameterstoseetheimpactofQoSrequirementontheperformanceofourscheme.WechooseWi;max=f0:001;0:005;0:01;0:05gs.AsobservedinFigure 2-4 (d),theincreaseofthemaximumaverageresponsetimegivesmoreopportunitytooptimizetheenergycost,sincefewernumberofserversneedstobeturnedontoservethesameamountofworkload. 2.5SummaryInthischapter,wehaveappliedtheLyapunovoptimizationtechniquetosolvetheproblemofoptimaltracdistribution,serverconguration,andbatterymanagement 49

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indatacentersforlocation-varyingandtime-varyingelectricitypricesunderwholesaleelectricitymarkets.Thealgorithmwehaveproposedmatchestheintuitionofdistributingmoretracintodatacenterswithlowerelectricitypriceandchargingwhenelectricitypriceislowwhiledischargingwhenelectricitypriceishigh.Moreover,itiseasytoimplementonlineandcangiveanalyticboundontheperformance.Withtheincreaseofbatterycapacity,theproposedalgorithmcangetarbitrarilyclosetotheoptimalvalue.Numericalevaluationsbasedonreal-worldtracesshowthatouralgorithmcanresultinsignicantenergycostreductionwithoutscarifyingthecustomerQoSrequirements. 50

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CHAPTER3WORKLOADMANAGEMENTFORSUSTAINABLEDATACENTERSInternet-servicecompaniesareincreasinglyinterestedinbecoming\sustainable",whichrequiresthemtoreducetheenvironmentalimpact(i.e.,carbonfootprint)besidesthenancialimpact(i.e.,electricitycost)oftheirdatacenters.Asshownin[ 31 ],twothirdsoftheworldwideelectricitydrawnfromtheutilitygridisgeneratedbyfossil-fuelgenerators,suchascoal,orgasplants,whichemitmuchmorecarbonthanrenewablegeneratorssuchaswindturbinesandsolarpanels.Withthedecreasingcostofbuildingrenewablegenerators,theyarebecomingincreasinglyattractiveoptionsforpoweringdatacenters,especiallywhentherenewableenergyissupportedbygovernmentincentives.However,unlikethetraditionalbrownenergydrawnfromtheutilitygrid,greenenergyfromrenewablesources,especiallywindandsolar,isintermittentanduncontrollable,whichpresentsagreatchallengefordatacenterstoeectivelyutilizethem.Thechallengeis,inessence,thedicultyininstantaneouslybalancingofenergysupplyanddemand.Large-scaleelectricenergystorage,mainlybatteries,canresolvethisdiculty,butitisstillprohibitivelyexpensive.Tohelpintegrategreenenergyintodatacenters,geographicalloadbalancing[ 60 , 61 ]hasbeenproposedtoutilizetheagilityofgeographicallydistributeddatacentersbydirectingmoreuserrequeststoplaceswhererenewableenergyisabundant.Althoughgeographicalloadbalancingisuseful,therearetwomoreopportunitiesthatcanbeexploitedtofurtherfacilitaterenewableenergyintegrationindatacenters.OneobservationisthatdatacentersusuallysupportawiderangeofITworkloads,includingbothdelay-sensitive,interactiveapplicationssuchaswebbrowsingandsearching,anddelay-tolerant,batchapplicationssuchasscienticcomputationandmassivelyparallelanddataintensivecomputationaljobs.Theinteractiveworkloaddiersfromthebatchworkloadinthefollowingtwoaspects.First,thecomputationalrequirementoftheinteractiveworkloadisusuallysmall,whilethebatchworkloadrequiresmuch 51

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largercomputationalcapability.Second,whiletheperformancemetricappropriatetotheinteractiveworkloadistheresponsetime,forthebatchworkload,itisthetotalthroughputwithinsometimeperiod.Thedelay-tolerantpropertyofbatchworkloadscanbeexploitedtoincreasetherenewableenergyutilizationbydelayingtheirservicestoperiodswhenrenewablesourcesareabundantwithoutexceedingtheirexecutiondeadlines.Anotherobservationisthatalargeportionofthepowerconsumptioninadatacentercomesfromthecoolinginfrastructure.Althoughlarge-scaleelectricenergystorage,suchasbatteries,isveryexpensive,thermalstorageismuchcheaper,andcanbeleveragedtoreducethecoolingenergycost.Infact,ApplehasalreadydeployedachilledwaterstoragesystemasthethermalstoragefacilityinitsgreendatacenterinMaiden,NC[ 5 ].Withthetime-varyingpropertiesofwholesaleelectricitypriceandrenewableenergygeneration,thermalstoragecanstoresomegreenenergyfromrenewablegeneratorsorcheapbrownenergyfromtheutilitygridrst.Later,whentheelectricitypriceishighorthegreenenergyisunavailable,thestoredenergycanbereleasedtohelpcoolthedatacenter,therefore,reducingtheelectricitybill.Withtheaboveobservationsascontext,weexploretheproblemofjointgeographicalloadbalancing,delay-tolerantworkloadscheduling,andthermalstoragemanagementforgreenenergyintegrationingeographicallydistributeddatacenters.Inadditionaltothebrownenergycost,wealsotakeintoaccountthebandwidthcostbetweencloudusersanddatacenters.Theobjectiveistominimizethetotaloperatingcostofservingdelay-tolerantworkloads.Totackletherandomnessinrenewablegenerations,workloadarrivals,andelectricityprices,weformulatetheproblemasastochasticprogramandproposeanecientonlinealgorithm,calledStochasticCostMinimizationAlgorithm(SCMA),withprovableperformanceguaranteebasedontheLyapunovoptimizationframework[ 67 ].Insummary,thischaptermakesthefollowingcontributions: 52

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Bytakingintoaccountthedelay-tolerantworkloadsandthermalstorage,weformulateastochasticoptimizationproblemtominimizethetotalenergyplusbandwidthcostofgeographicallydistributeddatacenterswithrenewablegeneration. WeproposeanonlinedistributedcontrolalgorithmSCMAtosolvetheproblemwithouttherequirementsofknowingthedetailedstatisticsofunderlyingrandomness. Ourproposedalgorithmenablesanexplicittrade-obetweenworkloaddelayandcostsaving,whichcanbeexiblyadjustedbyacontrolparameterV,makingitanattractivecontrolpolicyfordatacenteroperatorswithdierentapplications. Throughextensivenumericalevaluationsusingreal-worldtracesofrenewablegeneration,workloadarrival,andwholesaleelectricityprice,wedemonstratetheeectivenessofSCMA.Therestofthischapterisorganizedasfollows.InSection 3.1 ,modelsonworkloads,renewablegeneration,thermalstorage,andtotaloperatingcostarerstpresentedandthen,astochasticoptimizationproblemisformulated.WeproposeanalgorithmcalledSCMAtosolveitinSection 3.2 .TheanalyticalperformanceresultsofSCMAaredescribedinSection 3.3 .Wepresentthenumericalevaluationresultsbasedonreal-worldtracesinSection 3.4 .ThischapterendswithasummaryinSection 3.5 3.1ModelingandFormulationWeconsideracloudserviceprovider(CSP)havingmultiplegeographicallydistributeddatacenters,eachwithon-siterenewablegeneratorsandthermalstorage.ThetypicalcloudnetworkarchitectureofaCSPisdepictedinFigure 3-1 ,inwhichthereareseveralfront-endproxiesneartheclientsandmultipleback-endremotedatacentersinthecloud.AssumethatthereareMproxiesandeachofthemisresponsibleforageographicallyconcentratedsourceofrequestssuchasacity.TheproxydirectsuserrequeststoNdatacentersoftheCSPinthecloud.Duetothespatialdiversity,tracbetweendierentpairsofproxyanddatacentergoesthroughdierentISPsandtherefore,incursdierentbandwidthcosts.ThesustainabledatacenterweconsiderinthispaperisillustratedinFigure 3-2 ,whichisexplainedindetailasfollows.Weconsideradiscrete-timesystemwithtimedenotedbyt=0;1;2;:::. 53

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Figure3-1. TypicalcloudnetworkarchitectureofaCSP Figure3-2. Theblockdiagramofasustainabledatacenter 54

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3.1.1TheWorkloadModelTherearemanydierentworkloadsindatacenters.Ingeneral,theycanbedividedintothefollowingtwocategories:delay-sensitiveinteractiveworkloadanddelay-tolerantbatchworkload[ 59 ].Delay-sensitiveinteractiveworkloadssuchaswebservicesusuallyprocessreal-timeuserrequests,whichhavetobecompletedwithinacertaintime,i.e.,thereisamaximumresponsetime.Somebatchworkloadssuchasscienticapplications,simulations,orMapReducejobs[ 28 ]areoftendelay-tolerantinthesensethattheycanbescheduledtorunatanytimeaslongasthejobsarecompletedbeforethedeadline,i.e.,thereisamaximumcompletiontime.Sinceinteractiveworkloadshavehigherpriority,theyareusuallyprovisionedrst.Inthispaper,wefocusoncomputation-intensivebatchworkloadmanagement,assumingthatthemanagementofinteractiveworkloadshasbeendeterminedbypreviousschemes[ 61 , 79 ].ConsiderCtypesofjobsorservicerequestsinthedelay-tolerantworkloads.Eachtypemaycorrespondtoaspecicapplication.Assumethatalljobsarecomputation-intensive,andtheCPUresourceisthebottleneckresource.Thatis,ajobisexecutedwhenevertheCPUresourceisallocatedtoit.Ajobisrepresentedbyatuple:(c;dc;nc),wherecdenotestheapplicationtype,dcdenotescomputationdemand(i.e.,joblength)intermsoftheprocessorcycles,andncdenotesthecommunicationdemandintermsofthetransmitteddatasizebetweenthecloudandtheclient.WeassumethatjobsofdierenttypeshavedierentITresourcerequirements(e.g.,CPU,memory,storage,andnetwork)andjobsofthesametypehavethesameITresourcerequirements.Ajoborservicerequestrstarrivesatthefront-endproxyj.Theproxyisneartheclientsandactsasaworkloadrouter.Theproxywoulddecidewhichback-enddatacenterthejobrequestshouldberoutedtoforprocessing.Weassumenodatabueringattheproxysothatwheneverarequestarrivesattheproxy,itwouldberoutedtoadatacenterforprocessingimmediately.DenotethenumberoftypecjobsarrivingatproxyjintimetasWcj(t).ThejobarrivalratevectorattimetisdenotedasW(t)=(Wcj(t);8c;j)andthe 55

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time-averagerateofsuchanarrivalvectorisdenotedas!=EfW(t)g.WeassumethatthetotalarrivalrateoftypecjobsisboundedbyanitepositiveconstantWcmax.Thatis, JXj=1Wcj(t)Wcmax;8c;t:(3{1)Weusecij(t)todenotethenumberoftypecjobsthatisroutedfromproxyjtodatacenteriintimet,andusecj(t)=(cij(t);8i)todenotetheroutingvectorfortypecjobsatproxyj.Ineverytimeperiodt,cj(t)mustdrawfromsomefeasibleroutingsetcj(t),whichincludes,butisnotlimitedto,thefollowingconstraints: NXi=1cij(t)=Wcj(t);8c;j;t(3{2) cij(t)0;8c;i;j;t:(3{3)Additionalconstraintscanbeaddedintothefeasiblesetcj(t)tomodelotherpracticalconsiderations.Forexample,ifjobsofapplicationcfromproxyjcanonlyberoutedintoasetofdatacentersIcjduetosecurityconcern,thenwehavecij(t)=0;8i62Icj.Ifajobcontainsseveraltasks,whichneedtocommunicatewitheachotherduringtheprocessing,wemayneedtoplacethewholejobinsideonedatacentertoreducetheinter-DCtrac,whichismuchcostlierthantheintra-DCtrac.Then,cij(t)shouldbeanintegervalue.Otherpracticalconstraintscanbeformulatedintothesetcj(t)similarly.Denotethequeuelengthoftypecjobsattheback-enddatacenterDCiasQci(t).Then,wehavethefollowingqueuedynamics: Qci(t+1)=maxfQci(t))]TJ /F4 11.955 Tf 11.95 0 Td[(xci(t);0g+MXj=1cij(t);(3{4)wherexci(t)isthenumberoftypecjobsprocessedatdatacenteriintimet.Foreachdatacenteri,denotetheprocessingspeedoftheserverasiandthetotalnumberofserversforservingdelay-tolerantworkloadsasITi.Sincetheprocessedworkloadcannot 56

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exceedthemaximumavailablecomputingresources,wehave CXc=1xci(t)dcITii;8i;t:(3{5)Notethatintheformulationabove,weimplicitlyassumethatthejobscanbeperfectlyparallelizedandaretoleranttointerruptionduringrunningtime.ThejobsweconsiderinthisworkarethesameasthejobsthatcanbesupportedbytheAmazonEC2spotinstances,whicharetime-exibleandinterruption-tolerant.Weneedtocontrolthesystemsothatthequeuesinthesystemarestabilizedaccordingtothefollowingdenition: Q,limsupT!11 TT)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xt=0NXi=1CXc=1EfQci(t)g<1:(3{6) 3.1.2TheRenewableGenerationModelThereareseveralpossibleapproachesforInternet-servicecompaniestoutilizerenewableenergyintheirdatacenters[ 80 ],wherepowerpurchasingagreement(PPA)andon-siterenewablegenerationaretwocommonlyusedmethodsinindustrynow.Intherstapproach,thedatacenteroperatornegotiatesalong-termPPAwitharenewableenergyproducer,anddirectlypurchasesacertainamountofthegreenenergygeneratedbytheproduceratanegotiatedprice.Renewableenergycerticates(RECs)arekeptbythedatacenteroperatorastheproofofitsgreenenergyusage.Forexample,Googlehascontractedtopurchase114MWofwindpowerfor20yearsfromawindprojectinIowatopoweritsdatacenterthere.Thesecondapproachistobuildon-siterenewablegeneratorsneardatacenters,whichcanreducethetransmissionanddistributionlosses.Forexample,Appleisbuildingthenation'slargestenduser-ownedsolararray(40MW)andalso,thelargestnonutilityfuelcellinstallation(5MW)intheUSatitsnewdatacenterinMaiden,NC.Theseon-siterenewablegeneratorswillprovideover60percentofthecleanpoweritneeds.Inthispaper,wefocusonthesecondapproachbecauseithasamoredirectimpacton\greening"datacenters. 57

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Denotetheamountofon-siterenewableenergygeneratedatdatacenterDCiduringperiodtasri(t).Sincerenewableenergysources,mainlysolarorwind,arehighlyintermittent,time-varying,anduncontrollable,theymayvaryalotevenwithinoneperiod(e.g.,10mins)inourscenario.Inpractice,asobservedin[ 86 ],datacentersusuallyhaveexcessenergystoragecapabilityinUPSunits,whichcanprovidesucha\smoothing"function.Underthisassumption,therenewablegenerationcanberegardedasbeingconstantduringonetimeperiod. 3.1.3TheThermalStorageModelAsexplainedin[ 91 ],therearebasicallytwokindsofthermalstoragetechnologiesusedindatacenters.Oneistheinherentthermalmassesinadatacentersuchasthecoldairandtheraisedmetaloor.Theycanbeover-cooledtoalowertemperaturebytheCRACsystemrst,andabsorbheatlaterasacoolingunit.Theotheristhededicatedthermalstoragesystem.Thermalenergystoragesystemscommonlyusechilledliquidoricetoactasathermalbattery,enablingadatacenteroperatortorunairconditionersatnight(whenratesarelower)andduringtheday,pumpthechilledliquidaroundthefacilityforcooling.Whilethereisnoextracapitalcostfortherstapproach,itscapacityisusuallylimitedandtherefore,itisonlysuitableforshort-termstorage.Inthispaper,weconsiderthesecondapproach,whereeachdatacenterhasachilledliquid/icestoragesystembesidestheCRACcoolingsystem.Notethatourthermalstorage-basedapproachisorthogonalandsupplementarytootherapproaches,suchasDCpowerdistributionandseawatercooling,usedforreducingthecoolingcostofdatacenters.Foreachdatacenteri,denotebySmaxithecapacityofthethermalstorage,bySi(t)theenergylevelatperiodt,bys+i(t)theenergystored(i.e.,charged)intothethermalstoragesysteminperiodt,andbys)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t)theenergyreleased(i.e.,discharged)fromthethermalstoragesysteminperiodt.Inpractice,thereisconversionlossduringtheenergyconversionprocess.Withoutlossofgenerality,weassumetheconversionlossonlyhappensinthechargingprocessanddenotetheround-tripeciencyofthethermalstoragesystem 58

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atdatacenteriasi<1.Then,Si(t)woulddenotetheusableenergyinthethermalstorageandhasthefollowingdynamicsatdatacenteri: Si(t+1)=Si(t)+is+i(t))]TJ /F4 11.955 Tf 11.96 0 Td[(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t):(3{7)Also,eachthermalstorageusuallyhasanupperboundonthechargerate,denotedbys+i;max,andanupperboundonthedischargerate,denotedbys)]TJ /F5 7.97 Tf 0 -8.01 Td[(i;max.Thatis, 0s+i(t)s+i;max;(3{8) 0s)]TJ /F5 7.97 Tf 0 -8.02 Td[(i(t)s)]TJ /F5 7.97 Tf 0 -8.02 Td[(i;max:(3{9)Wealsodenes+max,maxis+i;maxasthemaximumchargerateofthermalstoragesystemsatalldatacenters.Withinonecontrolperiod,thethermalstoragecanbeeitherchargedordischarged,butnotboth[ 86 ].Thatis, s+i(t)>0)s)]TJ /F5 7.97 Tf 0 -8.02 Td[(i(t)=0;s)]TJ /F5 7.97 Tf 0 -8.02 Td[(i(t)>0)s+i(t)=0:(3{10)However,wewilltemporarilyignorethisconstraintanddecidetheoptimalcharge/dischargecontrolactions.Later,wewillconstructthecontroldecisionsthatcanmeetthatconstraintwithoutperformancedegradation.Foreachtimeperiod,weneedtoensurethatthethermalenergylevelindatacenterialwayssatisesthefollowing: 0Si(t)Smaxi:(3{11)Notethatsomethermalstoragesystemsmayhaveanonzerominimumenergylevelrequirementtoprotectthelifetimeoftheirsystem.Withoutlossofgenerality,weassumethattheminimumenergyleveliszerowhileSmaxidenotestheusablethermalstoragecapacity.TheinitialenergylevelindatacenteriisassumedtobeSi(0)2[0;Smaxi].Sincetheexcessiveusageofthermalstoragewouldimpactitslifetimeandreliability,aswith[ 74 ],thelossofthethermalstoragevalueismodeledasacostwhichisproportional 59

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totherechargedenergywithafactori.Thatis,theoperatingcostofusingthermalstorageatdatacenteriinperiodtisis+i(t). 3.1.4TheCostModelBesidesthecostofusingthermalstoragesystemsasdescribedbefore,therearetwootherpartsofthetotaloperatingcost:oneistheenergycostusedtoservetheworkloadindatacentersandtheotheristhebandwidthcostbetweentheclientsneartheproxiesandthedatacentersinthecloud.Toincentivizetheusageofgreenenergyfromrenewablegenerators,weassumethatthemarginalcostofrenewablegenerationiszerosothatthedatacentersshouldutilizeitasmuchaspossible.Thecostoftraditionalbrownenergydrawnfromtheutilitygriddependsonthewholesaleelectricitymarketandisbothspatiallyandtemporallyvarying.Denotebypi(t)thebrownenergypriceboughtfromthewholesaleelectricitymarketatdatacenterDCiinperiodt.Itisbothtime-varyingandlocation-dependent.Weassumethat0pi(t)pmaxiforallperiodstandpmaxi>i=i.1ThepowerconsumptionofaserverindatacentericanbeapproximatedtobelinearlyrelatedtotheaverageCPUutilizationasfollows[ 30 ]: (1)]TJ /F4 11.955 Tf 11.95 0 Td[()Pidlei+Pbusyi(3{12)wherePidleiisthepowerconsumptionwhentheserverisinidlestate,2[0;1]istheaverageCPUutilizationlevel,andPbusyiisthepowerconsumptionwhentheserverisbusy.Therefore,giventheserviceratexci(t)fortypecjobsatperiodtandthemaximumavailableactiveservernumbersITi,theITpowerconsumptionfordatacenteriis Ei(t)=ITiPidlei+Pcxci(t)dc i(Pbusyi)]TJ /F4 11.955 Tf 11.96 0 Td[(Pidlei):(3{13) 1Notethatthisassumptionrepresentsthatthereisopportunitytoutilizestorageforcostreducing. 60

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Denotebyfi(t)thecorrespondingcoolingenergyusageindatacenteriduringtimet.Sincewefocusonthethermalstorageinthispaper,weassumethatthedischargedpowerfromthethermalstoragecannotbegreaterthanthecoolingdemand,2i.e., s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))]TJ /F4 11.955 Tf 11.96 0 Td[(fi(t)0:(3{14)Inpractice,fi(t)maybeaconvexfunction,dependingonthespeciccoolinginfrastructuressuchasCRACandaircoolingsystems[ 59 ].Forsimplicityofanalysis,weassumethatfi(t)isalinearfunctionofthetotalITpowerconsumptioninthefollowingform: fi(t)=iEi(t);(3{15)whereiisafactortorepresentthepowerusageeciencyofdatacenter.Onaverage,iisaround1forthedatacenterindustry[ 4 ].Thatis,foreverywattofITpower,anadditionalwattisconsumedtocoolanddistributepowertotheITequipment.Althoughintensiveresearchhasbeendonetoreducethepowerusageeciencyofdatacenters,energystoragehasappearedasanattractivemechanismquiterecently[ 37 , 86 ].Notethatourframeworkisquitegeneralandcanincorporatemorepracticalcoolingmodelssuchas[ 59 ].Withtheabovemodels,theenergycostplusthethermalstorageoperatingcostofdatacenteriinperiodtisasfollows:Ei(t)=pi(t)(1+i)Ei(t)+s+i(t))]TJ /F4 11.955 Tf 11.96 0 Td[(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))]TJ /F4 11.955 Tf 11.96 0 Td[(ri(t)++is+i(t):Meanwhile,thereisbandwidthcostinvolvedforthecommunicationbetweenthejobsroutedintothedatacenterandtheclientneartheproxy.Inthispaper,weusethefollowinglinearbandwidthcostmodeltorepresentthebandwidthcostbetweentheclients 2Notethatforaelectricenergystorage,thedischargedpowercanalsobeusedtopowertheservers,therefore,eliminatingtheconstraint( 3{14 ).Ourframeworkandtheproposedtechniquesarestillapplicabletothecaseofelectricenergystoragesystemswithminormodication. 61

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andthecloud: Bij(t)=CXc=1bijcij(t)nc;(3{16)wherebijisthebandwidthcostcoecientbetweenproxyjanddatacenteri.Notethatncisthecommunicationdemandbetweenthecloudandtheclient.Dierentpairsofproxyanddatacenterhavedierentbandwidthcost.Morepracticalbandwidthchargingmodelbasedon95-thpercentilebandwidthusagemaybemodeledsimilarlyandwouldbeourfutureinvestigation.Wedenebmax,maxijbijasthemaximumbandwidthcostcoecientbetweenanypairofproxyanddatacenter.Thetotaloperatingcostofservingdelay-tolerantworkloadsforaCSPintimeperiodtisPNi=1Ei(t)+PNi=1PMj=1Bij(t). 3.1.5ProblemFormulationInthispaper,weareinterestedinminimizingthetime-averagetotaloperatingcostforservingthedelay-tolerantworkloadsoveralargetimehorizon.Therefore,thecontrolproblemcanbestatedasfollows:forthedynamicsystemdenedby( 3{4 )and( 3{7 ),designacontrolstrategywhich,giventhepastandthepresentrandomrenewablesupplies,workloadarrivals,andelectricityprices,choosestheworkloadroutingdecisions,thethermalstoragedecisionss+ands)]TJ /F1 11.955 Tf 7.09 -4.34 Td[(,andtheITresourceallocationdecisionsxsuchthatthetime-averagetotaloperatingcostforservingdelay-tolerantworkloadsisminimized.It 62

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canbeformulatedasthefollowingstochasticoptimization: minimize g=limsupT!11 TT)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xt=0E(NXi=1Ei(t)+NXi=1MXj=1Bij(t)) (3{17a)s.t.xci(t)0;CXc=1xci(t)dcITii;8c;i;t (3{17b)Si(t+1)=Si(t)+is+i(t))]TJ /F4 11.955 Tf 11.95 0 Td[(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t);8i;t (3{17c)0Si(t)Smaxi;8i;t (3{17d)0s+i(t)s+i;max;8i;t (3{17e)0s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t)s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i;max;8i;t (3{17f)s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))]TJ /F4 11.955 Tf 11.95 0 Td[(fi(t)0;8i;t (3{17g))]TJ /F4 11.955 Tf 5.48 -9.69 Td[(cij(t);8i2cj(t);8c;j;t (3{17h) Q<1: (3{17i)Here( 3{17b )meansthatthetotalallocatedITresourcescannotexceedtheITcapacity.( 3{17h )denotesthattheworkloadadmissionandroutingvectorsshouldbewithinthefeasibleset,whichdependsontherealapplication.( 3{17i )ensuresthattheaveragetotalqueuelengthforbueringdelay-tolerantjobsisnitesothatthedynamicsystemisstable.Onechallengeofsolvingtheproblemaboveistheconstraint( 3{17d ),whichbringsthe\time-coupling"propertytothecontroldecisions.Specically,thecurrentcontroldecisionss+i(t),s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t)willhaveanimpactonthefuturecontroldecisions.Inthelaterpart,wewilldesigna\virtualenergyqueue"toremovethis\time-coupling"propertywhilealsoensuringtheconstraint( 3{17d ). 3.2AlgorithmDesignInthissection,wedesignanonlinealgorithmbasedontheLyapunovoptimizationtechnique[ 67 ]tosolvethestochasticoptimizationproblemabove.Becauseofthetime-couplingconstraint( 3{17d ),Lyapunovoptimizationtechniquecannotbeapplieddirectly.Inthefollowing,werstconsiderarelaxedproblem,whichtsintothe 63

