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Stochastic Optimization and its Application to Communication Networks and the Smart Grid

Permanent Link: http://ufdc.ufl.edu/UFE0044527/00001

Material Information

Title: Stochastic Optimization and its Application to Communication Networks and the Smart Grid
Physical Description: 1 online resource (123 p.)
Language: english
Creator: Ding, Zongrui
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: optimization -- smart -- stochastic
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The scheduling of some practical systems is highly affected by the randomness of the system variables. Therefore, it is of great importance to model and optimize the system performance by stochastic optimization techniques. This work presents the results of stochastic optimization with application to the scheduling in two important areas: the multi-hop wireless networks and the Smart Grid. The wireless multi-hop networks are typically modeled as systems consisted of multiple queues. Our work is built under the Per-link (PL) queueing structure, which is more scalable compared with the per-destination (PD) queueing structure. We mainly focus on the scheduling of the PL queueing networks involving link scheduling, power allocation and routing under the network stability. For the Smart Grid systems, we study the integration of renewable resources for Distributed Energy Resources (DERs) in the hybrid power system. In Chapter 2 and 3, we consider the joint problem of congestion control and scheduling with multi-class Quality of Service (QoS) requirements in wireless multi-hop networks under both time invariant and time-varying channels. Generally, the joint problem is formulated as a Network Utility Maximization (NUM) problem and can be solved by the back-pressure algorithm under PD queueing. The PD queueing requires that each node maintains a separate queue for each destination. We consider the PL queueing, which reduces the queue number per node to the number of node neighbors. Based on this queueing model, we propose a Sliding Mode (SM) approach to designing a distributed controller for the congestion control problem while satisfying the multi-QoS constraints in the multi-path and multi-hop scenario. In Chapter 4, we study the capacity region and the dynamic stabilizing control policy for the PL queueing networks with time-varying channels. Based on the Lyapunov drift method, we propose a dynamic routing and power control policy, i.e. DRPC-PL, which stabilizes the network whenever the input rate is within the capacity region of the PL queueing network. The Smart Grid system is a future vision of the electric power system, which features interaction and communication among grid components, adaptation of renewable energy and reliability/self-healing, etc. In Chapter 5, we consider the integration of stochastic wind power generation in the Smart Grid under renewable portfolio standard (RPS) with deep wind penetration. For small-scale utilities installed with wind turbines and acting as distributed energy resources (DERs), wind energy can be potentially integrated to satisfy customer power demand or enter energy market. To exploit the temporal variation of system status and real-time electricity market, we propose a theoretical framework to dynamically determine the role of wind energy and provide long-term RPS guarantee. This approach yields a simple dynamic thresholding control under the assumption of system ergodicity, which maximizes the expected profit of the utility and is easy to be implemented online.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: 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.
Statement of Responsibility: by Zongrui Ding.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Wu, Dapeng.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044527:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044527/00001

Material Information

Title: Stochastic Optimization and its Application to Communication Networks and the Smart Grid
Physical Description: 1 online resource (123 p.)
Language: english
Creator: Ding, Zongrui
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: optimization -- smart -- stochastic
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The scheduling of some practical systems is highly affected by the randomness of the system variables. Therefore, it is of great importance to model and optimize the system performance by stochastic optimization techniques. This work presents the results of stochastic optimization with application to the scheduling in two important areas: the multi-hop wireless networks and the Smart Grid. The wireless multi-hop networks are typically modeled as systems consisted of multiple queues. Our work is built under the Per-link (PL) queueing structure, which is more scalable compared with the per-destination (PD) queueing structure. We mainly focus on the scheduling of the PL queueing networks involving link scheduling, power allocation and routing under the network stability. For the Smart Grid systems, we study the integration of renewable resources for Distributed Energy Resources (DERs) in the hybrid power system. In Chapter 2 and 3, we consider the joint problem of congestion control and scheduling with multi-class Quality of Service (QoS) requirements in wireless multi-hop networks under both time invariant and time-varying channels. Generally, the joint problem is formulated as a Network Utility Maximization (NUM) problem and can be solved by the back-pressure algorithm under PD queueing. The PD queueing requires that each node maintains a separate queue for each destination. We consider the PL queueing, which reduces the queue number per node to the number of node neighbors. Based on this queueing model, we propose a Sliding Mode (SM) approach to designing a distributed controller for the congestion control problem while satisfying the multi-QoS constraints in the multi-path and multi-hop scenario. In Chapter 4, we study the capacity region and the dynamic stabilizing control policy for the PL queueing networks with time-varying channels. Based on the Lyapunov drift method, we propose a dynamic routing and power control policy, i.e. DRPC-PL, which stabilizes the network whenever the input rate is within the capacity region of the PL queueing network. The Smart Grid system is a future vision of the electric power system, which features interaction and communication among grid components, adaptation of renewable energy and reliability/self-healing, etc. In Chapter 5, we consider the integration of stochastic wind power generation in the Smart Grid under renewable portfolio standard (RPS) with deep wind penetration. For small-scale utilities installed with wind turbines and acting as distributed energy resources (DERs), wind energy can be potentially integrated to satisfy customer power demand or enter energy market. To exploit the temporal variation of system status and real-time electricity market, we propose a theoretical framework to dynamically determine the role of wind energy and provide long-term RPS guarantee. This approach yields a simple dynamic thresholding control under the assumption of system ergodicity, which maximizes the expected profit of the utility and is easy to be implemented online.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: 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.
Statement of Responsibility: by Zongrui Ding.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Wu, Dapeng.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044527:00001


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STOCHASTICOPTIMIZATIONANDITSAPPLICATIONTOCOMMUNICATIONNETWORKSANDTHESMARTGRIDByZONGRUIDINGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012ZongruiDing 2

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ToZongruiDing 3

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ACKNOWLEDGMENTS FirstIwouldliketothankmyadvisorDr.DapengWuandco-advisorDr.YuguangFangfortheirenormouscontributiontomyPhDresearch.ThewideknowledgeofDr.Wuintheeldofcross-layerdesigninwirelessnetworkshasguidedmetoconstructtheframeworkofmyPhDresearch.Dr.Wuhasbeenveryaccessiblefortechnicalhelp.Besidesthegreathelpintechnicalaspects,Dr.Wualsotaughtmetheimportanceofcommunicatingideaswithpeopleandprovidedmechancesofpresentingworksinacademicconferencesandinteractingwiththepeoplefromindustry.ThegreatvisionofDr.FangintheeldofqueueingtheoryandtheSmartGridhascontributedandinspiredmanyofmyresearchworks.Hepresentedthewayofreliablyandefcientlytacklingresearchtopics.TheexperienceasaPhDstudentwithDr.WuandDr.Fangwillbenetmywholelife.Also,IwouldliketothankDr.PramodKhargonekar,Dr.MyThaiforservingonmycommittee.Thetimewithmyformerandcurrentcolleagues,ChiZhang,YangSong,MiaoPan,YuanxiongGuo,YakunHu,YuejiaHe,Zhengyuan,BaohuaSunandShijieLi,isverypleasant.Ourdiscussionsandcollaborationsareveryhelpfultoadvanceresearchprogress.IthasbeenverysweetandcomfortingtohavegreatsupportfrommyhusbandYijianOuyangandotherfamilymembers.AndIwouldliketoextendmysincerethankstothem.Lastly,Ithankourfundingagencies:NationalScienceFoundationandtheUSOfceofNavalResearch. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 1.1ProblemDescriptionandContributionsintheSchedulingofWirelessMulti-hopNetworksbasedonPer-LinkQueueing .............. 13 1.1.1TheJointCongestionControlandSchedulingunderPLqueueing 14 1.1.2NetworkCapacityRegionandDynamicControlunderPLQueueing 16 1.2ProblemDescriptionandContributionsinRenewableResourceIntegrationintheSmartGrid ................................ 17 1.3Outline ...................................... 20 2THEPER-LINKQUEUEBASEDCONGESTIONCONTROLANDSCHEDULINGINWIRELESSMULTI-HOPNETWORKSWITHTIMEINVARIANTCHANNELS 22 2.1NetworkModelandTheNUMProblem .................... 23 2.2TheSolutiontoTheNUMProblem ...................... 26 2.2.1Decomposition ............................. 26 2.2.2LagrangianPrice ............................ 27 2.2.3SMBasedSolutiontotheCongestionControlProblem ....... 28 2.2.4SolutiontotheSchedulingProblem .................. 28 2.2.5OptimalityandConvergenceofTheSolution ............. 29 2.2.6TheDiscrete-timeAlgorithm ...................... 33 2.3SimulationResults ............................... 34 2.3.1SimulationSetting ........................... 34 2.3.2PerformanceofAlgorithm 1 ...................... 35 2.4ChapterSummary ............................... 37 3THEPER-LINKQUEUEBASEDCONGESTIONCONTROLANDSCHEDULINGINWIRELESSMULTI-HOPNETWORKSWITHTIMEVARYINGCHANNELS 39 3.1NetworkModelunderPer-linkQueueing ................... 39 3.1.1TheQueueingModel .......................... 39 3.1.2QoSRequirements ........................... 41 3.1.3CapacityRegionofaPer-linkQueueingNetwork .......... 41 3.2TheNUMproblemwithTime-varyingChannels ............... 42 3.3PerformanceAnalysis ............................. 46 5

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3.3.1PerformanceofAlgorithm 2 ...................... 46 3.3.2PerformanceEnhancementMechanism:VIPQueues ....... 47 3.4SimulationResults ............................... 48 3.4.1SimulationSettings ........................... 49 3.4.2PerformanceofAlgorithm 2 ...................... 50 3.4.2.1ThroughputPerformance .................. 50 3.4.2.2SupportingMultimediaServices .............. 53 3.4.2.3RobustnessagainstLinkFailure .............. 55 3.5ChapterSummary ............................... 59 4CAPACITYREGIONANDDYNAMICCONTROLOFWIRELESSNETWORKSUNDERPER-LINKQUEUEING .......................... 60 4.1PLQueueingNetworkModel ......................... 60 4.1.1ThePLQueueingModel ........................ 60 4.1.2TheNetworkModel ........................... 62 4.2StabilizingControlPolicy ............................ 64 4.2.1LyapunovDriftMethod ......................... 64 4.2.2PDPLNetwork ............................. 65 4.2.3TheDRPC-PLAlgorithm ........................ 66 4.3PerformanceAnalysis ............................. 68 4.3.1PerformanceoftheDRPC-PDPLAlgorithm ............. 68 4.3.2PerformanceoftheDRPC-PL ..................... 69 4.3.3TheCapacityRegionofPLQueueingNetwork ........... 74 4.4ChapterSummary ............................... 74 5INTEGRATINGDISTRIBUTEDWINDENERGYINTOTHESMARTGRIDBASEDONELECTRICITYMARKET ....................... 76 5.1SystemModelofWindEnergyIntegration .................. 78 5.1.1DERandRenewableEnergyUtilitiesinElectricityMarket ..... 78 5.1.2WindEnergyIntegrationModelinGreenUtilities .......... 79 5.1.3SystemCostModel ........................... 82 5.2ProtMaximizationoftheGreenUtility .................... 85 5.3Long-termPerformanceEvaluation ...................... 87 5.4CaseStudyandSimulationResults ..................... 91 5.4.1SimulationSetting ........................... 91 5.4.2PerformanceAnalysis ......................... 92 5.5ChapterSummary ............................... 95 6CONCLUSIONS ................................... 96 6.1SummaryoftheDissertation ......................... 96 6.2FutureWork ................................... 97 6

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APROOFOFTHEOREM3.1AND3.2 ........................ 98 A.1ProofofTheorem 3.1 ............................. 98 A.2ProofofTheorem 3.2 ............................. 101 BPROOFSINCHAPTER4 .............................. 103 B.1ProofofTheorem 4.1 ............................. 103 B.1.1Necessity ................................ 103 B.1.2Sufciency ................................ 105 B.2ProofofTheorem 4.3 ............................. 105 B.2.1PerformanceoftheSTAT-PDPLAlgorithm .............. 105 B.2.2PerformanceoftheFRAME-PDPLAlgorithm ............ 109 CPROOFOFTHEOREM5 .............................. 113 REFERENCES ....................................... 117 BIOGRAPHICALSKETCH ................................ 123 7

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LISTOFTABLES Table page 2-1Notations. ....................................... 24 2-2Simulationparameters ................................ 35 3-1Performanceevaluationsetting ........................... 50 5-1Simulationsetting .................................. 92 8

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LISTOFFIGURES Figure page 1-1TheSmartGridvision[source:U.S.DepartmentofEnergy] ........... 18 2-1Chaintopology .................................... 35 2-2Looptopology .................................... 35 2-3TransmissionrateforchaintopologywithoutQoS ................. 36 2-4TransmissionrateforchaintopologywithQoS .................. 36 2-5TransmissionrateforlooptopologywithoutQoS ................. 37 2-6TransmissionrateforlooptopologywithQoS ................... 37 3-1Dumbbelltopology .................................. 48 3-2Chaintopology .................................... 48 3-3Looptopology .................................... 49 3-4TransmissionrateofdumbbelltopologyunderAlgorithm1 ............ 51 3-5QueuelengthofdumbbelltopologyunderAlgorithm1 .............. 51 3-6TransmissionrateofdumbbelltopologyunderMaxWeight ............ 52 3-7QueuelengthofdumbbelltopologyunderMaxWeight .............. 52 3-8TransmissionrateforchaintopologyunderconstantchannelswithoutQoSrequirements ..................................... 54 3-9TransmissionrateforchaintopologyunderconstantchannelswithQoSrequirements 54 3-10Transmissionrateforchaintopologyundertime-varyingchannelswithoutQoSrequirements ..................................... 55 3-11Transmissionrateforchaintopologyundertime-varyingchannelswithQoSrequirements ..................................... 56 3-12Transmissionrateforlooptopology ......................... 56 3-13Queuelengthforlooptopology ........................... 57 3-14Transmissionrateforlooptopologyunderlinkfailure ............... 57 3-15Queuelengthforlooptopologyunderlinkfailure ................. 58 4-1InputmodelforPLqueue .............................. 61 9

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4-2OutputmodelforPLqueue ............................. 61 5-1Systemmodelofagreenutility ........................... 80 5-2ERCOTmarketclearingpriceforregulationdown ................. 82 5-3Windenergyintegrationdiagram .......................... 84 5-4ParameterinthecontrolpolicyQvs.integrationratio .............. 93 5-5Real-timedecisionofpowerintegration ...................... 93 5-6Real-timepowerintegrationratio .......................... 94 5-7Timeaveragedcost ................................. 94 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophySTOCHASTICOPTIMIZATIONANDITSAPPLICATIONTOCOMMUNICATIONNETWORKSANDTHESMARTGRIDByZongruiDingAugust2012Chair:DapengWuMajor:ElectricalandComputerEngineering Theschedulingofsomepracticalsystemsishighlyaffectedbytherandomnessofthesystemvariables.Therefore,itisofgreatimportancetomodelandoptimizethesystemperformancebystochasticoptimizationtechniques.Thisworkpresentstheresultsofstochasticoptimizationwithapplicationtotheschedulingintwoimportantareas:themulti-hopwirelessnetworksandtheSmartGrid.Thewirelessmulti-hopnetworksaretypicallymodeledassystemsconsistedofmultiplequeues.OurworkisbuiltunderthePer-link(PL)queueingstructure,whichismorescalablecomparedwiththeper-destination(PD)queueingstructure.WemainlyfocusontheschedulingofthePLqueueingnetworksinvolvinglinkscheduling,powerallocationandroutingunderthenetworkstability.FortheSmartGridsystems,westudytheintegrationofrenewableresourcesforDistributedEnergyResources(DERs)inthehybridpowersystem. InChapter 2 and 3 ,weconsiderthejointproblemofcongestioncontrolandschedulingwithmulti-classQualityofService(QoS)requirementsinwirelessmulti-hopnetworksunderbothtimeinvariantandtime-varyingchannels.Generally,thejointproblemisformulatedasaNetworkUtilityMaximization(NUM)problemandcanbesolvedbytheback-pressurealgorithmunderPDqueueing.ThePDqueueingrequiresthateachnodemaintainsaseparatequeueforeachdestination.WeconsiderthePLqueueing,whichreducesthequeuenumberpernodetothenumberofnodeneighbors.Basedonthisqueueingmodel,weproposeaSlidingMode(SM)approachtodesigning 11

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adistributedcontrollerforthecongestioncontrolproblemwhilesatisfyingthemulti-QoSconstraintsinthemulti-pathandmulti-hopscenario.InChapter 4 ,westudythecapacityregionandthedynamicstabilizingcontrolpolicyforthePLqueueingnetworkswithtime-varyingchannels.BasedontheLyapunovdriftmethod,weproposeadynamicroutingandpowercontrolpolicy,i.e.DRPC-PL,whichstabilizesthenetworkwhenevertheinputrateiswithinthecapacityregionofthePLqueueingnetwork. TheSmartGridsystemisafuturevisionoftheelectricpowersystem,whichfeaturesinteractionandcommunicationamonggridcomponents,adaptationofrenewableenergyandreliability/self-healing,etc.InChapter 5 ,weconsidertheintegrationofstochasticwindpowergenerationintheSmartGridunderrenewableportfoliostandard(RPS)withdeepwindpenetration.Forsmall-scaleutilitiesinstalledwithwindturbinesandactingasdistributedenergyresources(DERs),windenergycanbepotentiallyintegratedtosatisfycustomerpowerdemandorenterenergymarket.Toexploitthetemporalvariationofsystemstatusandreal-timeelectricitymarket,weproposeatheoreticalframeworktodynamicallydeterminetheroleofwindenergyandprovidelong-termRPSguarantee.Thisapproachyieldsasimpledynamicthresholdingcontrolundertheassumptionofsystemergodicity,whichmaximizestheexpectedprotoftheutilityandiseasytobeimplementedonline. 12

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CHAPTER1INTRODUCTION Indifferentnetworksandsystems,thewordresourcehasdifferentmeaning.Forexample,time,bandwidthandspacecanbeconsideredasresourcesinwirelesscommunicationsystemsandrawmaterialsareregardedasresourceinmanufacturalscenarios.Sincetheseresourcesarevaluableandlimited,theschedulingtoallocateresourcesiscrucialforsystemefciency.Themathematicalabstractionoftheschedulingintheseresource-basedsystemsusuallyinvolvesrandomness,whichmakestheseproblemschallengingtosolve.Inthiswork,wetrytoapplythestochasticoptimizationtechniquestotheschedulingprobleminmulti-hopwirelessnetworksandtheSmartGrid. Stochasticoptimizationtechniquesareusedinavastnumberofareas,includingaerospace,medicine,transportation,andnance,tonamebutafew[ 1 ].Stochasticoptimizationsareoptimizationmethodsthatgenerateanduserandomvariablesintheirformulations.Differentfromdeterministicmethodsfordeterministicproblems,therandomvariablesappearintheformulationoftheoptimizationproblem,whichinvolvesrandomobjectivefunctionsorrandomconstraints[ 2 ]. Stochasticoptimizationplaysasignicantroleintheanalysis,design,andoperationofmodernsystems.Methodsforstochasticoptimizationprovideameansofcopingwithinherentsystemnoiseandcopingwithmodelsorsystemsthatarehighlynonlinear,highdimensional,orotherwiseinappropriateforclassicaldeterministicmethodsofoptimization. 1.1ProblemDescriptionandContributionsintheSchedulingofWirelessMulti-hopNetworksbasedonPer-LinkQueueing Incellularnetworks,communicationoccursonlyonthelastlinkbetweenthebasestationandthemobilestation.Thiscommunicationinfrastructurehasbeenverysuccessfulinmeetingourdailycommunicationneeds.However,thereareseveralapplicationsthatrequirelowpowercommunication[ 3 ][ 4 ][ 5 ],easydeploymentand 13

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nocentralmanagementunits,etc.Themulti-hopwirelessnetworkisproposedasonesolutiontothesecommunicationneeds.Differentfromthecellularnetworks,thereareoneormoreintermediatenodesalongthepathtoreceiveandforwardpacketsinthemulti-hopwirelessnetworks[ 6 ].Examplesarewirelessad-hocnetworksandwirelessmeshnetworksusedinthebattleeldandmilitaryoperatingscenario.Thebenetsofthisnetworkstructurearethreefold.Comparedtonetworkswithsinglelinks,multi-hopwirelessnetworksextendthecoverageofthenetworkandimproveconnectivity.Inaddition,transmissionrangeinonehopisshort,thustransmissionconsumeslessenergythanlonglinks.Asaresult,theseshortlinktransmissionsusinglowpowerreducetheinterferenceamongdevicesandincreasethethroughputofthenetwork. Sincethewirelessresourceislimited,theresourceallocationinmulti-hopwirelessnetworksiscrucialtoitscommunicationperformance.Becauseoftheuctuationofwirelesschannels,multi-hopwirelessqueueingnetworkisstudiedwiththeintuitionthatbufferingthedatamayhelpcombatchannelvariation.Theschedulinginmulti-hopwirelessnetworksarechallengingduetothefollowingreasons.First,theschedulingofthewirelesslinksinmulti-hopwirelessnetworksiscomplicatedduetotheinterferenceamonglinks.Second,anodemaybecongestedbyitsneighborsduetoactingasrelaynodeforothernodesatthesametime.Thecongestioncontrolproblemandtheschedulingproblemareusuallycoupledwitheachother. 1.1.1TheJointCongestionControlandSchedulingunderPLqueueing In[ 7 ],thecongestioncontrolandschedulinginthemulti-hopwirelessnetworksisstudiedandtheback-pressurealgorithm[ 8 ][ 7 ]isdevelopedtosolveitoptimallyundertime-invariantchannels.Theclassicback-pressurealgorithmprovidesanicedecompositionbetweenthecongestioncontrolandschedulingproblem,whichleadstoaniterativealgorithm.However,theback-pressurealgorithmcannotefcientlysolveschedulingproblemwithQoSconstraints,e.g.,requirementsontransmissionrates.Moreover,theclassicback-pressurealgorithmrequiresthateachnodemaintainsa 14

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queueforeachdestination,whichresultsinlargeoverhead.Therearetypicallytwoqueueingmodels:thePer-Destination(PD)queueingandPer-Link(PL)queueing.InthePDqueueingnetworks,eachnodeisrequiredtomaintainaqueueorvirtualqueueforeachowdestination,thusthenumberofqueuespernodecanbeaslargeasthenumberofowdestinations.Thisoverheadisnotdesirableespeciallyinwirelessmulti-hopnetworksbecauserelaynodesmaynotbeavailableorwillingtooffersuchresourceforotherusers.Therefore,itisdifculttoapplythePDqueuestructureinlarge-scalenetworks.Therefore,itisdifculttoapplythePDqueuestructureinlarge-scalenetworks.Incontrast,inthePLqueueing,packetsdestinedtothesamenexthopnodeareputintothesamequeue.Thusthenumberofqueuesforanodetomaintainisthenumberofitsneighborswithinonehop. Alotofworkshavestudiedthedynamiccontrolandschedulingofthewirelessmulti-hopnetworksunderthePDqueueing,suchas[ 9 ],[ 10 ],[ 11 ],[ 12 ],[ 13 ],[ 14 ][ 15 ]and[ 16 ],etc. Inthiswork,weconsiderthecongestioncontrolandschedulingofaPLqueueingwirelessnetworkwithmulti-classQoSrequirementsunderbothconstantchannelsandtime-varyingchannels.ThePLqueueingstructuredramaticallyreducesthequeueingoverheadcomparedtothePDqueueingnetworks.InChapter 2 ,weformulatethejointcongestioncontrolandschedulingwithmultipleQoSconstraintsunderPLqueueingasaNetworkUtilityMaximization(NUM)inwirelessmulti-hopnetworks.AniterativealgorithmbasedontheSlidingMode(SM)controlisproposedtooptimallysolvetheNUMproblemunderconstantchannels. DifferentfromtheSMcontrollerin[ 17 ][ 18 ]designedfortheInternet,thewirelesscontextinducesaninteractionbetweencongestioncontrolonLayer4andschedulingonLayer2,sothattheconvergenceconditionsarenaturallysatisedandnonetworkfeedbackisnecessary,whichisagoodexampleoflayeringasoptimizationdecomposition[ 19 ][ 20 ].Undertheper-next-hopqueueingmodel,weapplytheSMcontroltechnique 15

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tothecongestioncontrolproblemanddevelopadistributedscheme.InChapter 3 ,weextendourworkinChapter 2 tothescenariowithtime-varyingchannelsandevaluatetheperformanceoftheSMbasedalgorithm.Weprovethat1)thedistributedcontrolschemeconvergesandachievesoptimalityundercontinuous-timesystemparameters;and2)itconvergestoaboundedneighborhoodoftheoptimalsolutionunderdiscretetimeandtime-varyingchannels. 1.1.2NetworkCapacityRegionandDynamicControlunderPLQueueing Wirelessnetworksareactingasubiquitousinfrastructureformanyapplicationsandthedemandforalargercapacitykeepsincreasing.Accordingly,derivingthecapacityregionofawirelessnetworkisimportantbecause1)fairandeffectivenetworkingtechniquescanbedesignedtotakefulladvantageoftheresourceandsystemcapabilities;2)thecapacityregionprovidesimportantinformationforadmissioncontrol;and3)evenunderstochasticsystemparameters,thecapacityregioncanhelptondadynamiccontrolpolicytoimprovetheperformanceofthenetwork,i.e.throughputandnetworkutility. Inqueueingnetworks,thestabilityofanetworkisdenedbythestabilityofthequeues[ 21 ][ 9 ].Asafore-mentioned,therearetwocommonlyusedqueuestructures:thePDqueueasin[ 9 ][ 22 ][ 10 ][ 7 ]andthePLqueueasin[ 22 ][ 23 ][ 24 ].Correspondingly,itisshownin[ 24 ]thatthePLqueuestructureinvolveslessoverheadandpreservessomegooddesignfeaturessuchasdecomposition[ 22 ]. WorksonthedynamiccontrolunderthePDqueuestructurecanbefoundin[ 21 ][ 9 ][ 25 ][ 26 ][ 12 ][ 14 ][ 13 ][ 27 ][ 28 ].Theseminalwork[ 21 ]rstdenesthestabilityregionforawirelessnetworkandproposesathroughputoptimalalgorithmMaxWeight,a.k.a.,back-pressurealgorithm,whichhasbeenstudiedextensivelyintheliterature,e.g.,[ 13 ][ 27 ][ 28 ].In[ 9 ],Neely,et.al.furtherexplorethecapacityregionofastochasticnetworkwithtime-varyingchannelsandderivethenecessaryandsufcientconditionsofthecapacityregionbasedontheLyapunovdriftmethod[ 12 ].Theproposeddynamicrouting 16

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andpowercontrol(DRPC)[ 9 ]algorithmcanstabilizethenetworkinthecapacityregion.Song,et.al.in[ 25 ]proposeaminimumenergyscheduling(MES)whichachievesbothminimumenergyconsumptionandthroughputoptimality.In[ 23 ],Bui,et.al.proposeashadowalgorithmunderPLqueues,whichusescounterstomimicthePDqueuedynamicsbasedonstaticrouting.However,itisdifculttodesignagoodPLroutingalgorithmunderBui'sframework.Therefore,thecapacityregionaswellasthedynamiccontrolpoliciesforthePLqueueingnetworksstillneedexploring. Differentfromtheexistingworks,westudythecapacityregionanddynamicroutingandpowercontrolofthePLqueueingnetworks.Oursystemmodelsharessomesimilaritieswiththeroute-dependentcasein[ 22 ][ 23 ].AlltheexistingworksconsiderstaticroutingunderPLqueueingmodel.InChapter 4 ,weconsiderallpossiblepowerallocationandroutingschemesthatcanstabilizethenetwork.DifferentfromthePDqueueingstructure,thePLqueueinghasbiggerschedulinggranularitybecausetheowscontainsthepacketsdestinedtodifferentnodes.AlthoughtheLyapunovdriftmethodprovidesapowerfulsufcientconditionfornetworkstabilityin[ 12 ],propertiesoftheinputandoutputowsofthePLqueuesaredifculttocharacterize,whichmakesthisproblemchallenging.Byintroducingauxiliaryowvariables,wederivethethesufcientandnecessaryconditionsofthecapacityregionforaper-destinationper-link(PDPL)queueingnetworkundertime-varyingchannels.ForthePDPLnetwork,Weproposeastabilizingroutingandpowercontrolalgorithm,namely,DRPC-PDPL.Thenadynamiccontrolpolicy(i.e.,DRPC-PL)forthePLnetworkisproposedbyusingtheinformationofthePDPLcounters,whichkeeptrackofthePDshadowqueuelengths. 1.2ProblemDescriptionandContributionsinRenewableResourceIntegrationintheSmartGrid Thenationalpowerdemandisexpectedtotripleby2050,whichposesgreatchallengestothecurrentelectricpowergriddesignedover100yearsago.Asaresult,theconceptoftheSmartGridisproposedtoimproveenergyefciencyandmake 17

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Figure1-1. TheSmartGridvision[source:U.S.DepartmentofEnergy] energydistributionandgenerationmoreefcient,costeffective,andsecure.Generallyspeaking,theSmartGridisthevisionofthefutureelectricpowergrid,whichcanintelligentlyprovidepowerinasustainablemanner.ThefeaturesoftheSmartGridisannouncedbytheU.S.DepartmentofEnergy(DoE)[ 29 ]asbelow: Self-healingfrompowerdisturbanceevents; Enablingactiveparticipationbyconsumersindemandresponse; Operatingresilientlyagainstphysicalandcyberattack; Providingpowerqualityfor21stcenturyneeds; Accommodatingallgenerationandstorageoptions; Enablingnewproducts,services,andmarkets; Optimizingassetsandoperatingefciently. TheSmartGridtechnologiesareillustratedinFigure 1-1 ,wheretheSmartGridisregardedasthesumofservices,devices,andsoftware.Thesevalue-addedcomponentscanaddalayerofintelligenceontotoday's20th-centuryelectricitygrid 18

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suchthatthecomponentscantalktoeachothertofacilitatedecisionmakingandoptimizetheprocessfromthegenerationoftheelectricitytothedeliverytocustomers. Thetraditionalpowergridisdividedintothreeparts:generation,transmissionanddistribution[ 30 ].InFigure 1-1 ,theboundariesofthesepartsareblurredintheSmartGrid,wherepowergenerationcanhappenatthedistributionside,suchasrenewableresources.Theappliancesareabletocommunicateinaplug-inandplayway.Theadvancedmeteringsystemenablescommunicationwiththesupplyside,managementoftheapplianceandoptimizationofthehouseholdpowerusage. OneimportantconceptintheSmartGridisdemandresponse[ 31 ][ 32 ][ 33 ],whichallowsconsumerinteractionwiththegridbyprovidingreal-timeinformationabouttheirelectricityusage,thestatusofthepowergrid,eventheelectricitymarket.Smartgridtechnologiescanalsopotentiallyreduceelectricitycosts,increasereliability,andefciency.Additionally,smartgridtechnologiescanalsoplayanimportantroleasintermittentrenewableenergysourcesandplug-inhybridelectricvehicles(PHEVs)becomemorewidespread. AnotherimportantfeatureisthattheSmartGridwillaccommodateasmuchrenewableenergyaspossible.Clean,greenandrenewableenergyisoneofthebiggestdriversoftheSmartGrid.Intherenewables2011globalstatusreport[ 34 ]ofrenewableenergypolicynetworkforthe21stcentury(REN21),renewableenergysuppliedanestimated16%ofglobalnalenergyconsumptionanddeliveredcloseto20%ofglobalelectricityin2010.Despitethechallengesofharnessingrenewableenergysourcesandthechallengesofdeliveringrenewablepowertopowersupplygrids,windandsolararepushedasthesolutionstoweaningtheglobaleconomyofffossilfuels.However,mostoftherenewableresourcesarerandomandintermittent,e.g.,windandsolar.Currently,thecostofsolarpanelsisstilltoohigh,whichpreventsitsapplicationinlargearea[ 35 ].Thecostofwindpowerisalsohigherthanconventionalgenerationsuchaspowerproducedbynaturalgasandcoal.Duetotherandomnessofwindenergyproduction, 19

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ancillaryservice(AS)isneededinlargeamounttoregulatewindpowerandprovidesystemstabilityandreliability.Asaresult,thecostofwindpowerdependsontheASmarketandsometimescanbequitehigherthanconventionalgeneration.Therefore,itischallengingtoeconomicallyintegratewindenergyandsatisfyRPS.Therefore,efcientlyintegratinggreenenergyintothecurrentpowersystemisnotonlychallengingbutcrucialtothedevelopmentoftheSmartGrid.Withthecommunication,sensingandadvancedcontrolfeatureslaunchedintheSmartGrid,theinteractionamongthegridcomponentswillfacilitatesolvingthisproblem. Therenewableresourceintegrationisgainingimportanceasthefossilfuelsaredepleted.Therearemanyworksdiscussingaboutintegratingrenewableenergiesbytheplug-inhybridelectricvehicle(PHEV)[ 36 ][ 37 ][ 38 ][ 39 ].TherearealsomanydiscussingthewindpowerintegrationintheSmartGrid[ 40 ][ 41 ][ 42 ][ 43 ][ 44 ].Thechallengesinsolvingthisproblemlieintherandomnessandintermittenceofwindpower.Differentfromthepreviousworks,weconsidertheeconomicfeasibilityofwindpowerintegrationbasedontheelectricitymarketandtrytomakeuseoftherandomnessinwindpowergeneration. InChapter 5 ,weconsiderstatisticallyintegratingwindpowerundertheconstraintofRenewablePortfolioStandard(RPS).TheprotmaximizationproblemofagreenutilityintheSmartGridisformulatedasastochasticoptimizationproblemandsolvedbyastationarycontrollerundertheassumptionofsystemergodicityinChapter 5 1.3Outline Inthiswork,thecongestioncontrolandschedulinginmulti-hopwirelessnetworksunderinvariantandtime-varyingchannelsisstudiedinChapter 2 andChapter 3 ,respectively.InChapter 4 ,wedenethecapacityregionofthePLqueueingnetworkandproposeadynamicroutingandpowercontrolalgorithm,i.e.,DRPC-PL,whichcanstabilizethenetworkwithinthecapacityregionofthePLnetwork.InChapter 5 ,a 20

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marketbasedstatisticalintegrationofwindpowerisstudiedtomaximizetheprotofahybridwindpowerutilityundertheconstraintoftheRPS. 21

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CHAPTER2THEPER-LINKQUEUEBASEDCONGESTIONCONTROLANDSCHEDULINGINWIRELESSMULTI-HOPNETWORKSWITHTIMEINVARIANTCHANNELS Futureadhocnetworksaimtoprovidehighdatarate,andmulti-classserviceswithQualityofService(QoS).Howtosolvethecongestioncontrolandschedulingproblemefcientlyiscrucialtoachievethesekeyfeatures.Generally,thisproblemischallengingbecausethetwoproblemsarecoupledwitheachother,andthecomplexityoroverheadistoohigheventhroughdualdecomposition[ 19 ],especiallywithQoSrequirementsimposed. In[ 7 ][ 22 ],theauthorsmodelthejointcongestioncontrolandschedulingproblemasaNetworkUtilityMaximization(NUM)problemanddecomposeitintotwoseparateproblems,i.e.,acongestioncontrolproblemandaschedulingproblem,whichproducestheclassicprimal-dualalgorithminmulti-hopwirelessnetworks.Intheirworks,everynodeneedstomaintainaqueueorvirtualqueueforeachowdestination,soastoutilizetheback-pressurequeuelengthtotrackthechangeofutilityandoptimizelinkscheduling.Thisoverheadisnotdesirableespeciallyinadhocnetworksbecauserelaynodesmaynotbeavailableorwillingtooffersuchresourceforotherusers.Moreover,theydonotconsiderQoSrequirementssuchasminimumtransmissionratesforcertainusers.Generally,theQoSconstraintswillmakeitdifculttoimplementtheprimalalgorithminadistributedmanner. Forqueueingnetworks,theseminalwork[ 21 ]rstdenesastabilityregionandproposesamaximumweightscheduler.In[ 9 ],theauthorsfurthergeneralizethestabilityregionforastochasticnetworkbydeningthestabilityofqueuesandproposedtheDynamicRoutingPowerControl(DRPC)includingpowercontrolandroutingalgorithms,whichcanstabilizethesysteminthecapacityregion.Morerelatedworkscanbefoundin[ 12 ].Tomakeitfocused,wedonotconsidertime-varyingwirelesschannelsinthischapter,butthemethoddescribedherecanbereadilyextendedtotime-varyingwirelesschannelsbyapplyingtheLyapunovdriftmethod[ 12 ]. 22

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Inadhocnetworks,distributedalgorithmsaredesirableduetotheunavailabilityofglobalinformation.SlidingMode(SM)controltheory[ 45 ]providesapowerfultooltodesigndistributedalgorithmsforsolvingconvexoptimizationproblemswithalargenumberofconstraints,andhasbeenappliedtocongestioncontrolproblemsintheInternet[ 17 ].In[ 17 ],theauthorsproposeanadaptivecontrollawformulti-hop,multi-pathwirednetworkswithmulti-classservices.However,theSMcontrollercannotbeappliedtowirelessnetworksdirectlybecause,inadhocnetworks,theinterferenceamongusersmakesthecongestioncontrolproblemcoupledwiththeschedulingproblem.Thedecompositionbetweenthejointcongestioncontrolandschedulingproblem[ 22 ]intheNUMproblemfacilitatesustoapplytheSMcontroltechniquetocongestioncontrolinadhocnetworks. Differentfromtheexistingworks,weconsideraPLqueuingsystemwithmulti-classQoSrequirements.ByPL-queue,wemeanthatpacketsdestinedtothesamenexthopnode,areputintothesamequeue.Sothenumberofqueuesthatanodehastomaintainisthenumberofitsneighborswithinonehop.Withoutconfusion,wecallitthehop-basedqueueinthischaptercomparedtothedestination-basedqueuein[ 7 ][ 22 ].BasedonthePL-queuesystem,wearethersttoapplytheSMcontroltechniquetosolvingthecongestioncontrolprobleminwirelessnetworks;wealsoprovetheoptimalityandconvergenceoftheSMbaseddistributedsolutiondevelopedinthischapter. Therestofthechapterisorganizedasfollows.InSection 2.1 ,wedescribethesystemmodelandformulatetheNUMproblem,followedbythesolutiontotheNUMprobleminSection 2.2 .Section 2.3 presentssimulationresultsandSection 2.4 concludesthechapter. 2.1NetworkModelandTheNUMProblem Inthiswork,weconsiderawirelessmulti-hopmulti-pathadhocnetworkwiththetransmissionrateQoSrequirements.Itisassumedthatamulti-pathroutingalgorithmisappliedaccordingtotheinformationsuchaslocationandSignaltoInterferenceplus 23

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Table2-1. Notations. F:thesetofowindex. N:thesetofnodeindex. jAj:thecardinalityofsetA. NNi:theshortrepresentationof8i2N,j2Ni. Ni:thesetoftheneighborsofnodei(i2N). li,j:thelinkfromtransmitnodeitoreceivenodej (NNi). i,j:thechannelcapacityofli,j(NNi). pi,j:thetransmissionpowerforli,j(NNi). P:powerallocationmatrixP=[pi,j](i,j2N). P:thesetoffeasiblepowerallocationmatrixP. xs:thetotaltransmissionrateofows(s2F)over multipath. x:thetransmissionratevectorforalltheows x=[x1,x2,,xjFj]T. bs:thesourcenodeofows(s2F). ni:thenexthopnodeofnodei(i2N). xsi,j:thetransmissionrateofowsoverli,j(s2F, bs=i,ni=j,8i2N,j2Ni). R:thecapacityregionofagivenwirelessnetwork. Fii,j:theinowrateforli,j(i,j2N),denedin( 2 ). Foi,j:theoutowrateforli,j(i,j2N),denedin( 2 ). Us(xs):theutilityofowswithratexs(s2F). i,j:theLagrangianpriceforli,j(NNi). :Lagrangianpricematrix=[i,j](i,j2N). Hs:thesetoftheindicesoftheconstraintfunctionsfor ows(s2F). NoiseRatio(SINR)ateachnode.ThenotationsarelistedinTable 2-1 ,wherematrixA=[ai,j]isatensornotationformatrixA. TodescribethetransmissionrateQoSrequirements,denetheQoSconstraintfunctionsforowsby: h1(xs)=smin)]TJ /F3 11.955 Tf 11.96 0 Td[(xs (2a)h2(xs)=xs)]TJ /F9 11.955 Tf 11.96 0 Td[(smax (2b) wheresmin,andsmaxareconstants.TheQoSrequirementscanbedescribedbyusinginequalityconstraintsonthefunctionsin( 2 ).Forexample,theservicethatguarantees 24

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minimumtransmissionrateforowscanbedenedash1(xs)0.Fornotationsimplicity,weuseh1(xs)0andh2(xs)0torepresentequalityconstraint,wheresmin=smax. Thetotaltransmissionratexsofowsovermulti-pathisgivenby: xs=Xi=bsXj2Nixsi,j(2) Tostabilizethequeuesinthesystem,thetransmissionvectorshouldlieinRdenedin[ 9 ],andsatisfytheowconservationconstraintforeachqueue.TheinowandoutowratesFii,jandFoi,jforli,jaredenedasbelow: Fii,j=Xk2Nirk,i,j (2) Foi,j=i,j (2) wherethetransienttrafcraterk,i,j,differentfromk,i,istheinjectiondataratefromlk,iwithjasthenext-hopnode,whichdoesnotincludetrafcdestinedtonodei.SoFii,jisaquantitythatcanbemeasuredlocally.Here,anarbitrarymultipleaccessschemescanbeused. Inthischapter,westudythejointcongestioncontrolandschedulingproblemandformulateitasanNUMproblem: maxfx,PgXs2FUs(xs) (2a)s.t.:x2R (2b)Xs:bs=ixsi,j)]TJ /F3 11.955 Tf 11.96 0 Td[(Foi,j+Fii,j0,8i2N,j2Ni (2c)P2P (2d)h0 (2e) 25

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where( 2c )istheowconservationconstraint;( 2d )istheconstraintonpowerandeachelementofwhichissubjecttoconstraintsonminimumSINRandmaximumtransmissionpower;( 2e )speciestheQoSconstraintsandvector h=[h1(xs1),h2(xs2),,hM(xsM)]T, wherehm(xsm)(sm2F,m2f1,2,,Mg)canbeanyfunctiondenedin( 2 ),andHs=fmjhm(xs)isanelementofhg.Forexample,inanetworkwhereonlyFlow1hasaraterequirement10x120,thenh=[x1)]TJ /F7 11.955 Tf 13.07 0 Td[(20,10)]TJ /F3 11.955 Tf 13.08 0 Td[(x1]TandH1=f1,2g,Hs=;(s6=1). Inthenextsection,wepresentoursolutionto( 2 ). 2.2TheSolutiontoTheNUMProblem Thissectionisorganizedasbelow:in 2.2.1 ,theNUMproblem( 2 )isdecomposedintothecongestioncontrolproblem( 2 )andtheschedulingproblem( 2 ),given.Followingtheclassicprimal-dualalgorithm[ 7 ],theLagrangepriceisderivedin 2.2.2 .Then( 2 )and( 2 )aresolvediterativelyin 2.2.3 and 2.2.4 .TheoptimalityofthecontinuoustimecontrollawsisgivenbyTheorem 2.1 in 2.2.5 .Thediscretetimealgorithmsaregivenin 2.2.6 2.2.1Decomposition Tosolveproblem( 2 ),weusethedualdecompositionmethod[ 7 ][ 22 ].TheLagrangiandualfunctionof( 2 )withrespecttoconstraint( 2c )isD(): D()=maxfx,PgXs2FUs(xs))]TJ /F10 11.955 Tf -75.83 -29.26 Td[(XNNii,j Xs:bs=ixsi,j)]TJ /F3 11.955 Tf 11.95 0 Td[(Foi,j+Fii,j!=maxfxg Xs2FUs(xs))]TJ /F10 11.955 Tf 15.38 11.36 Td[(XNNii,jXs:bs=ixsi,j!+maxfPgXNNii,j)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Foi,j)]TJ /F3 11.955 Tf 11.96 0 Td[(Fii,j (2) 26

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wherei,jistheLagrangianpriceforconstraint( 2c ).Itisimpliedby( 2 )thatgiventhelinkpricei,j,thecongestioncontrolproblemandtheschedulingproblemcanbesolvedseparately,wherethecongestioncontrolproblemisgivenby: maxfx2R,h0g Xs2FUs(xs))]TJ /F10 11.955 Tf 15.39 11.35 Td[(XNNii,jXs:bs=ixsi,j!(2) andtheschedulingproblemisgivenby: maxfP2PgXNNii,j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Foi,j)]TJ /F3 11.955 Tf 11.96 0 Td[(Fii,j(2) Next,weexplainhowtodeterminelinkpriceij. 2.2.2LagrangianPrice Toobtaini,j,considerthedualproblemof( 2 ): minD()(2) Since( 2 )isaconvexfunctionofi,j[ 46 ],takethepartialderivativewithrespecttoi,j,weobtain: @D() @i,j=Foi,j)]TJ /F3 11.955 Tf 11.96 0 Td[(Fii,j)]TJ /F10 11.955 Tf 18.97 11.36 Td[(Xs:b(s)=ixsi,j(2) AccordingtoSlater'stheorem[ 46 ],thereisnodualitygapbetween( 2 )and( 2 ),sothelagrangianpricecanbeupdatediterativelyby: _i,j(t)= Xs:bs=ixsi,j(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Foi,j(t)+Fii,j(t)!+i,j(t),8i2N,j2Ni (2) wheretheprojection(y)+zisdenedas: (y)+z=8><>:y,ifz00,o.w.(2) 27

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2.2.3SMBasedSolutiontotheCongestionControlProblem Inthissubsection,theSMcontrollertothecongestioncontrolproblem( 2 )isdevelopedforgiven.TheSMcontroltheory[ 45 ][ 47 ]providesanoptimaldistributedmethodtosolveconvexoptimizationproblemswithalargenumberofconstraints.Furthermore,theSMcontrollerinvolvessomeadjustableparameterscorrespondingtotheperformanceofthecontrollersuchassteady-stateerrorandrateofconvergence.OurSMbasedcontrollerfor( 2 )isgivenby: _xsi,j(t)=@Us(xs) @xsi,jxs(t))]TJ /F9 11.955 Tf 11.96 0 Td[(i,j(t)+(t)u)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(xsi,j(t)+Xm2Hsvm(t)@hm(xs) @xsi,jxs(t),8i2N,j2Ni (2) where@Us(xs) @xsi,jxs(t)isthepartialderivativeofUs()withrespecttoxsi,j(t)andevaluatedatxs(t);u(x)isdenedin( 2 )tomakesurethatthetransmissionrateisnon-negative;(t)isauser-speciedfunctionforcontrolpurpose,whichisusuallyaconstant;andvm(t)isdenedin( 2 ): u(x(t))=8><>:1,ifx(t)<00,ifx(t)0 (2) vm(x(t))=8><>:m(t),ifhm(x(t))>00,ifhm(x(t))0 (2) wherem(t)isalsouser-speciedfunctionscorrespondingtohm(xs). 2.2.4SolutiontotheSchedulingProblem Generally,theschedulingpart( 2 )istightlyrelatedtotheinterferencemodelofthenetworksettingbecausethechannelcapacitydependsontheSINR.SimilartotheDynamicRoutingPowerControl(DRPC)[ 9 ],thesolutiontotheschedulingpartattimet 28

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is: ^P(t)=argmaxP2PXNNii,j(t))]TJ /F3 11.955 Tf 5.47 -9.68 Td[(Foi,j(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Fii,j(t)(2)Fii,j(t)in( 2 )isaparameterthatcanbemeasuredlocallyinnodei,andFoi,j(t)canbeobtainedbyestimatingthechannelofli,j.Problem( 2 )canbesimpliedaccordingto[ 7 ][ 22 ],andturnsouttobeagraphcoloringproblem,whichhasbeenstudiedintensively.Giventhelinkpricei,j,( 2 )isgenerallyNPhard,evenifthepowercontrolison-offwithtwovalues.Ifsuboptimalperformanceisallowed,somealgorithmssuchascolumngenerationcanbeusedtosolvethegraphcoloringproblem. 2.2.5OptimalityandConvergenceofTheSolution Inthissection,weprovetheoptimalityandconvergencepropertyofthesolutionabove,i.e.,( 2 ),( 2 )and( 2 ),fortheNUMproblem( 2 ),asstatedinTheorem 2.1 : Theorem2.1. Assumethat:1)x(0)isafeasibleratevector;2)eachutilityfunctionUs()isconcavew.r.t.thetransmissionratexs; 3)( 2 ),( 2 )and( 2 )arethegoverninglawsofthesystem. Thenthesystemstate(x(t),(t))willconvergetotheoptimal,denotedby(^xs,^),where^xsistheoptimalratevectorto( 2 ),and^istheoptimalpricematrix. Proof. Toprovetheconvergenceandoptimalityofthealgorithm,weconstructtheLyapunovfunction( 2 ): W(x,)=1 2Xs,j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(xsi,j)]TJ /F7 11.955 Tf 12.14 0 Td[(^xsi,j2+1 2Xi,ji,j)]TJ /F7 11.955 Tf 12.24 2.65 Td[(^i,j2(2)^i,jand^xsi,jaretheoptimalsolutions(supposedtoexist,oritismeaninglesstosolveit)toproblem( 2 ),sotheyareconstants.Withalittleconfusion,weomitthe(t)ofthe 29