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frameworkofLyapunovoptimization.Then,wedesignouralgorithmbasedontheinsightsprovidedbythisrelaxedproblem. 3.2.1RelaxedProblemDenotethetime-averageexpectedchargeanddischargerateofthermalstoragei,respectively,asfollows: s+i=limsupT!11 TT)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0Efs+i(t)g;(3{18) s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i=limsupT!11 TT)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0Efs)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t)g:(3{19)Accordingtothedynamicsofthermalstorageenergylevel( 3{7 ),inordertoensure0Si(t)Smaxiforallt,wemusthavethefollowingequation: i s+i= s)]TJ /F5 7.97 Tf 0 -8.02 Td[(i:(3{20)Therefore,wehavethefollowingrelaxedproblem: min;x;s+;s)]TJ /F1 11.955 Tf 9.58 13.97 Td[(: g=limsupT!11 TT)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xt=0E(NXi=1Ei(t)+NXi=1MXj=1Bij(t));(3{21)subjecttoconstraints( 3{17b ),( 3{17e ),( 3{17f ),( 3{17g ),( 3{17h ),( 3{17i )and( 3{20 ).TheoptimalsolutiontotherelaxedproblemaboveiseasytocharacterizebasedontheframeworkofLyapunovoptimization,whichisdescribedinthefollowingtheorem.Theorem 3.1 (below)showsthatwecanachievetheminimumtimeaverageoperatingcostforagivenworkloadarrivalratevector!usingastationary,randomizedalgorithm.Thealgorithmonlychoosescontroldecisionsaccordingtoaxedprobabilitydistributionthatdependsonthesystemstate(ri(t);pi(t);Wcj(t);8i;j;c),butisindependentof(Qci(t);Ei(t);8i;c).InTheorem 3.1 ,denotesthecapacityregionofthesystem,whichistheclosureofsetsofrates!forwhichthereexistsajointgeographicalloadbalancing,workloadscheduling,andstoragemanagementalgorithmthatcanensurethequeuestability( 3{6 ). 64

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Theorem3.1. Ifthevector(ri(t);pi(t);Wcj(t);8i;j;c)isi.i.d.overperiods,then,foranyarrivalratevector!,EfW(t)g2,thereexistsastationary,randomizedcontrolpolicythatchoosescontroldecisions~cij(t),~xci(t),~s+i(t)and~s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t),basedsolelyonthevalueof(ri(t);pi(t);Wcj(t);8i;j;c)irrespectiveofqueueinformationwhilesatisfyingallconstraintsoftherelaxedproblemandprovidingthefollowingguarantees: E(MXj=1~cij(t))]TJ /F1 11.955 Tf 12.68 0 Td[(~xci(t))=0;8i;c;t(3{22) Ei~s+i(t)=E~s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i;8i;t(3{23) E(NXi=1~Ei(t)+NXi=1MXj=1~Bij(t))= grel(!);8t(3{24)wheretheexpectationsarew.r.t.therandomnessin(ri(t);pi(t);Wcj(t);8i;j;c)andpossibly,randomizedcontroldecisions,and grel(!)istheoptimalobjectivevalueoftherelaxedproblem( 3{21 )givenanarrivalratevector!. Proof. TheresultfollowsfromTheorem4.5of[ 67 ]andisprovedbyusingtheCaratheodory'stheorem.Itisomittedhereforbrevity. Denotetheoptimalobjectivevalueoftheoriginalproblem( 3{17 )as g(!)givenanarrivalratevector!.Obviously, grel(!) g(!).LetA1,CXc=1Wcmaxbmaxnc+NXi=1(NXi=1pmaxi(1+i)ITiPbusyi+(pmaxi+i)s+i;max): (3{25)Fromtheboundsweassumedbefore,wehave gA1foranyfeasiblecontrolpolicysubjecttoconstraints( 3{17b ),( 3{17e ),( 3{17f ),and( 3{17h ).Insteadofsolvingtherelaxedproblem,wewillusetheexistenceofsuchanoptimalpolicytohelpusdesignourcontrolpolicythatmeetsallconstraintsoftheoriginalproblem( 3{17 ),andderivetheperformanceofouralgorithm. 65

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3.2.2TheStochasticCostMinimizationAlgorithm(SCMA)TheideaofouralgorithmistoconstructaLyapunov-basedcontrolalgorithmfordeterminingtheoptimalworkloadrouting,scheduling,andthermalstoragemanagementscheme.First,wedeneaLyapunovfunctionasfollows: L(t),1 2NXi=1"CXc=1(Qci(t))2+(Si(t))]TJ /F4 11.955 Tf 11.96 0 Td[(i)2#;(3{26)whereiisaconstanttobespeciedlater.NowdeneK(t),(Qci(t);Si(t);8i;c),anddeneaone-periodconditionalLyapunovdriftasfollows: 4(t),EfL(t+1))]TJ /F4 11.955 Tf 11.96 0 Td[(L(t)jK(t)g:(3{27)Heretheexpectationistakenovertherandomnessofworkloadarrival,electricityprice,andrenewablegeneration,aswellastherandomnessinchoosingthecontrolactions.Then,followingtheLyapunovoptimizationframework,weaddafunctionoftheexpectedcostoveroneperiod(i.e.,thepenaltyfunction)to( 3{27 )toobtainthefollowingdrift-plus-penaltyterm: 4V(t),4(t)+VE(NXi=1Ei(t)+NXi=1MXj=1Bij(t)jK(t));(3{28)whereVisapositivecontrolparametertobespeciedlater.Then,wehavethefollowinglemmaregardingthedrift-plus-penaltyterm: Lemma3. Foranyfeasibleactionunderconstraints( 3{17b ),( 3{17e ),( 3{17f ),( 3{17g ),and( 3{17h )thatcanbeimplementedatperiodt,wehave4V(t)A2+NXi=1E(Si(t))]TJ /F4 11.955 Tf 11.96 0 Td[(i)(is+i(t))]TJ /F4 11.955 Tf 11.96 0 Td[(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))jK(t)+NXi=1CXc=1E(Qci(t)(MXj=1cij(t))]TJ /F4 11.955 Tf 11.95 0 Td[(xci(t))jK(t))+VNXi=1EfEi(t)g+VNXi=1MXj=1E(bijCXc=1cij(t)ncjK(t)); (3{29) 66

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whereA2istheconstantgivenbythefollowing:A2,NXi=1maxf(is+i;max)2;(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i;max)2g 2+NXi=1CXc=1[(Wcmax)2+(ITii=dc)2] 2: (3{30) Proof. From( 3{7 ),subtractingbothsidesbyi,andsquaringbothsides,weobtainforeachdatacenteri,(Si(t+1))]TJ /F4 11.955 Tf 11.95 0 Td[(i)2)]TJ /F1 11.955 Tf 11.95 0 Td[((Si(t))]TJ /F4 11.955 Tf 11.95 0 Td[(i)2 2=(is+i(t))]TJ /F4 11.955 Tf 11.95 0 Td[(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))2 2+(is+i(t))]TJ /F4 11.955 Tf 11.95 0 Td[(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))(Si(t))]TJ /F4 11.955 Tf 11.95 0 Td[(i):Moreover,wehavethefollowinginequality:(is+i(t))]TJ /F4 11.955 Tf 11.96 0 Td[(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))2 2maxf(is+i;max)2;(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i;max)2g 2,a1:Takingexpectationsofbothsidesof( 3{7 )givenK(t),andsummingoveralldatacentersi,wecangetthefollowingupperboundfortheLyapunovdriftforSi(t))]TJ /F4 11.955 Tf 11.95 0 Td[(i:(Si(t+1))]TJ /F4 11.955 Tf 11.96 0 Td[(i)2)]TJ /F1 11.955 Tf 11.96 0 Td[((Si(t))]TJ /F4 11.955 Tf 11.95 0 Td[(i)2 2a1+(is+i(t))]TJ /F4 11.955 Tf 11.96 0 Td[(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))(Si(t))]TJ /F4 11.955 Tf 11.96 0 Td[(i):Also,from( 3{4 ),squaringbothsides,andusingthefollowinginequality: maxfQci(t))]TJ /F4 11.955 Tf 11.96 0 Td[(xci(t);0g+MXj=1cij(t)!2 MXj=1cij(t)!2+(Qci(t))2+(xci(t))2+2Qci(t) MXj=1cij(t))]TJ /F4 11.955 Tf 11.96 0 Td[(xci(t)!;weobtain(Qci(t+1))2)]TJ /F1 11.955 Tf 11.95 0 Td[((Qci(t))2 2a2+Qci(t) MXj=1cij(t))]TJ /F4 11.955 Tf 11.95 0 Td[(xci(t)!;wherea2,[(Wcmax)2+(ITii=dc)2]=2.Combiningthesetwoboundstogether,summingoveralliandc,takingtheexpectationw.r.t.K(t)onbothsides,andaddingpenaltytermVEnPNi=1Ei(t)+PNi=1PMj=1Bij(t)jK(t)otobothsidesoftheaboveinequality,wearriveattherequiredconclusion. 67

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WenowpresenttheSCMAformulation.ThemaindesignprincipleofouralgorithmistochoosecontrolactionsthatgreedilyminimizetheR.H.S.of( 3{29 ).Ouralgorithmcanbenaturallydecomposedintotwoparts:workloadroutingandjointworkloadschedulingandstoragemanagement,asfollows:StochasticCostMinimizationAlgorithm(SCMA): InitializeVandi;8i.Ateachperiodt,observe(Wcj(t);ri(t);pi(t);8i;j;c)andK(t),anddo: WorkloadRouting: Foreachproxyj,choosetheroutingvector((cij);8i)fortypecjobsasthesolutiontothefollowingproblem:minimizeNXi=1(Qci(t)+Vbijnc)cij(t)s.t.)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(cij;8i2cj(t): (3{31) WorkloadSchedulingandStorageManagement: Foreachdatacenteri,choosetheworkloadschedulingvectorf(xci(t));8cgandthermalstoragedecisions(s+i(t))and(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))asthesolutiontothefollowinglinearoptimizationproblem:minimize)]TJ /F5 7.97 Tf 17.29 14.95 Td[(CXc=1Qci(t)xci(t)+(Si(t))]TJ /F4 11.955 Tf 11.96 0 Td[(i)(is+i(t))]TJ /F4 11.955 Tf 11.96 0 Td[(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))+Vyi (3{32)s.t.yipi(t)(1+i)"PCc=1xci(t)dc i(Pbusyi)]TJ /F4 11.955 Tf 11.95 0 Td[(Pidlei)+ITiPidlei#+(pi(t)+i)s+i(t))]TJ /F4 11.955 Tf 11.96 0 Td[(pi(t)(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t)+ri(t)); (3{33)yiis+i(t); (3{34)s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t)i"ITiPidlei+PCc=1xci(t)dc i(Pbusyi)]TJ /F4 11.955 Tf 11.96 0 Td[(Pidlei)#; (3{35)0s+i(t)s+i;max;0s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t)s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i;max; (3{36)xci(t)0;8c;CXc=1xci(t)dcITii; (3{37)whereyiisaslackvariableusedtotransformthenonlinearoperator[]+intolinearones. QueueUpdate: UpdateK(t)accordingtothedynamics( 3{4 )and( 3{7 ).Notethatwhensolvingtheproblem( 3{32 ),theresultingoptimalcharge/dischargesolutionmaynotsatisfytheconstraint( 3{10 ).Inthiscase,letH,i(s+i(t)))]TJ /F1 11.955 Tf 12.26 0 Td[((s)]TJ /F5 7.97 Tf 0 -8.02 Td[(i(t)) 68

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andwedenetheactualthermalstoragechargeanddischargeratesasfollows:(s+i(t))0=8>><>>:H=iifH0,0otherwise. (3{38)(s)]TJ /F5 7.97 Tf 0 -8.02 Td[(i(t))0=8>><>>:)]TJ /F4 11.955 Tf 9.3 0 Td[(HifH<0,0otherwise. (3{39)Wehavethefollowinglemmaregardingtheoptimalityoftheactualthermalstoragechargeanddischargerates: Lemma4. Thethermalstoragechargeanddischargerates(s+i(t))0and(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))0aboveisalsoanoptimalsolutiontotheproblem( 3{32 ). Proof. Obviously,theactualthermalstoragechargeanddischargesolutionaboveisfeasiblesincealltheconstraintsinproblem( 3{32 )aresatised.Then,bysubstitutetheconstructedsolutionintotheobjectivefunctionof( 3{32 ),wecanseethattheterm(Si(t))]TJ /F4 11.955 Tf 11.96 0 Td[(i)(is+i(t))]TJ /F4 11.955 Tf 11.96 0 Td[(s)]TJ /F5 7.97 Tf 0 -8.02 Td[(i(t))isthesame.Moreover,whenH0,(s+i(t)))]TJ /F1 11.955 Tf 11.95 0 Td[((s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))(s+i(t)))]TJ /F1 11.955 Tf 14.8 8.09 Td[(1 i(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))=H i=(s+i(t))0)]TJ /F1 11.955 Tf 11.95 0 Td[((s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))0:WhenH<0,(s+i(t)))]TJ /F1 11.955 Tf 11.95 0 Td[((s)]TJ /F5 7.97 Tf 0 -8.02 Td[(i(t))i(s+i(t)))]TJ /F1 11.955 Tf 11.96 0 Td[((s)]TJ /F5 7.97 Tf 0 -8.02 Td[(i(t))=H=(s+i(t))0)]TJ /F1 11.955 Tf 11.95 0 Td[((s)]TJ /F5 7.97 Tf 0 -8.02 Td[(i(t))0:Therefore,inbothcases,wealwayshave(s+i(t)))]TJ /F1 11.955 Tf 11.95 0 Td[((s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))>(s+i(t))0)]TJ /F1 11.955 Tf 11.96 0 Td[((s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))0: 69

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Consequently,pi(t)"CXc=1xci(t)(1+i)+(s+i(t))0)]TJ /F1 11.955 Tf 11.95 0 Td[((s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))0)]TJ /F4 11.955 Tf 11.95 0 Td[(ri(t)#pi(t)"CXc=1xci(t)(1+i)+(s+i(t)))]TJ /F1 11.955 Tf 11.96 0 Td[((s)]TJ /F5 7.97 Tf 0 -8.02 Td[(i(t)))]TJ /F4 11.955 Tf 11.96 0 Td[(ri(t)#:Moreover,onecanobservethati(s+i(t))0i(s+i(t))alwaysholds.Therefore,theobjectivefunctionvaluewouldnotincreasebyreplacing(s+i(t));(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))with(s+i(t))0;(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))0.Since(s+i(t)),(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))isanoptimalsolution,(s+i(t))0,(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))0mustalsobeanoptimalsolutiontotheproblem( 3{32 ). Undertheaboveactualcharge/dischargedecisions,wepresentthefollowingtwopropertiesofthestructureoftheoptimalsolutionto( 3{32 )thatisusefulintheperformanceanalysis. Lemma5. Theoptimalsolutionto( 3{32 )withtheadditionalconstraint( 3{10 )hasthefollowingproperties: 1. IfSi(t))]TJ /F4 11.955 Tf 11.96 0 Td[(i>)]TJ /F4 11.955 Tf 9.3 0 Td[(Vi=i,then(s+i(t))=0. 2. IfSi(t))]TJ /F4 11.955 Tf 11.96 0 Td[(i<)]TJ /F4 11.955 Tf 9.3 0 Td[(Vpi(t),then(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))=0. 1. Weproveitbycontradiction.Supposethereisanoptimalsolution(s+i(t))0>0whenSi(t))]TJ /F4 11.955 Tf 12.3 0 Td[(i>)]TJ /F4 11.955 Tf 9.3 0 Td[(Vi=i.Then,wecompareittothecasethats+i(t)=s)]TJ /F5 7.97 Tf 0 -8.02 Td[(i(t)=0.Bysubstitutingthemintotheobjectivefunctionin( 3{32 ),wehave[(Si(t))]TJ /F4 11.955 Tf 11.96 0 Td[(i)i+Vi](s+i(t))0>0andVpi(t)"Xc(1+c)xci(t)+(s+i(t))0)]TJ /F4 11.955 Tf 11.96 0 Td[(ri(t)#+Vpi(t)"Xc(1+c)xci(t))]TJ /F4 11.955 Tf 11.95 0 Td[(ri(t)#+:Byaddingthetwoinequalityabovetogether,wecanseethattheobjectivevalueof(s+i(t))0islargerthanthatofs+i(t)=s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t)=0.Therefore,itcontradictswiththeoptimalityof(s+i(t))0. 70

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2. Similarly,weproveitbycontradiction.Supposethereisanoptimalsolution(s)]TJ /F5 7.97 Tf 0 -8.02 Td[(i(t))0>0whenSi(t))]TJ /F4 11.955 Tf 12.84 0 Td[(i<)]TJ /F4 11.955 Tf 9.3 0 Td[(Vpi(t).Then,wecompareittothecasethats+i(t)=s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t)=0.Bycomparingtheirobjectivefunctionsin( 3{32 ),wehave(Si(t))]TJ /F4 11.955 Tf 11.96 0 Td[(i)()]TJ /F1 11.955 Tf 9.3 0 Td[((s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))0)>Vpi(t)s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t)0andVpi(t)("Xc(1+c)xci(t))]TJ /F1 11.955 Tf 11.95 0 Td[((s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))0)]TJ /F4 11.955 Tf 11.95 0 Td[(ri(t)#+)]TJ /F7 11.955 Tf 11.95 20.44 Td[("Xc(1+c)xci(t))]TJ /F4 11.955 Tf 11.95 0 Td[(ri(t)#+)Vpi(t)()]TJ /F1 11.955 Tf 9.3 0 Td[((s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))0)Byaddingthetwoinequalityabovetogether,wecanseethattheobjectivevalueof(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t))0islargerthanthatofs+i(t)=s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t)=0.Therefore,itcontradictswiththeoptimalityof(s)]TJ /F5 7.97 Tf 0 -8.02 Td[(i(t))0. 3.2.3InterpretationofSCMAThedetailedcontroldecisionstakenbySCMAareasfollows: Thecomplexityofsolvingtheworkloadroutingproblem( 3{31 )dependsonthefeasiblesetcj(t).Usually,ithasathreshold-basedsolution.Forexample,supposethatthefeasiblesetcj(t)onlycontainsconstraints( 3{2 ),( 3{3 ),cij(t)=0;8i62Icj,andcij(t)2Z.Theoptimalsolutionisthefollowingthreshold-basedpolicy:Let i=argmini2Icj(Qci(t)+Vbijnc):(3{40)Then, (cij(t))=(Wcj(t)ifi=i,0ifi6=i.(3{41)Itmeansthatallthejobswouldberoutedtothedatacenterwiththeshortestqueuelengthorthelowestbandwidthcost.TheweightsofthequeuelengthandthebandwidthcostareadjustedbytheparameterV. Fromtheproblemformulation( 3{32 ),wecanseethatSCMAwillalwaysusetherenewableenergyri(t)asmuchaspossibletoservequeuedworkloadsirrespectiveofqueuelengthsandelectricitypricessothatthersttermintheobjectiveisminimizedwhilethethirdtermintheobjectiveisunchanged.WhenVpi(t)(1+i)dcPbusyi)]TJ /F4 11.955 Tf 11.95 0 Td[(Pidlei=i
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enoughorthequeuelengthfortypecjobsishighenough,SCMAwillalsousesomebrownenergytoservejobsoftypecifneeded.Forthermalstoragemanagement,whenSi(t))]TJ /F4 11.955 Tf 13.2 0 Td[(i>0,thestoredenergyinthermalstoragewillbeusedtocoolthedatacenter,sincethereisenoughenergystoredinit.Also,ifthecurrentelectricitypriceislowenoughsuchthatpi(t)<(i)]TJ /F4 11.955 Tf 11.96 0 Td[(Si(t))i=V)]TJ /F4 11.955 Tf 12.34 0 Td[(i,thethermalstoragewillstoreenergyasmuchaspossibleforlaterusetoleveragetheopportunityofcurrentlowelectricityprice.ThethresholdsofchargingordischargingdependonthecurrentstoredenergylevelaswellastheparameterV.NotethatSCMAonlyrequirestheknowledgeoftheinstantaneousvaluesofsystemdynamicsandcanoperateonlinewithoutrequiringanyknowledgeofthestatisticsofthesestochasticprocesses.Moreover,eachproxyordatacentersolvesitsownoptimizationproblemdistributively,whereonlythequeuelengthinformationofdatacentersneedstobeexchangedbetweendatacentersandproxies.Therefore,SCMAiseasytoimplementinpractice. 3.3PerformanceAnalysisInthissection,wepresenttheanalyticalperformanceresultsforSCMA.Detailednumericalresultsaredescribedinthenextsection.First,wepresenttheresultswhen(ri(t);pi(t);Wcj(t);8c;i;j)isi.i.d.stochasticprocess.NotethataccordingtotheframeworkofLyapunovoptimization[ 67 ],ourresultscanalsobeextendedtothemoregeneralsettingwhere(ri(t);pi(t);Wcj(t);8c;i;j)evolvesaccordingtosomenitestateirreducibleandaperiodicMarkovchain.Furthermore,ournumericalsimulationresultsinthenextsectionarebasedonthereal-worldtraceswithoutanyspecicdistributionassumption. Theorem3.2. Supposethat0
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2. Ifthevector(ri(t);pi(t);Wcj(t);8c;i;j)isi.i.d.overperiods,andifthereexistsaconstantsuchthat!+12,thenthetotalbatchworkloadqueuelengthsatisesthefollowingunderanyarbitrary(ri(t);pi(t);Wcj(t);8c;i;j)process: QA1V+A2 :(3{43)Thetime-averageexpectedtotaloperatingcostundertheSCMAalgorithmiswithinboundA2=Voftheoptimalvalue: gSCMAg+A2=V;(3{44)wheregistheoptimalcostachievedbyanyfeasiblecontrolpolicythatcanstabilizethequeues,andA1,A2areconstantsgivenby( 3{25 )and( 3{30 ),respectively. Proof. Inthefollowing,weproveTheorem 3.2 . 1. Weprovetheresultbyinduction.First,weprovetheupperbound,i.e.,Si(t)Smaxi;8t.Whent=0,Si(0)Smaxi.Nowsupposethattheboundaboveholdsfortimet.Weneedtoshowthatitalsoholdsfortimet+1.IfSi(t))]TJ /F4 11.955 Tf 12.37 0 Td[(i>)]TJ /F4 11.955 Tf 9.3 0 Td[(Vi=i,accordingtoLemma 5 ,SCMAwillchooses+i(t)=0.Therefore,thethermalstoragewouldnotchargeandwehaveSi(t+1)Si(t)Smaxi.IfSi(t))]TJ /F4 11.955 Tf 12.09 0 Td[(i)]TJ /F4 11.955 Tf 22.28 0 Td[(Vi=i,wehaveSi(t+1)Si(t)+is+i;maxi)]TJ /F4 11.955 Tf 11.96 0 Td[(Vi=i+is+i;maxSmaxi;wherewehaveusedthefactthatVVmax.Second,weprovethelowerbound,i.e.,Si(t)0;8t.Similarly,whent=0,Si(0)0.Nowsupposethattheboundaboveholdsfortimet.Weneedtoshowthatitalsoholdsfortimet+1.IfSi(t))]TJ /F4 11.955 Tf 12.61 0 Td[(i<)]TJ /F4 11.955 Tf 9.3 0 Td[(Vpi(t),accordingtoLemma 5 ,SCMAwillchooses)]TJ /F5 7.97 Tf 0 -8.01 Td[(i(t)=0.Therefore,Si(t+1)Si(t)0.IfSi(t))]TJ /F4 11.955 Tf 10.72 0 Td[(i)]TJ /F4 11.955 Tf 21.91 0 Td[(Vpi(t), 73

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wehaveSi(t+1)Si(t))]TJ /F4 11.955 Tf 11.95 0 Td[(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i;maxi)]TJ /F4 11.955 Tf 11.96 0 Td[(Vpi(t))]TJ /F4 11.955 Tf 11.95 0 Td[(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i;max0;wherewehaveusedthefactsthatVVmaxandpi(t)pmaxi.Thiscompletestheproof. 2. Aswehavementionedbefore,SCMAisalwaystryingtogreedilyminimizetheR.H.S.oftheupperbound( 3{29 )ofthedrift-plus-penaltytermateverytimetoverallpossiblefeasiblecontrolpoliciesincludingtheoptimalandstationarypolicygiveninTheorem 3.1 .Since!+12,itcanbeenshownusingTheorem 3.1 thatthereexistsastationaryandrandomizedpolicythatachievesthefollowing:E(MXj=1~cij(t))=Ef~xci(t)g)]TJ /F4 11.955 Tf 20.59 0 Td[(;8i;c;tEi~s+i(t)=E~s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i;8i;tE(NXi=1~Ei(t)+NXi=1MXj=1~Bij(t))= grel(!+1);8t:Therefore,bypluggingthispolicyintotheR.H.S.oftheinequality( 3{29 ),weobtainthefollowing:4V(t)A2+V grel(!+1))]TJ /F4 11.955 Tf 11.96 0 Td[(NXi=1CXc=1Qci(t):Takingtheexpectationofbothsides,wegetEfL(t+1))]TJ /F4 11.955 Tf 11.96 0 Td[(L(t)g+VE(NXi=1~Ei(t)+NXi=1MXj=1~Bij(t))A2+V grel(!+1))]TJ /F4 11.955 Tf 11.96 0 Td[(EfNXi=1CXc=1Qci(t)g: (3{45) 74

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Usingthelawofiterativeexpectationandthefactthat grel(!+1)A1,summingovert2f0;1;2;:::;T)]TJ /F1 11.955 Tf 11.96 0 Td[(1g,wehave1 TT)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xt=0EfNXi=1CXc=1Qci(t)gA2+VA1 :LettingTgoestoinnity,wearriveatthefollowing: QA2+VA1 :Toprove( 3{44 ),from( 3{45 ),wehaveEfL(t+1))]TJ /F4 11.955 Tf 11.96 0 Td[(L(t)g+VE(NXi=1~Ei(t)+NXi=1MXj=1~Bij(t))A2+V grel(!+1): (3{46)Summingovert2f0;1;2;:::;T)]TJ /F1 11.955 Tf 12.2 0 Td[(1g,dividingbothsidesbyT,andusingthefactsthatEfL(0)garenite,EfL(t)garenonnegative,and grel(!+1) g(!+1),wehave1 TT)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xt=0E(NXi=1~Ei(t)+NXi=1MXj=1~Bij(t)) g(!+1)+A2 V:LettingTgotoinnity,usingtheLebesgue'sdominatedconvergencetheorem,andlettinggotozero,wearriveatthefollowingperformanceguarantee: gSCMA,limsupT!11 TT)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0E(NXi=1~Ei(t)+NXi=1MXj=1~Bij(t)) g+A2=Vwheregistheoptimalobjectivevalue,A2istheconstantgivenby( 3{30 ),andVisacontrolparameterwhichhasthemaximumvalueVmaxasdenedbefore. 3.4CaseStudiesIntheremainderofthepaper,weevaluatetheperformanceoftheSCMAunderrealistictraces.Ourgoalisthreefold:(i)toillustratethebenetsbyjointlyconsideringthethermalstorage,delay-tolerantworkloads,andgeographicalloadbalancingindatacenterstoreducingtheoperatingcost;(ii)tounderstandtheimpactsofvarious 75