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quantitiesbelow.DifferentiatingtheLyapunovfunctionwithrespecttotime,weobtain: _W(x,)=Xs,j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(xsi,j)]TJ /F7 11.955 Tf 12.14 0 Td[(^xsi,j @Us(xs) @xsi,jxs)]TJ /F9 11.955 Tf 11.95 0 Td[(i,j+u(xsi,j)+Xm2Hs@hk @xsi,jxsvm!+Xi,ji,j)]TJ /F7 11.955 Tf 12.23 2.66 Td[(^i,j Xs:bs=ixsi,j+Fii,j)]TJ /F3 11.955 Tf 11.95 0 Td[(Foi,j!+i,j (2) Xs,j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(xsi,j)]TJ /F7 11.955 Tf 12.14 0 Td[(^xsi,j @Us(xs) @xsi,jxs)]TJ /F9 11.955 Tf 11.95 0 Td[(i,j+u(xsi,j)+Xm2Hs@hm @xsi,jxsvm!+Xi,ji,j)]TJ /F7 11.955 Tf 12.23 2.65 Td[(^i,j Xs:bs=ixsi,j+Fii,j)]TJ /F3 11.955 Tf 11.95 0 Td[(Foi,j! (2) Thelastinequalityfollowsbynotingthat( 2 )and( 2 )areequaliftheprojectionin( 2 )isinactive,andiftheprojectionisactive,theprojectionin( 2 )iszero.As^xsi,jistheoptimalsolutiontotheNUMproblem,atthattime,thecontrolu^xsi,jandvkarezero,sothefactcomesin( 2 ): ^qi,j=@Us(xs) @xsi,j^xs(2) 30

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Substituting( 2 )into( 2 ): _W(x,)Xs,j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(xsi,j)]TJ /F7 11.955 Tf 12.14 0 Td[(^xsi,j @Us(xs) @xsi,jxs)]TJ /F9 11.955 Tf 14.96 8.08 Td[(@Us @xsi,j^xs)]TJ /F9 11.955 Tf 11.95 0 Td[(i,j+^i,j+u(xsi,j)+Xm2Hsvm@hm(xs) @xsi,jxs!+Xi,ji,j)]TJ /F7 11.955 Tf 12.23 2.66 Td[(^i,j Xs:bs=ixsi,j)]TJ /F10 11.955 Tf 15.26 11.36 Td[(Xs:bs=i^xsi,j!+Xi,ji,j)]TJ /F7 11.955 Tf 12.23 2.66 Td[(^i,j Xs:bs=i^xsi,j)]TJ /F3 11.955 Tf 11.95 0 Td[(Fii,j)]TJ /F3 11.955 Tf 11.96 0 Td[(Foi,j! (2) =Xs,j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(xsi,j)]TJ /F7 11.955 Tf 12.14 0 Td[(^xsi,j @Us(xs) @xsi,jxs)]TJ /F9 11.955 Tf 14.34 8.08 Td[(@Us(xs) @xsi,j^xs!+Xs,j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(xsi,j)]TJ /F7 11.955 Tf 12.14 0 Td[(^xsi,j^i,j)]TJ /F9 11.955 Tf 11.95 0 Td[(i,j+Xs,j)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(xsi,j)]TJ /F7 11.955 Tf 12.14 0 Td[(^xsi,j u(xsi,j)+Xm2Hs@hm @xsi,jxsvm!+Xi,ji,j)]TJ /F7 11.955 Tf 12.23 2.66 Td[(^i,j Xs:bs=ixsi,j)]TJ /F10 11.955 Tf 15.26 11.36 Td[(Xs:bs=i^xsi,j!+Xi,ji,j)]TJ /F7 11.955 Tf 12.23 2.65 Td[(^i,j Xs:bs=i^xsi,j)]TJ /F3 11.955 Tf 11.96 0 Td[(Fii,j)]TJ /F3 11.955 Tf 11.95 0 Td[(Foi,j! (2) =Xs,j)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(xsi,j)]TJ /F7 11.955 Tf 12.14 0 Td[(^xsi,jXs,j)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(xsi,j)]TJ /F7 11.955 Tf 12.14 0 Td[(^xsi,j @Us(xs) @xsi,jxs)]TJ /F9 11.955 Tf 14.95 8.09 Td[(@Us @^xsi,j^xs!+Xs,j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(xsi,j)]TJ /F7 11.955 Tf 12.14 0 Td[(^xsi,j u(xsi,j)+Xm2Hs@hm(xs) @xi,jjxsvm!+Xi,ji,j)]TJ /F7 11.955 Tf 12.23 2.66 Td[(^i,j Xs:bs=i^xsi,j)]TJ /F3 11.955 Tf 11.96 0 Td[(Fii,j)]TJ /F3 11.955 Tf 11.95 0 Td[(Foi,j! (2) 31

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Thesecondtermtheforthtermin( 2 b)canceleachotherbecausefordifferentowfromthesamenodewhosharethesamenexthop,theyyieldthesamei,j,then Xs,j)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(xsi,j)]TJ /F7 11.955 Tf 12.14 0 Td[(^xsi,j(^i,j)]TJ /F9 11.955 Tf 11.95 0 Td[(i,j)=Xi,j^i,j)]TJ /F9 11.955 Tf 11.95 0 Td[(i,j Xs:bs=ixsi,j)]TJ /F10 11.955 Tf 15.26 11.36 Td[(Xs:bs=i^xsi,j! (2) AsUsareconcavewithxs,andxsi,jareinterchangeableforj,sowehave: Xs,j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(xsi,j)]TJ /F7 11.955 Tf 12.14 0 Td[(^xsi,j @Us(xs) @xsi,jxs)]TJ /F9 11.955 Tf 14.34 8.09 Td[(@Us(xs) @^xsi,j^xs!=Xs Xj:j2Nixsi,j)]TJ /F10 11.955 Tf 15 11.35 Td[(Xj:j2Ni^xsi,j! @Us @xsi,jxs)]TJ /F9 11.955 Tf 14.95 8.08 Td[(@Us @^xsi,j^xs! (2) =Xs(xs)]TJ /F7 11.955 Tf 12.14 0 Td[(^xs) @Us(xs) @xsi,jxs)]TJ /F9 11.955 Tf 14.35 8.09 Td[(@Us(xs) @^xsi,j^xs! (2) 0 (2) where( 2 )followsthatas@Us @xsi,jxsisfreeoftheindividualtransmissionrate,foranidenticalows,theterm@Us(xs) @xsi,jxs)]TJ /F14 7.97 Tf 14.35 5.47 Td[(@Us(xs) @^xsi,j^xsisthesameforanyj2Ni. Fromconstraint( 2c ), Xs:bs=i^xsi,j^Foi,j)]TJ /F7 11.955 Tf 13.18 2.66 Td[(^Fii,j(2) wherethequantitieswithhatsaretheoptimalsolutiontotheNUMproblem,thus Xi,ji,j(Xs:bs=i^xsi,j+Fii,j)]TJ /F3 11.955 Tf 11.96 0 Td[(Foi,j)0(2) sincetheratesFii,jandFoi,jsolve( 2 )and^Fii,jand^Foi,jarefeasiblesolutionstotheoutowandinowratesatnodeiwithnexthopj.FromtheKKTconditions: i,j Xs:bs=i^xsi,j+^Fii,j)]TJ /F3 11.955 Tf 11.96 0 Td[(Foi,j!=0(2) 32

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and Xi,j)]TJ /F7 11.955 Tf 9.58 2.65 Td[(^i,j0@Xs:b(s)=i^xsi,j+Fii,j)]TJ /F3 11.955 Tf 11.95 0 Td[(Foi,j1A=Xi,j)]TJ /F7 11.955 Tf 9.58 2.66 Td[(^i,jFii,j)]TJ /F7 11.955 Tf 13.17 2.66 Td[(^Fii,j)]TJ /F3 11.955 Tf 11.96 0 Td[(Foi,j+^Foi,j (2) 0 (2) so: Xi,ji,j)]TJ /F7 11.955 Tf 12.23 2.65 Td[(^i,j Xs:bs=i^xsi,j+Fii,j)]TJ /F3 11.955 Tf 11.96 0 Td[(Foi,j!0(2) Forthesecondpart,from( 2 ),xssatises( 2 ) u(xi,j)+Xm2Hsvm@hm @xsi,jjxs:8><>:=0,ifhm0,m2Hs0,o.w.(2) So, _W0(2) whichimpliesthatthecongestioncontrolschemeisoptimalandconvergent. 2.2.6TheDiscrete-timeAlgorithm Accordingtothecontinuoustimesolutionin( 2 ),( 2 )and( 2 ),wedesignadiscretetimealgorithmasbelow,wherenisthestepindexandisthetimeintervalofastep. SMbasedcongestioncontrolandschedulingalgorithm. Algorithm1. Initialization:n=0,ndafeasibleratevectorasx(0),set(0)=0,Fii,j(n)=0,andFoi,j(n)=0,andspecifyaconcaveutilityfunctionforeveryow. Attimen+1,update(n+1)accordingto( 2 );andestimateFii,j(n+1)andFoi,j(n)tosolve( 2 )bychoosingtheoptimaltransmissionpowermatrix^P(n+1); Eachowadjuststhetransmissionrateaccordingto( 2 ); 33

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Goto2. i,j(n+1)= i,j(n)+ Xs:bs=ixsi,j(n))]TJ /F3 11.955 Tf 11.96 0 Td[(Foi,j(n)+Fii,j(n)!!+ (2) xsi,j(n+1)= xsi,j(n)+1 @Us(xs) @xsi,jxs(n))]TJ /F9 11.955 Tf 11.96 0 Td[(i,j(n)+Xm2Hsvm(n)@hm(xs) @xsi,jxs(n)!!+ (2) ^P(n+1)=argmaxP2PXNNii,j(n+1))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Foi,j(n+1))]TJ /F3 11.955 Tf 11.95 0 Td[(Fii,j(n+1) (2) wherey=(z)+isdenedasmax(z,0). NotethatalltheinformationthateachnodeneedsforcomputingAlgorithm 1 islocal.Except( 2 ),( 2 )and( 2 )aredistributedwithlowcomputationalcomplexity. 2.3SimulationResults Inthissection,wesimulatea4-useradhocnetworkwithchainandlooptopologyunderTDMAchannelstodemonstratetheconvergenceandoptimalityof 1 .Section 2.3.1 describesthesimulationsetting,whileSection 2.3.2 illustratestheperformanceofouralgorithm. 2.3.1SimulationSetting Wesimulateadiscretetimesystem.ThetopologywesimulateisdepictedinFigure 2-1 and 2-2 ,whereFigure 2-1 showsachaintopologyandFigure 2-2 showsalooptopology.Thedistancebetweenanytwoneighboringnodesisthesame.ThearrowdirectioninFigure 2-1 and 2-2 indicatesthedirectionofalink.Withoutlossofgenerality,onlyone-hopinterferenceisconsidered,i.e.,eachnodecannottransmitandreceivedatasimultaneously. Weuseauidmodel,i.e.,thesizeofapacketisinnitesimal.Thetimeintervalisauser-speciedparameter.Thechannelcapacityofeachlinkisxedwithi,j= 34

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Figure2-1. Chaintopology Figure2-2. Looptopology 50packets=(8i2N,j2Ni).Theutilityfunctionofeachowis: Us(xs)=wslog(xs)(2) wherewsisauserspeciedparameter,andwechooseidenticallyws=1here.Table 2-2 liststheparametersusedinthesimulation. 2.3.2PerformanceofAlgorithm 1 Figure 2-3 andFigure 2-4 illustratethetransmissionratevectorxvs.iterationstepnforthechaintopologywithout/withQoS.Figure 2-5 andFigure 2-6 showthoseunderlooptopologywithout/withQoS.InFigure 2-4 ,Flow1hasaminimumraterequirementof18packets/andinFigure 2-6 ,Flow1hasaminimumraterequirementof33packets/.ItisobservedunderAlgorithm 1 thatthetransmissionrateforalltheowsinthefourscenariosconvergesfastforthetwotopologies.AsshowninFigure 2-3 through Table2-2. Simulationparameters Chain Loop Flow1:fl1,2,l2,3g Flow1:fl1,2g Flow2:fl2,3,l3,4g Flow2:fl2,3g Flow3:fl3,4g Flow3:fl3,4g QoS:x118 Flow4:fl4,1g vm=1/3 QoS:x133 vm=1/2 35

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Figure2-3. TransmissionrateforchaintopologywithoutQoS Figure2-4. TransmissionrateforchaintopologywithQoS Figure5,theNetworkutilitiesundernoQoSconstraintsarelargerthanthoseundersomeQoSconstraints,whichisreasonable. Inthechaintopology,l1,2andl3,4canbeactivesimultaneously.l2,3isthebottleneckbecauseFlow1andFlow2passl2,3whichhastobescheduledwithnootheractivelinks.Sox2isexpectedtobesmaller.Thelooptopologyissymmetric,sothetransmissionratesarealsoexpectedtobesymmetricwithx1=x3andx2=x4inFigure 2-5 andFigure 2-6 .Asallthedestinationsfortheowsinlooptopologyarewithinone-hop,theoptimaltransmissionratesforallnodesareexpectedtobe25packets/,whichmatches 36

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Figure2-5. TransmissionrateforlooptopologywithoutQoS Figure2-6. TransmissionrateforlooptopologywithQoS wellwithoursimulationresultsinFigure 2-5 .ThisdemonstratesthatAlgorithm 1 isabletoachieveoptimality. 2.4ChapterSummary Inthischapter,westudiedthejointcongestioncontrolandschedulingproblemformulti-hop,multi-pathadhocnetworkswithQoSconstraintsandweformulateditasanNUMprobleminthescenarioofthehop-basedqueuingnetworks.AfterdecompositionoftheNUMproblem,thecongestioncontrolproblemissolvediterativelybytheSMbasedcontrollerandtheschedulingproblemturnsouttobeagraphcoloringproblem. 37

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Thentheoptimaltransmissionratecanbeobtainedbythesystemupdatelaws,i.e.( 2 ),( 2 )and( 2 ).Inourfuturework,wewillextendourresultstothetimevaryingwirelesschannelsbyapplyingtheLyapunovdriftmethod.WewillalsocomparethecapacityregionofourPLqueuingsystemtothatoftheper-destinationqueuingsystem. 38

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CHAPTER3THEPER-LINKQUEUEBASEDCONGESTIONCONTROLANDSCHEDULINGINWIRELESSMULTI-HOPNETWORKSWITHTIMEVARYINGCHANNELS InChapter 2 ,wepresentthePLqueuebasedNUMproblemanditssolutionundertimeinvariantwirelesschannels.Inthischapter,weextendtheformulationandsolutiontothePLqueueingnetworkwithtime-varyingwirelesschannels.Therestofthechapterisorganizedasfollows.Section 3.2 extendsthesolutioninSection 2.2 towirelessnetworksundertime-varyingchannels.Section 3.3 presentsperformanceanalysisandSection 3.4 presentssimulationresults.Section 3.5 summarizesthechapter. 3.1NetworkModelunderPer-linkQueueing Inthissection,wepresentthemodelofamulti-hopmulti-pathper-linkqueueingwirelessnetwork.Section 3.1.1 describestheper-linkqueueingmodel.Section 3.1.2 presentstheQoSrequirementsforheterogenousservicesandSection 3.1.3 describesthecapacityregionofaper-linkqueueingnetwork. 3.1.1TheQueueingModel Foraper-linkqueueingwirelessnetwork,letFdenotethesetofowsandNdenotethesetofnodes.Alinkli,jinthenetworkhasatransmittingnodei2Nandareceivingnodej2Ni,whereNiisthesetofnodei'snexthopnodes.L=fli,jgdenotesthesetofalllinks.Theselinksaredirected;weassumeconnectivitybetweenanytwonodesissymmetricandthetopologyisassumedtobestatic.Eachows2Fhasatotaltransmissionratexsovermultiplepaths.Letxsi,jdenotethetransmissionrateofowsoverli,jwithsourcenodebs=i.Thenthetotaltransmissionratexsovermulti-pathis xs=Xi=bsXj2Nixsi,j.(3) Theratepowerfunctiondenotedby(G,P)istheofferedphysical-layercapacitymatrixunderchannelstatematrixG=[gi,j]andpowerallocationmatrixP=[pi,j]2P,wherePisthesetoffeasiblepowerallocationmatrixP.MatrixX=[xsi,j]denotesthemulti-pathtransmissionratesforallows.Throughoutthispaper,A=[ai,j]isatensor 39

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notationformatrixA.Awirelesschannelisasharedmedium,wherelinksinterferedwitheachothercontendforexclusiveaccesstothechannel.Aconictgraph[ 48 ]cancapturethecontentionamonglinks;theconictgraphdependsonGandP.Theconvexhullofthecorrespondingratevectorsoftheindependentsetsoftheconictgraphdeterminesthefeasiblerateregionatlinklayer.Itisassumedthataproperroutingalgorithmisappliedtothenetworkaccordingtoinformationsuchasdistanceorsignal-to-interference-plus-noise-ratio(SINR),whichresultsinaroutingtabledenotedbyR.Thesource-destinationpairsonlinklayerareconsideredasows.Inschedulingandrouting,weconsiderthepacket-leveloperationandtheoverheadofpacketizationisassumedtobezero. Sinceweusetheper-linkqueueingmodel,eachnexthopofli,jhasadedicatedqueue(atnodei)denotedbyqi,j.Therefore,thenumberofthequeuesinnodeiisatmostthesameasthenumberofitsnexthops.Ifli,jisactivated,thedatainqi,jistransmitted.Attimet,theoutputofqi,jconsistsofthetrafcdestinedtonodejandthetransittrafcdestinedtonodesotherthannodej.Thuswehave i,j(t)=Xk i,j,k(t)+i,j(t)(3) wherei,j(t)isthedataratedeliveredtonodejthroughli,j;andi,j(t)isashorthandnotationfori,j(G(t),P(t)),i.e.,thephysical-layercapacityofferedonli,j;and i,j,k(t)isthetransmissionratefromqi,jtoqj,kthroughli,jattimet.Here,atripletnodeindex(i,j,k)isusedtodenethetwocascadingqueuesqi,jandqj,kinvolvedinaper-linktransmission. 40

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3.1.2QoSRequirements TodescribetheQoSrequirementsofheterogenousapplications,denetheQoSconstraintfunctionsforowsby: h1(xs)=smin)]TJ /F3 11.955 Tf 11.96 0 Td[(xs (3a)h2(xs)=xs)]TJ /F9 11.955 Tf 11.96 0 Td[(smax (3b) wheresmin,andsmaxareconstants.TheQoSrequirementscanbedescribedbyusinginequalityconstraintsonthefunctionsin( 3 ).Forexample,theservicethatguaranteesminimumtransmissionrateforowscanbedenedash1(xs)0.Fornotationsimplicity,weuseh1(xs)0andh2(xs)0torepresentequalityconstraint,wheresmin=smax. 3.1.3CapacityRegionofaPer-linkQueueingNetwork Inqueueingnetworks,thejointcongestionandschedulingproblemshouldbesolvedundertheconstraintofnetworkstability,whichisdenedasthestabilityofthequeues.Lettheexogenousarrivalatqi,jbedenedby xi,j=Xs:bs=ixsi,j. (3) LetFii,j(t)andFoi,j(t)denotethesteadystateinowandoutowrateofqi,jattimet.Intuitively,wehave Fii,j(t)=Pt)]TJ /F8 7.97 Tf 6.59 0 Td[(1=0Pm2Ni m,i,j() tFoi,j(t)=Pt)]TJ /F8 7.97 Tf 6.59 0 Td[(1=0Pk2Nji,j,k() t. (3) Thentheaverageinowandoutowrate,denotedbyFii,jandFoi,jcanbewrittenasFii,j=limt!1Fii,j(t)Foi,j=limt!1Foi,j(t) 41

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DeneXasthecapacityregionofaper-linkqueueingnetworkundertime-varyingchannels,thenthenecessaryandsufcientconditionsarederivedas xi,jFoi,j)]TJ /F7 11.955 Tf 13.17 2.66 Td[(Fii,j,8(i,j)2L(3) undertheassumptionsofconvergentandboundedarrivalsandconvergenttime-varyingchannelstatessimilarto[ 9 ].Notethatanetworkwithconstantarrivalsandconstantwirelesschannelsisaspecialcaseforthisassumption.Usually,( 3 )isalsoknownastheowconservationconstraintforqueueingnetworks[ 7 ][ 22 ],whichisrecognizedasthenetworkstabilitycondition[ 9 ]. Notethatthecapacityregionofanetworkunderper-linkqueueingmaybedifferentfromthatunderper-destinationqueueing.Intuitively,theper-destinationqueueingyieldssmallergranularityinschedulingthanper-linknetworks.In[ 49 ],westudiedthecapacityregionoftheper-linkqueueingnetworkwithper-destinationbackloginformation.Pleasereferto[ 9 ][ 49 ]fordetailsaboutthederivationanddynamiccontrolinqueueingnetworks. 3.2TheNUMproblemwithTime-varyingChannels ThesystemmodelandassumptionsarethesameasdescribedinChapter 2 .Inthissection,theNUMproblemofawirelessnetworkisconsideredunderslottedtime-varyingchannels.Basedonthecontrollawsof( 2 ),( 2 )and( 2 ),wedesignthecongestioncontrolandschedulingalgorithms,whichasymptoticallysolvetheNUMproblemundertime-varyingchannels. Forsimplicity,weconsiderawirelessnetworkwithslottedtime-varyingchannels,thechannelgainsofwhicharei.i.d.convergent[ 50 ].Differentfromthetransmissionrateunderconstantchannels,theinstantaneoustransmissionrateundertime-varyingchannelschangesfromslottoslot,thustheoptimizedvariablesare[xsi,j],denedinatime-averagesense.Accordingly,theQoSconstraintsarealsodenedonthetime-averagedtransmissionrate.Thenetworkutilityforaowisafunctionofthe 42

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averagetransmissionrateandanoptimaldynamiccontrolpolicyisdenedoverallpossiblepowercontrol,routingandschedulingthatcanrealizetheoptimalaveragetransmissionrate^X. Similarly,theNUMproblemundertime-varyingchannelsisformulatedas maxfX2X,PgXs2FUs(xs) (3a)s.t.:P2P (3b)xi,j)]TJ /F3 11.955 Tf 11.95 0 Td[(Foi,j+Fii,j0,8(i,j)2L (3c)h0 (3d)xs=Xi=bsXj2Nixsi,j, (3e) where( 3b )istheconstraintonpowerandeachelementofwhichissubjecttoconstraintsonminimumSINRandmaximumtransmissionpower.( 3 )allowsmanytypesoftrafcsuchasconstantbitrate(CBR),variablebitrate(VBR)andelastictrafc. Although( 3 )seemssimilarto( 2 ),itcannotbedirectlysolvedbydualdecomposition,whichmayresultinanunfeasiblelinkassignmentortransmissionrate.Instead,wedirectlyextendthecontrollaw,i.e.,( 2 ),( 2 )and( 2 ),tothetime-varyingsystemsettings. TheSMbasedjointcongestioncontrolandschedulingalgorithmundertime-varyingchannelsisgiveninthefollowingalgorithmwhereisthesamplingintervalforthediscretetimesystem. Algorithm2. Initialization:X(0)2X,Foi,j(0)=0,Fii,j(0)=0,(0)=0; Attimet+1,updatethesystemstatesaccordingto: Price:theLagrangianpriceisupdatedaccordingto i,j(t+1)=yi,j(X,t))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(xi,j(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Foi,j(t)+Fii,j(t)+i,j(t)]+; (3) 43

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CongestionControl:eachsourceupdatesitsmulti-pathtransmissionrateby xsi,j(t+1)=b[xsi,j(t)+zsi,j(t))]TJ /F3 11.955 Tf 5.47 -9.69 Td[(Usi,j(t))]TJ /F9 11.955 Tf 11.96 0 Td[(i,j(t)+vbsi,j(t)+k; (3) Scheduling:performAlgorithm 3 togettheoptimalpowercontrol. ThetimeaverageinowrateFii,j(t)andoutowrateFoi,j(t)attimetcanberepresentedas Fii,j(t)=PmRm,i,j(t)]TJ /F7 11.955 Tf 11.96 0 Td[(1)+Pm m,i,j(t) t,Foi,j(t)=PkDi,j,k(t)]TJ /F7 11.955 Tf 11.95 0 Td[(1)+i,j(t) t, (3) where m,i,j(t)satises( 3 ). Substituting( 3 )into( 2 ),weobtaintheoptimalpowercontrolasbelow P(t)=argmaxP2PXi,ji,j(t) i,j(t))]TJ /F10 11.955 Tf 11.96 11.36 Td[(Xm m,i,j(t)!=argmaxP2PXi,j(i,j(t)i,j(t))]TJ /F10 11.955 Tf 11.29 11.36 Td[(Xk i,j,k(t)j,k(t)!, (3) where( 3 )isduetothefactthatthedecisionattimetdoesnotaffectPmRm,i,j(t)]TJ /F7 11.955 Tf 11.7 0 Td[(1)andPkDi,j,k(t)]TJ /F7 11.955 Tf 12.44 0 Td[(1).Itisrecognizedthat( 3 )isamaximumweightedindependentset(MWIS)problemwithi,j(t)i,j(t))]TJ /F10 11.955 Tf 12.47 8.96 Td[(Pk i,j,k(t)j,k(t)assignedastheweightWi,jforeachli,j.Then( 3 )canbesolvedinthreestepsbasedonapredenedroutingalgorithmR,asstatedinAlgorithm 2 ,whereLkm(i,j)(t)isthelengthofthemthpacketinqi,jattimet,whichwillbetransmittedfromnodeitonodejandputintoqj,k;and)]TJ /F6 7.97 Tf 6.77 -1.8 Td[(i,j(t)=fjPm=8k,m=1Lkm(i,j)i,j(t)gisthesetofpacketindexfortransmission,whichalsodependsonPandG. 44

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Algorithm3. Attimetforeachqi,j(8(i,j)2L),decidethenexthopnodekofeachpacketthatnotdestinedtonodejby k=argminn:(j,n)2Rj,n(t), thenget i,j,k(t)=Pm2)]TJ /F11 5.978 Tf 4.82 -1.4 Td[(i,j(t)Lkm(i,j)(t); TheweightWi,j(t)foranylinkli,jiscalculatedby: Wi,j(t)=i,j(t)i,j(t))]TJ /F10 11.955 Tf 11.96 11.36 Td[(Xk i,j,k(t)j,k(t); AllocatetransmissionpowerPaccordingto: P(t)=argmaxP2PXi,jWi,j(G(t),P);(3) ItiseasytojustifythatAlgorithm 3 givesanoptimalsolutionto( 3 )basedonroutingschemeR. TheLagrangianprice( 3 )andthetransmissionrate( 3 )ateachnodeareupdatedinadistributedmannerwithlowcomputationalcomplexityandtheinformationusedinAlgorithm 2 islocalratherthanglobal.Giventhelinkprice[i,j],( 3 )isgenerallyNPhard,evenifthepowercontrolison-offwithtwovalues.Ifsuboptimalperformanceisallowed,somealgorithmssuchascolumngeneration[ 51 ]canbeusedtosolvetheMWISproblem.TheMWISproblemcanalsobesolvedinadistributedmannerwithlowcomplexitybythealgorithmmentionedin[ 48 ],whichusuallyachievesaperformancewithinabout4/5oftheoptimalperformance.Therearealsosomeworksstudyingtheimpactoftheimperfectschedulinganditsboundsuchas[ 52 ][ 53 ]. Algorithm 2 cannotbederiveddirectlyfromthedualdecompositionoftheNUMproblem( 3 ).However,( 3 )canbeusedasareferencesystemtocharacterizetheperformanceofAlgorithm 2 .TheperformanceofAlgorithm 2 isstudiedinthenextsection. 45

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3.3PerformanceAnalysis Inthissection,Section 3.3.1 showstheperformanceofAlgorithm 2 .ToenhancetheperformanceofAlgorithm 2 intermsofdelay,weproposeaVeryImportantPacket(VIP)queueingstructureinSection 3.3.2 3.3.1PerformanceofAlgorithm 2 Thesystemstatesunderthediscrete-timecontrolpolicyevolveasaMarkovchainandweneedtoshowthatthisMarkovchainisstable.BecauseweapplytheooroperationtotheLagrangianpriceandthetransmissionrate,itiseasytocheckthattheMarkovchainhasacountablestatespace,butisnotnecessarilyirreducible.Thus,weconsiderthepartitionofthestatespaceasthetransitstatesetandtherecurrentstateset.ItisdenedtobestableifallrecurrentstatesarepositiverecurrentandtheMarkovprocesshitstherecurrentstateswithprobabilityone[ 21 ]astapproachesinnity.ThiscanguaranteethattheMarkovchainisabsorbed/reducedintosomerecurrentclass,andthepositiverecurrenceensurestheperiodicityoftheMarkovchainoverthisclass.Thenweevaluateitsstabilitybythefollowingtheorem: Theorem3.1. TheMarkovchaindescribedby( 3 )( 3 )isstable. Theorem3.2. ThenetworkutilityproducedbyAlgorithm 2 convergesstatisticallytotheneighborhoodoftheoptimalutilitywithradiusK2jSj(V20+D20+Z1+Z2) 2K11 2,i.e., "XsEfUs(xs(1))g)]TJ /F10 11.955 Tf 20.59 11.36 Td[(Xs^Us#2K2jFj(V20+D20+Z1+Z2) 2K1 (3) where( 3 )istheresultunderyi,j(X,t)=yi,k(X,t)=zs1i,j(X,t)=zs2i,j(X,t)=1(8(i,j)2L,s16=s2,k6=j)andsimilarresultholdsforyi,j(X,t)=yi,k(X,t)=zs1i,j(X,t)=zs2i,j(X,t)6=1(8(i,j)2L,s16=s2,k6=j);letusi,jdenotethepartialderivativeofUs(xs)w.r.t.xsi,jandxs(1)denotethetransmissionrateoftheMarkovchaininsteadystate, 46

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andletV0andD0betheupperboundsoffunctiongi,j(t)=(usi,j)]TJ /F9 11.955 Tf 11.95 0 Td[(i,j+vbsi,j)+ (3)di,j(t)=Foi,j(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Fii,j(t))]TJ /F3 11.955 Tf 11.95 0 Td[(xi,j(t); (3) andK2=maxsusi,j(xs(1))2andK1>0isthelargestlowerboundofusi,j)]TJ /F8 7.97 Tf 6.75 0 Td[(^usi,j xi,j(t))]TJ /F8 7.97 Tf 6.69 0 Td[(^xi,j. TheproofsofTheorem 3.1 and 3.2 willbeincludedinanotherversion. Theorems 3.1 and 3.2 implythattheexpectationofthenetworkutilityconvergestoasmallneighborhoodoftheoptimalutility.Thus,theiterationsoftransmissionratesandtheLagrangianpricesconvergesimultaneously.Theremaybeotherdecompositionpossibilitiesasstudiedin[ 20 ]. Here,weprovidesomeintuitionaboutourslidingmodebasedjointcongestioncontrolandschedulingdesign.Inqueueingnetworks,theNUMproblemcanbedecomposedintoaschedulingproblemandacongestioncontrolproblem,whichproducestheclassicprimal-dualalgorithm[ 8 ].Thedualalgorithmturnsouttobetheback-pressurerouting/scheduling,whichcanstabilizethenetworkwheneverthesourcearrivalrateiswithinthecapacityregionofthewirelessnetwork[ 21 ][ 9 ].SincetheQoSrequirementsinourproblemarefunctionsofsourcetransmissionrate,theseconstraintswillonlyaffectthesolutiontothecongestioncontrolproblemratherthanthestabilityofthenetworkwithinthecapacityregion.Sincetheslidingmodecontroltheoryprovidesadistributedoptimalsolutiontoconvexoptimizationproblems,itcanbeappliedtosolvingthecongestioncontrolproblem. 3.3.2PerformanceEnhancementMechanism:VIPQueues Intheabovesystemmodel,wedonotconsidertheserviceswithstringentdelayrequirements.Inthissection,weproposeaframeworkfortheco-existenceoftheper-next-hopqueues(non-VIPqueues)andtheVIPqueuessothatdelay-sensitiveservicescanbesupported.TheVIPqueuesandnon-VIPqueuesaremaintainedseparatelyalongtheroutesofdelay-sensitiveapplicationsandtheVIPqueueswill 47

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Figure3-1. Dumbbelltopology Figure3-2. Chaintopology bescheduledbysomedelay-awarealgorithms.Sincetheseapplicationsmayhavemuchlessnumberofdestinations,theper-destinationqueuescanalsobeused.OneapplicationisforinteractivemultimediasuchasVoIP,onlinemultiplayergames,whichhavestringentdelayrequirements.ThepacketsfrominteractivemultimediaapplicationsaremarkedasVIPsandwillbeputintoVIPqueuesandtransmittedwithhigherpriorityovernon-VIPs.Anotherapplicationisthetransmissionoflayeredvideo,wherethebase-layerpacketscanbeplacedintotheVIPqueuesandtransmittedwithhighpriority,andtheenhancementlayerpacketsareplacedintothenon-VIPqueuesandtransmittedwithlowpriority.Therearesometheoreticalanalysisoftheperformanceregardingpriorityqueues[ 54 ]. 3.4SimulationResults Inthissection,wedemonstratetheperformanceofAlgorithm 2 bysimulatingadhocnetworksinthreetopologieswithtypicaltrafcpatterns.Section 3.4.1 describesthesimulationsettings,andSection 3.4.2 presentsthesimulationresultsofAlgorithm 2 underconstantandtime-varyingtimedivisionmultipleaccess(TDMA)channels. 48

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Figure3-3. Looptopology 3.4.1SimulationSettings TheadhocnetworktopologythatwesimulateisdepictedinFigure 3-2 -Figure 3-1 ,whereFigures 3-2 3-3 and 3-1 showadumbbelltopology,achaintopologyandalooptopologyrespectively.ThelinesinFigure 3-2 3-3 and 3-1 indicatebi-directionallinksandthedistancebetweenanytwoneighboringnodesisthesame.Eachconstantwirelesschannelhasacapacityof1.4Mb/s.Thetime-varyingchannelstatecanbeoneofthevestates/rates,i.e.,f0.28,0.84,1.40,1.96,2.52gMb/s,withprobability1=5.Thus,theexpectationofthetransmissionrateforatime-varyingchannelis1.4Mb/s.Withoutlossofgenerality,onlyone-hopinterferenceisconsidered,i.e.,eachnodecannottransmitandreceivesimultaneously. Theutilityfunctionofeachowis: Us(xs)=wsln(xs+1)(3) wherewsisauserspeciedparametertoindicatedifferentweightoftheservice,andwechooseidenticallyws=2000here.InAlgorithm1and2,designparametervandthescalariterativestep-sizearealsouser-speciedparametersandlistedinTable 3-1 ,whereCCisshorthandnotationforconstantchannelsandTVCisshorthandnotationfortime-varyingchannels. 49

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Table3-1. Performanceevaluationsetting Topology Services QoS Parameters s1:fl1,3g NoQoS =0.01(TVC) s2:fl1,3,l3,2g Dumbbell s3:fl1,3,l3,4g s4:fl1,3,l3,4,l4,5g s5:fl1,3,l3,4,l4,6g s1:fl1,2,l2,3g x2350kbps =1 Chain s2:fl2,3,l3,4g x4=100kbps v=200 (CC) s3:fl3,4g s4:fl1,2,l2,3,l3,4g s1:fl1,2,l2,3g x2350kbps =0.01 Chain s2:fl2,3,l3,4g x4=100kbps v=200 (TVC) s3:fl3,4g s4:fl1,2,l2,3,l3,4g s1:fl1,2,l2,3g x1350kbps =0.01(TVC) Loop andfl1,4,l4,3g v=300 s2:fl2,3g 3.4.2PerformanceofAlgorithm 2 3.4.2.1ThroughputPerformance Inthissection,westudythethroughputperformanceofAlgorithm 2 .Thesetupofthesimulationisthefollowing:ina6-nodeadhocnetworkindumbbelltopology,asshowninFigure 3-1 ,therearevebesteffort(BE)servicesfromnode1tonode2,3,4,5,6respectively.Thetime-varyingchannelstatesarei.i.d.uniformlydistributedwithtransmissionratef40,120,200,280,360gkb/s.Forcomparison,itissimulatedunderAlgorithm 2 withper-next-hopqueueingandundertheMaxWeightalgorithm[ 21 ]withper-destinationqueueing.ItisknownthattheMaxWeightachievesthethroughputoptimalunderper-destinationqueueing.TogettheoptimalthroughputunderAlgorithm 2 ,wechoosetransmissionrateastheutilityfunctionforeachnode.TheroutingtableappliedtoAlgorithm 2 islistedinTable 3-1 ,whiletheMaxWeightalgorithmusestheback-pressurerouting. Figure 3-4 toFigure 3-7 illustratethetime-averagedratesofdatareachingeachdestination,andqueuelengthvs.timeunderbothqueueingmodels.FromFigure 3-4 50

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Figure3-4. TransmissionrateofdumbbelltopologyunderAlgorithm1 Figure3-5. QueuelengthofdumbbelltopologyunderAlgorithm1 wecanseethetime-averageddatarateateachdestinationconvergestoabout22kb/s,whichmeanstheper-next-hopqueuesinthedumbbellnetworkisabletosupportsuchdatarates.Thetime-averagedqueuelengthalsoconvergesfasttonitevalueasshowninFigure 3-5 .Thisdataratecanalsobesupportedbytheper-destination 51

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Figure3-6. TransmissionrateofdumbbelltopologyunderMaxWeight Figure3-7. QueuelengthofdumbbelltopologyunderMaxWeight queue/MaxWeight(notshowninsimulationresults).However,withatransmissionrate23kb/sforeachservice,thenetworkunderper-destinationqueueingisunstable.ItisseenfromFigure 3-6 thatthetime-averagedataratesatsomedestinationnodesareobviouslybelow23kb/s.Accordingly,thequeuesarenotstablein 3-7 becausetherate 52

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ofdataenteringthenetworkismorethantherateofdatareachingthedestination.Intheper-next-hopqueueing,node3istheonlynexthopnodeofnode1,thusthesourceratesofnode1willconvergetothesamerateaccordingtoAlgorithm 2 underthesameutilityfunction,whichisalsovalidatedbyFigure 3-4 Sinceitisverydifculttogetananalyticalresultsofthecapacityregionofawirelessnetwork,wehavetoresorttosimulationtoroughlycomparethethroughputofanetworkundertheper-next-hopqueueingandtheper-destinationqueueing.Itisseenthatwithreasonableroutingalgorithmapplied,Algorithm 2 canachieveathroughputperformanceclosetoMaxWeightbutusingmuchlessqueues.Thereare66=36queuesundertheper-destinationqueueingwhereasonly10queuesundertheper-next-hopqueueing. 3.4.2.2SupportingMultimediaServices Inthissection,weevaluatewhetherAlgorithm 2 cansimultaneouslysupportvariousQoSrequiredbydifferentmultimediaservices.Bothelasticandinelasticservicesaresimulatedina4-nodeadhocnetworkinchaintopology,asshowninFigure??B.Therearefourservicesinthenetwork.s1ands3areBEowssuchasemailandletransferprotocol(FTP).s2hasaminimumtransmissionraterequirementof350kb/s(intime-averagesenseundertime-varyingchannels),whichcanprovidetheVBRowssuchasvideooverIP.s4isaCBRowtransmittingat100kb/s,whichrepresents10VoIPstreams(eachwith10kb/s). Figure 3-8 andFigure 3-9 showsthetimeaveragetransmissionratesvs.iterationstepswithout/withQoSrequirementsunderconstantchannels(CC)inthe4-nodechainadhocnetwork.IfQoSrequirementisnotconsidered,thetransmissionrateforx1ismuchlowerthan350kbps,asshowninFigure 3-8 .Incomparison,theQoSrequirementsaresatisedinFigure 3-9 forallowsunderAlgorithm 2 .ComparingthenetworkutilityundernoQoSwiththatunderQoSrequirements,theformerislargerthanthelatter,whichisreasonablebecausethefeasibleregionissmallerunderQoS 53

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Figure3-8. TransmissionrateforchaintopologyunderconstantchannelswithoutQoSrequirements Figure3-9. TransmissionrateforchaintopologyunderconstantchannelswithQoSrequirements 54

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Figure3-10. Transmissionrateforchaintopologyundertime-varyingchannelswithoutQoSrequirements constraints.Inthischaintopology,l1,2andl3,4canbeactivatedsimultaneously.s1,s2ands4passl2,3,whichhastobeactivatedwithnootheractivelinks.Thus,l2,3isthebottleneckandx2isexpectedtobesmallerwithoutQoSconstraints,whichalsomatchesoursimulationresults.SimilarresultsareobservedinFigures 3-10 and 3-11 undertime-varyingchannelswheres2istheVBRow.Insummary,allthesesimulationresultsdemonstratethatAlgorithm 2 iscapableofsupportingmultimediatrafcwithmulti-classQoSrequirements. 3.4.2.3RobustnessagainstLinkFailure Inthissection,westudytherobustnessofAlgorithm 2 againstlinkfailureundertime-varyingchannels.ToemphasizethatAlgorithm 2 cansupportmulti-path,a4-nodeloopadhocnetworkissimulated,asshowninFigure 3-3 .s1isavideo-over-IPowwithminimumtransmissionraterequirementof350kb/sands2isBEow.Thereismulti-pathroutingidentiedfors1throughl1,2andl1,4respectively. Figure 3-12 toFigure 3-15 illustratethemulti-pathtransmissionratesvs.iterationstepsunderlooptopologyshowninFigure 3-3 undertime-varyingchannels.InFigures 55

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Figure3-11. Transmissionrateforchaintopologyundertime-varyingchannelswithQoSrequirements Figure3-12. Transmissionrateforlooptopology 56

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Figure3-13. Queuelengthforlooptopology Figure3-14. Transmissionrateforlooptopologyunderlinkfailure 57

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Figure3-15. Queuelengthforlooptopologyunderlinkfailure 3-12 and 3-13 ,theQoSrequirementissatisedfors1bythesumofthetransmissionratesovertwopaths.Thetime-averagequeuelengthalsoconverges.Sinces1ands2havetosharel2,3,thetransmissionratex11,4overl1,4isexpectedtobelargerthanx11,2overtheotherpath,whichmatchesthesimulationresultsinFigure 3-12 .Forcomparison,Figures 3-14 and 3-15 showthetransmissionrateandqueuelengthinthecaseofthelinkfailureofl1,4.Whenencounteringalinkfailure,itisshowninFigure 3-15 thatq1,4correspondingtol1,4increasesquickly,whichdecreasesthetransmissionratex11,4quicklytozero.Inthiscase,theQoSrequirementisnotsatisedandthethetransmissionratex11,2oftheotherpathincreasesaccordingto( 3 ).FinallytheQoSrequirementissatisedasshowninFigure 3-14 andthenetworkisstable.ThesesimulationresultsshowthatAlgorithm. 2 isrobustagainstlinkfailurebecausethemulti-pathtransmissionadaptivelydoesloadbalancingtosatisfyQoSrequirementswithoutchangingtheroutingtable. 58

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3.5ChapterSummary Inthischapter,westudythejointcongestioncontrolandschedulingproblemformulti-hop,multi-pathper-next-hopqueueingwirelessnetworkswithQoSconstraints.ItisformulatedasanNUMproblemunderbothconstantandtime-varyingchannels,whichisdecomposedintoacongestioncontrolproblemandaschedulingproblem.Then,adistributedSMbasedcontrollerisdesignedtoiterativelysolvethecongestioncontrolproblem,whichcanprovidemulti-pathrateadaptiontosatisfyQoSconstraints.Theschedulingproblemisidentiedasamaximumweightedindependentsetproblemandcanalsobesolvedinadistributedmanner.Weextendthisalgorithmtothecasewithtime-varyingchannelsandprovethatthedynamiccontrollaw,i.e.,Algorithm 2 isstableandthenetworkutilityconvergestoaboundedneighborhoodoftheoptimal.SimulationresultsshowthatAlgorithm 2 iscapableofprovidingheterogenousmultimediaserviceswithdifferentQoSrequirements.Becauseofthemulti-pathloadbalancingfeatureofthisalgorithm,itisrobustagainstnetworkanomaliessuchaslinkfailure. TheapplicationoftheSMcontroltheoryinwirelessnetworksistightlyrelatedtothedecompositionpropertyoftheprotocolstack.Sincethejointcongestioncontrolandschedulingisacross-layeroptimization,thesetwosubproblemsinteractwitheachotherthroughtheLagrangianprices.Therefore,thesourcesadaptthetransmissionratetotheprice,whichmasksthelinkstatussuchastime-varyingchannelcapacitiesanddynamicscheduling.Atthesametime,theschedulingproblemisindependentofthesourcetransmissionrates.ThispropertymakesitpossibletoapplySMcontroltheory.Furthermore,theconvergenceconditionsoftheSMcontrollerarealsorelaxedbecausethedecomposedsub-problemsprovideawayofnegativefeedbacktoeachotherinwirelessnetworks. 59

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CHAPTER4CAPACITYREGIONANDDYNAMICCONTROLOFWIRELESSNETWORKSUNDERPER-LINKQUEUEING Inthischapter,wecharacterizethenetworkcapacityregionunderthePLqueueingnetworkandproposeadynamiccontrol,i.e.,DRPC-PLtostabilizethenetworkwhenevertheinputrateiswithinthecapacityregionofthenetwork.Thischapterisorganizedasfollows.InSection 4.1 ,wedescribethesystemmodelforthePLqueueingnetworks.InSection 4.2 ,wedenethecapacityregionandderivethenecessaryandsufcientconditions,followedbyastabilizingcontrolpolicyDRPC-PL.Section 4.3 theoreticallyanalyzestheperformanceoftheproposedDRPC-PLandSection 5.5 concludesthechapter. 4.1PLQueueingNetworkModel Inthischapter,weconsideranetworkmodeledbyagraph,G=(N,L),whereNisthesetofnodesandListhesetoflinks.Alink(i,j)denotedbyli,jexistsifitisinL.ThesetoftheneighboringnodesofnodeiisdenotedbyNi.Weassumeaslottedsystemandthetimeslotisdenotedbyt.Ineverytimeslot,letsi,j(t)representtheinstantaneouschannelstateofli,j2L,whereiandjaretransmitterandthereceiverofthelink.ThechannelstatematrixofthenetworkcanbedenotedbyS(t)=[si,j(t)]. LetFbethesetofowsconsistedinthenetworkwhereeachofthemisindexedbyf=1,2,...,jFjandjjisthecardinalityofaset.Thedestinationnodeofowfisdenotedbyd(f).Inthenetwork,letlrepresentthetransmissionrateonlinkl,thenaschedule=(1,2,...,jLj)isthelinkratesthatcanbesupportedsimultaneouslybythenetwork.Weusethestrongstabilitydenedin[?]todenotethenetworkstability. 4.1.1ThePLQueueingModel ThePLqueueingmodelisillustratedinFigure 4-1 andFigure 4-2 .Solidlinesarelinksbetweentwonodesanddashlinesindicatetheroutebetweentwoqueuestoexchangedata.Inthesetwogures,qi,j,denedonlinkli,j,holdsthedatainnodeiwith 60

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Figure4-1. InputmodelforPLqueue Figure4-2. OutputmodelforPLqueue nodejasthenexthopnode.Therefore,thenumberofthequeuesinnodeiisatmostthenumberofitsneighbors.Ifli,jisactive,packetsinqi,jaretransmitted. Let i,j(t)denotetheallocatedinputdatarateofqi,jand m,i,j(t)denotetheallocateddataratefromqm,itoqi,jattimet.Let i,j(t)representtheallocatedoutputdatarateofqi,jandi,j,k(t)representtheallocateddataratefromqi,jtoqi,kattimet. 61