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Figure3-3. 10-minaverageworkloadarrivalsforoneday[ 26 ]. Figure3-4. Hourlyelectricitypricesinday-aheadmarketsforonedayatfourlocations[ 6 , 7 ]. parametersonthecontroldecisionsmadebySCMA;and(iii)tounderstandthetrade-osamongcostreduction,workloaddelay,andthermalstoragecapacityenabledbytheSCMA. 76

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Figure3-5. 10-minaveragesolarandwindenergygenerationforoneweek[ 2 ]. 3.4.1ExperimentalSetupInthispart,weintroducethedefaultsettingsthatareusedthroughouttheevaluationsunlessotherwisestated.Thelengthofacontrolperiodis10minutesandthetime-horizonintheevaluationsis4000periods.DataCenterDescriptions.Weconsiderfourdatacenters,oneatthegeographiccenterofeachcitythatisknowntohaveGoogledatacenters:NewYork,PaloAlto,Chicago,andHouston.Moreover,weassumethatthereisaproxylocatedneareachdatacenter.Thebandwidthcostbijbetweenproxiesanddatacentersissettobeproportionaltothedistancesbetweencitiesandcomparabletotheenergycost.ThenumberofavailableactiveserversineachdatacenteristakentobeITi=350.TheenergyconsumptionofeachserverduringoneperiodatidleandbusystateissettobePidlei=100W1=6h;8iandPbusyi=250W1=6h;8i,respectively.Withoutlossofgenerality,theprocessingspeedofeachserverisassumedtobei=1;8i.Thecoolingeciencyofeachdatacenterissettobetheaveragevalueofthedatacenterindustryasi=1;8i.Noticethathere,weassumethehomogenoussettingsofdatacentersinordertomaketheanalysisoftheimpactsofotherfactors(e.g.,energyprices,renewableavailability,bandwidthcost)moreexplicitly. 77

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WorkloadDescription.Aswith[ 93 ],wechooseMapReduce[ 28 ],whichisapopulartypeofcomputation-intensiveworkloadsindatacenters,astherepresentativeofdelay-tolerantworkloads.WeusethehistoricalHadoop(anopensourceimplementationofMapReduce)tracesona600-machineclusteratFacebook[ 26 ]tocalculatetheaverage10-minworkloadarrivals.AportionoftheworkloadtraceduringonedayisshowninFigure 3-3 .Theworkloadarrivalstoeachproxyareshiftedaccordingtothetimezone.Weassumetherearetwotypesofjobs,withjoblengthdc=f1;0:5gandcommunicationdemandnc=f1;0:5g.Weassumethathalfofthearrivingrequestsbelongtotype1andtheotherhalfbelongtotype2.Theworkloadtracesarescaledsuchthatthepeakdemandcanbesupportedentirelybyitsowndatacenterwithoutdelay.EnergyPriceDescription.Weusetheday-aheadhourlylocationalmarginalprices(LMPs)inwholesaleelectricitymarketsattheabovefourdatacenterlocations.Theyareobtainedfromthepubliclyavailablegovernmentsources[ 6 , 7 ].Aportionofthehourlyelectricitypricesduringtherst24hoursattheselocationsisshowninFigure 3-4 .RenewableEnergyDescription.Weconsideron-sitewindgenerationattwolocations(NewYorkandChicago)andon-sitesolargenerationattheothertwolocations(PaloAltoandHouston).Thetracesofwindandsolarsourcesareobtainedfrom[ 2 ]thathaswindspeedandsolarirradiancemeasurementsevery10minutes.Thetracesarescaledproperlysothattheaveragerenewableproductioncanmeethalfoftheaveragepowerconsumptionateachdatacenter.AportionofsolarandwindenergygenerationattwolocationsduringthersttwodaysisdepictedinFigure 3-5 .ThermalStorageDescription.Weassumethateachdatacenterhasinstalledathermalstoragesystem.Themaximumcharge(discharge)rates+i;max(s)]TJ /F5 7.97 Tf 0 -8.01 Td[(i;max)issettobethepeakcoolingenergyconsumptionduringoneperiod.Theround-tripchargingeciencyiissettobe0:8.Thestorageoperatingcostfactor~iandthestoragecapacitySmaxiareparametersofwhichtheimpactontheperformanceofSCMAwillbeinvestigated. 78

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AlgorithmBenchmarks.ToprovidebenchmarksfortheperformanceofSCMA,wecompareitwiththefollowingthreebaselinesthateitherapproximatethecurrentpractice[ 77 ],orareproposedbysomerecentwork[ 82 , 93 ]. Baseline1(B1):Noworkloadscheduling,nostorage.Inthisapproach,theworkloadsareroutedtothenearestdatacenterandservedimmediatelywithoutanydelay.Thisschemeisemployedbymanycompaniesinpracticesoastoservealltheincomingworkloadsassoonaspossiblewithoutanyconsiderationonenergypriceorrenewableenergyavailability[ 77 ]. Baseline2(B2):Renewable-obliviousworkloadscheduling,nostorage.Thisapproachisverysimilartothatproposedintherecentwork[ 93 ],whichinvestigatesjointlyroutingandschedulingdelay-tolerantworkloadsinmultipledatacenterstoleveragetheopportunityoftime-varyingenergyprice.However,norenewableenergyorthermalstorageistakenintoaccountinthisscheme. Baseline3(B3):Renewable-awareworkloadscheduling,nostorage.Thisapproachisproposedintherecentwork[ 82 ]forcostminimizationinasingledatacenter.Renewableenergyavailabilityandtime-varyingenergypriceareconsideredbutwithoutthermalstorage.Wemodifyitsalgorithmtoincorporateroutingdecisions. 3.4.2ExperimentalResultsTheevaluationofSCMAwillbeorganizedasthefollowingaspects. 3.4.2.1CostSavingsNotethatpriorstudiesmainlyfocusonreducingenergycostwithoutconsideringthebandwidthcostforworkloadrouting.ToevaluatetheenergycostsavingduetoouralgorithmSCMAbyleveragingdelay-tolerantworkloads,thermalstorage,andgeographicalloadbalancing,werstassumethatthebandwidthcostbij=0;8i;jsothatwecanfocusontheenergycost.SincetheperformancesofSCMA,B2,andB3alldependontheparameterV,forfaircomparison,wechoosetheparameterVindierentschemessuchthattheaveragedelayofqueuedworkloadsintheseschemesareequal.NotethatB1hasnodelay.Moroever,theSCMAisunderthefollowingparametersettings:thestorageoperatingcostfactori=10;8iandthestoragecapacitySmaxiisassumedtobeabletosupporttheaveragecoolingdemandofadatacenterfor10hours.Theresultisshown 79

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Figure3-6. Theaverageenergycost(inunitofdollars)comparisonbetweenSCMAandbaselineschemes. inFigure 3-6 .Fromthegure,wecanobservethatSCMAoutperformsallbenchmarkschemes.Specically,bycomparingSCMAwithB3,wecanobservethatthermalstoragecanindeedhelpreducethetotalelectricitycost.Moreover,althoughB2considersthetime-varyingelectricityprice,itisrenewable-obliviousandtriestoserveworkloadsonlywhentheelectricitypriceislowenough.Therefore,itwastesalotofrenewableenergyandperformstheworst.Thisshowstheimportanceofrenewable-awareworkloadmanagementindatacenterswithon-siterenewablegeneration.Finally,bycomparingB3withB1,wecanseetheadvantageofdelay-tolerantworkloadsinimprovingrenewableenergyutilizationandreducingelectricitycost.Then,wecompareouralgorithmSCMAwiththebaselineschemesabovewhileconsideringthebandwidthcost.Obviously,thebandwidthcostofB1iszerobyourassumptionbecauseitalwaysroutesallworkloadstothenearestdatacenters.TheresultoftheaverageoperatingcostfortheotherthreealgorithmsisshowninFigure 3-7 .By 80

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Figure3-7. Theaverageoperatingcost(inunitofdollars)comparisonbetweenSCMAandbaselineschemes. takingintoaccountthedierentbandwidthcostsbetweenproxiesanddatacenters,ouralgorithmcanachievethelargesttotaloperatingcostsaving.Theimportanceofnetwork-awarenessisclearfromthegureabove.NotethatB1andB3havesimilaroperatingcostsincethebandwidthcostofB1isminimumalthoughitsenergycostishigherthanthatofB3.B2hastheworstperformanceofoperatingcostsinceboththeenergycostandthebandwidthcostarethehighestamongallschemes. 3.4.2.2Trade-obetweenCostandDelayInthispart,wefocusonthetrade-osamongdelay,totaloperatingcost,andthermalstoragecapacityinSCMA.WechoosedierentVandobservethecorrespondingtotaloperatingcostandaverageworkloaddelayinSCMA.TheresultisshowninFigure 3-8 .Aswecanobservefromthegure,withtheincreaseoftheparameterV,SCMAcangetlowertotaloperatingcostwithatrade-oinworkloaddelay,whichvalidatestheanalyticalperformanceresultsinTheorem 3.2 .NotethatbyselectingalargerV,SCMAwouldbe 81

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Figure3-8. Theaveragetotaloperatingcost(inunitofdollars)anddelayperformance(inunitofcontrolperiods)ofSCMAwithdierentV. moreaggressivelyminimizingtheoperatingcost,whichmaydelaymorejobstobeservedlaterwhenenoughrenewableenergyisavailableorenergypriceislow,causinglargerqueuingdelay. 3.5SummaryInthischapter,wehavestudiedtheproblemofjointnetwork-awareworkloadrouting,delay-tolerantworkloadscheduling,andthermalstoragemanagementtoimprovetherenewableenergyutilizationandreducethetime-averagetotaloperatingcostindatacenters.WehavedesignedanonlinealgorithmcalledSCMAanddemonstrateditseectivenessthroughbothanalyticalanalysisandnumericalevaluations.Moreover,SCMAprovidesanexplicittrade-obetweencostsavingandworkloaddelay. 82

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CHAPTER4SMARTHOMEElectrication,asmadepossiblebytheelectricpowergrid,hasbeenselectedasthemostsignicantengineeringachievementofthe20thCenturybytheU.S.NationalAcademyofEngineering[ 88 ].However,notuntilrecentlyhastheimperativeofrevitalizingU.S.electricinfrastructurebeenrealizedandalotofeortsarebeingputintothegridmodernization,whichtransformsourcentury-oldpowergridsto\SmartGrids".Thistransformationhasbeenmotivatedbythefollowingnotabledrivers:(1)Thephysicalinfrastructureoftheelectricelectricgridisagingandover-burdened.Theelectricitydemandcontinuestorisewhiletheinvestmentsinthepowertransmissionanddistributioninfrastructurehavebeendwindling.(2)Concernsoverglobalclimatechange(e.g.,globalwarming)andcarbonemissionshaveforcedustoaimatmoreaggressivegoalsfordeepintegrationoflargeamountsofrenewablegeneration,especiallywindandsolar,tomeetourelectricenergyneeds.Asstatedin[ 50 ],smartgridswouldbetheenablerformeetingenvironmentaltargets,accommodatingagreateremphasisondemandresponse,andsupportingwidespreadplug-inhybridelectricvehicles(PHEVs)aswellasdistributedgenerationandstoragecapabilities.Torealizesuchvisionforthesmartgrid,wehavetoseekhelpsfromadvancedcommunication,information,andcontroltechnologiesinconjunctionwithadvancesinrenewableenergygeneration,energystorage,materials,sensors,andpower-electronics.Althoughsmartgridsarenotarealityasyet,wecanarticulatehowwecandesignthemtoachievethevision.Weenvisionthatasmartgridwillconsistofthefollowingessentialcomponentsinadditiontothetraditionalpowergrid: 1. Renewableenergygeneration:Giventhesignicantconcernsregardingclimatechange,greenenergysuchassolarandwindenergyhasbecomecriticalforourfutureenergysustainability.MoststatesintheU.S.havedevelopedtheirownrenewableportfoliostandards(RTPs),whichrequireapre-determinedamountof 83

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astate'senergyportfoliotocomeexclusivelyfromrenewablesourcesbythenextdecade[ 87 ].However,theintermittencyandvariabilityofwindandsolarenergyimposeasignicantchallengetothegridoperationwhenthepenetrationlevelincreases[ 89 ]. 2. Energystorage:Energystoragecanalleviatetheneedtogeneratepoweratthetimewhenneededandcansmoothoutthevariationsofenergyutilityduetorandompowerdemandanduncertainenergysupply,whichisdesirableduetoeconomicconsiderationandincorporationofmoreintermittentrenewablesources.Examplesofutility-scaleenergystorageincludecompressedair,pumpedhydro,ultra-capacitors,ywheels,fuelcellsaswellasbatteries[ 83 , 85 ].Inresidentialareas,energystorageispracticallysynonymouswithbatteries.DuetothepopularityoftheemergingPHEVs,thebatteriesinthemmaybetreatedastemporaryelectricitystoragesforresidentialcustomers[ 62 ]. 3. Demandsidemanagement:Whiletraditionalpowersystemisdesignedforhighlycontrollablesupplytomatchalargelyuncontrolleddemand,anewenergybalancingparadigminthesmartgridisneededinordertofacilitategreaterpenetrationofvariablerenewableenergysources.Asnotedin[ 46 ],over10%dailyenergyconsumptionintheU.S.isfromtheusageofappliancessuchaswaterheater,air-conditioners,clothdryers,anddishwashers,whichareenvisagedtobecomesmartandhavethefollowingdistinctfeatures:theirenergydemandsareelasticanddelay-tolerant,i.e.,aslongastheirenergydemandsaremetwithincertaintimelimits,thecustomerswouldbesatised.Wecalltheseappliancestheenergyconsumerswhileotherappliances,whichshouldbepoweredwheneverneeded,arereferredtoasthepowerconsumers.Byutilizingtheexibilityintheconsumersideandprovidingappropriateincentives(suchasreal-timepricing),wecanreducethepeakdemandintheelectricpowergridandlowertheneedforexpensivepeakloadgenerators. 84

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Withinthesmartgrid,advanceddynamiccontrolwillberequiredforsimultaneousmanagementofreal-timepricing,exibleloads,electricvehiclerecharging,solar,wind,andotherdistributedgenerationsources,manyformsofenergystorageaswellasmicrogridmanagement[ 13 ].Inthesmartgrid,weexpectmoreandmorerenewablegeneratorswillbelocatedatthesideofresidentialcustomersduetoenvironmentalconcerns.Moreover,bygeneratingpowerclosetothepremiseswheretheenergyisneeded,wecangreatlyreducethetransmissionlinelossesassociatedwithpowerdeliveryfromremotepowerplantsandsignicantlyrelievethecongestionduetolimitedtransmissioncapacityofpowerlines.Forexample,residentialcustomersmaybeinterestedinrooftopphotovoltaic(PV)systembecauseofitsleastenvironmentalimpact,scalablecapacity,aswellasdecreasingcost.AbasicPVcellconvertssunlightofcertainwavelengthsintodirectcurrent(DC)usingthe\photoelectriceect".Unfortunately,abasicPVcelltypicallygeneratesonlyasmallamountofpower,whichmaynotbeenoughtopowerawholehousehold.However,duetotheirmodularityandportability,PVcellscanbeeasilyinterconnectedtoformaPVpaneltomeetanyelectricalrequirement,nomatterhowlargeorsmallitis.AlthoughthecurrentcostonPVsystemsisstillhigh(around$10perwattinstalled),itisexpectedtoreducebytwoorthreefoldsinthefuture[ 50 ].Therefore,inthispaper,weselectaPVsystemastherenewableenergysourceforourstudy.However,ourmodelisquiteexibleandcanbeeasilyadaptedtootherformsofrenewableenergysources.ApracticalsystemmodelconsistingoftheaforementionedessentialcomponentsinthesmartgridforresidentialcustomersisshowninFig. 4-1 .Sincesolarenergycannotbedispatchedandtheuctuationsinsolarirradiancemayoccurinaminute-to-minutetimescale,theenergygeneratingproleofaPVsystemdoesnotcoincidewithresidentialenergydemandproleformostofthetime.TheremaybeelectricityspillageatdaytimewhenthePVelectricitygenerationishighandelectricityshortageatnighttimewhenthePVelectricitygenerationislow.Tocopewiththismismatch,batterystoragemayhavetobeused.Bystoringsomeexcessgenerated 85

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Figure4-1. APV-utilitygridsystemwithbatterystorageforresidentialcustomersinthesmartgrid. electricityatdaytime,itcanbereleasedatnighttimetosupplementthepowerusageforahousehold.Intuitively,throughthismethod,thetotalamountofelectricitydrawnfromtheelectricutilitygridcanbereduced.Unfortunately,batterychargesanddischargeswillimpacttheoperationallifeofabattery.Inordertoprotectthebatteryfromoverchargeandoverdischarge,acontrollerisneededtoregulatethecharginganddischargingprocess[ 1 ].Becauseofthenitecapacityofbatterystorage,somePVgeneratedelectricitymaystillhastobespilled.AsthepowergeneratedbyPVpanelsisDC,aninverterisneededtoconvertDCintoalternatingcurrent(AC)beforeitcanbeusedbyhouseholdappliances.Moreover,asynchronizationdeviceisrequiredtoadjustthevoltagephaseandmagnitudeoftheoutputpowerfromtheinverter,sothattheoutputpowercanbecombinedsmoothlywiththepowerdrawnfromtheelectricutilitygridtosupplyelectricitytohouseholdappliancestogether.Thiscombinationisusuallycompletedatthemainfusionbox.Inthesmartgrid,customerswouldbeenrolledinreal-timeelectricitypricingenvironment,wheretheelectricitypriceistime-varying[ 64 ].Theelectricitydrawnfromthepowergridcanalsobestoredinthebatterythroughthebatterychargersothatitcanbereusedlater.Intuitively,thetotalelectricitycostcanbereducedbyrecharging 86

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thebatteryfromtheelectricpowergridwhentheelectricitypriceislowwhiledischargingitduringtheperiodsofhighelectricityprices.Obviously,itischallengingtomanagetheuseofboththetraditionalenergyandtheharvestedrenewableenergytoovercomethevariabilityintheenergysupplyfromtherenewableenergysourceandtheenergydemandfromresidentialcustomerswhileminimizingthecostoftraditionalenergyusage.Inthispaper,wetaketherststeptoinvestigatetheoptimalpowermanagementforresidentialcustomersinasmartgridbyutilizingthebatterystoragefacilities.Weconsidertwodierenttypesofenergydemand:inelasticenergydemandandelasticone.BasedontheLyapunovoptimizationtechniques[ 67 ],wedevelopsomeinterestingalgorithmstosolveourproblems,respectively.Wedemonstratethatouralgorithmscanachieveclose-to-optimalperformancewithtradeobetweenbatterycapacityandcostsaving.Moreover,inthecaseforelasticenergydemands,weshowthatouralgorithmcanguaranteeniteworst-casedelayforanybueredelasticenergydemand.Insummary,thischaptermakesthefollowingcontributions: Fromtheperspectiveofresidentialcustomers,weproposeacomprehensivemodeltoincorporatealmostallessentialcomponentsofthesmartgrid,includingdistributedrenewableenergygeneration,energystorage,demandsidemanagement,andsmartappliances. Wepresentalgorithmstoapproximatelyachievetheminimumtime-averageexpectedelectricitycostforresidentialcustomersforbothinelasticandelasticenergydemandswithouttheknowledgeofthestatisticsofrelatedstochasticmodels. Throughtheoreticalanalysis,weshowthatouralgorithmscanachievebettertrade-obetweencostsavingandenergystoragecapacity.Moreover,throughextensivesimulationsandbyusingpracticaldatasets,wevalidatetheeectivenessofourproposedalgorithms.Therestofthischapterisorganizedasfollows.InSection 4.1 ,wedescribethemodelsforenergystorage,renewableenergysource,andelectricitymarketweuseinthispaper.Besides,ourcontrolobjectiveispresented.WerstconsiderthecaseforinelasticenergydemandsinSection 4.2 andproposeanalgorithmtoapproximately 87

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solvetheoptimizationproblem.ThenweexaminethecaseforelasticenergydemandsinSection 4.3 .InSection 4.4 ,weemployrealdatasetstoevaluatetheproposedalgorithms.ThischapterendswithasummaryinSection 4.5 . 4.1ModelingandFormulationInthissection,wedescribethemathematicalmodelsforrenewableenergygeneration,energystorage,andelectricitymarketweuseinthispaper.Wealsopresentthecontrolobjective,whichistominimizethelong-termtime-averageexpectedelectricitycost.Inthenexttwosections,weconsidertwodierenttypesofenergydemand,i.e.,theinelasticenergydemandandtheelasticone,andfurtheranalyzetheelectricitycostminimizationproblem.1 4.1.1RenewableGenerationLetS(t)denotetheamountofrenewableenergygeneratedinslottandweassumethatthisenergyisrststoredinbatterybeforeitcanbeusedinthenexttimeslot.Acontrolleristoregulatetheportion(t)ofthegeneratedenergystoredintobatteryforeachslottinordertopreventbatteryoverow.Theotherportionisspilled.Hence,wehave 0(t)1:(4{1)Moreover,thereisamaximumvalueSmaxforS(t),thatis, 0S(t)Smax:(4{2) 4.1.2EnergyStorageInpractice,thebatteryisnotidealandhasthefollowingphysicalpropertiesasanalyzedin[ 52 , 86 ].First,thebatterylifetimedependsonboththenumberoftimesitundergoescharging/dischargingandthedepthofdischargeduringitsoperation.This 1Weassumeatime-slottedsystemas[ 69 ]. 88

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relationshipisillustratedasbatterylifetimechart[ 58 ].Second,thereisanenergyconversionlossduringthechargingordischargingprocess,whichismeasuredascharging/dischargingeciency.Finally,asthetimegoes,thebatterywillleaksomeenergystoredinit.Tosimplifyouranalysis,weassumeabatterymodelwithoutanyineciencyinchargingordischarging.Asthetimeslotwechooseistypicallysmall(e.g.,5-minuteperiod),energyleakageovertimecanbeneglected.However,morecomplicatedbatterymodelcanbeeasilyincorporatedintoourmodelwithoutsignicantimpactonouranalysis.Weassumethatineachtimeslott,energyamountGb(t)canbedrawnfromthetraditionalpowergrid(orsimplypowergrid)torechargethebatteryinordertoutilizethetime-diversityofelectricityprice.Theintuitionisthatifwerechargethebatterywhenelectricitypriceislow,theoverallelectricitycostmaybereducedwithproperdesign.Thestateofcharge(SOC)levelB(t)inthebatteryevolvesaccordingtothefollowingequation: B(t+1)=B(t))]TJ /F4 11.955 Tf 11.96 0 Td[(D(t)+(t)S(t)+Gb(t);(4{3)whereD(t)istheamountofenergythatisdischargedfrombatterytosupplydemandinslott.Obviously,weshouldhavethefollowing\energy-availability"andnitecapacityconstraintforeachtimeslott: D(t)B(t)Bmax;(4{4)whereBmaxisthebatterycapacity.ThereisamaximumdischargerateDmaxofthebatteryforonetimeslot,i.e., 0D(t)Dmax:(4{5)TheenergyamountthatcanbedrawnfromtheelectricpowergridtorechargebatteryforonetimeslotisalsoboundedbyGb;max,i.e., 0Gb(t)Gb;max:(4{6) 89

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4.1.3ElectricityMarketAsshownin[ 77 ],electricitypriceinthereal-timeelectricitymarkethasbothtime-diversityandlocation-diversity.Inthispaper,astheproofofconcept,weconcentrateononesingleresidentialcustomer(onehousehold),whoissubjecttoatime-varyingelectricityprice.Assumethatthetime-varyingelectricityprice,C(t),issenttothecustomer'ssmartmeterbytheutilitycompanyatthebeginningofeachslott.Thecostofusingrenewableenergygeneratedbythecustomeritselfisassumedtobezero.DenoteGl(t)asthepowerdrawnfromtheelectricpowergridtodirectlysupplytheenergydemandinslott.SincethetotalelectricitydrawnfromtheelectricpowergridisGb(t)+Gl(t),theelectricitycostforeachtimeslottis(Gb(t)+Gl(t))C(t).Inouranalysis,theunitelectricitypriceC(t)doesnotdependonthetotalamountofenergydrawnfromthepowergrid.However,iftheunitelectricitypricedependsonthetotalpowerconsumed,suchasincliningblockratein[ 64 ],itcanbestillintegratedeasilyintoourmodelasin[ 86 ]. 4.1.4ControlObjectiveInthispaper,weareinterestedinlong-termelectricitycost.Hence,ourobjectivehereistominimizethelong-termtime-averageexpectedelectricitycostasdescribedbelow: P=limT!11 TT)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0EfC(t)(Gl(t)+Gb(t))g;(4{7)wheretheexpectationisw.r.t.possiblyrandomizedcontrolactionsaswellasthedistributionofelectricitypriceC(t). 4.2InelasticEnergyDemandInthesmartgrid,someenergydemandsofresidentialhouseholdareinelastic,suchaslighting,TVwatching,aswellascomputers.Forthiskindofenergydemands,theenergyrequestsmustbemetexactlyatthetimetwhenneeded.Basedonthemodelspresentedintheprevioussection,wecomeupwithaschematicofpowermanagementforinelasticenergydemandsdepictedinFigure 4-2 ,whereboththepowerowand 90

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theinformationowareshown.Thesplittercontrollermanagesthepowerdrawnfromthepowergridtorechargethebatteryordirectlysupplythedemand.ThebatterySOClevel,energydemand,electricitypriceandrenewableenergygenerationcanbedirectlymonitoredbythesplittercontroller.Anthercontrollerisusedtodeterminetheportionofrenewableenergytobestoredintothebattery,whichalsomonitorsthestatusofrenewableenergygenerationandbatterySOClevel.Duetotheinformationandcommunicationinfrastructuredeployedinthesmartgrid,ahomeareanetwork,eitherwiredorwireless,wouldbeformedinaresidentialhouse,whichenablescommunicationbetweenthecomponentsaboveforinformationgatheringanddissemination. Figure4-2. Aschematicofpowermanagementwithinelasticenergydemandsinthesmartgrid. WeassumetheinelasticenergydemandgeneratedintimeslottisAine(t).Foreachtimeslott,wehave Gl(t)+D(t)=Aine(t):(4{8) 91