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Wehave i,j(t)=Xm m,i,j(t) i,j(t)=Xki,j,k(t)+i,j(t), wherei,j(t)isthedataratedeliveredfromnodeitonodejatslott. Inaddition,sinceeachqueuecontainsthetrafcfordifferentows,wecanwritei,j,k(t)and m,i,j(t)intermsofthetheowrates.Thenwehavei,j,k(t)=Xf:d(f)6=jfi,j,k(t) m,i,j(t)=Xf:d(f)6=ifm,i,j(t), wherefi,j,k(t)istheallocatedrateforowfonli,jforqj,kattimet. 4.1.2TheNetworkModel WeconsiderawirelessnetworkinconsecutiveKtimeslots.ThePLqueueingmodelischaracterizedbythefollowingproperties. 1)Convergentwirelesschannels. LetTS(t,K)bethesetoftime-slotsatwhichthechannelstatematrixS(t)=Sduringtheinterval0tK)]TJ /F7 11.955 Tf 12.52 0 Td[(1.ThewirelesschannelprocessS(t)isassumedtobeconvergentwithanitenumberofchannelstatesfSgandstateprobabilitiesS.Theconvergenceinterval[ 9 ]Kisthenumberoftime-slots,suchthatforgivenvalue1>0,wehave: XfSgEfjTS(t,K)jg K)]TJ /F9 11.955 Tf 11.96 0 Td[(S1,(4) whereEfgisexpectation. 2)Boundedandconvergentarrivalrates. 62

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Forgiven2>0,anarrivalprocessafi,j(t)convergentwiththeexogenousarrivalratefi,jwithinintervalKsatises: fi,j)]TJ /F7 11.955 Tf 14.93 8.09 Td[(1 KK)]TJ /F8 7.97 Tf 6.58 0 Td[(1Xt=0Efafi,j(t)g2.(4) Besides,thesecondmomentofexogenousarrivalsateachnodeisboundedeverytime-slotbysomenitemaximumvalueafmaxregardlessofpasthistory,sothatforanyi2N,j2Ni: Enafi,j(t)2o)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(afmax2E8<:"Xfafi,j(t)#29=;(amax)2, whereafmaxandamaxareconstants. 3)Uppersemi-continuouspower-ratefunction. Let i,j(P(t),S(t))denotethepower-ratefunctionundersomepowerallocationmatrixP(t)=[pi,j(t)],P(t)2PandchannelstatematrixS(t),wherePisthesetoffeasiblepowerallocation.Eachelementpi,j(t)istheallocatedpoweronli,jattimet.Thetransmissionratesareboundedforeverytime-slottbymax,sothat max=maxj2Ni,S,P2P i,j(P,S), whichcanalsobededucedbythefactthatthepower-ratefunctionisboundedwithnitetransmissionpower. 4)Queueupdatedynamics. LetUi,j(t)denotethebacklogofqi,jattimet.Thenthequeueingdynamicsinthenetworksatisfy Ui,j(t+1)Ui,j(t))]TJ ET q .478 w 223.24 -611.93 m 230.28 -611.93 l S Q BT /F9 11.955 Tf 223.24 -618.75 Td[(i,j(t)++ i,j(t)+Xfafi,j(t), (4) 63

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where[x]+=max(x,0)anditisaninequalityinsteadofanequalitybecausethearrivalsmaybelessthantheallocatedoutputdataiftheneighboringnodeshavelittleornodatatotransmit[ 9 ]. 4.2StabilizingControlPolicy Inthissection,wedesignadynamiccontrolpolicy,namely,DRPC-PL,tostabilizethePLqueueingnetworks.Toprovidesomeintuitionsonhowtodesignthestabilizingalgorithms,werstgiveabriefintroductiontotheLyapunovdriftmethodnext. 4.2.1LyapunovDriftMethod AsufcientconditionfornetworkstabilityusingtheLyapunovtheoryisgiveninLemma2in[ 50 ].AssumethesetofqueuesinasystemisdenotedbyXandU(t)=[Ux(t)]isthequeuebacklogvectorformultiplequeuesandL(U(t))=PxU2x(t)istheLyapunovfunctionforthequeuestates.Notethatthismodelisapplicabletoanysystemthatcontainsmultiplequeues.IfthereexistsapositiveintegerKsuchthatforeverytime-slot,theLyapunovfunctionevaluatedKstepsintothefuturesatises EfL(U(K+t)))]TJ /F3 11.955 Tf 11.95 0 Td[(L(U(t))jU(t)gB1)]TJ /F10 11.955 Tf 11.95 11.36 Td[(XxxUx(t)(4) forsomepositiveconstantsB1,[x],andifEfU(t)g1fort2f0,1,,K)]TJ /F7 11.955 Tf 12.22 0 Td[(1g,thenthenetworkisstable,and limsupK!11 KK)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xt=0"XxxEfUx(t)g#B1. Similarto( 4 ),thequeuebacklogforKslotsintothefuturecanbeboundedintermsofthecurrentunnishedworkthat Ux(t+K)"Ux(t))]TJ /F6 7.97 Tf 11.95 14.95 Td[(K)]TJ /F8 7.97 Tf 6.58 0 Td[(1Xt=0x(t)#++K)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xt=0ax(t) wherex(t)istheoutputrateofqueuexandax(t)isthequeueinputrateincludingboththeexogenousarrivalsandthetrafcfromotherqueues.Squaringbothsidesof 64

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inequality( 4 )andtakingconditionalexpectationsw.r.t.U(t),wehave EfU2x(t+K))]TJ /F3 11.955 Tf 11.96 0 Td[(U2x(t)jU(t)gK2(2max+a2max))]TJ /F7 11.955 Tf 11.95 0 Td[(2KXx(x)]TJ /F9 11.955 Tf 11.95 0 Td[(x)Ux(t) (4) wheremax=maxxx(t)andamax=maxxax(t),andtheservicerate(t)andthearrivalratea(t)areconvergenttotherate=[x]and=[x]similarto( 4 ). Comparing( 4 )and( 4 ),wecanseeifthereexists>0,suchthat x)]TJ /F9 11.955 Tf 12.4 0 Td[(x,theconstantxcanbefoundtostabilizeallthequeues.Therefore,astabilizingalgorithmshouldbedesignedtokeeptheaverageserviceratelargerthanthearrivalrateforeachqueueinthenetworkduringKslots. 4.2.2PDPLNetwork WerstdeneaPDPLqueueingnetworkwhereeachnodekeepsaqueueforeachsource-destination-linkcombination.Thus,anodeinaPDPLnetworkwithjFjowsandjLjlinkswillhavejFjjLjPDPLqueues.ThisnetworkcontainsmorequeuesthanthePDqueueingandwillusedasareferencenetworkinourfollowinganalysis. Letqfi,jdenotethequeueforowfonli,jandlet fi,j(t)and fi,j(t)denotetheinstantaneousinputandoutputrateofqfi,jattimettoobtain fi,j(t)=Xmfm,i,j(t) (4a) fi,j(t)=Xkfi,j,k(t). (4b) 65

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DenepdplforthePDPLnetworkasthesetofallinputratematrices[fi,j]suchthatthereexistowratevariablesdfi,j,ksatisfying: dfi,j,k0,8i,j,k2N (4a)dfi=d(f),j,k=0 (4b)Xkdfi,j,k)]TJ /F10 11.955 Tf 11.95 11.36 Td[(Xmdfm,i,jfi,j,8d(f)6=j (4c)[Gi,j]"Xkdfi,j,k#, (4d) wheretheinequalityof( 4d )isconsideredentrywiseanddfi,j,kisderivedin( B )later.Notethattheinputratematrix[fi,j]isdifferentfromthelinkratematrixG=[Gi,j],andisinatime-averagesense. ThefollowingtheoremgivesthenecessaryandsufcientconditionofthecapacityregionofanetworkunderPDPLqueueingstructure. Theorem4.1. CapacityRegionforthePDPLQueueingNetwork a)Anecessaryconditionfornetworkstabilityis[fi,j]2pdpl. b)Asufcientconditionfornetworkstabilityisthat[fi,j]isstrictlyinteriortopdpl. TheproofisinAppendix B.1 Notethatthecapacityregionofawirelessnetworkdoesnotdependonacertainroutingorschedulingalgorithm. 4.2.3TheDRPC-PLAlgorithm FollowingthecluesprovidedbytheLyapunovdriftmethod,wedevelopacontrolpolicythatstabilizesthePLqueueingnetworkwhenevertheinputratematrix[fi,j]isinsidethenetworkcapacityregionpl. First,wedenethePDPLshadowqueues.ThePDPLshadowqueuesarecountersthatkeeptrackofthePDtrafcforeachowinthePLqueues.LetVfi,j(t)denotethequeuebacklogforowfinqi,jattimet.ThenthePDPLshadowqueuelengthcanbe 66

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updatedaccordingtoVfi,j(t+1)=Vfi,j(t))]TJ ET q .478 w 226.03 -41 m 233.07 -41 l S Q BT /F9 11.955 Tf 226.03 -47.82 Td[(fi,j(t)++afi,j(t)+ fi,j(t), (4) where fi,j(t)and fi,j(t)isdenedin( 4 ). DRPC-PLAlgorithm Foreverytime-slott,theDRPC-PLschedulesthenetworkasfollows 1)Foreachlinkli,j,theweightWi,j(t)iscalculatedby: Wi,j(t)= i,j(t)maxfVfi,j(t))]TJ /F7 11.955 Tf 11.95 0 Td[(minkVfj,k(t), where i,j(t)istransmissionrateonli,jattimetunderpowercontrol. 2)Powerallocationtoli,jfollowing:P(t)=argmaxP2PXi,jWi,j(P(t),S(t)). (4) 3)Transmitthepacketsinqi,jaccordingtotheallocatedrateyieldedbythepowerallocationin( 4 )androutepacketsgoingthroughli,jforowftoqj,k,wherek=argmink)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Vfj,k(t). (4) Thencalculate[Vfi,j(t+1)]accordingto( 4 ),where fi,j(t)and fi,j(t)canbecalculatedaccordingto( 4 )bymonitoringtheowbaseddatatransmittedoneachlinkli,j. DifferentfromtheDRPCalgorithmproposedin[ 9 ],DRPC-PLtransmitsthepacketsfordifferentowsfromthePLqueuesifthelinkisactive.DRPC-PLessentiallychoosesasetoflinkswithoutinterference,whichgivesthemaximumweightofthenetworkbasedonthePDPLshadowqueuedifference.Thetrafcforaowisdynamicallyroutedtochoosethenexthop.Therefore,theDRPCalgorithmschedulesli,jandperformsroutingaccordingtothemaximumPDqueuedifferencebetweentwonodes,whereastheDRPC-PLalgorithmschedulesli,jandperformsroutingaccordingtothemaximumPDPLshadowqueuedifferencebetweentwolinks.From( 4 ),wecansee 67

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thepowerallocationcanalsoberegardedasaMaxWeightproblemsimilarinDRPC.MuchprogresshasbeenmadeineasingthecomputationalcomplexityandderivingdecentralizedsolutionsforthecentralizedMaxWeightalgorithm[ 52 ][ 27 ][ 28 ][ 55 ][ 56 ][ 57 ][ 58 ][ 59 ][ 60 ][ 61 ][ 62 ].TheDRPC-PLalgorithmisalsodifferentfromthePDshadowqueueschemeintroducedin[ 23 ].ThePDPLshadowqueueskeepreal-timebacklogforowsinthePLqueueswhilethePDshadowqueuesin[ 23 ]donotreectthereal-timebacklogofdifferentows. TheperformanceoftheDRPC-PLalgorithmisstatedasfollows. Theorem4.2. TheproposedDRPC-PLalgorithmisthroughputoptimal,i.e.,foranarbitrarynetworkadmissionratevector[fi,j]insideofthenetworkcapacityregiondenotedbypl,theDRPC-PLalgorithmstabilizesthenetwork. Theproofispresentedinthenextsection. 4.3PerformanceAnalysis Inthissection,weevaluatetheperformanceofDRPC-PLbycomparingitwiththedynamiccontrolpolicyforPDPLqueueingnetworks,i.e.,DRPC-PDPLalgorithm.InSection 4.3.1 ,wepresenttheDRPC-PDPLalgorithmthatcanstabilizethePDPLqueueingnetworkwithinthecapacityregionpdpl.Then,weprovethattheDRPC-PLalgorithmstabilizesthePLqueueswhenevertheinputrateiswithinthecapacityregionofthePDPLnetworkpdplin 4.3.2 .InSection 4.3.3 ,weprovethatthecapacityregionofanetworkunderPDPLqueueingandPLqueueingisthesametocompletetheproofofTheorem 4.2 4.3.1PerformanceoftheDRPC-PDPLAlgorithm UnderthePDPLnetworkmodel,theDRPC-PDPLalgorithmschedulesthenetworkasfollows: 1)Foreachlinkli,j,chooseowftocalculatethelinkweightWi,j(t),wheref=argmaxfVfi,j(t))]TJ /F7 11.955 Tf 11.96 0 Td[(minkVfj,k(t), 68

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andWi,j(t)= i,j(t)Vfi,j(t))]TJ /F7 11.955 Tf 11.95 0 Td[(minkVfj,k(t). 2)Thepowerallocationtoli,jfollowsP(t)=argmaxP2PXi,jWi,j(P(t),S(t)). 3)Routethepacketsofowfgoingthroughli,jtoqfj,kifd(f)6=j,wherek=argminkVfj,k(t). Here,were-useVfi,jtodenotethequeuelengthofthephysicalPDPLqueueqfi,j. NotethatDRPC-PDPLisdifferentfromtheDRPCalgorithm[ 9 ]duetothequeueingmodel.InDRPC,thepacketsforaowistransmittedandroutedbetweentwonodes,whereasthepacketsofaowistransmittedandroutedbetweentwolinks.TheperformanceoftheDRPC-PDPLisstatedinthefollowingtheorem. Theorem4.3. TheproposedDRPC-PDPLalgorithmisthroughputoptimal,i.e.,foranarbitrarynetworkadmissionratevector[fi,j]whichisinsideofthenetworkcapacityregionpdpl,DRPC-PDPLstabilizesthenetwork. TheproofisinAppendix B.2 4.3.2PerformanceoftheDRPC-PL ToproveTheorem 4.2 ,werstprovethatfor8[fi,j]2pdpl,theDRPC-PLalgorithmcanstablizethenetwork.Then,weprovethatthecapacityregionofanetworkunderPDPLqueueingisthesameasthatunderPLqueueinginSection 4.3.3 BycomparingthePDPLshadowqueuelengthsandthePLqueuelengthsintheDRPC-PL,wecaneasilyget Ui,j(t)=XfVfi,j(t) Thus,wewanttoevaluatethestabilityofthePLqueuesbyevaluatingthestabilityofthePDPLshadowqueues. 69

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TheonestepLyaponuvdriftofthePDPLshadowqueuescanbewrittenas)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Vfi,j(t+1)2)]TJ /F10 11.955 Tf 11.95 9.69 Td[()]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Vfi,j(t)2)]TJ /F7 11.955 Tf 5.48 -9.69 Td[((max)2+(max+amax)2)]TJ /F7 11.955 Tf 11.95 0 Td[(2Vfi,j(t) fi,j(t))]TJ /F9 11.955 Tf 11.96 0 Td[( fi,j(t))]TJ /F3 11.955 Tf 11.95 0 Td[(afi,j(t). (4) Next,wesum( 4 )overthewholenetworkonallshadowqueuesandobtainXf;i,j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Vfi,j(t+1)2)]TJ /F10 11.955 Tf 11.96 11.36 Td[(Xf;i,j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Vfi,j(t)2B)]TJ /F7 11.955 Tf 11.95 0 Td[(2Xf;i,jVfi,j(t) fi,j(t))]TJ /F9 11.955 Tf 11.95 0 Td[( fj,k(t))]TJ /F3 11.955 Tf 11.96 0 Td[(afi,j(t), (4) where B=jLjjFj)]TJ /F7 11.955 Tf 5.48 -9.68 Td[((max)2+(max+amax)2(4) isaconstant. Takingtheconditionalexpectationof( 4 )w.r.t.V(t)=[Vfi,j(t)],wehaveE Xf;i,j)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Vfi,j(t+1)2V(t)!)]TJ /F17 11.955 Tf 11.96 0 Td[(E Xf;i,j)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Vfi,j(t)2V(t)!B)]TJ /F7 11.955 Tf 11.96 0 Td[(2Xf;i,jVfi,j(t)E fi,j(t))]TJ /F9 11.955 Tf 11.96 0 Td[( fi,j(t))]TJ /F3 11.955 Tf 11.95 0 Td[(afi,j(t)jV(t). (4) Denetheroutingschemeattimetassociatedwitheachowonli,jby[fi,j,k(t)](d(f)6=j),wherefi,j,k(t)2[0,1]andXkfi,j,k(t)=1,(d(f)6=j). Thenwecanrewritetheoutputrateofeachowintermsoftheroutingparametersas fm,i,j(t)=fm,i,j(t) fm,i(t) (4) 70

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Substituting( 4 )into( 4 ),wehaveE Xf;i,j)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Vfi,j(t+1)2V(t)!)]TJ /F17 11.955 Tf 11.96 0 Td[(E Xf;i,j)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Vfi,j(t)2V(t)!B)]TJ /F7 11.955 Tf 11.95 0 Td[(2Xf;i,jVfi,j(t)E fi,j(t))]TJ /F10 11.955 Tf 11.96 11.36 Td[(Xmm,i,j(t) fm,i(t)jV(t)!)]TJ /F17 11.955 Tf 11.96 0 Td[(E)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(afi,j(t)jV(t)=B)]TJ /F17 11.955 Tf 11.95 0 Td[(E)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(afi,j(t)jV(t))]TJ /F7 11.955 Tf 11.96 0 Td[(2Xf;i,jE fi,j(t) Vfi,j(t))]TJ /F10 11.955 Tf 11.96 11.35 Td[(Xkfi,j,k(t)Vfj,k(t)!jV(t)! (4) Nowdenethemaximumqueuedifferenceforeachowfandli,jasVfi,j,k(t)=Vfi,j(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Vfj,k(t)Vmaxi,j(t)=maxf;kVfi,j,k(t)=maxfVfi,j(t))]TJ /F7 11.955 Tf 11.96 0 Td[(minkVfj,k(t). DeneJPLV(t)=2E Xf;i,j fi,j(t))]TJ /F7 11.955 Tf 5.48 -9.68 Td[(Vmaxi,j(t)+Vfi,j,k(t)jV(t)!, and JPL(t)=E)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(JPLV(t). Next,weaddbothsidesbyJPLV(t)to( 4 ),andhaveE Xf;i,j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Vfi,j(t+1)2V(t)!)]TJ /F17 11.955 Tf 11.96 0 Td[(E Xf;i,j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Vfi,j(t)2V(t)!+JPLV(t)B+JPLV(t))]TJ /F7 11.955 Tf 11.96 0 Td[(2Xf;i,jVfi,j(t)E fi,j(t))]TJ /F9 11.955 Tf 11.96 0 Td[( fi,j(t))]TJ /F3 11.955 Tf 11.95 0 Td[(afi,j(t)jV(t). (4) DenotetheR.H.S.of( 4 )as,whichcanberewrittenasB)]TJ /F17 11.955 Tf 11.96 0 Td[(E)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(afi,j(t)jV(t))]TJ /F7 11.955 Tf 11.95 0 Td[(2Xf;i,jE fi,j(t) Vfi,j(t))]TJ /F10 11.955 Tf 11.95 11.36 Td[(Xkfi,j,k(t)Vfj,k(t)!jV(t)!+2Xf;i,jE)]TJ ET q .478 w 96.82 -605.62 m 103.86 -605.62 l S Q BT /F9 11.955 Tf 96.82 -612.45 Td[(fi,j(t))]TJ /F7 11.955 Tf 5.48 -9.69 Td[(Vmaxi,j(t)+Vfi,j,k(t)jV(t). (4) 71

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Thelasttwotermsof( 4 )canbesimpliedas2Xi,jE Xf fi,j(t)!Vmaxi,j(t)!)]TJ /F7 11.955 Tf 11.95 0 Td[(2Xf;i,jE fi,j(t) Vfi,j(t))]TJ /F10 11.955 Tf 11.95 11.36 Td[(Xkfi,j,k(t)Vfj,k(t))]TJ /F7 11.955 Tf 11.96 0 Td[(Vfi,j,k(t)!jV(t)!. Itisofgreatimportancetoobservethat,theDRPC-PLalgorithmessentiallyminimizestheR.H.S.of( 4 )overallpossibleschedulingalgorithms. Since[fi,j]liesintheinteriorofthePDPLnetworkcapacityregionpdpl,itimmediatelyfollowsthatthereexistsasmallpositiveconstant>0suchthat [fi,j]+2pdpl WecangetsimilarresultstotheCorollary3.9in[ 12 ]underPDPLqueueingthatthereexistsarandomizedschedulingpolicy,denotedbyRAthatstabilizesthePDPLnetworkwhileprovidingadatarateof fi,j(t))]TJ /F9 11.955 Tf 11.96 0 Td[( fj,k(t)=fi,j+, and fi,j(t)and fj,k(t)arethelinkdataratesinducedbytheRAalgorithm.Thus,wehaveRA=B)]TJ /F7 11.955 Tf 11.95 0 Td[(2Xf;i,jVfi,j(t)+Xf;i,jE)]TJ ET q .478 w 204.65 -443.68 m 211.69 -443.68 l S Q BT /F9 11.955 Tf 204.65 -450.5 Td[(fi,j(t))]TJ /F7 11.955 Tf 5.48 -9.68 Td[(V maxi,j(t)+V fi,j,k(t)jV(t), (4) wherethelasttermin( 4 )isthequeuedifferenceundertheRAalgorithmduringslott.Therefore,thePDPLqueuesarebounded,sodothePDPLqueuedifferencesV maxi,j(t)andV fi,j,k(t).Thus,thelasttermof( 4 )isalsoboundedandwedenotetheboundbyJmax.ThenwehaveRA=B)]TJ /F7 11.955 Tf 11.96 0 Td[(2Xf;i,jVfi,j(t)+Jmax. 72

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Inlightof( 4 ),itisobtainedthatE Xf;i,j)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Vfi,j(t+1)2V(t)!)]TJ /F17 11.955 Tf 11.96 0 Td[(E Xf;i,j)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Vfi,j(t)2V(t)!+JPLV(t)DRPC)]TJ /F6 7.97 Tf 6.59 0 Td[(PLRAB)]TJ /F7 11.955 Tf 11.95 0 Td[(2Xf;i,jVfi,j(t)+Jmax. (4) Takingexpectationw.r.t.V(t)to( 4 ),wehaveE Xf;i,j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Vfi,j(t+1)2!)]TJ /F17 11.955 Tf 11.96 0 Td[(E Xf;i,j)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(Vfi,j(t)2!+ JPL(t)B)]TJ /F7 11.955 Tf 11.96 0 Td[(2Xf;i,jE)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(Vfi,j(t)+Jmax. (4) Summing( 4 )overtimeslots0toK)]TJ /F7 11.955 Tf 11.95 0 Td[(1yieldsE Xf;i,j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Vfi,j(t+1)2!)]TJ /F17 11.955 Tf 11.96 0 Td[(E Xf;i,j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Vfi,j(t)2!+K)]TJ /F8 7.97 Tf 6.58 0 Td[(1Xt=0 JPL(t)KB2)]TJ /F7 11.955 Tf 11.95 0 Td[(2K)]TJ /F8 7.97 Tf 6.58 0 Td[(1Xt=0Xf;i,jE)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Vfi,j(t)+KJmax. (4) Then,wedivide( 4 )byKandmanipulatetheresultandhave21 KK)]TJ /F8 7.97 Tf 6.58 0 Td[(1Xt=0Xf;i,jE)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Vfi,j(t)B+1 KE Xf;i,j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Vfi,j(0)2!+Jmax)]TJ /F7 11.955 Tf 14.93 8.09 Td[(1 KE Xf;i,j)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Vfi,j(K)2!)]TJ /F7 11.955 Tf 14.93 8.09 Td[(1 KK)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xt=0 JPL(t). (4) Notethatthelasttwotermsof( 4 )arebothnon-positive.BytakinglimsupK!1tobothsidesof( 4 ),weobtainlimsupK!11 KK)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xt=0Xf;i,jE)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Vfi,j(t)B+Jmax 2<1. (4) RememberthatthePLqueuelengthsarethesumofanitenumberofthePDPLshadowqueuelengths.Thenwehave limsupK!11 KK)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xt=0Xi,jE(Ui,j(t))1, 73

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whichimpliesthestabilityofthePLqueues. Therefore,weprovethattheDRPC-PLalgorithmstabilizesboththePDPLshadowqueuesandthePLqueueswithinthePDPLcapacityregionpdpl. 4.3.3TheCapacityRegionofPLQueueingNetwork Next,westudytherelationshipbetweenthecapacityregionofanetworkunderPDPLqueueingandthatunderPLqueueing,asfollows. Theorem4.4. LetpdpldenotethecapacityregionofawirelessnetworkunderthePDPLqueueingandpldenotethecapacityregionofthenetworkunderPLqueueing,thenpdpl=pl. Proof. First,weprovethatpl2dpdl.Forany[fi,j]2pl,thereexistsaschedulingalgorithmRthatcanstabilizethenetwork.Atslott,RyieldsapowerallocationmatrixP0(t)forthelinksandatransmissionratematrix[0fi,j,k(t)]fortheowsunderPLqueueing.SupposeinitialqueuestateofthenetworkunderPDPListhesameasthePDPLshadowqueuestateofthenetworkunderPDPLqueueing.Then,byallocatingthepoweraccordingtoP0(t)tothelinksunderPDPLqueueingandschedulethePDPLqueuesthesameasthePDPLshadowqueuesunderPLqueueing.Thetransmissionrate[0fi,j,k(t)]canbeachievedunderPDPLqueueing.Thisimpliesthatfi,j2pdpl.Second,any[fi,j]2pdplcanbesupportedbyDRPC-PLalgorithminSection 4.2.3 underPLqueueing.Therefore,pdpl2pl.Insummary,wehavepdpl=pl. 4.4ChapterSummary Inthischapter,westudythecapacityregionofwirelessnetworkswithPLqueueingstructure,whichsignicantlyreducesthequeueingcomplexityofeachnodecomparedtothewell-studiedPDqueueingstructure.WerstcharacterizethecapacityregionofthePLnetworkandproposeadynamiccontrolpolicydenotedbyDRPC-PLtoschedulethelinksandroutethepacketstostabilizethenetwork.WeshowthatDRPC-PLcanstabilizethenetworkwhenevertheinputratematrixisinsidethecapacityregionof 74

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thePLqueueingnetworks.Inourfuturework,wewillcomparethecapacityregionsofwirelessnetworksunderbothPLandPDqueueingstructure. 75

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CHAPTER5INTEGRATINGDISTRIBUTEDWINDENERGYINTOTHESMARTGRIDBASEDONELECTRICITYMARKET Clean,greenandrenewableenergyisoneofthebiggestdriversoftheSmartGrid.Amongtheserenewableresources,windenergyisgrowingrapidlyandpromisingtobeintegratedintotheSmartGrid.In2009,globalnewinstallationsofwindpowersystemshaveprovidedover38GWandrankedrstamongallsourcesfornewelectricityproductioncapacity[ 40 ]. Althoughwindenergyisavailableinlargeareasandcanbepotentiallyusedasacleanrenewableresource,howtoefcientlyandeconomicallyintegrateitintothecurrentpowersystemischallengingbecauseoftherandomnessandintermittenceofwindpower.TakingCaliforniaasanexample,thereal-timewindpowergenerationcanbefoundin[ 63 ],whichuctuatesconsiderablyindifferenttimescales.Intheelectricpowergrid,thedemandandsupplyshouldbebalancedinreal-time[ 64 ].Sincewindpowerisusuallynon-dispatchable,theshortfallofpowerincreasestheprobabilityofblackoutwhilethesuddenincreaseofwindpowergenerationresultsinalargeamountofspillageorevendamagetotheelectricpowergrid.Asaresult,ancillaryservices(ASs)indifferenttimeframesarerequiredforsystemstabilityandreliability.Withdeepwindpenetration,thereservecapacitycouldfaceunforeseenchallenges,andtheASpricecouldbeevenhigher.Takingthisintoaccount,thecostofusingwindenergymaybequitehighalthoughwindenergyappearstobeawindfall. Integratingwindenergyintotheelectricpowersystemhasbeenanimportantresearchsubjectforbothacademiaandindustries[ 40 ][ 64 ][ 65 ][ 42 ][ 66 ][ 67 ].In[ 40 ],Bitaret.al.modelthecontractgoalasmaximizingthetotalprotofawindfarminthemarketofconventionalgeneration.Incomparison,Botterudet.al.proposeanoptimalday-aheadbiddingmethodinthewholesaleelectricalmarketbasedonthewindenergypriceprediction[ 66 ].In[ 42 ],lotsofrecentresearchresultsincurrentwindpowersystemsarepresented.In[ 65 ],real-timepricingintheelectricitymarketsarestudiedto 76

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shiftthepeakdemandtooff-peakhours.Brookset.al.pointoutthattheplug-inhybridelectricvehicles(PHEVs)canpossiblyactasanAStobalancethepowerdemandandsupply[ 67 ]. Therenewableportfoliostandard(RPS)hasbeenproposedtofurthergreentheelectricpowergrid,whichmakestheintegrationofwindenergymorechallenging.ARPSrequireselectricutilitiesandotherretailelectricproviderstosupplyaspeciedminimumamountofcustomerloadwithelectricityfromeligiblerenewableenergysources.TomeettheRPSrequirements,smallwindsystemsareinstalledinlargecapacity,whichisexpectedtogrowinmanystates,e.g.,California[ 68 ].Thesesystemscouldcreatebusinessopportunitiesforsmall-scaleutilitiesthatcombinesrenewableenergywithconventionalgenerationandactasdistributedenergyresources(DER). Considerthescenarioofthewindenergyintegrationforanaforementionedsmall-scaleutility,whichcombineswindenergyandconventionalenergy.TofullltheRPS,windturbinesareinstalledforprovidingrenewableenergy.ThiscouldbethemosteconomicwaytogetRPSrenewablecreditsbecausewindenergyispricedhighincurrentmarket.Toleveragetheeconomiesofscale,theconventionalenergycomesfromtheelectricitymarket.Inthisscenario,thewindpowerintegrationinvolvesschedulingconstraintsintwotimescales,i.e.,thereal-timecustomerloadprovisionandthelong-termRPScompliance.Asmentionedabove,thewindpowergenerationisarandomvariable,thecostofwhichdependsonthereal-timeASmarket[ 69 ].Withthetwo-waycommunicationinfrastructureinthefutureSmartGrid,theinformationofthereal-timemarketpricesandsystemstatuscanbeavailablewithveryshortdelay.Toaddresstheeconomicfeasibilityofthewindenergyintegration,wefocusondesigningamarketbasedcontrolthatmaximizestheexpectationoftheprotfortheutility. Fromtheperspectiveoftheutility,windpowercanbepotentiallyusedforcustomerloadorparticipatingenergymarket.Thisexibilityleadstoamarket-basedtemporalintegrationstrategy.Forexample,undertheassumptionthatthewindcapacityislarger 77

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thantheminimumrequirementsofRPS,theutilitycanselltheexcesswindenergytotheenergymarket.Whenisthebesttimetoselltheexcessenergycanbeoptimizedaccordingtotheinformationsuchasthewindpowergeneration,theASpriceandtheenergymarketprice,etc.Followingthislineofthoughts,weformulatetheprotmaximizationoftheutilityasastochasticoptimizationproblemandproposeatheoreticalframeworktodynamicallydecidewhentosellthewindenergytotheenergymarket.ThisschemecanprovideperformanceguaranteetostatisticallyfullltheRPS.Besides,ourschemecanbecombinedwithanyreal-timepricing(RTP)scheme[ 65 ],whichisabletoshapethedemandandshifttheloadofpeakhourstonon-peakhours. Therestofthechapterisorganizedasfollows.InSection 5.1 ,Wedescribethewindenergyintegrationmodelofthemarket-basedutilityandanalyzeitscostforprovidingcustomerload.InSection 5.2 ,weformulatetheprotmaximizationoftheutilityasastochasticoptimizationproblem.DuetothehighcomplexityofthisprobleminSection 5.2 ,wesolveitundertheassumptionofsystemstatusergodicitybyastationarydynamiccontrolpolicyinSection 5.3 .Section 5.4 presentsthenumericalandsimulationresults,andSection 5.5 concludesthechapter. 5.1SystemModelofWindEnergyIntegration Inthissection,wedescribeourwindenergyintegrationmodelofautilitybasedonreal-timeelectricitymarket.Sincethesesmall-scaleutilitiesmainlyactasDERsintheSmartGrid,abriefintroductionisgivenaboutDERandrenewableenergyutilitiesinSection 5.1.1 .InSection 5.1.2 ,wedescribethemodelofwindpowerintegrationforsuchautilityandincorporatetherelatedelectricitymarket.BasedonthemarketmodelinSection 5.1.2 ,thecostoftheutilityisanalyzedandformulatedinSection 5.1.3 5.1.1DERandRenewableEnergyUtilitiesinElectricityMarket IntheSmartGrid,theconceptofMicrogridemphasizesdistributedgeneration(DG),whichmakesthesmall-scaleutilityanimportantentity[ 70 ].AsaDER[ 71 ],thepowergenerationsourcesareusuallysmall-scaleandlocatedclosetowhereelectricity 78

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isused(e.g.,ahomeorbusiness),providinganalternativetooranenhancementofthetraditionalelectricpowergrid. DERtechnologiesconsistprimarilyofenergygenerationandstoragesystemsplacedatornearthepointofuse.Distributedenergyencompassesarangeoftechnologiesincludingfuelcells,micro-turbines,reciprocatingengines,loadreduction,andotherenergymanagementtechnologies.DERalsoinvolvespowerelectronicinterfaces,aswellascommunicationsandcontroldevicesforefcientdispatchandoperationofsinglegeneratingunits,multiplesystempackages,andaggregatedblocksofpower.Theprimaryfuelformanydistributedgenerationsystemsisnaturalgas,buthydrogenmaywellplayanimportantroleinthefuture.Renewableenergytechnologies,suchassolarelectricity,biomasspower,andwindturbines,arealsopopular. TheRPSprovidesamechanismtoincreaserenewableenergygenerationusingacost-effective,market-basedapproachthatisadministrativelyefcient.ThegoalofaRPSistostimulatemarketandtechnologydevelopmentsothat,ultimately,renewableenergywillbeeconomicallycompetitivewithconventionalformsofelectricpower.TocomplywithRPS,small-scalewindturbinesareinstalledinlargecapacitytoproviderenewableenergy.Thevariabilityofproductionfromasmallnumberofwindturbinescanbehigh.Thus,conventionalgenerationareusedincombinationtomaintainareliableoutput.IntheNationalRenewableEnergyLaboratory(NREL)greenpowermarketingreport[ 72 ],therehavebeenalargenumberofrenewableenergyutilitiesparticipatingtherenewableenergymarketasDERs.Mostoftheseutilitiescombine2ormorekindsofenergysourcesandthegeneratingunitsaretypicallysmall-scale,i.e.,intherangeof3kWto50MW.Inthischapter,wenametheutilitiescomplyingwithRPSasgreenutilities. 5.1.2WindEnergyIntegrationModelinGreenUtilities Weconsiderthewindenergyintegrationinagreenutilitywithtwo-waycommunicationinfrastructureavailable,asshowninFigure 5-1 .Inthisgure,thepowerowsare 79

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Figure5-1. Systemmodelofagreenutility illustratedbysolidlineswhiletheinformationowsoftwo-waycommunicationbetweensystemcomponentsareillustratedbydashlines.Thedirectionofasolidlineindicatesthedirectionofthepowerow.Thetotalpoweroutputcomesfromtheconventionalpowerboughtfromtheelectricitymarketandthegreenpowerproducedbywindturbines. Thewindenergyhasalreadybeenanoptionalpowersourceforalongtime.Duetoitsrandomandenergy-limitednature,conventionalgeneratorsareusedsimultaneouslytoprovidepower.Whenthewindpowerisnotappropriatetobeusedforprovidingcustomerloadduetoitsavailabilityandquality,theexcesspowercanbesoldtoaseparateenergymarket.Theenergymarketcanbeagreenenergymarket,usedforchargingPHEVs,etc.Inthiscase,conventionalenergymaybeusedtoprovideallcustomerloadforacertainperiodoftime.Atpresentlevelofwindpenetrationintheelectricpowergrid,storingwindenergyisnotapriorityofconsiderationduetohighcostofstorage,assummedupbytheNREL.Thus,wedonotconsiderstorageinthismodel,butitisanimportantoptionworthexploringinwindenergyintegration. Withthetwo-waycommunicationinfrastructure,theinformation,e.g.,real-timewindpowergeneration,powerdemandandmarketprices,canbedeliveredwithveryshortdelay.Therefore,ne-grainedcontrolofintegratingwindenergyisexpectedtobedoneattheorderofminutes.Weconsideratimeslottedsystemandlett2f0,1,2,...,T)]TJ /F7 11.955 Tf 11.97 0 Td[(1gdenotetheindexofoperatingslotsinaday. 80

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Theconventionalenergyofthegreenutilityisboughtfromaconventionalpowerpoolparticipatinginelectricitymarketsthatareclearedandsettledbythemarketoperator,suchasanIndependentSystemOperator(ISO)orRegionalTransmissionOrganization(RTO).Acommonelectricitymarketconsistsoftwosuccessiveex-antemarkets:aday-ahead(DA)forwardmarketandreal-time(RT)market[ 73 ].ThesystempriceintheDAmarketis,inprinciple,determinedbymatchingoffersfromgeneratorstobidsfromconsumersateachnodetodevelopaclassicsupplyanddemandequilibriumprice,usuallyonanhourlyinterval.TheschedulesclearedintheDAmarketaretheinitialoperationstobalanceloadandsupply,whicharesubjecttodeviationpenalties.Duetothesystemstatusforecasterrorsandcontingencies,RTmarketisemployedtoensurethebalanceofloadandsupplyinreal-timebyallowingmarketparticipantstoadjusttheirDAschedulesbasedonmoreaccuratewindandloadforecasts.TheRTmarketiscleared5to15minutesbeforetheoperatinginterval,whichisontheorderofveminutes. EnergyrelatedcommoditiesmanagedbymarketoperatorstoensurereliabilityareconsideredASs,includingspinningreserve,non-spinningreserve,operatingreserves,responsivereserve,regulationup,regulationdown,andinstalledcapacity.SomeoftheelectricitymarketsandASmarketsareintegratedwhilesomeareseparatelymanaged.Inthischapter,wedonotdifferentiatemarketmanagementscheme,butusepricetocharacterizethemarketstatistics.Thegreenutilityisassumedtobeapricetakingparticipantduetoitssmall-scale,thusthemarketclearingprice(MCP)isnotaffectedbythebidoftheutilityintheRTelectricityandASmarkets. IntheUnitedStates,controlareaoperators/balancingauthoritiesfollowtwocontrolledperformancestandards,i.e.,CPS-1andCPS-2.Duetotherandomnessofwindenergyproduction,windenergyshouldbecompensatedbyASsiffeedingintotheelectricgrid.TheASsareinthreetimeframes:regulationinsecondstominutes,load-followingintensofminutestohoursandschedulinginhoursoraday 81

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Figure5-2. ERCOTmarketclearingpriceforregulationdown [ 74 ].Pricesshouldbemorevolatileinrealtimethandayaheadbecauseofalltheunexpectedeventsthatcanoccurinrealtime,includingforcedoutagesofgenerationandtransmissionequipmentandsuddenweatherchanges[ 65 ].ThehourlyregulationdownpricefortwodaysinERCOTareshownasexamplesinFigure 5-2 [ 69 ].Inthisgure,largeuctuationisobservedfordifferenthoursinadayandthesametimeperiodindifferentdays. Theintegrationofwindenergyshouldconsidereconomicreturnsandfeasibility.DeepwindpenetrationwillfurtherincreasethedemandofASs,asstudiedin[ 75 ][ 76 ][ 43 ].ItisintuitivelynoteconomictousewindpowertoprovideloadduringthepricepikesasshowninFigure 5-2 .Withhighpenetrationofwindpower,theASpricecouldbeevenhigher. 5.1.3SystemCostModel Inthissection,wemodelthecostoftheaforementionedgreenutility.Sincewindpowerisconsideredasnon-dispatchableincurrentpowersystems,weconsidertheintegrationschemeofthewindpowerasa0-1control,i.e.,todecidewhethertousewindpowertoprovideloadorsellittotheenergymarketforeachslottbasedontheinformationofmarketpricesandsystemstatus.LetQrepresentarbitrarycontrol 82

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schemeandfQ=cgdenotethatwindpowerissoldtotheenergymarketwhilefQ=wgdenotesthatwindpowerisintegratedaspartofthesystemoutput.ThediagramofthewindenergyintegrationisillustratedinFigure 5-3 .Inthisdiagram,D(t)isthetotalcustomerloadofthegreenutilityatthebeginningofslottandassumedtobeunchangedforaslot.ItisassumedthatD(t)israndomvariableacrossdifferenttimeslots,butcanbeknownbeforethebeginningofslott.ThewindpowergenerationforecastisdenotedbyW(t)andthetruevalueofwindpowergenerationisdenotedbyWb(t).TheconventionalpowerboughtfromtheRTmarketisdenotedbyG(t),whichisasystemvariableneededtobedetermined. Thecustomerloadneedstobebalancedbythepoweroutputofthegreenutilityforeveryslot.Therefore,ifthewindenergyisnotusedforprovidingloadatslott,theconventionalpowerG(t)=D(t).Otherwise,wehaveD(t)=G(t)+W(t)forslott.Toavoidtrivialcases,weassumethatD(t)W(t)alwaysholds.Thenwehave D(t)=W(t)IfQ=wg(t)+G(t),(5) whereIA2f0,1gistheindicatorfunctionofeventA,andIfQ=wg(t)=1denotesthatwindenergywillbeusedtoservepartofcustomerloadattimeslott,whichisalsoasystemvariableneededtobedeterminedinthewindenergyintegration.Similarly,IfQ=cg(t)=1representsthatwindenergyisgoingtotheenergymarketatslott.Obviously,wehave IfQ=cg(t)+IfQ=wg(t)=1.(5) Itisintuitivethatthecostofwindenergyrelatestotheuctuationofwindenergyproduction,whichcanbereectedbythewindforecasterror.OneusefulmodelforwindenergyandsystemanalysisistheSIVAELmodel[ 42 ].IntheupdatedversionofSIVAEL,thestochasticwindenergydescriptionisincludedtosimulatetheneedofregulation.Tomakeourchapterfocused,weuseasimplermodeltocharacterizethewindpowerforecasterror.Tocapturetherandomnessofwindgenerationand 83

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Figure5-3. Windenergyintegrationdiagram prediction,itisassumedthatWb(t)andW(t)arerandomvariablesacrossdifferenttimeslots,butstayunchangedinaslot.Thewindpowerpredictionerroriscapturedbythenormalizedabsoluteerrort2[0,1],whichistheabsoluteerrormeasuredasapercentageofthewindpowerforecast.Thenthereal-timewindenergygenerationsatises Wb(t)=(1t)W(t),(5) wheretisassumedtobeindependentofW(t).In[ 77 ],theauthorsshowthatthedistributionoftcanbeobtainedbycombiningbetadistributionsandtheparametercanbeestimatedaccordingtothehistoricdata.Alargerwindforecasterrorcanbeduetolargevariationinwindpoweroutput,sothedistributionoftcontainsinformationaboutthereal-timeuctuationofwindpower. Thecostofusingwindpowercanbeevaluatedasthecosttocompensateforecasterrorofwindpowergeneration.Forexample,ifWb(t)>W(t),thewindenergyoutputneedsaregulation-downserviceandviceversa.Forsimplicity,wedonotdifferentiatetheregulation-upandregulation-downserviceprice.Letqtdenotetheunitcostofregulationrequiredbywindpower.Thecostofthewindenergyismodeledtobeproportionaltotheabsolutevalueofthewindforecasterrort.Forthepowerboughtfromtheelectricitymarket,theunitpriceisdeterminedbythebidsofmarketparticipants,whichisarandomvariableacrossdifferenttimeslotsanddenotedbypt.Letktbetheenergymarketpricethatthewindenergycanbesoldatifnotusedfor 84

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providingload.Itisassumedthatthemarketpricespt,qtandktareknownpriortothebeginningofslott.ThenthecostoftotalpowergenerationintimeslottcanbecalculatedasptG(t)+qttW(t)IfQ=wg(t))]TJ /F3 11.955 Tf 11.96 0 Td[(ktW(t)IfQ=cg(t), (5) wherethelasttermof( 5 )istheincomefromsellingwindenergy,whichisconsideredasanegativecost.Notethatktmaybedifferentfromptbecausetheymaybedifferentenergymarkets. Substituting( 5 )and( 5 )into( 5 ),weobtainedthetotalcostofthegreenutilityas ptG(t)+qttW(t)IfQ=wg(t))]TJ /F3 11.955 Tf 11.95 0 Td[(ktW(t)IfQ=cg(t)=ptG(t))]TJ /F3 11.955 Tf 11.96 0 Td[(ktW(t)+(qtt+kt)W(t)IfQ=wg(t)=ptW(t)IfQ=cg(t)+(qtt+kt)W(t)IfQ=wg(t)+pt[D(t))]TJ /F3 11.955 Tf 11.96 0 Td[(W(t)])]TJ /F3 11.955 Tf 11.95 0 Td[(ktW(t). (5) Inthischapter,theoperationalcostofthewindturbinesareregardedasconstantduringalongperiodoftime,whichcanalsobeoptimizedindependentlyofthewindenergyintegrationprocess. 5.2ProtMaximizationoftheGreenUtility Inthissection,theprotmaximizationofthegreenutilityisformulatedasastochasticoptimizationproblemundertheconstraintsofRPS.Currently,theaverageunitcostofwindenergyisstillhigherthanthatofconventionalpowergeneration.TheRPSrequirestheintegrationpercentageofrenewableresourcetothetotalpowerconsumptionaboveathreshold.However,itmaynotbeeconomicallyfeasibletosatisfythisrequirementinashorttimeduetothevariationofwindenergygeneration.Intuitively,byintegratingwindenergywhenitisavailablewithlowercost,thehybrid 85

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windutilitycompanymayincreaseprotaswellasmeetingtherequiredintegrationpercentageofwindenergyinalongrun. Followingthisintuition,wemodelthiscontrolprocessasastochasticoptimizationproblemtostatisticallyexploitthetemporalopportunitiesinthevariationofwindenergyandASmarket.Denetheoptimizedvariableas~I=[IfQ=wg(t)](t2f0,1,2,...,T)]TJ /F7 11.955 Tf 11.95 0 Td[(1g),where[]isthetensornotation.Sincetheincomeoftheutilityisindependentofthecontrolscheme,tomaximizetheexpectationoftheprotisequivalenttominimizingtheexpectationofthecostofthegreenutility.Notethatthetermpt[D(t))]TJ /F3 11.955 Tf 12.11 0 Td[(W(t)])]TJ /F3 11.955 Tf 12.12 0 Td[(ktW(t)of( 5 )isindependentofthevariablesin~I,thustominimizetheexpectationofthegenerationcost,weonlyneedtominimizetheexpectationofthersttwotermin( 5 ). SinceW(t)andD(t)arerandomvariables,ourobjectiveistominimizethetimeaveragedexpectationofthecostofpowergenerationinalong-termtimeperiodofTasbelow P1:minQlimT!11 T(T)]TJ /F8 7.97 Tf 6.58 0 Td[(1Xt=0EptW(t)IfQ=cg(t)+(qtt+kt)W(t)IfQ=wg(t)] (5) s.t.PT)]TJ /F8 7.97 Tf 6.58 0 Td[(1t=0EW(t)IfQ=wg(t) PT)]TJ /F8 7.97 Tf 6.59 0 Td[(1t=0E[D(t)]r (5) IfQ=wg(t)+IfQ=cg(t)=1, where( 5 )istheconstraintontheintegrationpercentageofwindenergywithrespecttototalpowerconsumptionandr2[0,1)isaparameterindicatingtheminimumintegrationpercentageofwindenergyrequiredbytheRPS.Alargerrmeansastrongerattempttousewindenergy. P1ischallengingtosolveduetothefollowingtworeasons:1)thewindenergygenerationisarandomvariable,andtheavailablewindenergycannotbepredictedpreciselyforalongperiodoftime.Thebiasoflong-termwindenergyforecastcanbelarge,whichmakesitdifculttosolveP1byapproachessuchasdynamicprogramming;2)theRPSconstraint( 5 )introducestemporalcorrelationamongtimeslots,which 86