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Thus,ourproblemcanbeformulatedasthefollowingstochasticoptimization,calledProblemOne:minimizeP1=limT!11 TT)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0EfC(t)(Gl(t)+Gb(t))g (4{9)s.t.B(t+1)=B(t))]TJ /F4 11.955 Tf 11.96 0 Td[(D(t)+(t)S(t)+Gb(t); (4{10)D(t)B(t)Bmax; (4{11)Gl(t)+D(t)=Aine(t); (4{12)0D(t)Dmax; (4{13)0Gl(t)Gl;max;0Gb(t)Gb;max; (4{14)0(t)1: (4{15)DenetheoptimalobjectivevalueoftheoptimizationproblemaboveasP1.Inthefollowing,weapplytheLyapunovoptimizationtechniques[ 33 , 67 ]tondanapproximatesolution,whichattainsananalyticalperformanceguaranteewithinO(1=V)oftheoptimalobjectivevalue,whereVisatunablecontrolparameterrelatedtothebatterycapacity.Theproblemaboveischallengingmainlybecauseofthetime-couplingpropertybroughtbythebatteryconstraint( 4{4 ).Tobespecic,inourproblem,thecurrentcontrolactionmayimpactthefuturecontrolactionsinthesensethatacurrentactionmayoverusethebatteryandleaveinsucientenergyforfutureuse,orthecurrentactionmayleavelessavailablecapacityandthefuturegeneratedrenewableenergycannotbeutilizedeciently.Previousmethodstohandlethistime-couplingproblemareusuallybasedondynamicprogramming,whichsuersfromthe\curseofdimensionality"problem[ 20 ]andrequiresdetailedknowledgeofstatisticsofC(t),S(t)andAine(t)inourproblem.However,inreality,thestatisticsofC(t),S(t)andAine(t)maybeunknownordiculttoobtain,andweneedtodesignanoptimalcontrolalgorithmunderuncertainty.WeusetherecentlydevelopedLyapunovoptimizationtechniques[ 67 ]andndamodiedLyapunov 92

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functiontodevelopouralgorithm.Asalientfeatureofouralgorithmisthatitdoesnotneedanyfutureknowledgeofthesystemstatesandcanbeimplementedinreal-time.Inthenextsection,insteadofsolvingtheabovestochasticoptimizationproblemexactly,westudyarelaxedproblem,whosesolutioniseasytocharacterizebasedontheframeworkofLyapunovoptimization[ 33 , 67 ]. 4.2.1RelaxedProblemWedenethetime-averageexpectedvalueofutilizedrenewableenergy,charginganddischargingrateunderanyfeasiblecontrolpolicyofProblemOne,respectively,asfollows. S=limT!11 TT)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0Ef(t)S(t)g; Gb=limT!11 TT)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xt=0EfGb(t)g; D=limT!11 TT)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xt=0EfD(t)g:SincethebatterySOClevelevolvesaccordingto( 4{3 ),summingoverallt2f0;1;2;:::;T)]TJ /F1 11.955 Tf 11.95 0 Td[(1gandtakingexpectationonbothsides,wehaveEfB(T)g)]TJ /F4 11.955 Tf 20.59 0 Td[(B0=T)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0Ef(t)S(t)+Gb(t))]TJ /F4 11.955 Tf 11.96 0 Td[(D(t)g;whereB(0)=B0istheinitialbatterySOClevel.As0B(t)Bmaxforanytimeslott,dividingbothsideswithTandtakingT!1,wehave D= S+ Gb.Hence,weobtain 93

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thefollowingrelaxedproblem,calledProblemTwo:minimizeP1=limT!11 TT)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0EfC(t)(Gl(t)+Gb(t))g (4{16)s.t. D= S+ Gb; (4{17)Gl(t)+D(t)=Aine(t); (4{18)0D(t)Dmax; (4{19)0Gl(t)Gl;max;0Gb(t)Gb;max; (4{20)0(t)1: (4{21)DenotetheoptimalobjectivevalueofProblemTwoasP1;rel.Fromthediscussionabove,weobservethatanyfeasiblesolutiontoProblemOneisalsoafeasiblesolutiontoProblemTwo,i.e.,ProblemTwoislessconstrainedthanProblemOne.Therefore,P1;relP1.ItiseasytondtheoptimalsolutiontoProblemTwoduetotheremovalofdependencebetweenbatterySOClevelsacrosstimeslots.Asgivenbythefollowinglemma,theoptimalsolutiontoProblemTwocanbeobtainedbyarandomized,stationarycontrolpolicythatonlychoosesD(t),Gl(t),Gb(t)and(t)everyslotpurelyasa(possiblyrandomized)functionofC(t),S(t)andAine(t).ThatmeansthecontrolpolicyisindependentofbatterySOClevel.Thisfactisstatedasbelow: Lemma6. IfthefAine(t);C(t);S(t)garei:i:d:overslots,thenthereexistsastation-aryandrandomizedpolicythattakescontroldecisions^Dine(t),^Gl;ine(t),^Gb;ine(t)and^ine(t)everyslottpurelyasafunction(possiblyrandomized)ofcurrentsystemstatesfAine(t);C(t);S(t)gwhilesatisfyingtheconstraintsaboveandprovidingthefollowingguarantees: Ef^Dine(t)g=Ef^ineS(t)g+Ef^Gb;ine(t)g;(4{22) EfC(t)(^Gl;ine(t)+^Gb;ine(t))g=P1;rel;(4{23) 94

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wheretheexpectationsarew.r.t.thestationarydistributionoffAine(t);C(t);S(t)gandrandomizedcontroldecisions.Theproofissimilartothatin[ 49 , 86 ]andfollowsdirectlyfromtheframeworkofLyapunovoptimizationin[ 33 , 67 ],whichisomittedhereforbrevity.Toderivesuchapolicy,weneedtoknowthestatisticaldistributionofallcombinationsoffAine(t);C(t);S(t)g,whichusuallyhasthe\curseofdimensionality"problem[ 20 ]ifsolvedbydynamicprogramming.Moreover,thiscontrolpolicymaynotbeafeasiblesolutiontoProblemOne.Instead,weusetheexistenceofsuchapolicytohelpusdesignourcontrolpolicythatmeetsallconstraintsofProblemOneandderivetheperformanceresultsforouralgorithm. 4.2.2OurProposedAlgorithmBeforepresentingouralgorithm,wedeneanothervariableXine(t)asashiftedversionofbatterySOClevelB(t)foreachtimeslottasfollows: Xine(t)=B(t))]TJ /F4 11.955 Tf 11.95 0 Td[(VineCmax)]TJ /F4 11.955 Tf 11.95 0 Td[(Dmax;(4{24)whereVineisacontrolparametertobespeciedlater.Xine(t)isusedtoensurethattheconstraint( 4{4 )ofbatterySOClevelissatisedinouralgorithm.TheintuitionbehindXine(t)istoconstructthealgorithmbasedonaquadraticLyapunovfunction,butcarefullyperturbtheweightsusedfordecisionmaking,soastopushtheSOClevelinbatterytowardscertainnonzerovaluestoavoidunderow.Accordingto( 4{3 )ofB(t),wehavethesameupdateequationforXine(t), Xine(t+1)=Xine(t))]TJ /F4 11.955 Tf 11.95 0 Td[(D(t)+(t)S(t)+Gb(t):(4{25)Inthelatterpartofthispaper,wewillprovethatthroughouralgorithm,Xine(t)isboundedinsomerangesothattheconstraint( 4{4 )onB(t)isalwayssatisedforeachslott. 95

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TheproposedalgorithmforinelasticenergydemandisshowninAlgorithm 3 .ThealgorithmisdesignedbasedontheLaypunovoptimizationtechniquedevelopedin[ 33 , 67 ].Theideaofthealgorithmistogreedilyminimizeaupperboundofthedrift-plus-penaltyfunctionin( 4{29 ).NotethatthealgorithmonlyusesthecurrentsystemstatesXine(t),C(t),S(t)andAine(t),anddoesnotrequireanyknowledgeofthestatisticsofrenewableenergygeneration,electricityprice,andenergydemandarrivalprocess. Algorithm3:Powermanagementwithinelasticenergydemands foreachTimeslottdo 1MeasurethesystemstatesXine(t),C(t),S(t)andAine(t). 2ChoosecontroldecisionsDine(t),Gl;ine(t),Gb;ine(t)andine(t)asthesolutiontothefollowingoptimizationproblem,calledProblemThree:minimize[VineC(t)+X(t)]Gb(t)+[X(t)S(t)](t)+[VineC(t)]Gl(t))]TJ /F4 11.955 Tf 11.96 0 Td[(X(t)D(t)s.t.D(t)+Gl(t)=Aine(t);0Gl(t)Gl;max;0Gb(t)Gb;max;0D(t)Dmax;0(t)1: 3X(t+1)=X(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Dine(t)+ine(t)S(t)+Gb;ine(t). 4.2.3AlgorithmicPropertiesInthissubsection,wesummarizethepropertiesofourproposedalgorithmasfollows. Theorem4.1. AssumethatGl;max+Gb;maxAmaxine,thenforanyparameterVinesatisfying0
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andX(t)Bmax)]TJ /F4 11.955 Tf 11.96 0 Td[(VineCmax)]TJ /F4 11.955 Tf 11.96 0 Td[(Dmax: (2) Allcontroldecisionsarefeasible. (3) IfS(t),C(t)andAine(t)arei:i:d:overslots,thenthetime-averageexpectedcostunderouralgorithmiswithinboundB1=Vineoftheoptimalvalue,i.e., limT!11 TT)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0EfC(t)(Gl(t)+Gb(t))gP1+B1=Vine;(4{27)whereB1isaconstantgivenby B1,[(Gb;max+Smax)2;D2max] 2:(4{28) Proof. Inthefollowing,weproveTheorem 4.1 . 1. ItisobviousthattheoptimalsolutiontoProblemThreehasthefollowingproperties: IfXine(t)>)]TJ /F4 11.955 Tf 9.3 0 Td[(VineCmin,Gb;ine(t)=0andDine(t)=minfAine(t);Dmaxg. IfXine(t)<)]TJ /F4 11.955 Tf 9.3 0 Td[(VineCmax,Gb;ine(t)=Gb;maxandDine(t)=maxf0;Aine(t))]TJ /F4 11.955 Tf -378.78 -14.45 Td[(Gl;maxg.Wenowuseinductiontoprovethisresult.Whent=0,asXine(0)=B0)]TJ /F4 11.955 Tf -386.61 -23.91 Td[(VineCmax)]TJ /F4 11.955 Tf 12.58 0 Td[(Dmaxand0B0Bmax,wehave)]TJ /F4 11.955 Tf 9.3 0 Td[(VineCmax)]TJ /F4 11.955 Tf 12.58 0 Td[(DmaxXine(0)Bmax)]TJ /F4 11.955 Tf 11.95 0 Td[(VineCmax)]TJ /F4 11.955 Tf 11.96 0 Td[(Dmax.Nowsupposethattheaboveboundholdsfortimeslott.Weneedtoprovethatitalsoholdsfortimeslott+1.First,suppose)]TJ /F4 11.955 Tf 9.3 0 Td[(VineCmax)]TJ /F4 11.955 Tf 12.77 0 Td[(DmaxXine(t)<)]TJ /F4 11.955 Tf 9.3 0 Td[(VineCmax,thenXine(t+1)=Xine(t)+Gb;ine(t)+S(t))]TJ /F4 11.955 Tf 12.33 0 Td[(Dine(t))]TJ /F4 11.955 Tf 22.86 0 Td[(VineCmax)]TJ /F4 11.955 Tf -416.15 -23.9 Td[(Dmax+Gb;max)]TJ /F1 11.955 Tf 13.16 0 Td[(maxf0;Aine(t))]TJ /F4 11.955 Tf 13.16 0 Td[(Gl;maxg.AsGl;max+Gb;maxAmaxine,wehaveXine(t+1)Xine(t))]TJ /F4 11.955 Tf 24.77 0 Td[(VineCmax)]TJ /F4 11.955 Tf 13.1 0 Td[(Dmax.Moreover,Xine(t+1)Xine(t)+Gb;max+Smax)]TJ /F4 11.955 Tf 22.64 0 Td[(VineCmin+Gb;max+SmaxBmax)]TJ /F4 11.955 Tf 12.24 0 Td[(VineCmax)]TJ /F4 11.955 Tf 12.24 0 Td[(Dmax, 97

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wherewehaveusedVineBmax)]TJ /F4 11.955 Tf 11.96 0 Td[(Dmax)]TJ /F4 11.955 Tf 11.95 0 Td[(Gb;max)]TJ /F4 11.955 Tf 11.95 0 Td[(Smax Cmax)]TJ /F4 11.955 Tf 11.95 0 Td[(Cmin:Second,suppose)]TJ /F4 11.955 Tf 9.3 0 Td[(VineCmaxXine(t))]TJ /F4 11.955 Tf 23.68 0 Td[(VineCmin,then)]TJ /F4 11.955 Tf 9.3 0 Td[(VineCmax)]TJ /F4 11.955 Tf 12.66 0 Td[(DmaxXine(t))]TJ /F4 11.955 Tf 10.74 0 Td[(DmaxXine(t+1)Xine(t)+Gb;max+Smax)]TJ /F4 11.955 Tf 21.92 0 Td[(VineCmin+Gb;max+SmaxBmax)]TJ /F4 11.955 Tf 13.15 0 Td[(VineCmax)]TJ /F4 11.955 Tf 13.15 0 Td[(Dmax,wherewehaveusedthesameboundofVineasthecaseabove.Third,suppose)]TJ /F4 11.955 Tf 9.3 0 Td[(VineCmin)]TJ /F4 11.955 Tf 9.3 0 Td[(VineCmax)]TJ /F4 11.955 Tf 12.96 0 Td[(Dmax.Moreover,Xine(t+1)Xine(t)+SmaxSmaxBmax)]TJ /F4 11.955 Tf 12.9 0 Td[(VineCmax)]TJ /F4 11.955 Tf 12.89 0 Td[(Dmax,wherewehaveusedtheupperboundofVineandGb;maxVmaxineCmin.Finally,suppose0
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As0D(t)Dmaxand0(t)S(t)+Gb(t)Gb;max+Smax,wehave(D(t))]TJ /F4 11.955 Tf 11.95 0 Td[((t)S(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Gb(t))2 21 2max(Gmaxb+Smax)2;D2max:Therefore,wecanobtainthefollowingupperboundontheLyapunovdriftforXine(t):X2ine(t+1))]TJ /F4 11.955 Tf 11.96 0 Td[(X2ine(t) 21 2max[(Gmaxb+Smax)2;D2max])]TJ /F4 11.955 Tf 11.96 0 Td[(Xine(t)(D(t))]TJ /F4 11.955 Tf 11.95 0 Td[((t)S(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Gb(t)):Takingexpectationw.r.t.Xine(t)andaddingthepenaltytermVineEfC(t)(Gl(t)+Gb(t))jXine(t)gtobothsidesoftheinequalityabove,weobtainthefollowinginequality:4(Xine(t))+VineEfC(t)(Gl(t)+Gb(t))jXine(t)gB1)]TJ /F4 11.955 Tf 11.95 0 Td[(Xine(t)EfD(t))]TJ /F4 11.955 Tf 11.95 0 Td[((t)S(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Gb(t)jXine(t)g+VineEfC(t)(Gl(t)+Gb(t))jXine(t)g; (4{29)whereB1isdenedasB1,[(Gb;max+Smax)2;D2max] 2:ComparingwiththeobjectiveofProblemThree,itisobviousthatouralgorithmalwaysattemptstogreedilyminimizetherighthandside(R.H.S.)oftheinequalityaboveforeachtimeslottoverallpossiblefeasiblecontrolpoliciesincludingtheoptimal,stationarypolicygiveninLemma 6 .Pluggingthispolicy(^Dine(t);^Gl;ine(t);^Gb;ine(t);^ine(t))intotheR.H.S.oftheinequalityaboveandusingthefactthatthispolicyisindependentofqueuestateXine(t),weobtainthefollowing:4(Xine(t))+VineEfC(t)(Gl(t)+Gb(t))jXine(t)gB1)]TJ /F4 11.955 Tf 11.96 0 Td[(Xine(t)Ef^Dine(t))]TJ /F1 11.955 Tf 12.39 0 Td[(^ine(t)S(t))]TJ /F1 11.955 Tf 14.62 3.02 Td[(^Gb;ine(t)jXine(t)g+VineEfC(t)(^Gl;ine(t)+^Gb;ine(t))jXine(t)gB1+VineP1;relB1+VineP1; 99

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wherethefollowingfactshavebeenused:En^Dine(t))]TJ /F1 11.955 Tf 12.39 0 Td[(^ine(t)S(t))]TJ /F1 11.955 Tf 14.62 3.02 Td[(^Gb;ine(t)jXine(t)o=0;EfC(t)(^Gl;ine(t)+^Gb;ine(t))jXine(t)g=P1;rel:TheequationsabovefollowfromLemma 6 .Takingtheexpectationonbothsides,usingthelawofiterativeexpectation,andsummingovert2f0;1;2;:::;T)]TJ /F1 11.955 Tf 12.16 0 Td[(1g,weobtainVineT)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xt=0EfC(t)(Gl(t)+Gb(t))B1T+VineTP1)]TJ /F9 11.955 Tf 11.95 0 Td[(EfL(Xine(T))g+EfL(Xine(0))g:DividingbothsidesbyT,lettingT!1,andusingthefactsthatEfL(Xine(0))gareniteandEfL(Xine(t))garenonnegative,wenallyarriveatthefollowing:limT!11 TT)]TJ /F6 7.97 Tf 6.58 .01 Td[(1Xt=0EfC(t)(Gl(t)+Gb(t))gP1+B1=Vine;whereP1istheoptimalobjectivevalue,B1isaconstantgivenby( 4{28 ),andVineisacontrolparameterwhichhasamaximumvaluegivenby( 4{26 ).Thiscompletestheproofof3). 4.3ElasticEnergyDemandWhiletheprevioussectiondealswithinelasticenergydemand,somehouseholdappliancesinthesmartgridcanbemade\smart",meaningthattheycanbecontrolledtoadjustthetimesoftheiroperationsandtheamountoftheirenergyusage.Inotherwords,aslongastheirenergyrequirementsaremetwithincertaindeadlines,theresidentialcustomerswillbesatised.Sometypicalexamplesincludedishwasher,waterheater,airconditioning,andchargingofPHEVs.AschematicforthepowermanagementwithelasticenergydemandsisshowninFig. 4-3 . 100

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Figure4-3. Aschematicforpowermanagementwithelasticenergydemandsinthesmartgrid. WeassumethattheamountofelasticenergydemandsrequestedatslottareAela(t).Theseenergydemandsarestoredinanenergydemandqueue.Ineveryslott,theenergydischargedfromthebatteryisdenotedasD(t)andtheenergyamountdrawndirectlyfromthepowergridisdenotedasGl(t),whicharedeterminedfromthebueredenergydemands.TheenergydemandsareservedinaFirst-In-First-Out(FIFO)manner.LetQ(t)denotethetotalenergydemandsinthequeuefortimeslott,wehavethefollowingqueuingequation: Q(t+1)=maxfQ(t))]TJ /F4 11.955 Tf 11.96 0 Td[(D(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Gl(t);0g+Aela(t):(4{30)Aslongasthewaitingtimeofanybueredenergydemandinthisdemandqueuedoesnotexceedacertainmaximumdeadlinemax,theutilityperceivedbycustomersdoesnotdecrease.Weusethesamebatterymodel( 4{3 )asinthecaseforinelasticenergydemands.Ourproblemhereistominimizethetime-averageexpectedelectricitycostsubjecttoallconstraintsdiscussedbeforeandtoensureniteaveragedelayforany 101

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bueredenergydemand,whichcanbestatedbelowandiscalledProblemFour:minimizeP2=limT!11 TT)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xt=0EfC(t)(Gl(t)+Gb(t))g (4{31)s.t.B(t+1)=B(t))]TJ /F4 11.955 Tf 11.95 0 Td[(D(t)+(t)S(t)+Gb(t); (4{32)D(t)B(t)Bmax; (4{33)Q(t+1)=maxfQ(t))]TJ /F4 11.955 Tf 11.96 0 Td[(D(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Gl(t);0g+Aela(t); (4{34)0D(t)Dmax; (4{35)0Gb(t)Gb;max;0Gl(t)Gl;max; (4{36)0(t)1; (4{37) Q<1; (4{38)where Qisthetime-averageexpectedenergydemandqueuebacklogdenedas: Q=limT!11 TT)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xt=0EfQ(t)g: 4.3.1RelaxedProblemSimilartothecasefortheinelasticenergydemands,wedenearelaxedproblem,calledProblemFive,whichcanbestatedasfollows:minimizeP2=limT!11 TT)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xt=0EfC(t)(Gl(t)+Gb(t))g (4{39)s.t. D= S+ Gb; (4{40)Q(t+1)=maxfQ(t))]TJ /F4 11.955 Tf 11.96 0 Td[(D(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Gl(t);0g+Aela(t); (4{41)0D(t)Dmax; (4{42)0Gb(t)Gb;max;0Gl(t)Gl;max; (4{43)0(t)1; (4{44) Q<1: (4{45) 102

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DenotetheoptimalobjectivevalueofProblemFourandProblemFiveasP2andP2;rel,respectively.Similarly,anyfeasiblesolutiontoProblemFourisalsoafeasiblesolutiontoProblemFive,therefore,P2;relP2.SimilartoLemma 6 ,wehavethefollowingresult: Lemma7. IffAela(t);C(t);S(t)garei:i:d:overslots,thenthereexistsastation-aryandrandomizedpolicythattakescontroldecisions^Dela(t),^Gl;ela(t),^Gb;ela(t)and^ela(t)everyslottpurelyasa(possiblyrandomized)functionofcurrentsystemstatesfAela(t);C(t);S(t)gwhilesatisfyingtheconstraintsaboveandprovidingthefollowingguarantees: Ef^Dela(t)g=Ef^elaS(t)g+Ef^Gb;ela(t)g;(4{46) Ef^Dela(t)+^Gl;ela(t)gEfAela(t)g;(4{47) EfC(t)(^Gl;ela(t)+^Gb;ela(t))g=P2;rel;(4{48)wheretheexpectationsarew.r.t.thestationarydistributionoffAela(t);C(t);S(t)gandtherandomizedcontroldecisions.Theproofissimilartothatin[ 49 , 86 ]andfollowsdirectlyfromtheframeworkofLyapunovoptimizationin[ 33 , 67 ],whichisomittedhereforbrevity.Notethattheconstraint( 4{38 )onlyensuresniteaveragedelaywithoutanyguaranteefortheworst-casedelay.Inthefollowing,weusethetechniqueof-persistentqueue[ 69 ]toguaranteetheniteworst-casedelayforanybueredenergydemandinthequeue. 4.3.2Delay-AwareVirtualQueueWeusethefollowingvirtualqueueZ(t)toprovidetheworst-casedelayguaranteeonanybueredenergydemandinQ(t): Z(t+1)=maxZ(t))]TJ /F4 11.955 Tf 11.95 0 Td[(D(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Gl(t)+1fQ(t)>0g;0;(4{49) 103

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where1fQ(t)>0gisanindicatorfunctionthatis1ifQ(t)>0or0otherwise;isaxedpositiveparametertobespeciedlater.TheintuitionbehindthisvirtualqueueisthatsinceZ(t)hasthesameserviceprocessasQ(t),buthasanarrivalprocessthataddswhenevertheactualbacklogisnonempty,thisensuresthatZ(t)growsifthereisenergydemandinthequeueQ(t)thathasnotbeenservicedforalongtime.ThefollowinglemmashowsthatifwecancontrolthesystemtoensurethatthequeuesQ(t)andZ(t)haveniteupperbounds,thenanybueredenergydemandisservedwithintheworst-casedelay. Lemma8. SupposewecancontrolthesystemtoensurethatZ(t)ZmaxandQ(t)Qmaxforallslotst,whereZmaxandQmaxaresomepositiveconstants,thentheworst-casedelayforallbueredenergydemandisupperboundedbymaxslotswhere max,(Qmax+Zmax) :(4{50) Proof. ConsideranyslottforwhichAela(t)>0.WewillshowthatthisenergyrequestAela(t)isservedonorbeforetimet+maxbycontradiction.Supposenot,thenduringslots2ft+1;:::;t+maxgitmustbethatQ()>0,otherwise,theenergyrequestAela(t)wouldhavebeenservedbefore.Therefore,1Q(t)>0=1,andfromtheupdateequation( 4{49 )ofZ(t),wehaveforall=ft+1;:::;t+maxg:Z(+1)Z())]TJ /F4 11.955 Tf 11.95 0 Td[(D(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Gl(t)+:Summingtheaboveover=ft+1;:::;t+maxgyields:Z(t+max+1))]TJ /F4 11.955 Tf 11.95 0 Td[(Z(t+1))]TJ /F5 7.97 Tf 23.91 14.94 Td[(t+maxX=t+1[D(t)+Gl(t)]+max:RearrangingthetermsandusingthefactsthatZ(t+1)0andZ(t+max+1)Zmaxyields:t+maxX=t+1[D(t)+Gl(t)]max)]TJ /F4 11.955 Tf 11.96 0 Td[(Zmax: 104

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SincetherequestAela(t)arequeuedinaFIFOmannerandQ(t+1)Qmax,itwouldbeservedonorbeforetimet+maxwheneverthereareatleatQmaxunitsofenergyservedduring2ft+1;:::;t+maxg.AswehaveassumedthattherequestAela(t)arenotservedbytimet+max,itmustbethatPt+max=t+1[D(t)+Gl(t)]max)]TJ /F4 11.955 Tf 11.96 0 Td[(Zmax;whichimpliesthatmax<(Qmax+Zmax)=,contradictingthedenitionofmaxin( 4{50 ). WewillshowthatthereindeedexistsuchconstantsZmaxandQmaxlater. 4.3.3OurProposedAlgorithmBeforepresentingouralgorithm,wedeneanothervariableXela(t)asashiftedversionofbatterySOClevelB(t)fortimeslottasfollows: Xela(t)=B(t))]TJ /F1 11.955 Tf 11.95 0 Td[(max)]TJ /F4 11.955 Tf 11.95 0 Td[(Dmax;(4{51)wheremaxisapositiveconstanttobespecied.Xela(t)isalsousedtoensurethattheconstraint( 4{4 )ofbatterySOClevelissatisedinouralgorithm.Accordingto( 4{3 ),weobtainthesameupdateequationforXela(t), Xela(t+1)=Xela(t))]TJ /F4 11.955 Tf 11.95 0 Td[(D(t)+(t)S(t)+Gb(t):(4{52)ThealgorithmforelasticenergydemandsisshowninAlgorithm 4 .ThealgorithmisdesignedbasedontheLaypunovoptimizationtechniquedevelopedin[ 33 , 67 ].Theideaofthealgorithmistogreedilyminimizeaupperboundofthedrift-plus-penaltyfunctionin( 4{59 ).Similarly,thealgorithmonlymakesuseofthecurrentsystemstates(Xela(t);Q(t);Z(t)),C(t),S(t)andAela(t),anddoesnotrequireanyknowledgeonthestatisticsoftherenewableenergygeneration,theelectricityprice,andtheenergydemandarrivalprocess. 105