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considerablyincreasethecomplexityofthisproblem.InSection 5.3 ,wesimplifyandsolveP1undertheassumptionofsystemergodicity,whichgenerallyholdsinpracticalsystems. 5.3Long-termPerformanceEvaluation Inthissection,wesimplifyP1undertheassumptionthatthewindenergygenerationWb(t),predictionW(t)anddemandD(t)areergodicacrosstimeslots,aswellasthemarketpricespt,qtandkt.Thisassumptionisusuallyvalidbecausethepowerload,windenergygenerationandmarketbehaviorsrepeatinsomedailyandseasonalpatterns.Wereferthisassystemergodicity. Here,wefocusonstationarycontrolschemes,whicharepracticalandeasytoimplementonline.Bystationarycontrolpolicy,wemeanthatthedecisionforslottonlydependsonthesystemstatusatcurrenttimeslot.Withtheergodicityassumption,P1canberewrittenas P2:minQE(qtt+kt)W(t)IfQ=wg(t)+ptW(t)IfQ=cg(t) (5) s.t.EW(t)IfQ=wg(t)rE[D(t)], (5) IfQ=wg(t)+IfQ=cg(t)=1. Tosimplifynotations,wedene: Utw=(qtt+kt)W(t),Utc=ptW(t), (5) and~Ut=[Utw,Utc].ThemeaningofP2isintuitivethatptistheunitcostofusingconventionalpowerandthepartqttinUtwistheunitcostofusingwindpower.Besides,ktcanberegardedastheopportunitycostofusingwindenergytoservetheworkloadinsteadofsellingtotheenergymarket.Thus,wedeneqtt+ktasthetotalunitcostofusingwindpower. 87

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P2isanalogoustoauser-selectionproblemoverasharedwirelesschannelunderperformanceguarantees[ 78 ].TosolveP2,wefollowthestrategyin[ 78 ]anddeneastationarycontrolpolicyQ.Astationarypolicyisamemorylesspolicywhosedecisiondoesnotdependontheslotindextexplicitlybutthevalueof~Ut.Dene Q(~Ut)=argmini(iUti),i2fw,cg(5) wherefig(i2[1,+1))arerealparameters,andwewillderivethemlater.From( 5 ),wecanseethatQcontrolthepowerintegrationaccordingtoafigmodiedversionof~Ut.Anexampleoftheparametersettingisthatc>1andw=1undertheassumptionthatE[pt]1,w=1ghelpsincreasetheintegrationpercentageofgreenenergy,soastosatisfytheconstraint( 5 ). Fromtheperspectiveoftheelectricitymarket,ourmodelreectstheroleofthemarketinbalancingthedemandandsupply.Forexample,whenASisneededinlargeamount,theASpriceqtishighandthegreenutilitytendstosellthewindpowerandbuymoreconventionalpowertomeetthecustomerload.Thisdecisionfurtherpreventstheuctuationbroughtbywindpowerservingtheload,andalleviatestherequirement 88

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ofAS.Ontheotherhand,whenASispricedlow,whichmeansthereislessdemandforAS.WindpowercanbeusedatlowercostandthegreenutilityscheduleittoservethecustomerloadandearnrenewablecreditsforRPSfulllment. Inaddition,theeffectofthequalityofwindpowercanalsobecharacterizedinourmodelbythepredictionerror.Largerwindpoweructuationusuallyincurslargerpredictionerror,thushigherregulationcost,asreectedbyt.Itisobviouslymoreefcienttoislanddistributedwindpowergeneratingunitsfromtheelectricpowergridwhentheiroutputsuctuatesignicantly,whichreducesthedemandofASandincreasestheprotofthegreenutilities.Itisalsoimpliedby( 5 )thattheadvanceofwindpredictionhelpsimprovetheefciencyofwindpowerintegration. Tocalculatetheparametersfig,i2fw,cg,( 5 )canbere-writtenas EW(t)IfQ=cg(t)E[W(t)])]TJ /F3 11.955 Tf 11.96 0 Td[(rE[D(t)]. (5) Observethat( 5 )isequivalenttocpt
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weobtain P(qtt+kt)c wpt=Zqtt+kt ptc w0fp,,q,k(x,w,u,v)dxdvdudw (5) 1)]TJ /F3 11.955 Tf 13.16 8.08 Td[(rE[D(t)] E[W(t)], wherefp,,q,k(w,u,v)isthejointdistributionofpt,t,qt,andkt. LetFt(a)denotetherighthandsideof( 5 ),then Ft(a)=Zqtt+kt pta0fp,,q,k(x,w,u,v)dxdvdudw,a2[1,+1). ItisnotdifculttondthatF(a)isincreasingw.r.t.a.LetF)]TJ /F8 7.97 Tf 6.59 0 Td[(1()denotetheinversefunctionofF(a),thentheoptimalparameterc wcanbewrittenas c w=F)]TJ /F8 7.97 Tf 6.58 0 Td[(11)]TJ /F3 11.955 Tf 13.15 8.09 Td[(rE[D(t)] E[W(t)](5) From( 5 ),wecanseethatc wdependsonthejointdistributionofpowerpricept,qt,tandktaswellastheintegrationpercentager.Tomakesurethattheintegrationpercentagercanbesatised,E[D(t)] E[W(t)]2[0,1 r]needstohold. Ourschemecanbereadilyextendedtothesituationwherethereareseveralkindsofrenewableresourcessuchassolarenergy,hydro-power,etc.,requiredtobeintegratedintotheSmartGridbypercentage.Inthiscase,theparameterfigcannotbecalculatedincloseform,butcanbeestimated.Wereferthereadersto[ 78 ]fordetails. Insummary,thewindpowerintegrationwithconventionalpowercanbecontrolledasfollows.Accordingtothejointdistributionofpt,qt,kt,calculatetheparametersc wbasedon( 5 ),whereE[D(t)],E[W(t)]andtheRPSlevelrareknown.BecauseD(t)andW(t)areassumedtobeergodic,theparameterc wonlyneedstobeestimatedonce.Beforethebeginningofslott,calculatecptandw(tqt+kt).Ifcptw(tqt+kt),usewindtoprovidecustomerloadatthisslot.Otherwise,sellthewind 90

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energyofthisslottotheenergymarket.Thiscalculationanddecisionhappenbeforethebeginningofslott. 5.4CaseStudyandSimulationResults Inthissection,wepresentthenumericalresultsoftheparameterestimationinthestationarycontrolQandthesimulationresultsofthecontrolpolicy.Section 5.4 -AdescribesthesimulationsettingwhileSection 5.4 -BdemonstratestheperformanceofthestationarycontrolpolicyQ. 5.4.1SimulationSetting Wechoosethetimeslotintervalasonehourandsimulatethehourlywindpowerintegration.Themarketpricesofregulationqt,thegreenenergyktandconventionalgenerationptareassumedtobemarketclearingprice(MCP),whicharecorrelatedasstudiedin[ 41 ].Tocapturetheirrandomnessandcorrelation,pt,qtandthektareassumedtobemulti-dimensionaltruncatednormaldistributionintheinterval[0,pm][0,qm][0,km]withparameter(,)asthemeanvectorandthecovariancematrix,wherepm,qmandkmarethemaximumsofpt,qtandkt.Let1,23denotethecorrelationcoefcientsbetweenptandandkt,qtandkt,andptandqt,respectively.ThepowerdemandfromthecustomersandtheforecastedwindenergyW(t)areassumedtobeuniformlydistributedwithE[D(t)] E[W(t)]=2.Theabsolutevalueofwindenergyforecasterrorisassumedtobebetadistributedwithparameterfa,bg.Thissimulationsettingbynomeanscapturesallthecharacteristicsofaliberalizedpowermarket.However,wewanttousethissimpleexampletoillustratetheintuitionofourcontrolscheme. Thestatisticalmodelofenergymarketpricehasbeeninvestigatedin[ 79 ][ 80 ].Wereferreaderstotheseworksforadetailedcharacterizationofenergymarketprices.TheparametersusedinthissimulationislistedinTable 5-1 ,where=[pkq].TheseparametersarechosenaccordingtottingthehourlypricedatainERCOT[ 69 ]from01/01/2011to03/31/2011bynormaldistribution.Here,wechoosetheslotinterval 91

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Table5-1. Simulationsetting q=$15/MWhq=$9/MWh k=$12/MWhk=$10/MWh p=$f5,10,15g/MWhp=$5/MWh E[W(t)]=E[D(t)]=0.51=0.2 2=0.43=0.4 a=2b=5 asonehourbecausethecurrenthourlypricesareavailableintheelectricitymarketssuchasERCOTandPJM.However,weexpecttheslotintervaltobeshorterwiththecommunicationinfrastructureintheSmartGrid. 5.4.2PerformanceAnalysis Theparameterw cisestimatedbysimulatingQforT=1000hoursandcalculatingtheprobabilityin( 5 ).Theparameterw cvs.risshowninFigure 5-4 ,wheretheconventionalenergypriceischosenas5,10,and15$/MWh,respectively.Itisintuitivethatw cdecreaseswiththewindenergyintegrationratior,asalargerrwillincreasetheprobabilityofusingwindenergytoservetheworkload.Tofulllthesamewindenergyintegrationpercentager,theparameterw cisincreasingwiththeconventionalenergyprice.Itisquiteintuitiveaccordingto( 5 )thatw cincreaseswithptwhenrisaconstant.Thelargestachievablewindenergyintegrationpercentageis50%underthesethreescenariosinFigure 5-4 ,whichagreeswiththepriorithatE[W(t)]=E[D(t)]=0.5inthissimulation.Thecasethatthereisnorequirementonwindenergyintegrationpercentageisequivalenttoc=1.Thisimpliestodeterminetheroleofwindenergyonlybyprice.Underthiscondition,theintegrationpercentageisabout20%withpt=$15/MWhandonly3%withpt=$5/MWh,indicatedbythecurvesinFigure 5-4 AccordingtothecurvesinFigure 5-4 ,withtheintegrationratior=0.3andtheconventionalenergypricep=$10/MWh,theparameterc wisabout0.48.Thenwesimulatethewindpowerintegrationprocedurefor200independenthours.Thiscanberegardedastheintegrationprocedureforseveralconsecutivedays.Thereal-timepowerpricesofconventionalgenerationandwindpowerareshowninFigure 5-5 intherst20 92

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Figure5-4. ParameterinthecontrolpolicyQvs.integrationratio Figure5-5. Real-timedecisionofpowerintegration hours.Figure 5-6 showsthereal-timeroleofwindpowerdeterminedbycontrolQ.Thebarsindicatetheslotswhenthewindpowerisusedtoservethecustomerload.Thecurveillustratesthereal-timeintegrationpercentageinFigure 5-6 .Wecanseethatafteraperiodoftime,say,30slots,thegreenenergyintegrationpercentageisfullled. ToevaluatethetimeaveragedsystemcostunderQ,wecompareitwitharandomizedcontrolschemeandadeterministicschemewherewindpowerareusedtoprovideloadforalltheslots.Boththerandomizedandthedeterministiccontrolschemes 93

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Figure5-6. Real-timepowerintegrationratio Figure5-7. Timeaveragedcost complywithRPS.Thenrandomizedcontrolschemechoosesarandomfractionofslotstosellthewindenergytothemarketandusetherestofslotstoservethecustomerload.Forexample,whenr=30%andE[W(t)]=E[D(t)]=0.5,therandomizedcontrolpolicyrandomlychooses40%oftheslotstosellwindpowertotheenergymarket.Thetime-averagedenergycostisshowninFig. 5-7 .Thesystemcostisreducedbyabout15%usingcontrolschemeQcomparedtotherandomizedscheme,whichdoesnotutilizetheopportunityofthetime-varyingmarketprices.Itisintuitivethat 94

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thedeterministicschemeresultsinthehighestcostbecauseitdoesnotleveragetheinformationofthewindpowergenerationandenergymarket. 5.5ChapterSummary Inthischapter,weconsidertheproblemofstatisticallyintegratingwindenergyintotheSmartGridforgreenutilitiescomplyingwithRPS.Theprotmaximizationofthegreenutilityisformulatedasastochasticoptimizationproblem.Byutilizingthereal-timeinformationofwindpowergenerationandtheelectricitymarket,weproposeastationarydynamicthresholdbasedschemetoalternatetheroleofwindpowerintheelectricpowersystem.Underthisscheme,windenergyisusedtoservethecustomerloadwhenitisavailablewithlowcost.Alternatively,windpowerissoldtotheenergymarketwhenitispricedhigh.Asaresult,theprotofthesystemismaximizedandtheRPSrequirementsarefullledstatistically,asshownbythesimulationresults. Ourresultsprovidethekeyinsightsintothetrade-offbetweenthewindenergyintegrationpercentageandthecostofwindenergyintegration.Besides,thisworkemphasizesthebenetsbroughtbytwo-waycommunicationandcontroltechniquesintheintegrationofrenewableresource,whicharekeyfeaturesoftheSmartGrid. 95

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CHAPTER6CONCLUSIONS 6.1SummaryoftheDissertation Inthiswork,weaddressthedynamiccontrolandschedulingofwirelessmulti-hopnetworksunderPLqueueingbythestochasticoptimizationtechniquesundernetworkstability.Besides,westudytheintegrationofDERsunderstochasticmarketandwindpowergenerationintheSmartGrid.InChapter 1 ,wereviewtheresearchbackgroundofcongestioncontrolandschedulinginwirelessmulti-hopnetworksunderPLqueueingandthemotivationforcharacterizingthecapacityregionofthePLqueueingnetworks.Inaddition,weintroducetheSmartGridconceptsandidentifytherenewableresourceintegrationproblemsbroughtbydeepstochasticrenewablegeneration. InChapter 2 ,westudythejointcongestioncontrolandschedulingproblemformulti-hop,multi-pathwirelessnetworkswithconstantwirelesschannelsandQoSconstraintsunderPLqueueing.ThisjointproblemcanbeformulatedasanNUMproblem,whichcanbedecomposedintoacongestioncontrolproblemandaschedulingproblem.WeproposeaSM-basedalgorithm,anoptimaltransmissionratecanbeobtainedbythesystemupdatelaws,i.e.( 2 ),( 2 )and( 2 ). InChapter 3 ,thejointcongestioncontrolandschedulingproblemisformulatedasastochasticoptimizationproblemundertime-varyingchannels.WeextendandmodifytheSMbasedalgorithmtothecasewithtime-varyingchannelsandprovethatthedynamiccontrollaw,i.e.,Algorithm 2 isstableandthenetworkutilityconvergestoaboundedneighborhoodoftheoptimal.SimulationresultsshowthatAlgorithm 2 iscapableofprovidingheterogenousmultimediaserviceswithdifferentQoSrequirements.Becauseofthemulti-pathloadbalancingfeatureofthisalgorithm,itisrobustagainstnetworkanomaliessuchaslinkfailure. InChapter 4 ,westudythedynamiccontrolofthewirelessPLqueueingnetworkwithtime-varyingchannels.Thisqueuestructuresignicantlyreducesthequeue 96

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numberpernodecomparedwiththecommonlyusedPDqueuestructure.TostudythecapacityregionofthePLnetworkandanalyzetheevolutionofthePLqueuedynamics,weusethePDPLqueueingnetworkwiththesamelinksandchannelstateforreference.FollowingtheLyapunovdriftmethod,adynamiccontrolpolicyDRPC-PLisproposedtoschedulethelinksandroutethepacketsaccordingtothePDPLshadowqueuelengths.ItisprovedthattheDRPC-PLcanstabilizethenetworkwhenevertheinputratematrixisinsidethecapacityregionofthePLqueueingnetworks. InChapter 5 ,weconsidertheproblemofstatisticallyintegratingwindenergyintotheSmartGridforgreenutilitiescomplyingwithRPS.Theprotmaximizationofthegreenutilityisformulatedasastochasticoptimizationproblem.Byutilizingthereal-timeinformationofwindpowergenerationandtheelectricitymarket,weproposeastationarydynamicthresholdbasedschemetoalternatetheroleofwindpowerintheelectricpowersystem.Underthisscheme,windenergyisusedtoservethecustomerloadwhenitisavailablewithlowcost.Alternatively,windpowerissoldtotheenergymarketwhenitispricedhigh.Asaresult,theprotofthesystemismaximizedandtheRPSrequirementsarefullledstatistically,asshownbythesimulationresults. 6.2FutureWork WiththecapacityregiondenedforthePLqueueingnetwork,itisnecessarytocomparethecapacityregionofanarbitrarywirelessmulti-hopnetworkunderboththePLqueueandthePDqueueingstructure.ItisalsointerestingtocomparethedelayperformanceoftheDRPCalgorithmandtheDRPC-PLalgorithmalthoughtheupperboundscanbeobtainedtheoreticallybyLyapunovmethod. 97

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APPENDIXAPROOFOFTHEOREM3.1AND3.2 A.1ProofofTheorem 3.1 Proof. TheLyapunovfunctionisthesameasdenedinAppendixA.Here,forsimplicity,itisassumedthatyi,j(X,t)=zsi,j(t)=1(8s2F,(i,j)2L,8X,t).Considertheexpectationoftheone-stepLyapunovdrift EfW(t+1))]TJ /F3 11.955 Tf 11.96 0 Td[(W(t)jX(t),(t)g=EfW1(t+1))]TJ /F3 11.955 Tf 11.96 0 Td[(W1(t)jX(t),(t)g+EfW2(t+1))]TJ /F3 11.955 Tf 11.95 0 Td[(W2(t)jX(t),(t)g=EfW1(X(t+1)))]TJ /F3 11.955 Tf 11.96 0 Td[(W1(X(t))jX(t),(t)g+EfW2((t+1)))]TJ /F3 11.955 Tf 11.95 0 Td[(W2((t))jX(t),(t)g (A) Bysubstituting( 3 )into( A ),thersttermof( A )is EfW1(t+1))]TJ /F3 11.955 Tf 11.96 0 Td[(W1(t)jX(t),(t)g=EW1\004[(t))]TJ /F7 11.955 Tf 11.96 0 Td[(d]+)]TJ /F3 11.955 Tf 11.95 0 Td[(W1((t))jX(t),(t)=EW1)]TJ /F7 11.955 Tf 5.48 -9.85 Td[([(t))]TJ /F7 11.955 Tf 11.96 0 Td[(d]+)]TJ /F9 11.955 Tf 11.96 0 Td[(1X(t),(t))]TJ /F17 11.955 Tf 11.95 0 Td[(EfW1((t))jX(t),(t)gEfW1((t))]TJ /F7 11.955 Tf 11.95 0 Td[(d))]TJ /F3 11.955 Tf 11.96 0 Td[(W1((t))jX(t),(t)g)]TJ /F17 11.955 Tf -219.97 -26.89 Td[(E1h((t))]TJ /F7 11.955 Tf 11.96 0 Td[(d)+)]TJ /F7 11.955 Tf 12.51 2.65 Td[(^i)]TJ /F7 11.955 Tf 13.15 8.08 Td[(1 2k1k22X(t),(t)EfW1((t))]TJ /F7 11.955 Tf 11.95 0 Td[(d))]TJ /F3 11.955 Tf 11.96 0 Td[(W1((t))jX(t),(t)g+1^+1 2k1k22, (A) whered=[di,j(t)]withdi,j(t)=Foi,j(t))]TJ /F3 11.955 Tf 12.88 0 Td[(Fii,j(t))]TJ /F3 11.955 Tf 12.88 0 Td[(xi,j(t)asdenedin( 3 )and1=[(t))]TJ /F7 11.955 Tf 11.96 0 Td[(d]+)]TJ /F10 11.955 Tf 12.23 9.68 Td[([(t))]TJ /F7 11.955 Tf 11.96 0 Td[(d]+;therstinequalitycomesfromexpandingthetermW1)]TJ /F7 11.955 Tf 5.48 -9.85 Td[([(t))]TJ /F7 11.955 Tf 11.95 0 Td[(d]+)]TJ /F9 11.955 Tf 11.95 0 Td[(1andrelaxtheprojectioninit.Notethat01<21,where0and1arezeroandunitmatrixrespectively.Thus,1^+1 2k1k222k^k22+1 24k1k22, 98

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2Z1.Substituteitinto( A ),andwehave EfW1(t+1))]TJ /F3 11.955 Tf 11.96 0 Td[(W1(t)jX(t),(t)gEfW1((t))]TJ /F7 11.955 Tf 11.95 0 Td[(d))]TJ /F3 11.955 Tf 11.96 0 Td[(W1((t))jX(t),(t)g+2Z1=)]TJ /F7 11.955 Tf 9.3 0 Td[(E(Xi,jdi,ji,j)]TJ /F7 11.955 Tf 16.51 2.66 Td[(^i,jX(t),(t))+1 22Ekdk22X(t),(t)+2Z1 (A) wherethesecondequalityfollowsfromexpandingthetermW1((t))]TJ /F7 11.955 Tf 11.95 0 Td[(d))]TJ /F3 11.955 Tf 11.96 0 Td[(W1((t)). Denotetheobjectivefunctionin( 2 )asV(X(t),(t))=Ps2SUs(xs))]TJ /F10 11.955 Tf -405.85 -14.94 Td[(P(i,j)2Li,jxi,j.Thepartialderivativew.r.t.xsi,jisVsi,j=usi,j)]TJ /F9 11.955 Tf 12.53 0 Td[(i,jandg=[Vsi,j],whereusi,jisthepartialderivativew.r.t.xsi,jofUs(xs).Substituting( 3 )intothesecondtermof( A )andfollowingthesamemanipulationsasabove,wehaveEfW2(t+1))]TJ /F3 11.955 Tf 11.96 0 Td[(W2(t)jX(t),(t)g=EW2\004[X(t)+(g+vB(t))]+X(t),(t))]TJ /F17 11.955 Tf 11.95 0 Td[(EfW2(X(t))jX(t),(t)g=EW2)]TJ /F7 11.955 Tf 5.48 -9.85 Td[([X(t)+(g+vB(t))]+)]TJ /F9 11.955 Tf 13.15 0 Td[(2)X(t),(t))]TJ /F17 11.955 Tf 11.95 0 Td[(EfW2(X(t))jX(t),(t)gEfW2(X(t)+(g+vB(t)))jX(t),(t)g)]TJ /F17 11.955 Tf 20.58 0 Td[(EW2(X(t)))]TJ /F7 11.955 Tf 13.16 8.09 Td[(1 2k2k22X(t),(t))]TJ /F17 11.955 Tf -426.6 -32.08 Td[(En2h(X(t)+(g+vB(t)))+)]TJ /F7 11.955 Tf 13.64 2.66 Td[(^XiX(t),(t)oEW2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(X(t)+(g+vB(t))+X(t),(t))]TJ /F17 11.955 Tf 11.96 0 Td[(EfW2(X(t))jX(t),(t)g+2^+1 2k2k22, (A) 99

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whereB(t)=[bsi,j(t)]and2=[X(t)+g]+)]TJ /F10 11.955 Tf 12.17 9.69 Td[([X(t)+g]+.Furthersimplify( A )by2^X+1 2k2k222k^Xk22+1 24k1k22,2Z2andweget EfW2(t+1))]TJ /F3 11.955 Tf 11.95 0 Td[(W2(t)jX(t),(t)gEW2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(X(t)+(g+vB(t))+X(t),(t))]TJ /F17 11.955 Tf 11.96 0 Td[(EfW2(X(t))jX(t),(t)g+2Z2=E(Xs,jgxsi,j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(xsi,j)]TJ /F7 11.955 Tf 12.14 0 Td[(^xsi,jX(t),(t))+Xs,jEvbsi,j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(xsi,j)]TJ /F7 11.955 Tf 12.14 0 Td[(^xsi,jX(t),(t)+1 22Ek(g+vB(t))+k22X,+2Z2E(Xs,jgxsi,j)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(xsi,j)]TJ /F7 11.955 Tf 12.14 0 Td[(^xsi,jX(t),(t))+1 22Ek(g+vB(t))+k22X,+2Z2 (A) wherethesecondequalityfollowsthedenitionofbsi,j(t).Substitute( A )and( A )into( A ),then EfW(t+1))]TJ /F3 11.955 Tf 11.96 0 Td[(W(t)jX(t),(t)g)]TJ /F7 11.955 Tf 28.56 0 Td[(E(Xi,jd(i,j)]TJ /F7 11.955 Tf 12.23 2.65 Td[(^i,j)X(t),(t))+E(Xs,jgxsi,j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(xsi,j)]TJ /F7 11.955 Tf 12.14 0 Td[(^xsi,jX(t),(t))+2(Z1+Z2+D20+V20), (A) wherethelastinequalityfollowsthattheconditionalexpectationofkdk22andk(g+vB(t))+k22areuniformlyboundedbyconstantsD0>0andV0>0.Fordi,j(t),theper-linkowvariables[Foi,j(t)],[Fii,j(t)]andthetransmissionrate[xi,j(t)]arebounded.Fork(g+vB(t))+k22,ifg+vB(t)0,k(g+vB(t))+k22kusi,j(t)k22+kvB(t)k22isbounded,otherwisek(g+vB(t))+k22=0. FollowsimilarmanipulationprocedureasinTheorem 2.1 from( 2 )andtakeconditionalexpectation,( A )canbesimpliedto EfW(t+1))]TJ /F3 11.955 Tf 11.95 0 Td[(W(t)jX(t),(t)gXsEf(us)]TJ /F7 11.955 Tf 12.2 0 Td[(^us)(xs(t))]TJ /F7 11.955 Tf 12.14 0 Td[(^xs)g+2(Z1+Z2+D20+V20), (A) 100