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Algorithm4:Powermanagementwithelasticenergydemands foreachTimeslottdo 1Measuresystemstates(Xela(t);Q(t);Z(t)),C(t),S(t)andAela(t) 2ChoosecontroldecisionsDela(t),Gl;ela(t),Gb;ela(t)andela(t)asthesolutiontothefollowingoptimizationproblem,calledProblemSix:minimize[VelaC(t)+Xela(t)]Gb(t)+[Xela(t)S(t)](t)+[VelaC(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Z(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Q(t)]Gl(t))]TJ /F1 11.955 Tf 11.96 0 Td[([Xela(t)+Z(t)+Q(t)]D(t)s.t.0D(t)Dmax;0Gl(t)Gl;max;0Gb(t)Gb;max;0(t)1: 3X(t+1)=X(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Dela(t)+ela(t)S(t)+Gb;ela(t) 4Z(t+1)=maxZ(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Dela(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Gl;ela(t)+1fQ(t)>0g;0 5Q(t+1)=maxQ(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Dela(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Gl;ela(t);0+Aela(t) 4.3.4AlgorithmicPropertiesIntuitively,byconstructingthequadraticLaypunovfunctionasthesquareofXela(t)andkeepingthevaluesmall,weindeedpushthevalueofB(t)towardstheshiftedvalueinthedenition( 4{52 )ofXela(t).Therefore,bycarefullyselectingtheshiftedvalue,wecanensurethatthebatterySOClevelalwayssatisesthenitecapacityconstraint.Wesummarizethepropertiesofourproposedalgorithmasfollows. Theorem4.2. AssumethatGl;maxmax[Amaxela;].IfQ(0)=Z(0)=0,thenforanyxedparameter0EfA(t)gandaparameterVelasuchthat0
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Further,Q(t)+Z(t)areupperboundedbymaxwhere max,VelaCmax+Amaxela+:(4{55) (2) Theworst-casedelayofanybueredenergydemandisgivenby: max=2VelaCmax+Amaxela+ :(4{56) (3) ThequeueXela(t)isalwayslowerandupperboundedforallslotstbythefollowing:)]TJ /F1 11.955 Tf 9.3 0 Td[(max)]TJ /F4 11.955 Tf 11.96 0 Td[(DmaxXela(t)Bmax)]TJ /F1 11.955 Tf 11.95 0 Td[(max)]TJ /F4 11.955 Tf 11.95 0 Td[(Dmax: (4) Allcontroldecisionsarefeasible. (5) IfS(t),C(t),andAela(t)arei:i:d:overslots,thenthetime-averageexpectedelectric-itycostunderouralgorithmiswithinboundB2=Velaoftheoptimalvalue,i.e., limT!11 TT)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xt=0EfC(t)(Gl(t)+Gb(t))gP2+B2=Vela;(4{57)whereB2isaconstantgivenbyB2,[(Dmax+Gl;max)2+A2ela;max] 2+max[(Dmax+Gl;max)2;2] 2+max[(Smax+Gb;max)2;D2max] 2: (4{58) Proof. Inthefollowing,weproveTheorem 4.2 . 1. First,weproveQ(t)Qmaxforeverytimeslott.Onceagain,wewilluseinductionmethod.Obviously,Q(0)Qmax.Supposeitholdsattimeslott,weneedtoshowthatitalsoholdsattimeslott+1.AsQ(t+1)=max[Q(t))]TJ /F4 11.955 Tf 10.35 0 Td[(D(t))]TJ /F4 11.955 Tf 10.35 0 Td[(Gl(t);0]+Aela(t),ifQ(t)VelaCmax,thenthemaximumamountofenergydemandarrivalisAmaxela,wehaveQ(t+1)VelaCmax+Amaxela.IfVelaCmax0,then,intimeslotttheamountofenergydemandbeingservedisatleastGl;max,whichislargerthanthe 107

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maximumamountofarrivalduringtimeslott.Hence,thequeuecannotincrease,i.e.,Q(t+1)Q(t)VelaCmax+Amaxela.IfQ(t))]TJ /F4 11.955 Tf 12.73 0 Td[(Dela(t))]TJ /F4 11.955 Tf 12.73 0 Td[(Gl;max0,thenQ(t+1)=Aela(t)AmaxelaVelaCmax+Amaxela.Therefore,wehaveprovedQ(t)VelaCmax+Amaxela.Next,weproveZ(t)Zmaxforeverytimeslott.Obviously,Z(0)Zmax.Supposeitholdsfortimeslott,weneedtoshowthatitalsoholdsintimeslott+1.AsZ(t+1)=max[Z(t))]TJ /F4 11.955 Tf 9.86 0 Td[(D(t))]TJ /F4 11.955 Tf 9.86 0 Td[(Gl(t)+1Q(t)>0;0],ifZ(t)VelaCmax,thenthemaximumamountofqueuingincreaseis,wehaveZ(t+1)VelaCmax+;ifVelaCmax0,then,intimeslotttheamountofenergydemandbeingservedisatleastGl;max,whichislargerthanthemaximumamountofarrivalduringtimeslott.Hence,thequeuecannotincrease,i.e.,Z(t+1)Z(t)VelaCmax+.IfZ(t))]TJ /F4 11.955 Tf 12.09 0 Td[(Dela(t))]TJ /F4 11.955 Tf 12.1 0 Td[(Gl;max0,thenZ(t+1)VelaCmax+.Therefore,wehaveprovedthatZ(t)VelaCmax+.Finally,weproveQ(t)+Z(t)max.Obviously,Q(0)+Z(0)max.SupposeQ(t)+Z(t)maxholdsfortimeslott.IfQ(t)+Z(t)VelaCmax,then,accordingtothequeuingequationsofQ(t)andZ(t),themaximumincreaseduringoneslotisAmaxela+.IfVCmax)]TJ /F4 11.955 Tf 9.3 0 Td[(VelaCmin,Gb;ela(t)=0. IfX(t)<)]TJ /F1 11.955 Tf 9.3 0 Td[([Q(t)+Z(t)]max=)]TJ /F1 11.955 Tf 9.3 0 Td[(max,Dela(t)=0. 108

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Inthefollowing,weprovethisresultbyinduction.Whent=0,asXela(0)=B0)]TJ /F1 11.955 Tf 12.9 0 Td[(max)]TJ /F4 11.955 Tf 12.9 0 Td[(Dmaxand0B0Bmax,wehave)]TJ /F1 11.955 Tf 9.3 0 Td[(max)]TJ /F4 11.955 Tf 12.9 0 Td[(DmaxX(0)Bmax)]TJ /F1 11.955 Tf 11.95 0 Td[(max)]TJ /F4 11.955 Tf 11.95 0 Td[(Dmax.Nowsupposethattheaboveboundholdsfortimeslott.Weneedtoshowthatitalsoholdsfortimeslott+1.First,suppose0
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follows:4(~K(t))=EfL(~K(t+1)))]TJ /F4 11.955 Tf 11.95 0 Td[(L(~K(t))j~K(t)g:Fromtheupdate( 4{52 ),squaringbothsides,weobtainX2ela(t+1))]TJ /F4 11.955 Tf 11.96 0 Td[(X2ela(t) 2=(D(t))]TJ /F4 11.955 Tf 11.96 0 Td[((t)S(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Gb(t))2 2)]TJ /F4 11.955 Tf 11.96 0 Td[(Xela(t)(D(t))]TJ /F4 11.955 Tf 11.95 0 Td[((t)S(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Gb(t)):As0D(t)Dmaxand0(t)S(t)+Gb(t)Gb;max+Smax,wehave(D(t))]TJ /F4 11.955 Tf 11.96 0 Td[((t)S(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Gb(t))2 21 2max[(Gb;max+Smax)2;D2max]:Therefore,wecangetthefollowingupperboundfortheLyapunovdriftforX(t):X2ela(t+1))]TJ /F4 11.955 Tf 11.96 0 Td[(X2ela(t) 21 2max[(Gb;max+Smax)2;D2max])]TJ /F4 11.955 Tf 11.95 0 Td[(Xela(t)(D(t))]TJ /F4 11.955 Tf 11.96 0 Td[((t)S(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Gb(t)):Fromtheupdate( 4{49 ),wehaveZ(t+1)max[Z(t))]TJ /F4 11.955 Tf 11.96 0 Td[(D(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Gl(t)+;0];then,Z2(t+1)(Z(t))]TJ /F4 11.955 Tf 11.95 0 Td[(D(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Gl(t)+)2;andweobtainthefollowinginequality:Z2(t+1))]TJ /F4 11.955 Tf 11.96 0 Td[(Z2(t) 2()]TJ /F4 11.955 Tf 11.95 0 Td[(D(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Gl(t))2 2+Z(t)()]TJ /F4 11.955 Tf 11.96 0 Td[(D(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Gl(t))max[(Dmax+Gl;max)2;2] 2+Z(t)()]TJ /F4 11.955 Tf 11.95 0 Td[(D(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Gl(t)):Fromtheupdate( 4{30 ),squaringbothsidesandusingthefollowinginequality:(max[Q(t))]TJ /F4 11.955 Tf 11.95 0 Td[(D(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Gl(t);0]+Aela(t))2A2ela(t)+Q2(t)+(D(t)+Gl(t))2+2Q(t)(Aela(t))]TJ /F4 11.955 Tf 11.95 0 Td[(D(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Gl(t)); 110

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weobtainQ2(t+1))]TJ /F4 11.955 Tf 11.96 0 Td[(Q2(t) 2[(Dmax+Gl;max)2+A2ela;max] 2+Q(t)(Aela(t))]TJ /F4 11.955 Tf 11.95 0 Td[(D(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Gl(t)):Combiningthesethreeboundstogetherandtakingtheexpectationw.r.t.~K(t)onbothsides,wearriveatthefollowinginequality:4(~K(t))B2+EfZ(t)()]TJ /F4 11.955 Tf 11.95 0 Td[(D(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Gl(t))j~K(t)g+EfQ(t)(Aela(t))]TJ /F4 11.955 Tf 11.95 0 Td[(D(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Gl(t))j~K(t)g)]TJ /F9 11.955 Tf 11.96 0 Td[(EfXela(t)(D(t))]TJ /F4 11.955 Tf 11.96 0 Td[((t)S(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Gb(t))j~K(t)g;whereB2=max[(Smax+Gb;max)2;D2max]=2+max[(Dmax+Gl;max)2;2]=2+[(Dmax+Gl;max)2+A2ela;max]=2.AddingpenaltytermVelaEfC(t)(Gl(t)+Gb(t))j~K(t)gtobothsidesoftheaboveinequality,weobtainthefollowinginequality:4(~K(t))+VelaEfC(t)(Gl(t)+Gb(t))j~K(t)gB2)]TJ /F4 11.955 Tf 11.95 0 Td[(Xela(t)EfD(t))]TJ /F4 11.955 Tf 11.95 0 Td[((t)S(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Gb(t)j~K(t)g+Z(t)Ef)]TJ /F4 11.955 Tf 11.96 0 Td[(D(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Gl(t)j~K(t)g+Q(t)EfAela(t))]TJ /F4 11.955 Tf 11.96 0 Td[(D(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Gl(t)j~K(t)g+VelaEfC(t)(Gl(t)+Gb(t))j~K(t)g: (4{59)ComparingwiththeobjectiveofProblemSix,itisobviousthatouralgorithmalwaysattemptstogreedilyminimizetheR.H.S.oftheaboveinequalityateachtimeslottoverallfeasiblecontrolpoliciesincludingtheoptimal,stationarypolicygiveninLemma 7 .Pluggingthispolicy(^Dela(t);^Gl;ela(t);^Gb;ela(t);^ela(t))intotheR.H.S.oftheinequalityaboveandusingthefactthatthispolicyisindependentofqueue 111

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state~K(t),weobtainthefollowing:4(~K(t))+VelaEfC(t)(Gl(t)+Gb(t))j~K(t)gB2)]TJ /F4 11.955 Tf 11.95 0 Td[(Xela(t)Ef^Dela(t))]TJ /F1 11.955 Tf 12.39 0 Td[(^ela(t)S(t))]TJ /F1 11.955 Tf 14.62 3.02 Td[(^Gb;ela(t)j~K(t)g+Z(t)Ef)]TJ /F1 11.955 Tf 14.71 3.02 Td[(^Dela(t))]TJ /F1 11.955 Tf 14.62 3.02 Td[(^Gl;ela(t)j~K(t)g+Q(t)EfAela(t))]TJ /F1 11.955 Tf 14.7 3.02 Td[(^Dela(t))]TJ /F1 11.955 Tf 14.62 3.02 Td[(^Gl;ela(t)j~K(t)g+VelaEfC(t)(^Gl;ela(t)+^Gb;ela(t))j~K(t)gB2+VelaP2;relB2+VelaP2;wherethefollowingfactshavebeenused: Ef^Dela(t))]TJ /F1 11.955 Tf 12.39 0 Td[(^ela(t)S(t))]TJ /F1 11.955 Tf 14.62 3.02 Td[(^Gb;ela(t)j~K(t)g=0;(4{60) EfAela(t))]TJ /F1 11.955 Tf 14.7 3.02 Td[(^Dela(t))]TJ /F1 11.955 Tf 14.62 3.02 Td[(^Gl;ela(t)j~K(t)g0;(4{61) Ef)]TJ /F1 11.955 Tf 14.7 3.02 Td[(^Dela(t))]TJ /F1 11.955 Tf 14.62 3.02 Td[(^Gl;ela(t)j~K(t)g0:(4{62)ThersttwoequationsfollowfromLemma 7 andthelastonefollowsfrom( 4{61 )togetherwithEfAela(t)g.Takingtheexpectationonbothsides,usingthelawofiterativeexpectation,andsummingovert2f0;1;2;:::;T)]TJ /F1 11.955 Tf 11.96 0 Td[(1g,weobtainVelaT)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0EfC(t)(Gl(t)+Gb(t))B2T+VelaTP2)]TJ /F9 11.955 Tf 11.96 0 Td[(EfL(~K(T))g+EfL(~K(0))g:DividingbothsidesbyT,lettingT!1,andusingthefactsthatEfL(~K(0))gareniteandEfL(~K(t))garenonnegative,wearriveatthefollowingresultforouralgorithm:limT!11 TT)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xt=0EfC(t)(Gl(t)+Gb(t))gP2+B2=Vela;whereP2istheoptimalobjectivevalue,B2isaconstantgivenby( 4{58 ),andVelaisatunablecontrolparameterwhichhasamaximumvaluegivenby( 4{53 ).Thiscompletestheproof. 112

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4.4CaseStudiesInthissection,weevaluatetheproposedalgorithmsusingpracticaldatasetsofelectricitypriceandrenewableenergygeneration.Weconsiderasinglehouseholdwithabattery,aPVpanelandvariousappliancessubjecttoreal-timepricing. 4.4.1ExperimentalSetup Figure4-4. Averagehourlyspotmarketpriceduringtheweekof01/01/2011to01/07/2011atLA[ 3 ]. Figure4-5. Averagehourlysolarirradianceproleduringtheweekof01/01/2011to01/07/2011atLA[ 2 ]. 113

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ThedatasetofelectricitypriceweuseisfromtheCaliforniaIndependentSystemOperator(CAISO)[ 3 ]forLosAngelesarea,whichconsistsof5-minintervalaveragespotmarketpriceC(t).Meanwhile,weusethe5-minintervalaveragesolarirradiancedataforLosAngelsareafromtheMeasurementandInstrumentationDatacenter(MIDC)[ 2 ]atNationalRenewableEnergyLaboratory.TheperiodweconsiderinthispaperishalfyearfromJanuary1,2011toJune30,2011.Intotal,thisdurationincludes181daysor52;1285-minslots.Thecontrolintervalischosentobe5-minute.AportionofaveragehourlyspotmarketelectricitypriceandsolarirradianceduringtherstweekofJanuary2011areplottedinFigure 4-4 andFigure 4-5 ,respectively.Weexecuteouralgorithmsin5-mintimeslotsandexperimentwithdierentvaluesofparametersV,Bmax,and.Inoursimulation,wesettheenergydemandarrival,eitherelasticorinelastic,duringeachtimeslottasuniformlydistributedfrom[1;24]KW-slotbasedonthepracticalhomeapplianceusagein[ 64 ].WextheparametersDmax=30KW-slot,Gl;max=30KW-slot,andGb;max=20KW-slot. 4.4.2ExperimentalResults Figure4-6. TotalCostwithi.i.d.A(t)anddierentbatterycapacityBmaxforinelasticenergydemands. First,weconsidertheimpactofstoragecapacityBmaxoncostsavinginthecaseforinelasticenergydemands.Wecompareouralgorithmagainstasimplealgorithmwithout 114

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Figure4-7. Comparisonoftotalcostforthecaseswhenthereisnobattery,thereisbatterywithinelasticenergydemands,andthereisbatterywithelasticenergydemands. Figure4-8. Comparisonoftotalcostfordierentwithelasticenergydemands. storage.Thesimplealgorithmusestherenewableenergygenerationtomeetthedemandasmuchaspossible;inthecaseofinsuciency,itwoulddrawsomepowerfromthepowergridtomeettheenergydemand.TheresultisillustratedinFigure 4-6 during6-monthperiodwithBmax=f100;150;200gKW-slotandV=Vmaxine.Fromthegure,itisclearthatthelargerthebatteryis,themoresavingourproposedalgorithmcanobtain.Thesavingcomesfromtwoaspects:oneisbystoringexcessiverenewableenergygeneratedincurrenttimeslotforuseatlatertimewhenrenewableenergygenerationisinsucient;theotherisbychargingthebatterywhenpriceislowwhiledischargingitwhenpriceishigh. 115

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Next,wecomparethecasesfortheinelasticenergydemandandtheelasticonewhenBmax=100KW-slot,=1,andV=Vmaxela.TheresultisshowninFigure 4-7 .Thecasewhenthebatteryisusedinconjunctionwithelasticenergydemandprovidesmorespacestooptimizethecostsaving,asillustratedinthegure.Thisresultisintuitiveassomeelasticenergydemandscanbedelayedtotimewhenfreerenewableenergyissucientortheelectricitypriceislow.Finally,weconsidertheimpactofontheperformanceofouralgorithmforthecaseforelasticenergydemands.Asexplainedbefore,isrelatedtotheworst-casedelayofqueuedenergydemands.Smallerimplieslargerdelay.WesetBmax=100KW-slotandV=Vmaxela,andselectdierent2f0;0:3;0:6;1g.AsobservedinFigure 4-8 ,thedecreaseingiveslowercostwiththetradeothattheworst-casedelayisincreased. 4.5SummaryInthischapter,wehaveproposedanapproachtoaddressingtheoptimalmanagementofreal-timepricing,inelasticandelasticenergydemands,renewableenergygeneration,andenergystoragetoreducetheelectricitycostforaresidentialcustomerinthesmartgrid.Theintuitionbehindourapproachistouseenergystoragetoharvestexcessiverenewablegenerationforlateruseandtochargethebatterywhenelectricitypriceislowwhiledischargingitwhenelectricitypriceishigh.ByusingtheLyapunovoptimizationtechniques,wecanachieveclosetooptimalcostwithincreasedbatterycapacity.Specically,ouralgorithmcanndasolutionachievingthecostdeviatednomorethanO(1=V)fromtheoptimalcostwhereVisatunablecontrolparameterdeterminedbythebatterycapacity.Moreover,wehavealsoinvestigatedthecaseforelasticenergydemandandusethetechniqueofvirtualqueuetoguaranteeniteworst-casedelayforanybueredenergydemand.Finally,weusepracticaldatasetstoevaluatetheeectivenessofouralgorithms. 116

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CHAPTER5SMARTNEIGHBORHOODThegrowingdemandsofelectricityandconcernsoverglobalclimatechangeandcarbonemissionhavemotivatedthegridmodernization,whichtransformsthecurrentpowergridstothefuture\smartgrids".Asstatedin[ 50 ],smartgridswillenabledeeppenetrationofrenewablegeneration,customerdrivendemandresponse,widespreadadoptionofelectricvehicles,andelectricenergystorage.Sensing,communication,computation,andcontroltechnologiesinconjunctionwithadvancesinrenewablegeneration,energystorage,powerelectronics,etc.arecriticaltorealizingthevisionandpromiseofasmartgrid.Withinsmartgrids,demandsidemanagement(DSM)isakeycomponent,whichcanhelpreducepeakload,increasegridreliability,andlowergenerationcost[ 70 ].Therearemainlytwotypesofdemandsidemanagementtechniques:directloadcontrol(DLC)anddemandresponsebasedontime-varyingpricing[ 11 ].InDLC,theloadservingentity,usuallyautilitycompany,entersintoacontractwiththeconsumersbeforehand,sothatcertainamountofenergyloadcanbecurtailedduringthepeakhoursinordertoreleasethecongestiononthepowergridortoavoidtheoperationofhighcostpeakgenerators.Currently,itismainlyemployedbylargeindustrialandcommercialcustomers.Ontheotherhand,thedemandresponsebasedontime-varyingpricingencouragesthecustomerstoeitherreduceorshifttheirnormalenergyconsumptionbasedonthepricingsignalissuedbytheloadservingentity(LSE)inreturnforsomebenets,suchaselectricitybillreduction.Severalpopularschemesalreadyexistinthisregard,suchascritical-peakpricing(CPP),time-of-use(TOU)pricing,andreal-timepricing(RTP).Withtheintroductionoftheadvancedmeteringinfrastructure(AMI),whichcanprovidetwo-waycommunicationbetweenutilitycompaniesandsmartmeters,itisexpectedthattherewillbeawidespreaddeploymentofsuchdemandresponseprogramsforresidentialandbusinesscustomersinthesmartgrid[ 29 ]. 117

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Meanwhile,nearly7%ofelectricityislostduringtransmissionanddistribution(T&D)fromremotepowerplantstodistanthomes[ 7 ].Distributedgeneration(DG)frommanysmallon-siteenergysourcesdeployedatindividualhomesandbusinessescanbeusedtodecreasebothT&Dlossesandcarbonemissions.Typicalexamplesofthesesmallon-siteenergysourcesincluderooftopsolarpanels,fuelcells,microturbines,andmicro-windgenerators.Distributedenergystoragedevicesareusuallyusedincombinationwiththeserenewablesourcestobetterutilizethem.Weenvisionresidentialhouseholdsinthesmartgridwhichuseon-siterenewablegeneration,modestenergystorage,andtheelectricgridtomeettheirenergydemands,withinwhichsomeareelasticandcanbeservedinaexiblemanner.Howtosimultaneouslymanagethesecomponentsforhouseholdswithinaneighborhoodinordertoreducethetotalenergycostaswellastheimpactonthedistributionnetworkofthepowergridsisachallengingproblem,especiallyconsideringtherandomdynamicsinthesystem.ThischapterextendsthesinglehouseholdcaseinChapter 4 tothatofmultiplehouseholdswithinasmartgridneighborhood.Inthiswork,notonlydoestheenergycostofanindividualhouseholdmatter,butalsothetotalenergycostinaneighborhoodisofequalimportance.Weshowthat,throughourcollaborativeanddistributedenergyconsumptionschedulingalgorithminmultiplehouseholds,theimpactofthehouseholdenergyconsumptiononthepowersystemandthetotalenergycostcanbegreatlyreduced.Duetothedistributedandonlinepropertiesofourproposedalgorithm,itcanbeeasilyimplementedinthesmartgrid.Insummary,thischaptermakesthefollowingcontributions: Inthesettingofmultiplehouseholdswithinaneighborhood,weproposeanewsystemarchitecturetoincorporatethefollowingessentialcomponentsinthesmartgrid:distributedrenewablegeneration,energystorage,demandresponse,andsmartappliances. Wedevelopadistributedonlinealgorithm,calledLyapunov-basedcostminimizingalgorithm(LCMA),toapproximatelyminimizetime-averagetotalenergycostfor 118

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householdswithinaneighborhoodwithouttheknowledgeofthestatisticsofrelatedstochasticmodels. Throughtheoreticalanalysis,weshowthatouralgorithmcanobtainanexplicittrade-obetweencostsavingandenergystoragecapacity.Moreover,throughextensivesimulationsbasedonreal-worlddatasets,wedemonstratetheeectivenessofourproposedalgorithms.Therestofthischapterisorganizedasfollows.InSection 5.1 ,wedescribeoursystemmodelandformulatetheproblemasastochasticprogrammingproblem.WedescribethedesignprinciplebehindouralgorithmandpresentanonlinealgorithminSection 5.2 .WethenanalyzeouralgorithminSection 5.3 .Wepresentnumericalresultsbasedonreal-worlddatainSection 5.4 .Finally,thischapterendswithasummaryinSection 5.5 . Figure5-1. AschematicdiagramofneighborhoodenergymanagementsystemintheSmartGrid. 5.1ModelingandFormulationInthissection,weprovidemathematicaldescriptionsfortheloadservingentity(LSE),energyload,energystorage,anddistributedrenewablegenerationinresidentialhouseholds.Basedonthesedenitions,wewillformulateourcontrolproblemasastochasticprogram.ConsiderasetofNhouseholds/customersthatareservedbytheLSEinasmartgridneighborhoodsettingasdepictedinFig. 5-1 .TheLSEmaybeautilitycompanyandthesmartgridneighborhoodmaycoverallhouseholdsconnectedtoastep-downtransformer 119

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inthedistributionnetwork.TheLSEmayparticipatesintowholesaleelectricitymarkets(day-ahead,hour-ahead,real-timebalancing,ancillaryservice)topurchaseelectricityfrompowergeneratorsandthensellittotheNcustomersintheretailmarket.Theelectricitypriceintheretailmarketistypicallysetataxedlevelthatreectsthebroadaverageofthehourlycoststoservecustomersoverayearorseason.However,itdoesnotencourageecientusageofelectricity,causinghighpeakdemandandlowloadfactor.Weconsideratime-slottedmodelwithaninnitehorizon.Eachslotrepresentsasuitableperiodforcontroldecisions(e.g.,1houror15min)andisindexedbyt=f0;1;:::g.1 5.1.1LoadServingEntityTheLSEservesasanagentthatisresponsibleforpurchasingenoughelectricityfromwholesaleelectricitymarketstoservetheenergydemandofthehouseholdsinitsservicearea.TheretailpriceissetinordertoatleastrecovertherunningcostoftheLSE.Inthefuturesmartgrid,aeldareanetwork(FAN)wouldbedeployed,whichcanprovideconvenientcommunicationsbetweenutilitycompaniesandsmartmetersofresidentialhouseholds.Forsimplicity,wemaketheassumptionthatthecostoftheLSEcanberepresentedbyacostfunctionC(D)thatspeciesthecostofprovidingamountDofelectricitytotheNcustomersduringoneperiod.WeassumethatthecostfunctionC(D)isincreasing,continuouslydierentiable,andconvexinDwithaboundedrstderivative.WeuseminandmaxtodenotetheminimumandthemaximumrstderivativesofC(D),respectively. 5.1.2EnergyLoadIngeneral,theenergyloadsinahouseholdcanberoughlydividedintotwocategories:inelasticandelasticloads.Examplesofinelasticenergyloadsincludelights,TVs, 1Inthispaper,allpowerquantitiessuchasri(t),si(t),yi(t),di;1(t),di;2(t)areintheunitofenergyperslot,sotheenergyproduced/consumedintimeperiodtisri(t),si(t),yi(t),di;1(t),di;2(t),respectively. 120

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microwaves,andcomputers.Forthistypeofenergyloads,theenergyrequestsmustbemetexactlyatthetimetwhenneeded.Incontrast,therearesomeenergyloadsinhouseholdsthatareelasticinthesensethattheycanbecontrolled(usingsmartappliances,forexample,)toadjustthetimesoftheiroperationsandtheamountoftheirenergyusagewithoutimpactingthesatisfactionofcustomers.Examplesincluderefrigerators,dehumidiers,airconditioners,andelectricvehicles.Asobservedin[ 17 ],whiletheelasticenergyloadscompriselessthan7.5%ofthetotalloadsinahousehold,theyaccountfor59%oftheaverageenergyconsumption.Therefore,thereisgreathiddenpotentialinexploitingtheinherentexibilityofsuchelasticloadsforvariousimportantindividualandsystemlevelobjectives.Insideahousehold,electricloadscancommunicatewiththesmartmeterviathehomeareanetwork(HAN),whichmaybeWi-FiorZigBee.Foreachhouseholdi2N,denotebydi;1(t)theinelasticenergyloads(inunitofkWh)andbydi;2(t)theelasticenergyloads(inunitofkWh)attimet.Asin[ 69 ],weassumethattheelasticenergyloadsare\buered"(i.e.,theenergyrequestsareheldordelayed)rstinaqueueQi(t)beforebeingserved.Denotebyyi(t)theamountofenergythatisusedforservingthequeuedenergyloadsattimet.ThenthedynamicsofQi(t)areasfollows: Qi(t+1)=maxfQi(t))]TJ /F4 11.955 Tf 11.95 0 Td[(yi(t);0g+di;2(t);8i:(5{1)Foreachi,weassumethat 0yi(t)ymaxi;(5{2)whereymaxidmaxi;2sothatthequeueQicanalwaysbestabilized.Foranyfeasiblecontroldecision,weneedtoensurethattheaveragedelayoftheelasticloadsinthequeueisnite.Inotherwords,theserviceofelasticenergyloadscannotbedelayedforarbitrarilylongtime.Thiscanbestatedasfollows: Qi:=limsupT!11 TT)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0EfQi(t)g<1:(5{3) 121

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5.1.3EnergyStorageInadditiontoenergyloads,eachhouseholdmayhavesomekindofenergystoragedevice,possiblyintheformofthebatteryinthePHEV.Foreachhouseholdi,wedenotebyEmaxithebatterycapacity,byEi(t)theenergylevelofthebatteryattimet,andbyri(t)thepowerchargedto(whenri(t)>0)ordischargedfrom(whenri(t)<0)thebatteryduringslott.Assumethatthebatteryenergyleakageisnegligibleandbatteriesathouseholdsoperateindependentlyofeachother.Thenwemodelthedynamicsofthebatteryenergylevelby Ei(t+1)=Ei(t)+ri(t):(5{4)Foreachhouseholdi,thebatteryusuallyhasanupperboundonthechargerate,denotedbyrmaxi,andanupperboundonthedischargerate,denotedby)]TJ /F4 11.955 Tf 9.3 0 Td[(rmini,wherermaxiand)]TJ /F4 11.955 Tf 9.3 0 Td[(rminiarepositiveconstantsdependingonthephysicalpropertiesofthebatteryaswellasthecharginginfrastructure.Therefore,wehavethefollowingconstraintonri(t): rminiri(t)rmaxi:(5{5)Thebatteryenergylevelshouldalwaysbenonnegativeandcannotexceedthebatterycapacity.Soineachtimeslott,weneedtoensurethatforeachhouseholdi, 0Ei(t)Emaxi:(5{6)However,thecostofbatteryusecannotbeignored.Inpractice,batteriescanonlybechargedanitenumberoftimes.Besides,conversionlossoccursbothincharginganddischargingprocesses.Storedenergyisalsosubjecttoleakagewithtime.Allthesefactorsdependonhowfast/much/oftenitischargedanddischarged.Insteadofmodelingthesefactorsexactly,weuseanamortizedtime-invariantcostfunctionFi(ri)(inunitofdollars)tomodeltheimpactofchargingordischargingoperationrionthebatteryduringoneslotforhouseholdi.EachbatterycostfunctionFi(ri)isassumedtobecontinuouslydierentiableinriwithaboundedrstderivativeandFi(0)=0.Weuseminiandmaxi 122

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todenotetheminimumandthemaximumrstderivativesofFi(ri)foreachhouseholdi,respectively. 5.1.4RenewableDistributedGenerationEachhouseholdimaypossessadistributedrenewablegeneratorinstalledonitssite,suchasrooftopPVpanelorsmallwindturbine.Sincerenewablesourcessuchaswindandsolar,areusuallyintermittent,uncertain,anduncontrollable,wemodeltherenewableenergygeneratedbytherenewableDGbyadiscretetimerandomprocesssi(t),whichhasthemaximumvaluegivenbyitsnameplatecapacitysmaxi.Therefore,wehave 0si(t)smaxi8i;t:(5{7)Notethatthepowergenerationfromarenewablegeneratorisusuallylowerthanthenormalpowerconsumptionofresidentialhouseholds.Residentialhouseholdsneedtoconnecttotheutilityelectricgridforbackuppowerand,therefore,aremostlygrid-tiedsystems.Inthispaper,weassumethatthemarginalcostofrenewableenergyiszeroandshouldbeutilizedasmuchaspossible. 5.1.5ProblemFormulationWiththeabovemodelsforthebatteryandthedistributedrenewablegenerator,ateachtimet,thetotalpowerdemandofhouseholdineededfromtheutilityelectricgridis gi(t):=maxfdi;1(t)+yi(t)+ri(t))]TJ /F4 11.955 Tf 11.96 0 Td[(si(t);0g:(5{8)Notethatintheformulaabove,wehaveassumedthatpowercannotbefedfromthehouseholdintotheutilityelectricgridthrough,forexample,netmetering.Sinceweassumethateachhouseholdhasanenergystoragedevice,excessrenewableenergygenerationcanbestoredintoitwithoutspillageaslongasthestoragehasenoughcapacity.Weplantoincorporatetheoptionoftwo-wayenergyowinourfutureinvestigation. 123

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Inthispaper,weareinterestedinminimizingtheLSE'stotalcostofprovidingtheelectricitytothewholesmartgridneighborhoodinasucientlylonghorizon.NotethatreducingthecostofsupplyingelectricityfortheLSEisbothbenecialtotheLSEaswellasindividualcustomerssincethecostwillbenallytransferredtothecustomer'selectricitybill.Therefore,thecontrolproblemcanbestatedasfollows:forthedynamicsystemdenedbyequations( 5{1 )and( 5{4 ),designacontrolstrategywhich,giventhepastandpresentrandomrenewablesupplies,thebatteryenergylevels,theenergydemands,andtheenergycostfunction,choosesthebatterycharge/dischargevectorrandtheelasticloadservingratevectorysuchthatthetime-averagetotalenergycostofthewholesmartgridneighborhoodisminimized.Itcanbeformulatedasthefollowingstochasticprogrammingproblem,calledP1: miny;r:limsupT!11 TT)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0E(C NXi=1gi(t)!+NXi=1Fi(ri(t)));(5{9)subjecttoconstraints( 5{1 ),( 5{2 ),( 5{3 ),( 5{4 ),( 5{5 ),and( 5{6 ).Heretheexpectationintheobjectiveisw.r.t.therandomrenewablegenerationsi(t),therandominelasticenergyloadsdi;1(t),andtherandomelasticenergyloadsdi;2(t)foreachhousehold.DeneP1astheinmumtimeaveragecostassociatedwithP1,consideringallfeasiblecontrolactionssubjecttothequeuestabilityandthenitebatteryenergylevel.Wewilldesignacontrolalgorithm,parameterizedbyaconstantV>0,thatsatisestheconstraintsaboveandachievestheaveragecostwithinO(1=V)oftheoptimalvalueP1,whileguaranteeingthattheworst-casedelayiswithinO(V). 5.2OnlineDistributedAlgorithmInthissection,wedesignalgorithmstosolveP1.Onechallengeofsolvingthestochasticoptimizationproblemaboveistheuncertaintyoffuturerenewablegeneration,time-varyingcostfunction,inelasticorelasticenergyloads.Moreover,theconstraintsonEi(t)bringthe\time-coupling"propertytothestochasticoptimizationproblemabove.Thatistosay,thecurrentcontrolactionmayimpactthefuturecontrolactions, 124

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makingitmorechallengingtosolve.OursolutionisbasedonthetechniqueofLyapunovoptimization[ 67 ]andrequiresminimuminformationontherandomdynamicsinthesystem. 5.2.1Delay-AwareVirtualQueueSincetheconstraint Qi<1onlyensuresniteaveragedelayfortheelasticenergyloadsinhouseholdi,worst-casedelayguaranteeisusuallydesiredinpractice.Forthispurpose,weleveragethetechniqueof\virtualqueue"intheLyapunovoptimizationframework.Specically,thefollowingvirtualqueuesZi(t);i=1;2;:::;Naredenedtoprovidetheworst-casedelayguaranteeonanybueredelasticenergyloadsinQi(t): Zi(t+1)=maxZi(t))]TJ /F4 11.955 Tf 11.95 0 Td[(yi(t)+i1fQi(t)>0g;0;(5{10)where1fQi(t)>0gisanindicatorfunctionthatis1ifQi(t)>0or0otherwise;iisaxedpositiveparametertobespeciedlater.TheintuitionbehindthisvirtualqueueisthatsinceZi(t)hasthesameserviceprocessasQi(t),buthasanarrivalprocessthataddsiwhenevertheactualbacklogisnonempty,thisensuresthatZi(t)growsifthereareenergyloadsinthequeueQi(t)thathavenotbeenservicedforalongtime.ThefollowinglemmashowsthatifwecancontrolthesystemtoensurethatthequeuesQi(t)andZi(t)haveniteupperbounds,thenanybueredenergyloadisservedwithinaworst-casedelayasfollows: Lemma9. SupposewecancontrolthesystemtoensurethatZi(t)ZmaxiandQi(t)Qmaxiforallslotst,whereZmaxiandQmaxiaresomepositiveconstants.Then,theworst-casedelayforallbueredenergyloadsinhouseholdiisupperboundedbymaxislotswhere maxi,(Qmaxi+Zmaxi) i:(5{11) Proof. TheproofhereissimilartothatofLemma 8 andomittedhereforbrevity. WewillshowthatthereindeedexistsuchconstantsZmaxiandQmaxiforallhouseholdsilater. 125

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5.2.2TheLyapunov-basedApproachTheideaofouralgorithmistoconstructaLyapunov-basedschedulingalgorithmwithperturbedweightsfordeterminingtheoptimalpowerusage.Bycarefullyperturbingtheweights,wecanensurethatwheneverwechargeordischargethebattery,theenergylevelinthebatteryalwaysliesinthefeasibleregion.First,wechooseaperturbationvector=(i;8i)(tobespeciedlater).WedeneaperturbedLyapunovfunctionasfollows: L(t):=1 2NXi=1(Ei(t))]TJ /F4 11.955 Tf 11.96 0 Td[(i)2+Q2i(t)+Z2i(t):(5{12)NowdeneK(t)=(Q(t);Z(t);E(t)),anddeneaone-slotconditionalLyapunovdriftasfollows: 4(t)=EfL(t+1))]TJ /F4 11.955 Tf 11.96 0 Td[(L(t)jK(t)g:(5{13)Heretheexpectationistakenovertherandomnessofloadarrivals,costfunction,andrenewablegeneration,aswellastherandomnessinchoosingthecontrolactions.Then,followingtheLyapunovoptimizationframework,weaddafunctionoftheexpectedcostoveroneslot(i.e.,thepenaltyfunction)to( 5{13 )toobtainthefollowingdrift-plus-penaltyterm: 4V(t):=4(t)+VE(C NXi=1gi(t)!+NXi=1Fi(ri(t))jK(t));(5{14)whereVisapositivecontrolparametertobespeciedlater.Then,wehavethefollowinglemmaregardingthedrift-plus-penaltyterm: Lemma10. Foranyfeasibleactionunderconstraints( 5{2 ),( 5{5 ),and( 5{6 )thatcanbeimplementedatslott,wehave4V(t)B+NXi=1Ef(Ei(t))]TJ /F4 11.955 Tf 11.96 0 Td[(i)ri(t)jK(t)g+NXi=1EfQi(t)(di;2(t))]TJ /F4 11.955 Tf 11.96 0 Td[(yi(t))jK(t)g+NXi=1EfZi(t)(i)]TJ /F4 11.955 Tf 11.96 0 Td[(yi(t))jK(t)g+VE(C NXi=1gi(t)!+NXi=1Fi(ri(t))jK(t)); (5{15) 126

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whereBisaconstantgivenbyB:=NXi=1maxf(rmini)2;(rmaxi)2g 2+maxf(ymaxi)2;2ig 2+(ymaxi)2+(dmaxi;2)2 2: (5{16) Proof. From( 5{4 ),subtractingbothsidesbyi,andsquaringbothsides,wehaveforeachhouseholdi, (Ei(t+1))]TJ /F4 11.955 Tf 11.95 0 Td[(i)2)]TJ /F1 11.955 Tf 11.96 0 Td[((Ei(t))]TJ /F4 11.955 Tf 11.96 0 Td[(i)2 2=r2i(t) 2+ri(t)(Ei(t))]TJ /F4 11.955 Tf 11.96 0 Td[(i):(5{17)Moreover,wehavethefollowinginequality: r2i(t) 2maxf(rmaxi)2;(rmini)2g 2:(5{18)Takingexpectationsofbothsidesof( 5{4 )givenK(t),andsummingoverallhouseholdsi,wecangetthefollowingupperboundfortheLyapunovdriftforEi(t))]TJ /F4 11.955 Tf 11.95 0 Td[(i:(Ei(t+1))]TJ /F4 11.955 Tf 11.96 0 Td[(i)2)]TJ /F1 11.955 Tf 11.95 0 Td[((Ei(t))]TJ /F4 11.955 Tf 11.95 0 Td[(i)2 2maxf(rmaxi)2;(rmini)2g 2+ri(t)(Ei(t))]TJ /F4 11.955 Tf 11.95 0 Td[(i): (5{19)Also,from( 5{1 ),squaringbothsides,andusingthefollowinginequality:(maxfQi(t))]TJ /F4 11.955 Tf 11.96 0 Td[(yi(t);0g+di;2(t))2d2i;2(t)+Q2i(t)+y2i(t)+2Q(t)(di;2(t))]TJ /F4 11.955 Tf 11.95 0 Td[(yi(t)); (5{20)weobtainQ2i(t+1))]TJ /F4 11.955 Tf 11.96 0 Td[(Q2i(t) 2(ymaxi)2+(dmaxi;2)2 2+Qi(t)(di;2(t))]TJ /F4 11.955 Tf 11.96 0 Td[(yi(t)): (5{21)Similarly,from( 5{10 ),wehave Z2i(t+1)(Zi(t))]TJ /F4 11.955 Tf 11.96 0 Td[(yi(t)+i)2:(5{22)Then,weobtainthefollowinginequality:Z2i(t+1))]TJ /F4 11.955 Tf 11.96 0 Td[(Z2i(t) 2(i)]TJ /F4 11.955 Tf 11.96 0 Td[(yi(t))2 2+Zi(t)(i)]TJ /F4 11.955 Tf 11.95 0 Td[(yi(t))maxf(ymaxi)2;2ig 2+Zi(t)(i)]TJ /F4 11.955 Tf 11.96 0 Td[(yi(t)): (5{23) 127

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Combiningthesethreeboundstogether,takingtheexpectationw.r.t.K(t)onbothsides,andaddingpenaltytermVEnCPNi=1gi(t)+PNi=1Fi(ri(t))jK(t)otobothsidesoftheaboveinequality,wearriveattheconclusionintheLemma. WenowpresenttheLCMAalgorithm.ThemaindesignprincipleofouralgorithmistochoosecontrolactionsthatapproximatelyminimizetheR.H.S.of( 5{15 ).Lyapunov-basedCostMinimizationAlgorithm(LCMA): Initialize(i;i);8iandV.Ateachslott,observe(di;1(t);di;2(t);si(t));8i,andK(t),anddo: Choosecontroldecisionsyandrastheoptimalsolutiontothefollowingoptimization,calledP3:minimize0yi(t)ymaxi;8i;rminiri(t)rmaxi;8iNXi=1f(Ei(t))]TJ /F4 11.955 Tf 11.96 0 Td[(i)ri(t)+VFi(ri(t)))]TJ /F1 11.955 Tf 11.95 0 Td[((Qi(t)+Zi(t))yi(t)g+VC NXi=1gi(t)!: UpdateK(t)accordingtothedynamics( 5{1 ),( 5{4 ),and( 5{10 ),respectively.Theintuitionbehindouralgorithmistryingtostoreexcessrenewableenergyforlateruse,rechargethebatteryduringtheperiodoflowelectricitypricewhiledischargingitduringtheperiodofhighelectricityprice,anddelayelasticenergyloadstolaterslotswithlowerelectricityprice.Notethatwedonotneedtoconsiderthetime-couplingconstraints( 5{6 )ofthebatteryenergylevelinthealgorithm,sincetheycanbeautomaticallysatisedduringouroperationofthequeues,asproveninTheorem 5.1 below.Moreover,thealgorithmonlyrequirestheknowledgeoftheinstantaneousvaluesofsystemdynamicsanddoesnotrequireanyknowledgeofthestatisticsofthesestochasticprocesses.However,thealgorithmaboveshouldbeabletoruninadistributedmannerinordertobeimplementedinpractice.Intheensuingsubsection,wedesignadistributedalgorithmtosolvetheoptimizationproblemP3. 128

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5.2.3DistributedAlgorithmtoP3First,weintroducethefollowingslackvariables:hi(t);8itoupperboundindividualgridpowerdemandandD(t)toupperboundthetotalgridpowerdemand.Then,wecantransformP3intothefollowingformulation,calledP4: minimizeNXi=1f(Ei(t))]TJ /F4 11.955 Tf 11.96 0 Td[(i)ri(t)+VFi(ri(t)))]TJ /F1 11.955 Tf 11.95 0 Td[((Qi(t)+Zi(t))yi(t)g+VC(D(t)); (5{24a)s.t.0yi(t)ymaxi;8i (5{24b)rminiri(t)rmaxi;8i (5{24c)hi(t)di;1(t)+yi(t)+ri(t))]TJ /F4 11.955 Tf 11.96 0 Td[(si(t);8i (5{24d)0hi(t)hmaxi;8i (5{24e)NXi=1hi(t)D(t)Dmax; (5{24f)wherethemaximumgridpowerconsumptionhmaxiisimposedbecauseofsecurityandreliabilityconsiderationsforhouseholdi,andDmaxisthetransformercapacity.SinceC()isastrictlyincreasingfunction,wecaneasilyprovebycontradictionthattheformulationaboveandP3areequivalentandhaveexactlythesameoptimalsolutionsintermsofrandy.SinceP4isaconvexoptimizationproblemandhasdecomposabilitystructures,itmotivatesustodesignthefollowingdistributedsubgradient-basedalgorithmtoiterativelysolveitbasedontheideaofconsistencypriceinnetworkutilitymaximization[ 71 ].Ineachtimeslott,thealgorithmimplementsthestepsasindicatedinAlgorithm 5 .TheowchartforAlgorithm 5 isshowninFig. 5-2 .Whentheconstantstep-sizeissmallenough,thealgorithmaboveconvergestotheoptimalsolution[ 19 ].Notethatothertypesofstep-sizecanalsobeusedwithdierentconvergenceproperties[ 24 ].AdesirablefeatureofourdistributedalgorithmisthattheLSEdoesnotneedtoknowthedetailedinformationabouttheenergyusageineachindividualhouseholdandonlyrequiresthetotalgridenergyusageforallNhouseholds.Byoperatinginthismanner,ouralgorithm 129

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canhelppreservetheprivacyofhomeowners,whoareshowntobeconcernedwithsomeprivacyissuesassociatedwiththethesmartgrid[ 23 ]. Figure5-2. FlowchartfortheAlgorithm 5 . NotethattheoptimalLagrangianmultiplier(t)issimilartothedynamicelectricityretailpricechargedbytheLSEtoeachhouseholdattimet. 5.3PerformanceAnalysisInthissection,weanalyzetheperformanceofLCMAunderthecasethattherenewableenergygenerationsi(t);8i,energyloadarrivalprocessesdi;1(t);8ianddi;2(t);8iarealli.i.d.,Notethatourresultscanalsobeextendedtothemoregeneralsettingwheresi(t);8i,di;1(t);8i,anddi;2(t);8iallevolveaccordingtosomenitestateirreducibleandaperiodicMarkovchainsaccordingtotheLyapunovoptimizationframework[ 67 ]. Theorem5.1. IfQi(0)=Zi(0)=0andi=V(max+maxi))]TJ /F4 11.955 Tf 12.28 0 Td[(rminiforallhouseholdsi,thenundertheLCMAalgorithmforanyxedparameters0iEfdi;2(t)g,and 130

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Algorithm5:DistributedAlgorithmtoP3 1Initialization:Set(0)(t)equaltosomenonnegativevalue,k=0 2foreachIterationkdo 3whileNotsatisfyingconvergencecriteriondo 4Thehomeenergymanagementsystem(HEMS)inhouseholdi'ssmartmeterupdatesr(k)i(t),y(k)i(t),andh(k)i(t)afterreceivingtheLagrangianmultiplier(k)(t)accordingtothesolutiontothefollowingoptimizationproblem: minimize(Ei(t))]TJ /F4 11.955 Tf 11.96 0 Td[(i)ri(t)+VFi(ri(t))+(k)(t)hi(t))]TJ /F1 11.955 Tf 11.96 0 Td[((Qi(t)+Zi(t))yi(t); (5{25a)s.t.hi(t))]TJ /F4 11.955 Tf 11.96 0 Td[(yi(t))]TJ /F4 11.955 Tf 11.95 0 Td[(ri(t)di;1(t))]TJ /F4 11.955 Tf 11.95 0 Td[(si(t); (5{25b)0hi(t)hmaxi; (5{25c)rminiri(t)rmaxi; (5{25d)0yi(t)ymaxi: (5{25e) 5TheLSEcollectsthepredicationsoftotalutilitypowerdemandsPNi=1hki(t)fromallhouseholdsiovertheFAN.Then,itsneighborhoodenergymanagementsystem(NEMS)obtainstheoptimalgenerationpowerandupdatestheLagrangemultiplierasfollows: D(k)(t)=argmin0D(t)DmaxVC(D(t)))]TJ /F4 11.955 Tf 11.95 0 Td[((k)(t)D(t);(5{26) (k+1)(t)="(k)(t))]TJ /F4 11.955 Tf 11.96 0 Td[( D(k)(t))]TJ /F5 7.97 Tf 16.81 14.95 Td[(NXi=1h(k)i(t)!#+;(5{27)where>0isaconstantstep-size,andthen,broadcasts(k+1)(t)toallhouseholdsovertheFAN. 6Setk k+1. 0
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1. ThequeuesQi(t)andZi(t)aredeterministicallyupperboundedbyQmaxiandZmaxiateveryslot,whereQmaxi:=Vmax+dmaxi;2; (5{29)Zmaxi:=Vmax+i: (5{30)Further,Qi(t)+Zi(t)areupperboundedbymaxiwhere maxi:=Vmax+dmaxi;2+i:(5{31) 2. Theworst-casedelayofanybueredelasticenergyloadisgivenby: maxi=2Vmax+dmaxi;2+i i:(5{32) 3. TheenergyqueueEi(t)satisesthefollowingforalltimeslotst: 0Ei(t)Emaxi:(5{33) 4. Allcontroldecisionsarefeasible. 5. Ifsi(t);8i,di;1(t);8i,anddi;2(t);8iarei.i.d.overslots,thenthetime-averageexpectedoperatingcostunderouralgorithmiswithinboundB=Voftheoptimalvalue,i.e.,limsupT!11 TT)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0E(C NXi=1gi(t)!+NXi=1Fi(ri(t)))P1+B V;whereBistheconstantspeciedin( 5{16 ). Proof. Theproofisalsoastraightforwardextensionoftheresultsinourpreviouswork[ 42 ]intothecaseofmultiplehouseholds.Weprovidethesketchofourproofasfollows. 1. Weprovetheresultsbyinduction.First,ifQi(t)Vmax,themaximumincreaseduringoneslotisdmaxi;2.Therefore,weobtaintheupperboundinthiscase.Second,ifVmax
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isnegative.Thearrivalamountcannotbelargerthantheservedamountbyourassumption.Therefore,thequeuelengthcannotincrease.Thiscompletestheproof.TheupperboundofZi(t)andQi(t)+Zi(t)canbeprovedsimilarly. 2. ThisfollowsdirectlyfromLemma 9 . 3. Onceagain,weprovetheresultbyinduction.If0Ei(t)
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A B CFigure5-3. (a)ConvergenceofAlgorithm1;(b)Comparisonofthetotalenergycostinthreeapproaches;(c)Histogramofdelayfortheelasticdemandsintheservicequeueforonehousehold. 5.4.1ExperimentalSetupWeconsiderasimplepowersystemconsistingofeighthouseholdsinoneneighborhoodthatsharethesameloadservingentityandhaveon-siterenewablegeneration,energystorage,andelasticandinelasticenergyloads.Thehouseholdsaredividedintotwocategories.Forthersttypeofhouseholds(indexedbyi=1;2;3;4),boththeelasticandinelasticenergyloadarrivalsduringoneslotarei.i.d.andtakevaluefrom[1;5]kWhuniformlyatrandom.Forthesecondtypeofhouseholds(indexedbyi=5;6;7;8),boththeelasticandinelasticenergyloadarrivalsduringoneslotarealsoi.i.d.andtakevaluefrom[1:5;7:5]kWhuniformlyatrandom.Fortherenewablegeneration,weusethehourlyaveragesolarirradiancedataforLosAngelesareafromtheMeasurementandInstrumentationDataCenter(MIDC)[ 2 ]atNationalRenewableEnergyLaboratory.TheperiodweconsiderinthispaperishalfyearfromJanuary1,2011toJune30,2011.Intotal,thisdurationincludes181daysor43441-hourslots.Thecontrolintervalischosentobe1-hour.Weusedierentscalingfactorstocharacterizetheheterogeneityofhouseholds.Specically,wechoosethescalingfactorssuchthattheaveragesolarenergyproductionduringoneslotisabout3kWhforthersttypeofhouseholdsand4.5kWhforthesecondtypeofhouseholds.Wexthemaximumchargeanddischargeratesofbatteriesinhouseholdsasfollows:fori2f1;2;3;4g,rmaxi=1kWh,rmini=)]TJ /F1 11.955 Tf 9.3 0 Td[(1kWh,and 134

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A B CFigure5-4. (a)Theimpactofbatterycapacityonthecostsaving;(b)Theimpactofbatterycostb1onthecostsaving;(c)Theimpactofionthecostsaving fori2f5;6;7;8g,rmaxi=1:5kWh,rmini=)]TJ /F1 11.955 Tf 9.3 0 Td[(1:5kWh.Also,wechooseymaxi=dmaxi;2foralli.As[ 55 ],thebatterycostisassumedtobeasimplequadraticfunctionasfollows: Fi(ri)=b1r2i;(5{34)whereb1isaconstantcoecient.Forthepurposeofsimpleillustration,wechoosethesamebatterycostfunctionforallhouseholdsiintheevaluations.FortheLSE,weassumethattheenergycostfunctionisasmoothquadraticfunctionasfollows: C(D)=c1D2+c2D+c3;(5{35)wherec1,c2,andc3areconstantcoecients.Wechoosec1=0:1,c2=0:1,andc3=0:2inourevaluations. 5.4.2ExperimentalResultsInordertoanalyzetheperformanceimprovementduetoourLCMA,wecompareitwiththefollowingtwoapproaches:(i)Nostorage,nodemandresponse(B1):Thehouseholdtriestousetherenewableenergyasmuchaspossible.Whentherenewableenergyisnotsucient,thehouseholddrawsenergyfromtheutilitygrid.Unusedrenewableenergyiswasted;(ii)Storage,nodemandresponse(B2):Thehouseholdusesrenewableenergyonlyasasupplementtothegridbyconsumingitwheneverit 135

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isavailable.Thehouseholdstoresanyextrarenewableenergyinitsbattery,butneverchargesthebatteryfromthegrid.Thestoredenergywouldbeusedtoservethefuturedemands.NotethatLCMAdiersfromtheapproachesaboveinthesensethatLCMAwouldactivelychargethebatterywhenthegridpowerischeapwhiledischargingitwhenthegridpowerisexpensive.Moreover,LCMAdierentiatesbetweeninelasticandelasticenergyloadsanddelaystheelasticenergyloadstolatertimewhenthegridpowercostislow.First,westartbyconsideringtheconvergenceofthedistributionalgorithm.Figure 5-3A illustratestheconvergenceofAlgorithm 5 .Weuseastepsize=0:1andthealgorithmshowsnosignoflackofconvergence.Then,wecompareouralgorithmwiththetwoapproachesaboveusingthereal-worldsolarpowergeneration.NotethattheperformanceofLCMAdependsonthebatterycapacity,thebatterycost,andthecontrolparametersVandi.Wechooseb1=0:5,Emaxi=20kWh;i2f1;2;3;4g,andEmaxi=30kWh;i2f5;6;7;8g.Theinitialbatteryenergylevelateachhouseholdischosentobezero.LetV=Vmaxandi=Efdi;2g;8i.AscanbeseeninFigure 5-3B ,ourproposedLCMAcanreducethetotalenergycostbyapproximately20%comparedwithB1and13%comparedwithB2inthesix-monthperiod.Also,theslopesofthelinesaredierent,meaningthatthesavingsareunboundedasthetimeincreases.Meanwhile,theLCMAhasonaverageamuchsmallerdelaythantheworst-caseguarantee( 5{11 ),asshowninFig. 5-3C .Inthefollowing,weconsidertheimpactofvaryingcontrolparametersontheperformanceofLCMA. ImpactofBatteryCapacity:Inthisevaluation,wevarythebatterycapacitiesofhouseholdswithotherparametersxed.WesetEmaxi=f20;30;40gkWhfori2f1;2;3;4g,Emaxi=f30;40;50gkWhfori2f5;6;7;8g,andV=Vmax.TheresultisillustratedinFigure 5-4A .Fromthegure,itisclearthatthelargerthecapacityis,themorecostsavingLCMAcanobtain,whichcoincideswiththealgorithmicperformanceresultsofouralgorithminTheorem 5.1 .Aswehavementionedbefore,thesavingcomesfromthefactthatouralgorithmchargesthe 136

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batterywhenthemarginalenergycostislow,whiledischargingitwhenthemarginalenergycostishigh. ImpactofBatteryCost:Currently,thebatteryisstillexpensive.Thechargingordischargingoperationwouldreducethelifetimeofthebattery.However,itisexpectedthatthecostofbatterywoulddecreasegreatlyinthenextdecade.Inthisevaluation,weestimatetheimpactofbatterycostonthecostsavingofouralgorithm.Wesetb1=f0;1;2;5;20;200gandkeepEmaxi=20;i2f1;2;3;4g,andEmaxi=30;i2f5;6;7;8gxed.TheresultisshowninFigure 5-4B .Notethatwhenthebatterycostperusageduringoneperiodb1isverylarge(e.g.,200$),ouralgorithmwouldnotchargeordischargethebatteryatall,soitisthesameastheapproachB1.Asthebatterycostincreases,thetotalcostsavingofLCMAcomparedwithB1woulddecreaseuntiltheyarethesamesincetheopportunitytoutilizethetemporalvariationofelectricitypricesissmaller. ImpactofWorst-caseDelayRequirement:Inthissetting,weadjusttheparametersiwhilexingotherparameterstoseetheimpactoftheworst-casedelayguaranteeforelasticenergyloadsontheperformanceofLCMA.Wechoosei=f0;1;2;3g;8i,respectively.AsobservedinFigure 5-4C ,theincreaseofi(i.e.,theworst-casedelay)givesmoreopportunitytooptimizetheenergycost,sincetheelasticenergyloadsaremorelikelytobeservedinthelowenergycostperiod. 5.5SummaryInthischapter,wehavepresentedanalgorithmcalledLCMAforthedistributedandcoordinatedstochasticoptimizationofexibleenergyresourcesinasmartgridsetting.Thetotalsystemcostcanbereducedifmoreenergyloadsareelasticandcantoleratebeingservedwithsomedelay.Ouralgorithmissimpleandhasbeenshowntobeabletooperatewithoutknowingthestatisticalpropertiesoftheunderlyingdynamicsinthesystem.Withtheincreaseofenergystoragecapacities,theperformanceofouralgorithmisprovedtobearbitrarilyclosetotheoptimalvalue.Moreover,ouralgorithmprovidesanexplicitrelationshipbetweenenergystoragecapacity,worst-casedelay,andcostsaving.Extensivenumericalevaluationsbasedonreal-worlddatasetsshowtheeectivenessofourapproach. 137

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CHAPTER6IMBALANCECOSTMINIMIZATIONFORVIRTUALPOWERPLANTSTheissueofglobalclimatechangeandthelimitedsupplyofcarbon-intensivefossilfuelshavecalledforutilizationofcleanrenewableenergysources(RESs)suchassolar,wind,andgeothermal.Manygovernmentsaroundtheworldhavesetuprenewableportfoliostandards(RPSs)topromotetheproductionofenergyfromRESs.Asoftoday,renewableenergyproducersareusuallyenrolledintothefeed-intari(FIT)programs,inwhichtheyreceiveguaranteedgridaccessandfavorableregulatedFITs.However,thisapproachmaynotbesustainablesinceFITshaveaxedexpirydate,mostly12-15years,afterwhichtherenewableenergyproducerbecomesanon-subsidizedproducerthatneedstoparticipateinelectricitymarkets[ 84 ].Moreover,theeectivenessofFITsonreducingthecarbonemissionisindoubtnow,asillustratedbytheGermanenergymarket[ 18 ].Duetothestochasticvariationsofenergyproducedfromnon-dispatchablerenewabledistributedgenerators(RDGs)suchassolarandwindpowerunits,theriskforasingleunittoparticipateintheelectricitymarketisveryhigh.Iftheactualgeneratedenergyissmallerthanthescheduledenergyoutput,therenewablepowerproducerhastobuyexpensivebalancingenergytocovertheshortage.Onthecontrary,iftheactualgeneratedenergyislargerthanthescheduledenergy,theexcessenergyhastobesoldatalesscompetitivepriceorevencurtailedinsomecircumstances.Theuncertaintyisoftentoohightomakethemarketparticipationimpossible.AnotherproblemwithRDGsisthattheircapacities(e.g.,inscaleofkW)areoftentoosmalltoparticipateintheelectricitymarket,wheretheminimumtradingamountofhourlycontractisinunitofMWforenergy.Todealwiththeseissues,theframeworkofvirtualpowerplant(VPP)[ 76 ]hasbeenproposedtofacilitatecost-ecientintegrationofdistributedenergyresources(DERs)intotheexistingpowersystems.DERs,includingdistributedgeneration,distributedstorage,andcontrollableloads,havethebenetsofreducingcarbonemissionsandimprovingthe 138

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powerqualityandreliabilityduetothepresenceofgenerationclosetodemand.VPPcanenhancethevisibilityandcontrolofDERstosystemoperatorsandothermarketplayersbyprovidinganappropriateinterfacebetweenthesesystemcomponents.ByaddingmanyDERunitsinoneclusterandconnectingthemwithaninformationnetwork,thestochasticvariationscanbebalancedbetweenlotsofsingleDERunits.Therefore,theclusterofDERsiscomparabletoaconventionalpowerplantandcanparticipateintheelectricitymarket.Inthischapter,weconsideraVPPconsistingofRDGsandendconsumers,whohavebothinelasticandelasticenergydemand.Specically,weinvestigatetheoptimalrenewablepowercurtailmentanddemandresponsestrategyfortheVPPinthebalancingmarketaftertheday-aheadbidhasbeensubmitted(i.e.,aftertheday-aheadmarkethasbeencleared)sothattheimbalancecostincurredinthebalancingmarketisminimized.Consideringthestochasticnatureofrenewablepowergeneration,demandarrivals,andmarketprices,theproblemisformulatedasastochasticprogram.BasedontheLyapunovoptimizationtechnique,weproposeacontrolalgorithmtosolveitapproximately.Theadvantagesofourproposedalgorithmarethatitissimpletoimplementandcanoperatewithouttheknowledgeofdetailedprobabilitydistributionsofrelatedrandomprocesses,whichmaybediculttoobtaininpractice.Ourworkiscomplementarytopreviousresearch[ 22 , 65 , 72 ]ontheoptimalday-aheadbiddingstrategyforrenewableenergyproducersorVPPsbyfocusingonthereal-timeoperationstrategyinthebalancingmarket.Also,dierentfrom[ 69 ],whichdoesnotconsideranyelectricitymarketoperationwhensellingrenewableenergytodelay-tolerantconsumers,ourworkfocusesonthemarketparticipationofavirtualpowerplantwithguaranteedlevelofrenewableenergyutilization.Therestofthischapterisorganizedasfollows:InSection 6.1 ,wedescribethemodelsweuseinthispaperandpresentthemathematicalformulationoftheproblem.Section 6.2 introducestheproposedcontrolalgorithmbasedontheLyapunovoptimizationtechnique. 139

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Figure6-1. StructureofaVPP PerformancepropertiesoftheproposedalgorithmisanalyzedinSection 6.3 .Weconductanumericalstudyofourproposedalgorithmonreal-worlddatasetsinSection 6.4 .Finally,thischapterendswithasummaryinSection 6.5 . 6.1ProblemFormulationWeconsideraVPPconsistingofnon-dispatchablerenewabledistributedgenerators(RDGs)andendconsumers(havingbothinelasticandelasticdemand),asshowninFigure 6-1 .Anexampleisthedistributioncompany(DISCO),whichownsbothDGsandhasanobligationtomeettheloadsinitsterritory.TheVPPwouldparticipateintoacompetitiveelectricitymarket,eitherasaconsumeroraproducer.Inthispaper,weconsiderasucientlylongschedulinghorizonwheret=0;1;:::;TforalargeT. 6.1.1RenewableDistributedGeneratorModelWeassumethatonedayahead,theVPPcanforecasttheaggregatedoutputofRDGs W(t)foreachtimeperiodtofthecomingday.However,theactualoutputmaynotbeexactlythesameastheforecastedvalueduetotheforecasterror.WemodelthiseectbyassumingthattheactualaggregatedrenewableoutputofRDGsisasfollows: W(t)= W(t)+Nw(t);(6{1) 140

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whereNw(t)isarandomvariablewithunknowndistribution.Anyforecastingmethodcanbeused.TherealvalueoftherenewablepowergenerationW(t)isassumedtobeknownonlyatthebeginningofthetimeperiodtduringtheoperationday.DenotethenameplatecapacityoftheaggregatedRDGsasWmax.Weknowthat0W(t)Wmaxforallt.Asanalyzedin[ 22 ],itmaybenecessarytocurtailsomeofthegeneratedrenewablepowerinordertoreducetheimbalancecostwhentransmissioncongestionexistsorthereisoverproductioninthewholesystem.Sincewedonotconsiderthestorageinthiswork,theunusedrenewablepowerisspilled.DenotetheactualutilizedrenewablepowerattimetasY(t).Obviously,wehave 0Y(t)W(t):(6{2)Tocomplywithsomerenewableenergyutilizationregulations(e.g.,collectingenoughrenewableenergycreditstosatisfyRPS),theVPPhastoensurethatatleastacertainfractionofactualgeneratedrenewablepowerfromRDGsisutilized[ 90 ].Tomodelit,weassumethatthetime-averagerenewableutilizationlevelmustbenosmallerthansomethresholdw(e.g.,90%).Therefore,theconstraintforrenewableenergyutilizationcanbestatedas limT!11 TT)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0Y(t)wlimT!11 TT)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0W(t):(6{3) 6.1.2DemandModelWedividetheenergydemandofendconsumersintheVPPintotwocategories:inelasticdemandandelasticdemand.Inelasticdemandisthebaseloadsthatcannotbedelayedandneedtobeservedimmediately.Wemodelitasthefollowingdiscrete-timestochasticprocessDb(t),representingthetotalamountofrequestedenergyattimet: Db(t)= Db(t)+Nb(t);(6{4)where Db(t)istheday-aheadforecastofinelasticdemandsandNb(t)istheforecasterrorfortimetofthecomingday.Onceagain,Nb(t)isassumedtobearandomvariablewith 141

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someunknowndistribution,whichcanbeobservedonlyatthebeginningoftimeperiodtduringtheoperationday.Moreover,weassumethatDb(t)Db;maxforallt.Ontheotherhand,elasticdemandsareexibleandcanbeadjustedinreal-timewhenneeded.TypicalexamplesoftheseelasticdemandsareEVchargingandairconditioning.Theseenergydemandsareelasticinthesensethataslongastherequiredamountofenergyissatisedwithinsomeperiod,itwouldnotimpacttheusercomfort.AssumethatthereareNtypesofelasticdemand,whereeachtypemayhasadierentrequirementonthemaximumtolerabledelay.Foreachtypei,thedemandsarriverandomlyaccordingtoaprocessDi(t)(beingtheamountofenergythatisrequestedontimet)characterizedasfollows: Di(t)= Di(t)+Ni(t);(6{5)where Di(t)canbeforecastedonedayaheadandNi(t)istheforecasterrorfortimeperiodtofthecomingday.Sinceweassumethatconsumersrequestingenergyareexibleandcantoleratetheirenergyrequestsbeingsatisedwithsomedelay,therequestsarebueredinaqueue.Foreachtypei,letQi(t)bethetotalenergyrequests(inunitofrequiredenergy)inthequeueattimet,wehavethefollowingupdateequation: Qi(t+1)=maxfQi(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Xi(t)+Di(t);0g;(6{6)whereXi(t)isacontrolvariable,representingtheamountofenergythatisallocatedtoservethebueredrequestsattimet.WeneedtoensurethequeueQi(t)isstabilizedsothatthebueredenergyrequestsarenotdelayedinnitelylong.WealsoassumeanupperboundXi;maxontheallocatedenergyamountduringonetimeperiodforeachtypeiofelasticdemands.ThemaximumamountofenergyrequestarrivalsfortypeiduringonetimeslotisdenotedasDi;max,andwefurtherassumethatXi;maxDi;maxsothatitisalwayspossibletostabilizethequeueQi(t).Therefore,for 142

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eachtypeiofelasticdemand,wehave 0Xi(t)Xi;max:(6{7) 6.1.3MarketModelAswith[ 22 ],weconsideraconventionaltwo-settlementmarketsystemconsistingofanex-anteday-aheadforwardmarketwithanex-postimbalancesettlementmechanismtopenalizeuninstructeddeviationsfrombidssubmittedex-ante.Inthismarket,theVPPneedstosubmithourlybiddingschedulesB(t)ofproduction(B(t)>0)orconsumption(B(t)<0)intheday-aheadmarket(e.g.,by10AMofthepreviousday)forthenextdaybasedonitsforecast.WeassumethatBminB(t)Bmax;8t.Theschedulesclearedintheday-aheadmarketarenanciallybindingandaresubjecttodeviationpenalties.Inthebalancingmarket,thesystemoperatorwouldemployasettlementmechanismtocomputetheimbalancepricesforpositiveandnegativedeviationsfromtheday-aheadbid.Notethatrenewablepowercurtailmentcanhelpreducethedeviationinsomecircumstances.However,theVPPcannotspilltoomuchrenewablepowerasitmaybesubjecttocertainregulationsthatrequirerenewableenergyaccountingforacertainshareoftheirtotalgenerationoutput.Moreover,theVPPcanexploittheelasticdemandsinitsportfoliotohelpreducetheimbalancecost.Foreachtimet,therelatedmarketpricesaredenotedasfollows: p(t):settlementpriceintheday-aheadmarket q+(t):positiveimbalancepriceinthebalancingmarket q)]TJ /F1 11.955 Tf 7.09 -4.33 Td[((t):negativeimbalancepriceinthebalancingmarketAsanalyzedin[ 65 ],theimbalancepricesandtheday-aheadpricessatisfythefollowingrelationshipinmostelectricitymarkets: )]TJ /F4 11.955 Tf 11.95 0 Td[(q+(t)p(t)q)]TJ /F1 11.955 Tf 7.09 -4.93 Td[((t):(6{8)Theaboveinequalityillustratesthefollowingtwofacts: 143

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theVPPcannotgainmoreprotbytradinginthebalancingmarketratherthantradingintheday-aheadmarket. thenegativeimbalancecostisnolessthanthecostofpurchasingelectricityintheday-aheadmarket.Moreover,weassumethatq+(t)q+maxandq)]TJ /F1 11.955 Tf 7.09 -4.33 Td[((t)q)]TJ /F5 7.97 Tf -.43 -7.29 Td[(max.Withthenotationsdenedabove,thepowerbalanceequationineachtimeperiodtisasfollows: Y(t)=B(t)+(t)+Db(t)+NXi=1Xi(t);(6{9)where(t)isthenetimbalancebetweentheday-aheadmarketandthebalancingmarket.Morespecically,wedenotethepositiveimbalance+(t)andthenegativeimbalance)]TJ /F1 11.955 Tf 7.09 -4.33 Td[((t),respectively,asfollows:+(t)=max"Y(t))]TJ /F4 11.955 Tf 11.96 0 Td[(B(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Db(t))]TJ /F5 7.97 Tf 16.8 14.94 Td[(NXi=1Xi(t);0#; (6{10))]TJ /F1 11.955 Tf 7.08 -4.93 Td[((t)=max"B(t)+Db(t)+NXi=1Xi(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Y(t);0#: (6{11)Obviously, (t)=+(t))]TJ /F1 11.955 Tf 11.96 0 Td[()]TJ /F1 11.955 Tf 7.09 -4.93 Td[((t):(6{12) 6.1.4VPPOperationModelInthispaper,weassumethatVPPisapricetakersinceitscapacityissmallrelativetothewholemarket.Oursystemworksasfollows:onedayahead,theVPPcanforecasttheoutputfromRDGsandtheinelasticandelasticdemandsforthenextday.Sincewefocusprimarilyonthereal-timeoperatingofVPP,weassumeinthispaperthattheVPPwillbidtheforecastedproductionminustheforecastedconsumptionintotheday-ahead 144

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marketforeachtimetofthecomingday1: B(t)= W(t))]TJ ET q .478 w 237.34 -37.98 m 251.18 -37.98 l S Q BT /F4 11.955 Tf 237.34 -47.82 Td[(Db(t))]TJ ET q .478 w 279.12 -37.98 m 293.27 -37.98 l S Q BT /F4 11.955 Tf 279.12 -47.82 Td[(De(t):(6{13)Inreal-timeattimet,aftertheVPPknowstheaccuraterenewablepoweroutput,demandarrivals,aswellastheimbalanceprices,itwoulddecidetheamountofcurtailedrenewablepowerandenergyallocatedtoserveelasticdemands.TheprotofVPPacquiredattimetcanbestatedas f(t)=p(t)B(t))]TJ /F4 11.955 Tf 11.96 0 Td[(q+(t)+(t))]TJ /F4 11.955 Tf 11.96 0 Td[(q)]TJ /F1 11.955 Tf 7.09 -4.93 Td[((t))]TJ /F1 11.955 Tf 7.08 -4.93 Td[((t):(6{14)SinceB(t)isgivenby( 6{13 ),theprotgainedfromtheday-aheadmarketisindependentofthecontroldecisionsinourproblem.Therefore,wefocusonminimizingthefollowingimbalancecostincurredinthebalancingmarketattimet: c(t)=q+(t)+(t)+q)]TJ /F1 11.955 Tf 7.08 -4.93 Td[((t))]TJ /F1 11.955 Tf 7.08 -4.93 Td[((t)(6{15)Consideringasucientlylongschedulinghorizon,theproblemofminimizingimbalancecostcanbeformulatedasfollows: minimizelimT!11 TT)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0Efc(t)g(6{16)subjecttoconstraints( 6{2 ),( 6{3 ),( 6{7 ),( 6{9 ),( 6{10 ),( 6{11 ),( 6{12 ),and Q,limT!11 TT)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0NXi=1EfQi(t)g<1:(6{17)Notethattheexpectationhereisw.r.t.therandomrenewablepoweroutput,imbalanceprices,demandarrivals,aswellaspossiblyrandomizedcontroldecisions.Notethattheaboveproblemisdenedonlyintermsofaqueuestabilityconstraintanddoesnotimposeanyadditionaldelayconstraintforthebueredelasticdemand 1Otherbiddingstrategiessuchas[ 22 ]arepossibleandourworkprimarilyfocusesonthereal-timeoperationstrategyratherthantheday-aheadbiddingstrategy. 145

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requests.Theaboveproblemischallengingtosolve,especiallyconsideringthatthefutureinformationisunknownandthestatisticalpropertiesofrenewablepoweroutput,marketprices,andenergydemandsmaybehardtoobtain.Toaddressthischallenge,wewillconstructanalgorithmwithaparameterVthatachievestheaveragecostwithinO(1=V)oftheoptimalvaluewhiletheworst-casedelayisnomorethanO(V)inthefollowing. 6.2ProposedAlgorithmFirst,weintroducethefollowingvirtualqueuetoensuretheconstraintforrenewableenergyutilizationlevel( 6{3 ): Zw(t+1)=maxfZw(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Y(t);0g+wW(t):(6{18)Wehavethefollowingresultregardingtheequivalencebetween( 6{3 )andqueuestability: Lemma11. IfwecancontrolthesystemsuchthatvirtualqueueZw(t)isstabilized,thenconstraint( 6{3 )issatised. Proof. Fromthequeueupdateequation( 6{18 ),wehave Zw(t+1)Zw(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Y(t)+wW(t):(6{19)Summingovert=0;1;:::;T)]TJ /F1 11.955 Tf 11.96 0 Td[(1foranyT>0onbothsides,weobtain Zw(T))]TJ /F4 11.955 Tf 11.95 0 Td[(Zw(0))]TJ /F5 7.97 Tf 24.09 14.94 Td[(T)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0Y(t)+wT)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xt=0W(t):(6{20)DividingbothsidesbyTandletT!1,wehavelimT!1Zw(T))]TJ /F4 11.955 Tf 11.96 0 Td[(Zw(0) T)]TJ /F1 11.955 Tf 27.3 0 Td[(limT!11 TT)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xt=0Y(t)+wlimT!11 TT)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xt=0W(t): (6{21)NotethatZw(0)isniteandifwecancontrolthesystemsuchthatthequeueZi(t)isratestable,i.e.,limT!1Zi(T)=0,then( 6{3 )issatised. 146

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Second,toguaranteetheniteworst-casedelayforelasticdemandtypei,weintroducethefollowingvirtualqueue: Zi(t+1)=maxZi(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Xi(t)+Di(t)+i1fQi(t)>0g;0:(6{22)Then,wehavethefollowingresultregardingtheworst-casedelayforanybueredrequestofelasticenergydemands: Lemma12. SupposewecancontrolthesystemtoensurethatQi(t)Qi;maxandZi(t)Zi;maxforallslotst,whereQi;maxandZi;maxaresomepositiveconstants.Then,theworst-casedelayforallbueredtypeienergydemandsisupperboundedbyislotswhere i,Qi;max+Zi;max i:(6{23) Proof. TheproofhereissimilartothatofLemma 8 andomittedhereforbrevity. FollowingtheframeworkofLyapunovoptimization,wedeneaLyapunovfunctionasfollows: L(t),1 2 Xi[Qi(t)]2+Xi[Zi(t)]2+[Zw(t)]2!:(6{24)LetS(t),(Zw(t);(Qi(t);Zi(t);8i)),thentheone-slotconditionalLyapunovdriftisasfollows: L(t),EfL(t+1))]TJ /F4 11.955 Tf 11.95 0 Td[(L(t)jS(t)g:(6{25)Here,theexpectationistakenovertherandomnessofmarketprices,renewablepowergeneration,demandarrivals,aswellastherandomnessinchoosingthecontrolactions.Weaddafunctionoftheexpectedimbalancecostoveroneslot(i.e.,thepenaltyfunction)to( 6{25 )toobtainthefollowingdrift-plus-penaltyterm: L(t)+VEq+(t)+(t)+q)]TJ /F1 11.955 Tf 7.09 -4.94 Td[((t))]TJ /F1 11.955 Tf 7.09 -4.94 Td[((t)jS(t);(6{26)whereVisapositivecontrolparametertobespeciedlater.Weobtainthefollowinglemmaregardingthedrift-plus-penaltyterm: 147

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Lemma13. Foranyfeasibleactionunderconstraints( 6{2 )and( 6{7 )thatcanbeimplementedatslott,wehaveL(t)+VEq+(t)+(t)+q)]TJ /F1 11.955 Tf 7.08 -4.94 Td[((t))]TJ /F1 11.955 Tf 7.08 -4.94 Td[((t)jS(t)A+E(Xi(Qi(t)Di(t)+iZi(t))+wZw(t)W(t)jS(t))+EV(q+(t)+(t)+q)]TJ /F1 11.955 Tf 7.09 -4.93 Td[((t))]TJ /F1 11.955 Tf 7.09 -4.93 Td[((t))jS(t))]TJ /F9 11.955 Tf 11.95 0 Td[(E(Xi(Qi(t)+Zi(t))Xi(t)+Zw(t)Y(t)jS(t)); (6{27)whereAisaconstantgivenbyA,(1+2w)W2max+maxfX2i;max;2ig+X2i;max 2: (6{28) Proof. From( 6{6 ),wehaveQ2i(t+1)(Qi(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Xi(t)+Di(t))2: (6{29)Hence,Q2i(t+1))]TJ /F4 11.955 Tf 11.95 0 Td[(Q2i(t) 2(Di(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Xi(t))2 2+Qi(t)(Di(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Xi(t))X2i;max 2+Qi(t)(Di(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Xi(t)): (6{30)From( 6{18 ),squaringbothsides,andusingthefollowinginequality:(maxfZw(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Y(t);0g+wW(t))2 (6{31)Z2w(t)+Y2(t)+2wW2(t)+2Zw(t)(wW(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Y(t));weobtainZ2w(t+1)Z2w(t)+Y2(t)+2wW2(t)+2Zw(t)(wW(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Y(t)): (6{32) 148

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Then,weobtainthefollowinginequality:Z2w(t+1))]TJ /F4 11.955 Tf 11.95 0 Td[(Z2w(t) 2(1+2w)W2max 2+Zw(t)(wW(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Y(t)): (6{33)FromtheZi(t)queueupdateequation( 6{22 ),wehave Zi(t+1)maxfZi(t))]TJ /F4 11.955 Tf 11.96 0 Td[(Xi(t)+i;0g;(6{34)andhence,Z2i(t+1))]TJ /F4 11.955 Tf 11.96 0 Td[(Z2i(t) 2(i)]TJ /F4 11.955 Tf 11.96 0 Td[(Xi(t))2 2+Zi(t)(i)]TJ /F4 11.955 Tf 11.96 0 Td[(Xi(t))maxfX2i;max;2ig 2+Zi(t)(i)]TJ /F4 11.955 Tf 11.95 0 Td[(Xi(t)): (6{35)Combiningthesethreeboundstogether,summingoveralli,takingtheexpectationw.r.t.S(t)onbothsides,andaddingpenaltytermVEfc(t)jS(t)gtobothsidesoftheaboveinequality,wearriveattheconclusioninthelemma. ThedesignprincipleofourcontrolalgorithmistominimizetheR.H.S.of( 6{27 )ineachtimeperiodtsuchthatthecostisminimizedwhilestabilizingthequeues.TherelativeimportanceoftheabovetwoobjectivesareadjustedbytheparameterV.Notethatconstraints( 6{10 )and( 6{11 )arenonlinear,whichadddicultyintotheoptimizationproblem.However,withthepricerelationship( 6{8 ),foragiventotalenergydeviation(t)=+(t))]TJ /F1 11.955 Tf 12.35 0 Td[()]TJ /F1 11.955 Tf 7.08 -4.34 Td[((t),theoptimalsolutionisguaranteedtobeachievedwithoneofthevariables+(t)or)]TJ /F1 11.955 Tf 7.09 -4.34 Td[((t)equalszero.OurcontrolalgorithmisdescribedinAlgorithm 6 . Notethattheproposedalgorithmonlyneedstosolvealinearprogramateachtimeperiodtandrequiresnodetailedstatisticsofrelatedrandomprocessesexceptthevalueatcurrenttime.Therefore,itismuchsimplertoimplementthanthedynamicprogrammingapproach.Beforeanalyzingtheperformanceofouralgorithm,wegivesomeresultsregardingtheoptimalsolutiontotheaboveoptimizationproblem. 149

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Algorithm6:ProposedAlgorithm Initilization:ChooseparametersV,andi;8i;SetQi(0);Zi(0);8i,andZw(0)tobezero;foreachtimeperiodtdo 1ObservethecurrentsystemstateS(t),q+(t),q)]TJ /F1 11.955 Tf 7.09 -4.34 Td[((t),W(t),Db(t),Di(t);8i. 2ChooseXi(t);8iandY(t)tominimizethefollowing:Vq+(t)+(t)+Vq)]TJ /F1 11.955 Tf 7.09 -4.94 Td[((t))]TJ /F1 11.955 Tf 7.09 -4.94 Td[((t))]TJ /F4 11.955 Tf 11.95 0 Td[(Zw(t)Y(t))]TJ /F5 7.97 Tf 16.8 14.95 Td[(NXi=1(Qi(t)+Zi(t))Xi(t) (6{36)subjectto( 6{2 ),( 6{7 ),( 6{9 ),( 6{12 ),and0+(t)W(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Bmin)]TJ /F4 11.955 Tf 11.96 0 Td[(Db(t); (6{37)0)]TJ /F1 11.955 Tf 7.09 -4.94 Td[((t)Bmax+Db(t)+XiXi;max: (6{38) 3UpdatequeuesQi(t);Zi(t);8i,andZw(t)accordingly.end Lemma14. Theoptimalsolutiontotheoptimizationproblem( 6{36 )satisesthefollowing: (1) IfQi(t)+Zi(t)>Vq)]TJ /F1 11.955 Tf 7.08 -4.33 Td[((t),Xi(t)=Xi;max. (2) IfZw(t)>Vq+(t),Y(t)=W(t). Proof. Inthefollowing,weprovethelemma. 1. Noteateachtimeslott,onlyoneof+(t)and)]TJ /F1 11.955 Tf 7.08 -4.34 Td[((t)canbenonzero.Werstassumethat+(t)>0and)]TJ /F1 11.955 Tf 7.08 -4.34 Td[((t)=0.Then,sinceQi(t)+Zi(t)>Vq)]TJ /F1 11.955 Tf 7.08 -4.34 Td[((t))]TJ /F4 11.955 Tf 9.3 0 Td[(Vq+(t),byobservingtheobjectivefunction( 6{36 ),wecanseethatXi(t)shouldbeaslargeaspossibleinordertominimizetheobjective.Therefore,Xi(t)=Xi;max.Ontheotherhand,if+(t)=0and)]TJ /F1 11.955 Tf 7.08 -4.34 Td[((t)>0,sinceQi(t)+Zi(t)>Vq)]TJ /F1 11.955 Tf 7.08 -4.34 Td[((t),Xi(t)shouldalsobeaslargeaspossiblebyobservingtheobjectivefunction( 6{36 ).Therefore,Xi(t)=Xi;maxwheneverQi(t)+Zi(t)>Vq)]TJ /F1 11.955 Tf 7.08 -4.34 Td[((t)foralltimeslotst. 2. Similarly,byobservingtheobjectivesaswellasthefactthat)]TJ /F4 11.955 Tf 9.3 0 Td[(q+(t)q)]TJ /F1 11.955 Tf 7.09 -4.34 Td[((t),wecanprovethatY(t)=W(t)wheneverZw(t)>Vq+(t). 150

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Thelemmaabovedescribessomeinsightsofourcontrolalgorithm.Specically,whenthelengthofbueredelasticdemandrequestsislarge,theelasticdemandrequestsaremorelikelytobeservedsoastoreducetheirwaitingtime.Also,whenthenegativeimbalancepriceislow,thatmeans,itincurslesspenaltytoproducelessthantheagreedpoweroutputintheday-aheadmarket.Therefore,itismorelikelytoservethebueredelasticenergydemandatthecurrenttime.TherelativenessbetweenthequeuelengthandthepriceisadjustedbytheparameterV.Similarly,whenthequeuelengthZw(t),whichcorrespondstomorespilledrenewableenergyinthepast,theactualrenewableutilizationismorelikelytobelargeatthecurrenttimesoastosatisfytherenewableenergyutilizationlevel.Also,whenthepositiveimbalancepriceislow,thatmeans,itincurslesspenalty(orgainsmoreprot)byproducingmorethantheagreedpoweroutputintheday-aheadmarket.Therefore,itismorelikelytofullyutilizethecurrentrenewablepowerproduction.Onceagain,therelativenessbetweenthequeuelengthandthepriceisadjustedbytheparameterV. 6.3PerformanceAnalysis Theorem6.1. (PerformanceAnalysis)SupposeXi;maxmax[Di;max;i].IfQi(0)=Zi(0)=0;8iandZw(0)=0.Then,underourcontrolalgorithmwithanyxedi0;8iandV>0,wehave (1) ThequeuesQi(t),Zi(t),Zw(t)aredeterministicallyupperboundedasfollows: 0Qi(t)Vq)]TJ /F5 7.97 Tf -.43 -7.89 Td[(max+Di;max;8i;t(6{39) 0Zi(t)Vq)]TJ /F5 7.97 Tf -.43 -7.89 Td[(max+i;8i;t(6{40) 0Zw(t)Vq+max+wWmax;8t:(6{41) (2) Theworst-casedelayforanyelasticdemandrequestis i=2Vq)]TJ /F5 7.97 Tf -.43 -7.29 Td[(max+Di;max+i i;8i:(6{42) 151

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(3) Ifthevector(q+(t);q)]TJ /F1 11.955 Tf 7.09 -4.34 Td[((t);Nw(t);Nb(t);(Ni(t);8i))isi:i:d:overperiods,andifiissettosatisfyiEfDi(t)g;8i,thenforalltimeslotsT>0,thetimeaverageimbalancecostsatises: 1 TT)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xt=0Efc(t)gc+A=V;(6{43)wherecistheminimumtimeaveragecostwiththeproblem( 6{16 )andAisaconstantgivenby( 6{28 ). Proof. Inthefollowing,weprovethetheorem. 1. Weprovetheresultsbyinduction.First,weprovethatQi(t)Vq)]TJ /F5 7.97 Tf -.43 -7.3 Td[(max+Di;max.Whent=0,Qi(0)=0Vq)]TJ /F5 7.97 Tf -.43 -7.29 Td[(max+Di;max.Supposetheboundaboveholdsfortimet,weneedtoshowthatitalsoholdsfortimet+1.IfQi(t)Vq)]TJ /F5 7.97 Tf -.43 -7.29 Td[(max,sincethemaximumincreaseduringonetimeslotisDi;max,wehaveQi(t+1)Qi(t)+Di;max;ifVq)]TJ /F5 7.97 Tf -.43 -7.3 Td[(max
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EfX0i(t)gEfDi(t)g;8i:(6{46)wherecistheminimumtimeaveragecostwiththeproblem( 6{16 ).Then,wecanusetheabovelemmatoprovetheresult( 6{43 ).Oneverytimet,ourproposedalgorithmgreedilyminimizestheR.H.S.of( 6{27 )overallcontrolpoliciesthatsatisfyconstraints( 6{2 )and( 6{7 ),includingthestationary,randomizedcontrolpolicygivenbytheabovelemma.Therefore,byplug-inthepolicygivenbyLemma 15 intotheR.H.S.of( 6{27 ),wehave:L(t)+VEfc(t)jS(t)gB+VEfc0(t)jS(t)g+XiQi(t)EfDi(t))]TJ /F4 11.955 Tf 11.95 0 Td[(X0i(t)jS(t)g+Zw(t)EfwW(t))]TJ /F4 11.955 Tf 11.95 0 Td[(Y0(t)jS(t)g+XiZi(t)Efi)]TJ /F4 11.955 Tf 11.96 0 Td[(X0i(t)jS(t)g: (6{47)Usingthefacts( 6{44 ),( 6{45 ),and( 6{46 )inLemma 15 togetherwithiEfDi(t)g;8i,wearriveatthefollowing: L(t)+VEfc(t)jS(t)gA+Vc:(6{48)Takingtheexpectationoftheaboveinequalityandusingthelawofiteratedexpectationyields: EfL(t+1)g)]TJ /F9 11.955 Tf 20.59 0 Td[(EfL(t)g+VEfc(t)gA+Vc:(6{49)Summingovert2f0;1;:::;T)]TJ /F1 11.955 Tf 11.96 0 Td[(1gforanyT>0,wehave EfL(T)g)]TJ /F9 11.955 Tf 20.59 0 Td[(EfL(0)g+T)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xt=0VEfc(t)gAT+VTc:(6{50)SinceL(0)=0andL(T)0,dividingbothsidesbyTVandrearrangingtheterms,wearriveattheresult: 1 TT)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xt=0Efc(t)gc+A=V:(6{51)foranyT>0. 153

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From Theorem6.1(1) and Theorem6.1(2) ,wecanseethatthereexistsanexplicittrade-obetweencostminimizationandworst-caseelasticdemanddelay.Thatis,withtheincreaseoftheparameterV,thecostdecreaseswhiletheworst-casedelayalsoincreases.Therefore,whenthedemandismoreelastic(i.e.,moredelay-tolerant),wecanachievesmallerimbalancecost.Notethattheproofsfor Theorem6.1(1) and Theorem6.1(2) arebasedonlyonthesamplepathswithoutanyassumptiononthedistributionofunderlyingrandomprocesses.Althoughtheanalyticalperformanceresultin Theorem6.1(3) requiresthei:i:d:assumption,wewanttoemphasizethattheresultisquitegeneralandcanbeextendedtotreatthenon-i.i.d.casetoowithminormodicationsbyusingtheuniversalschedulingresultsin[ 68 ].Thedetaileddiscussionisomittedhereforbrevity.Moreover,oursimulationsarebasedonthereal-worldtraceswithoutanyassumptionontheprobabilitydistributionsofunderlyingrandomprocesses. 6.4CaseStudiesWedemonstratetheperformanceofourproposedalgorithmthroughextensivenumericalevaluations.WesimulateaVPPconsistingofawindplantandfourtypesofelasticdemand.WeuseawindpowertimeseriesdatasetprovidedbytheBPA[ 8 ],whichcontainsbothactualwindpoweroutputandforecastedwindpoweroutputintheBPAcontrolareafromJan.1,2013toJan.31,2013.ThedatasetisscaleddownsuchthattheinstalledwindplantnameplatecapacityWmax=2MW.TheinelasticdemandDb(t)ineachtimeslottissettobeuniformlydistributedin[0,200]kWh.TheelasticpowerdemandDi(t)ineachtimeslottisuniformlydistributedin[0,160]kWhforeachtypeitosimulatethechargingdemandof10PHEVs.Dierenttypesofelasticpowerdemandmayhavedierentdelayrequirements,whichcanbesatisedbyadjustingtheparametersiforeachtypei,respectively.WesetXi;max=0:5MWh.Tobetterillustratetheimpactofrandomwindpoweroutputontheimbalancecost,weassumethatboth 154

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inelasticdemandsDb(t)andelasticdemandsDe(t)canbepredictedaccuratelyonedayahead,i.e.,Nb(t)=Ne(t)=0;8t.Weconsiderthefollowingtwotypesofimbalanceprices: TypeI:Forthistype,aswith[ 15 ],weassumethatdeviationsfromtheday-aheadbid,eitherpositiveornegative,inthebalancingmarketarealwayspenalized.Therefore,q+(t);q)]TJ /F1 11.955 Tf 7.08 -4.33 Td[((t)0anditisneverprotabletodeviatefrombidintheday-aheadmarket. TypeII:Forthistype,weassumethatinsomecircumstances,surplusrenewablepowerhaseconomicvalue(q+(t)<0),whileincasesofsystemicoverproduction,excesswindmustbecurtailedorhasnegativevalue(q+(t)>0).Theday-aheadmarketpricep(t)isobtainedfromtheNewYorkISO[ 9 ],alsofromJan.1,2013toJan.31,2013.Wesetthepositiveimbalancepriceq+(t)asfollows:q+(t)=)]TJ /F1 11.955 Tf 9.29 0 Td[((1++(t))p(t);where+(t)isuniformlydistributedin[-2,-1]forTypeIimbalanceprices,and[-2,0]forTypeIIimbalanceprices.Notethatwiththissetting,thesurplusrenewablepowerinthebalancingmarketisbenecialinsomecircumstanceswhenq+(t)<0inTypeIIimbalanceprices,whileq+(t)isalwaysnon-negativeforTypeIimbalanceprices.Similarly,wesetthenegativeimbalancepriceq)]TJ /F1 11.955 Tf 7.09 -4.33 Td[((t)asq)]TJ /F1 11.955 Tf 7.09 -4.94 Td[((t)=(1+)]TJ /F1 11.955 Tf 7.08 -4.94 Td[((t))p(t);where)]TJ /F1 11.955 Tf 7.08 -4.34 Td[((t)isuniformlydistributedin[0,1]forbothtypesofimbalanceprices.Withthissetting,theshortagepowerinthebalancingmarketischargedapricehigherthantheday-aheadprice.Theminimumrenewableenergyutilizationlevelwissettobe90%.Thetimeperioddurationis1hour.Withthesedefaultsettings,weevaluatetheperformanceofouralgorithminthefollowingwithdierentparametersundertheabovetwosetsofprices.Toshowtheeectivenessofouralgorithm,wechooseagreedyapproachwhichfullyutilizesalltherenewablepowergenerationandservesalltheelasticdemandswithoutdelayasthe 155

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benchmark.WealsoshowtheperformanceofouralgorithmwhenthereisnoDRresource,i.e.,alldemandsareconsideredasinelastic. 6.4.1PerformanceResultsUnderTypeIPrices AAveragecost BRenewableenergyutilization CAveragedelayFigure6-2. ComparisonbetweenouralgorithmandgreedyalgorithmwithdierentV. First,weevaluatetheperformanceofouralgorithmundertheimbalancepricesofTypeI.Toisolatetheimpactofrandomrenewablepoweroutput,weassumethatbothinelasticdemandsDb(t)andelasticdemandsDi(t);8icanbepredictedaccuratelyonedayahead,i.e.,Nb(t)=Ni(t)=0;8i.WeevaluatetheperformanceofouralgorithmunderdierentVandassumei=0;8i.AswecanseefromtheFigure 6-2A ,ourproposedalgorithmcanlargelyreducetheimbalancecostwiththeincreaseoftheparameterV.Specically,bycomparingouralgorithmwithoutDRwiththegreedybenchmark,wecanobservethatthereal-timerenewableenergycurtailmentcanhelpreducetheimbalancecost.Ontheotherhand,bycomparingouralgorithmswithDRandwithoutDR,wecanobservethegreatbenetsbroughtbyaggregatingtherenewablegeneratorsandelasticdemands.Meanwhile,fromtheFigure 6-2B ,ouralgorithmswithDRandwithoutDRcanbothachieveahighrenewableenergyutilizationlevelwhilereducingtheimbalancecost.Moreover,DRresourcescanimprovetherenewableenergyutilizationbesidesreducingtheimbalancecost.However,ouralgorithmwithDRdoeshaveatrade-o,asshowninFigure 6-2C .TheaveragedelayincreasesasparameterVincreases,whichmatchestheresultsinTheorem 6.1 .Sincetheaveragedelayisrelativelysmall(around2hours)even 156

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forthelargestvalueofVwechooseinthenumericalevaluations,itfurtheraugmentstheeectivenessofouralgorithm. 6.4.2PerformanceResultsUnderTypeIIPrices AAveragecost BAveragedelayFigure6-3. ComparisonbetweenouralgorithmandgreedyalgorithmwithdierentVandi=0;8i.Here,theaveragedelayismeasuredoveronetypeofelasticdemands. Next,weevaluatetheperformanceofouralgorithmundertheTypeIIofimbalanceprices,whichismorerealisticasshownin[ 65 ].Theday-aheadforecastvaluesofdemandarrivalsaresettobetheexpectedvalues,thatis, Db(t)=100kWh, Di(t)=80kWh;8i;t.AswecanseefromFigure 6-3A ,ourproposedalgorithmlargelyreducestheimbalancecostwiththeincreaseoftheparameterV.Specically,thegreedilyalgorithmincursanaveragepenaltyof5.0601dollarsperhourinthebalancingmarketduetotheimbalancebetweenday-aheadbidandreal-timeusage.Incontrast,withtheincreaseoftheparameterV,ourproposedalgorithmcanachieveaaverageprotofaslargeas2.4989dollarsperhourinthebalancingmarket.Thisisnotsurprisingsincethegreedyapproachpassivelyaccepttheimbalanceinreal-timewhileourapproachlearnstoactivelycurtailthewindpowerorservetheelasticdemandstoreducethepenaltywhenthepositiveimbalancepriceishigherthantypicalanddelaytheelasticdemandsinreal-timetoreducethenegativeimbalancewhenthenegativeimbalancepriceishigherthantypical.Sincethepositiveimbalancepriceisnegativeinsomecircumstances,ouralgorithmlearnstoutilizethesechancestogainsomeprotsbydelayingtheserviceofelasticdemands 157

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soastosellmorepower.However,ouralgorithmdoeshaveatrade-o,asshowninFigure 6-3B .TheaveragedelayincreaseswiththeincreaseofparameterV,whichmatchestheresultsinTheorem 6.1 . 6.4.3Worst-caseDelayPerformance Figure6-4. Delayperformanceoffourtypesofelasticdemands. Figure6-5. Imbalancecostunderdierenti;8i. Finally,weevaluateouralgorithmwhenweneedtoprovidesomeworst-casedelayguaranteeforeachtypeofelasticdemands.Forbrevity,weonlypresenttheresultsunderthetypeIIprices.Specically,wesetV=0:01andi=EfDi(t)g=80kWhfori=1;2andi=EfDi(t)g=2=40kWhfori=3;4.Figure 6-4 showstheresultsofdelayperformance.Itcanbeobservedthattheproposedalgorithmhasonaverageamuch 158

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smallerdelaythanthemaximumdelaygivenbyTheorem 6.1 ,which,forourparameters,is132hoursfori=1;2and262hoursfori=3;4.Therefore,theperformanceofouralgorithmmaybemuchbetterinpracticethantheresultsgivenbyTheorem 6.1 sincetheyarerobustresultsandtheboundsarederivedonlyfromtheboundassumptions.Wealsoevaluatethecostperformanceofouralgorithmwiththeinuenceofvaryingi;8i.Here,wechoosethesameiforalliandxV=0:01.Figure 6-5 showsthecostperformanceresults.Thecostdecreasesasidecreases.However,themaximumdelayguaranteegivenbyTheorem 6.1 alsoincreases.Therefore,whentheelasticdemandsarelesssensitivetodelay,ouralgorithmcanachieveasmallerimbalancecost. 6.5SummaryInthischapter,wehavesolvedtheproblemofminimizingtheimbalancecostofaVPPparticipatingintoatwo-settlementelectricitymarket.Thestochasticnatureofimbalanceprices,renewablepowergeneration,andenergydemandsisconsidered.Anecientonlinealgorithm,whichdoesnotrequirethestatisticalpropertiesoftherelatedrandomprocesses,hasbeenproposedandevaluatedbothanalyticallyandnumerically. 159

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CHAPTER7CONCLUSIONSANDFUTUREWORKInthisdissertation,wehaveconsideredtheenergyoptimizationandcontrolfordatacentersandsmartgrids.Fordatacenters,wehaveproposedtouseenergystorageanddelay-tolerantworkloadsincombinationwithprevioustechniquestoconservetheirenergyconsumption,reducetheirenergycost,andreducetheircarbonfootprint.Novelcontrolalgorithmshavebeendevelopedtomanagethesedatacenterswhileguaranteeingquality-of-serviceconstraints.Wehaveprovedthatthesecontrolalgorithmsyieldclose-to-optimalperformancewithoutrequiringapriorinformationonrandomelectricitypricesandtracworkloads.Forsmartgrids,wefocusontheproblemsofDERmanagement.Twointegratedframeworks,\SmartHome"and\SmartNeighborhood",havebeenproposedandnovelenergymanagementalgorithmshavebeendevelopedintheseframeworksundertheresidentialscenario.Ourworkinthisarearevealsdeepinsightsonhowtoeectivelymanagetheenergyusageinresidentialhouseholdstohelprealizethevisionofsmartgrid.Moreover,theoptimalreal-timeoperationstrategyofavirtualpowerplantconsistingofmultipleDERunitsinelectricitymarketshasbeendeveloped.Thereareseveralpossibleextensionstoourwork.First,wecanimprovetheperformanceofouralgorithmsviaprediction.Ourproposedalgorithmsrequireminimuminformationontheunderlyingrandomprocessesandtherefore,arerobustbutconservative.Inpractice,wemayknowsomefutureinformationviapredictionduetotheavailabilityofdata.Then,howtoimprovetheperformanceofourLyapunov-basedcontrolalgorithmswithsomeinformationaboutfutureisstillanopenquestion.Second,weneedtoinvestigatethecoordinationofdatacentersandsmartgrids.Sincedatacentersarelargepowerconsumers,theycouldplayanimportantroleinthefutureelectricitysupply.Ourworkhasshownthatenergydemandofdatacentersishighlyexible.Thispropertyenablesdatacenterstoparticipateintodemandresponse 160

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programs.However,identifyingthemostsuitabledemandresponseprogramsfordatacentersanddeterminingtheoperationstrategiesofdatacentersintheseprogramsarestillopenquestions.Ontheotherhand,manysensorsexistinsmartgridsandtheygeneratealotofdata,whichislikelytobemanagedinclouddatacenters.Therefore,thesetwosystemsareinterdependentandmayneedjointoptimization.Third,wecouldinvestigatetheDERmanagementinmicrogrids.Whileourfocusisontheresidentialhouseholdlevel,itmaybepossibletoextendourmethodologiestoamicrogrid.However,therearesomeimportantdierences.Inourwork,weignorenetworkconstraintssincewefocusonasmallarea.However,inamicrogrid,DERunitsmaybelocatedatdierentbuses,andpowerlossesandnetworkcongestionsshouldbeconsidered.Moreover,optimizationandcontrolobjectivesaredierentinmicrogrids. 161

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BIOGRAPHICALSKETCH Yuanxiong(Richard)GuowasborninWuhan,China,in1988.HereceivedtheB.E.degreeinelectronicsandinformationengineeringfromHuazhongUniversityofScienceandTechnology,Wuhan,China,in2009,andtheM.S.degreeinelectricalandcomputerengineeringfromtheUniversityofFloridain2012.HeiscurrentlyworkingtowardshisPh.D.degreeintheDepartmentofElectricalandComputerEngineeringattheUniversityofFloridaunderthesupervisionofDr.YuguangFangandDr.PramodP.Khargonekar.From2009to2010,hewasagraduateassistantintheDepartmentofComputerScienceandEngineeringattheChineseUniversityofHongKong.Hisresearchinterestsincludesmartgrid,renewablepowerandenergysystems,cyber-physicalsystems,cloudcomputing,sustainabledatacenters,andBigData.HeisarecipientoftheBestPaperAwardfrom2012IEEEGlobalCommunicationsConference(GLOBECOM'12),Houston,TX,USA. 170