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where^usisthepartialderivativeofUs(xs)w.r.t.xsi,jandevaluatedat^X. DeneA,suchthat A,(XE"Xi,j(us)]TJ /F7 11.955 Tf 12.19 0 Td[(^us)(^xs)]TJ /F3 11.955 Tf 11.95 0 Td[(xs(t))#)(V20+D20+Z1+Z2), whichisnotemptyandhasnitenumberofelementswithproperlychosen. Therefore,wehave: EfW(t+1))]TJ /F3 11.955 Tf 11.96 0 Td[(W(t)jX(t),(t)g1 22(V20+D20+Z1+Z2)IX2A)]TJ /F7 11.955 Tf 13.15 8.08 Td[(1 22(V20+D20+Z1+Z2)IX2Ac (A) whereIistheindicatorfunction.Thus,byTheorem3.1in[ 21 ],whichisanextensionofFoster'scriterion,theMarkovchainisstable. A.2ProofofTheorem 3.2 Proof. Takingexpectationof( A )w.r.t.(X,),wehave EfW(t+1))]TJ /F3 11.955 Tf 11.96 0 Td[(W(t)g1 22(V20+D20+Z1+Z2)+XsEf(us)]TJ /F7 11.955 Tf 12.2 0 Td[(^us)(xs(t))]TJ /F7 11.955 Tf 12.14 0 Td[(^xs)g (A) Summing( A )over=0,...,t)]TJ /F7 11.955 Tf 11.95 0 Td[(1,itisobtainedthat1 tt)]TJ /F8 7.97 Tf 6.58 0 Td[(1X=0 XsEf(us)]TJ /F7 11.955 Tf 12.19 0 Td[(^us)(xs(t))]TJ /F7 11.955 Tf 12.14 0 Td[(^xs)g!EfW(X(0),(0))g+t 22(V20+D20+Z1+Z2) t (A) Ifxs(t)6=^xs,( A )canbesimpliedas1 tt)]TJ /F8 7.97 Tf 6.59 0 Td[(1X=0 XsEusi,j)]TJ /F7 11.955 Tf 12.2 0 Td[(^usi,j xi,j(t))]TJ /F7 11.955 Tf 12.14 0 Td[(^xi,j(xi,j(t))]TJ /F7 11.955 Tf 12.14 0 Td[(^xi,j)2!EfW(X(0),(0))g+t 22(V20+D20+Z1+Z2) t. (A) 101

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SinceUs(xs)isastrictlyconcavefunctionofxsandthetransmissionratexs2[0,xsm],usi,j)]TJ /F8 7.97 Tf 6.74 0 Td[(^usi,j xi,j(t))]TJ /F8 7.97 Tf 6.69 0 Td[(^xi,jisnon-negativeandbounded.Thenusi,j)]TJ /F8 7.97 Tf 6.75 0 Td[(^usi,j xi,j(t))]TJ /F8 7.97 Tf 6.7 0 Td[(^xi,jK1holds,whereK1>0isthelargestlowerboundofusi,j)]TJ /F8 7.97 Tf 6.75 0 Td[(^usi,j xi,j(t))]TJ /F8 7.97 Tf 6.7 0 Td[(^xi,j.Lettingtgoingtoinnityin( A )produces XsE(xs(1))]TJ /F7 11.955 Tf 12.14 0 Td[(^xs)2(V20+D20+Z1+Z2) 2K1. (A) Thenitisobtainedthat "XsEfUs(xs(1))g)]TJ /F10 11.955 Tf 20.59 11.36 Td[(Xs^Us#2XshEfUs(xs(1))g)]TJ /F7 11.955 Tf 22.05 2.66 Td[(^Usi2XsEUs(xs(1)))]TJ /F7 11.955 Tf 13.42 2.65 Td[(^Us2XsEnjus(xs(1))j2(xs(1))]TJ /F7 11.955 Tf 12.14 0 Td[(^xs)2oK2jFj(V20+D20+Z1+Z2) 2K1, (A) whereK2=maxsjus(xs(1))j2isaconstant;andjFjisthecardinalityofF. 102

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APPENDIXBPROOFSINCHAPTER4 B.1ProofofTheorem 4.1 Inthissection,wesketchtheproofforTheorem 4.1 basedonthefollowingclues:rst,weprovethat[fi,j]2dpdlisthenecessaryconditionforthenetworkstability,thenweprovethatwhenevertheinputrateisinsidethecapacityregion,thereexistsapowercontrolandroutingalgorithmthatcanstabilizethenetwork. B.1.1Necessity ConsideraPDPLqueueingnetworkwithconvergentinputrates[fi,j]denedin( 4 ),andletAfi,j(t)representtheamountofpacketsthatexogenouslyenterthenetworkatnodeiwithnodejasthenexthopnodeduringtheinterval[0,t].Supposethesystemisstabilizablebysomeroutingandpowercontrolpolicy.LetQfi,j(t)representtheresultingunnishedworkforqfi,j.LetDm,i,j(t)representthetotalnumberofpacketsthatenterqfi,jfromnodemduringtheinterval[0,t]andDfi,j,k(t)representthetotalnumberofpacketsthatdepartfromqfi,jandgointoqfj,k.Wehavethefollowingforalltimet: Di,j,k(t)0 (Ba)XkDfi,j,k(t))]TJ /F10 11.955 Tf 11.95 11.36 Td[(XmDfm,i,j(t)=Afi,j(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Qfi,j(t) (Bb)Zt0fi,j(P(t),S(t))dtXkDfi,j,k(t), (Bc) where( Bb )followsthattheunnishedworkinanynodeisequaltothedifferencebetweenthetotalnumberofpacketsthathavearrivedanddeparted.Inequality( Bc )holdsbecausethetotalpacketstransferredoveranylinkislessthanorequaltotheofferedtransmissionrateintegratedoverthetimeinterval[0,t]and( Bb )holdsforthescenariothatDfi,j,k(t)containthepaddednullpacketsifthereisnotenoughtrafcfortransmission.Notethatalthoughsomecontrolpolicycanstabilizethenetwork,the 103

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powerallocationprocessP(t)isnotnecessarilyergodic,noraretheinternalbitstreamsproducedbyroutingdecisions. Sincethechannelprocessisconvergenttoanitestatespaceasdenedin( 4 ),whenmeasuredoveranysufcientlylargeinterval[0,t],thetimefractionofachannelstateSsatisesjjTS(t)jj=t!Swithprobability1.Besides,theinputprocessisrateconvergentto[fi,j]asdenedin( 4 ).FromtheLemma1in[ 50 ],weknowthatifthenetworkisstable,thenforany0,thereexistsanitevalueM,forwhicharbitrarilylargetimes~tcanbefoundsothatPr[Pi,jUfi,j(~t)]1)]TJ /F9 11.955 Tf 12.4 0 Td[(.Therefore,theremustexistsomenitevalueMsuchthatatarbitrarilylargetimes~t,theunnishedworkinallqueuesissimultaneouslylessthanMwithprobabilityatleast1 2.Hence,thereexistsatime~tsuchthatwithprobabilityatleast1 2,allofthefollowinginequalitiesaresatised: Qfi,j(~t)M (Ba)M ~t (Bb)Afi,j ~tfi,j)]TJ /F9 11.955 Tf 11.96 0 Td[( (Bc)jjTS(~t)jj ~tS+,8S. (Bd) Nowdene: dfi,j,k,Dfi,j,k(~t) ~t (B) Substituting( B )to( Bb ),itfollowsthatforallli,j (fi,j+)Xmdfm,i,j)]TJ /F10 11.955 Tf 11.96 11.36 Td[(Xkdfi,j,k. Thus,thenecessityisproved. 104

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B.1.2Sufciency TheproofinAppendix B.2 directlyprovesthattheDRPC-PDPLcanstabilizethenetworkwhentheinputrateiswithinthecapacityregionofthePDPLnetwork.ThustheSufciencyisproved. B.2ProofofTheorem 4.3 Inthissection,weprovethattheDRPC-PDPLalgorithmcanstabilizethenetworkwhenevertheinputrateiswithinthecapacityregiondenedinSection 4.2.2 B.2.1PerformanceoftheSTAT-PDPLAlgorithm TheSTAT-PDPLisastationaryrandomizedalgorithmassumingthatthevaluesof[Gi,j],[dfi,j,k]areknowntothescheduler.Supposetheinputratematrixsatisfyingfi,j+2pdplandthechannelprobabilitySareknowninadvance,thenasetoftheowvariables[dfi,j,k]andalinkratematrix[Gi,j]mustexistaccordingtoTheorem 4.1 Sincethephysicalchannelandinterferencemodelarethesameasthosein[ 9 ],theLemma8(GraphFamilyAchievability)alsoholdsfortheper-linkqueueingnetworks.Therefore,astationaryrandomizedpowerallocationpolicyP1,canbeimplementedyieldingatransmissionratematrix1(t),whichisentrywiseconvergentwithratematrix[Gi,j],sowehave 1 KK)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xt=0Ef1i,j(t)g)]TJ /F3 11.955 Tf 20.59 0 Td[(Gi,j (B) StationaryRandomizedPolicyforKnownSystemStatisticsunderper-destinationper-linkqueue(STAT-PDPL): 1)PowerAllocation:Everytime-slot,observethechannelstateS(t),andallocatethepoweraccordingtoP1[ 50 ]; 2)Schedulingandrouting:Foreveryli,j,transmitasingleowfrandomlychosenwithprobabilityPkdfi,j,k PfPkdfi,j,k,androutethepacketforowfrandomlytoqfj,kwithprobability 105

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dfi,j,k Pkdfi,j,k,thenwehave: 1fi,j(t)=8><>:1i,j(t)Pkdfi,j,k Gi,j,ifowfischosen0,o.w. (B) Ifanodedoesnothaveenough(orany)packetsofacertainowtosendoveritsoutputlinks,nullpacketsaredelivered,sothatlinkshaveidletimeswhicharenotusedbyotherows.Inthelightof( B ),wehaveEf1fi,j(t)j1i,j(t)g=1i,j(t)Pkdfi,j,k Gi,j, (B) andEf1fi,j,k(t)j1i,j(t)g=1i,j(t)dfi,j,k Gi,j, (B) TheperformanceoftheSTAT-PDPLalgorithmisstatedbelow. Lemma1. (StabilizingPolicyunderPer-destinationPer-linkqueueingforKnownStatistics)considerawirelessnetworkwithcapacityregionpdplandinputrates[fi,j]suchthat[fi,j+]2pdplforsome0.ThenthealgorithmSTAT-PDPLstabilizesthenetwork. Proof. Here,weprovethatiftheinputrate[fi,j]2,thejointpowercontrolandroutingalgorithmSTAT-PDPLcanstabilizethenetwork. Takeconditionalexpectationsw.r.t.1i,j(t)tobothsidesof( B )and( B )andsubstitutetheminto( B ),wehaveforsomeconvergentintervalK: 1 KPK)]TJ /F8 7.97 Tf 6.59 0 Td[(1t=0E1fi,j,k(t))]TJ /F3 11.955 Tf 11.96 0 Td[(dfi,j,k1 6 (B) 1 KPK)]TJ /F8 7.97 Tf 6.58 0 Td[(1t=0E1fm,i,j(t))]TJ /F3 11.955 Tf 11.95 0 Td[(dfm,i,j2 6, (B) 106

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where1>0and2>0arecarefullyselectedconstants.Summing( B )overkproduces: 1 KK)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xt=0"XkE1fi,j,k(t))]TJ /F10 11.955 Tf 11.96 11.36 Td[(Xkdfi,j,k#Xk1 KK)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xt=0E1fi,j,k(t))]TJ /F3 11.955 Tf 11.96 0 Td[(dfi,j,kjjNjjj1 6 6 (B) Summing( B )overmandfollowingthesimilarmanipulations,wehave:1 KK)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xt=0"XmE1fm,i,j(t))]TJ /F10 11.955 Tf 11.95 11.36 Td[(Xmdfm,i,j# 6 (B) Remembertheinputprocessisentrywiserateconvergenttotheinputrate[fi,j],sothat: fi,j)]TJ /F7 11.955 Tf 14.92 8.08 Td[(1 KK)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xt=0Efafi,j(t)g 6(B) Sincetheinputrateiswithinthecapacityregion,wehaveforanyli,jthat (fi,j+)Xkdfi,j,k)]TJ /F10 11.955 Tf 11.96 11.36 Td[(Xmdfm,i,j(B) Substituting( B )( B )and( B )into( B ),wehave:1 KK)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xt=0E(Xk1fi,j,k(t))]TJ /F10 11.955 Tf 11.96 11.36 Td[(Xm1fm,i,j(t))]TJ /F3 11.955 Tf 11.95 0 Td[(afi,j(t)) 2 (B) NowdenetheLyapunovfunctionasL(Q(t))=Pi,j;f:d(f)6=i)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Qfi,j(t)2.TheK-stepdynamicsofunnishedworksatisesthequeueupdatelawandwerewriteitw.r.t. 107

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1i,j,k(t)and1m,i,j(t)Qfi,j(t+K)"Qfi,j(t))]TJ /F6 7.97 Tf 11.96 14.94 Td[(K)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xt=0Xk1fi,j,k(t)#++K)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xt=0Xm1fm,i,j(t)+K)]TJ /F8 7.97 Tf 6.58 0 Td[(1Xt=0afi,j(t) (B) FollowingthesamemanipulationsasinSection 4.2.1 ,wehavetheK-slotLyapunovdrift)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Qfi,j(t+K)2)]TJ /F10 11.955 Tf 11.96 9.69 Td[()]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Qfi,j(t)2K224 ~afi,j+Xm~1fm,i,j!2+ Xk~1fi,j,k!235)]TJ /F7 11.955 Tf 11.95 0 Td[(2KQfi,j(t) Xk~1fi,j,k)]TJ /F10 11.955 Tf 11.96 11.36 Td[(Xm~1fm,i,j)]TJ /F7 11.955 Tf 11.86 0 Td[(~afi,j!K2B)]TJ /F7 11.955 Tf 11.95 0 Td[(2KQfi,j(t) (B) whereBisdenedin( 4 )and ~1fi,j,k,1 KK)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xt=01fi,j,k(t)~1fm,i,j,1 KK)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xt=01fm,i,j(t)~afi,j,1 KK)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xt=0afi,j(t) Summing( B )overallli,jandtakingconditionalexpectationsyields EfL(Q(t+K)))]TJ /F3 11.955 Tf 11.96 0 Td[(L(Q(t))jQ(t)gK2B)]TJ /F7 11.955 Tf 11.96 0 Td[(2KXi,jQfi,j(t). (B) ThenthenetworkisstableaccordingtoLemma2in[ 12 ].TheSTAT-PDPLalgorithmisdevelopedwiththenetworkstatisticsknowntoactonlyasabenchmarkfortheperformanceoftheDRPC-PLinSection 4.2 108

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WethenevaluatetheperformanceofDRPC-PDPLthroughasequenceoftwoLemmas.TherstlemmacomparestheLyapunovdriftoftheSTAT-PDPLalgorithmtothedriftofamodiedframe-basedDRPC-PDPLalgorithm,i.e.,FRAME-PDPL.ThesecondlemmacomparesthedriftofFRAME-PDPLtothatofDRPC-PDPL. B.2.2PerformanceoftheFRAME-PDPLAlgorithm WerewritetheKstepdriftbound( B )intermsofaquantity(Q(t)),whichcapturestheonlycomponentoftheboundthatdependsonthecontrolstrategy.Thenwehave EfL(Q(t+K)))]TJ /F3 11.955 Tf 11.96 0 Td[(L(Q(t))jQ(t)gK2B)]TJ /F7 11.955 Tf 11.96 0 Td[(2K[(Q(t)))]TJ /F9 11.955 Tf 11.96 0 Td[((Q(t))] (B) where (Q(t)),1 KK)]TJ /F8 7.97 Tf 6.58 0 Td[(1Xt=0Xi,j;fQfi,j(t)E("Xkfi,j,k(t))]TJ /F10 11.955 Tf 14.88 11.35 Td[(Xmfm,i,j(t)#Q(t))(Q(t)),1 KK)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xt=0Xi,j;fQfi,j(t)Eafi,j(t)jQ(t) WenowconsideraK-slot-basedmodicationoftheDRPC-PDPLpolicyFRAME-PDPL,whichmaximizesthe(Q(t))functionoverallpossiblecontrolpolicies.TheFRAME-PDPLalgorithmisdenedasfollows:Scheduling,powerallocationandroutingaredoneeverytime-slotexactlyasintheDRPC-PDPLalgorithm,withtheexceptionthatbacklogQ(t)updateseveryKslots.Specically,foranytime-slottwithinaKslotframef0,1,,K)]TJ /F7 11.955 Tf 12.14 0 Td[(1g,powerisallocatedtomaximizePi,jWi,j(t)denedin( 4 )evaluatedatQ(t)subjecttoP2P.Thus,currentchannelstateinformationbutoutofdatebackloginformationisusedeveryslotandfi,j,k(t)iscalculatedaccordingtotherealqueuestate. 109

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Lemma2. ThecontrolalgorithmFRAME-PDPLmaximizes(Q(t))overallpossiblepowerallocation,routing,andschedulingstrategies.Thatis: F)]TJ /F6 7.97 Tf 6.59 0 Td[(PDPL(Q(t))X(Q(t))(B) foranyotherstrategyX,includingtheSTAT-PDPLalgorithm. Proof. TheFRAME-PDPLalgorithmactsthesameastheDRPC-PDPLalgorithmwiththeexceptionthatqueuebacklogisupdatedonlyonframeboundariest=f0,K,2K,g.Thus,foreveryt2f0,1,,K)]TJ /F7 11.955 Tf 12.52 0 Td[(1g,thealgorithmFRAME-PDPLallocatesapowermatrixP(t)tomaximize: Xi,j;fQfi,j(t) Xkfi,j,k(t))]TJ /F10 11.955 Tf 11.96 11.36 Td[(Xmfm,i,j(t)!=Xi,j;fi,j(t))]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Qfi,j(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Qfj,k(t) (B) Takingconditionalexpectationsw.r.t.Q(t)aboveandsummingovertyieldsanalternativewaytoexpress(Q(t)):(Q(t))=1 KK)]TJ /F8 7.97 Tf 6.59 0 Td[(1Xt=0E(Xi,j;fi,j(t))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Qfi,j(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Qfj,k(t) (B) ThevalueofX(Q(t))isobtainedfrom( B )byusingthei,j(t)correspondingtosomepolicyX.Thus,theFRAME-PDPLessentiallymaximizes( B )toyieldF)]TJ /F6 7.97 Tf 6.58 0 Td[(PDPL(Q(t))bychoosingtheweightforalinkas( 4 ).TakingconditionalexpectationsandcompareF)]TJ /F6 7.97 Tf 6.59 0 Td[(PDPL(Q(t))withX(Q(t)),wehave: F)]TJ /F6 7.97 Tf 6.59 0 Td[(PDPL(Q(t))X(Q(t)) WenowcomparethealgorithmFRAME-PDPLandDRPC-PDPL. 110

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Lemma3. ThecontrolpolicyDRPC-PDPLproducesaD)]TJ /F6 7.97 Tf 6.59 0 Td[(PDPL(Q(t))satisfying: D)]TJ /F6 7.97 Tf 6.58 0 Td[(PDPL(Q(t))F)]TJ /F6 7.97 Tf 6.58 0 Td[(PDPL(Q(t)))]TJ /F7 11.955 Tf 13.15 8.09 Td[((K)]TJ /F7 11.955 Tf 11.96 0 Td[(1)jFjjLj~B 2(B) where ~B=2max(amax+2max). Proof. Foreverytimeslot,theDRPC-PDPLmaximizes( B )overallpossiblecontroldecisions.Hence: Xi,j;fD)]TJ /F6 7.97 Tf 6.59 0 Td[(DPDLi,j(t))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Qfi,j(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Qfj,k(t)Xi,j;fF)]TJ /F6 7.97 Tf 6.59 0 Td[(DPDLi,j(t))]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Qfi,j(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Qfj,k(t) (B) Rewritingthelefthandsideof( B ),wehave: Xi,j;fD)]TJ /F6 7.97 Tf 6.59 0 Td[(DPDLi,j(t))]TJ /F3 11.955 Tf 5.47 -9.68 Td[(Qfi,j(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Qfj,k(t)+Xi,j;ffi,j(t)2maxXi,j;fF)]TJ /F6 7.97 Tf 6.59 0 Td[(DPDLi,j(t))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Qfi,j(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Qfj,k(t) (B) Xi,j;fF)]TJ /F6 7.97 Tf 6.59 0 Td[(DPDLi,j(t))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Qfi,j(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Qfj,k(t))]TJ /F10 11.955 Tf -176.4 -21.81 Td[(Xi,j;ffi,j(t)2max (B) wherefi,j(t),Qfi,j(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Qfi,j(t),and( B )followsthefact: Xi,j;ffi,j(t))]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Qfi,j(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Qfj,k(t)Xi,j;ffi,j(t)2max 111

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Summing( B )overt2f0,1,,K)]TJ /F7 11.955 Tf 12.48 0 Td[(1gandtakingconditionalexpectationsw.r.t.Q(t)yields: D)]TJ /F6 7.97 Tf 6.58 0 Td[(PDPL(Q(t))F)]TJ /F6 7.97 Tf 6.58 0 Td[(PDPL(Q(t)))]TJ /F7 11.955 Tf -175.05 -20.22 Td[(2 KK)]TJ /F8 7.97 Tf 6.58 0 Td[(1Xt=0E(Xi,j;ffi,j(t))2max (B) Substitute( B )into( B )yieldsimplies( B ). NotethattheperformanceanalysisoftheaboveSTAT-PDPL,FRAME-PDPLandtheDRPC-PDPLalgorithmswhenthesameroutingalgorithmisapplied. 112

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APPENDIXCPROOFOFTHEOREM5 Proof. Rememberthattheparametersfig,i2fw,cgdenedinQsatisfyi2[1,+1).ToproveTheorem1,wersthavetodeneauxiliaryvariablesbelow: 1)Denethefollowingparameters mc=E[ptjIfQ=cg(t)=1](E[W(t)])]TJ /F3 11.955 Tf 11.96 0 Td[(rE[D(t)])mw=maxQE[UtwIfQ=wg(t)]; 2)Foralli,EUtiIfQ=ig(t)mi,i2fw,cg; 3)Foralli,ifEUtiIfQ=ig(t)
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Theexpectedsystemutilitysatises EUtQEUtQ+NXi=1(i)]TJ /F7 11.955 Tf 11.96 0 Td[(1))]TJ /F17 11.955 Tf 5.48 -9.68 Td[(EUtiIfQ=ig(t))]TJ /F3 11.955 Tf 11.95 0 Td[(mi=NXi=1EUtiIfQ=ig(t)+NXi=1(i)]TJ /F7 11.955 Tf 11.95 0 Td[(1)EUtiIfQ=ig(t))]TJ /F6 7.97 Tf 16.63 14.95 Td[(NXi=1(i)]TJ /F7 11.955 Tf 11.95 0 Td[(1)mi=NXi=1EiUtiIfQ=ig(t))]TJ /F6 7.97 Tf 17.3 14.94 Td[(NXi=1(i)]TJ /F7 11.955 Tf 11.96 0 Td[(1)mi, (C) wheretherstinequalitycomesfromthat(i)]TJ /F7 11.955 Tf 11.96 0 Td[(1))]TJ /F17 11.955 Tf 5.48 -9.69 Td[(EUtiIfQ=ig(t))]TJ /F3 11.955 Tf 11.96 0 Td[(mi0foranyi. BythedenitionofQin( 5 ),weobtainthat NXi=1iUtiIfQ=ig(t)NXi=1iUtiIfQ=ig(t). Thus,wehave EUtQNXi=1EiUtiIfQ=ig(t))]TJ /F6 7.97 Tf 17.3 14.95 Td[(NXi=1(i)]TJ /F7 11.955 Tf 11.95 0 Td[(1)mi=NXi=1(i)]TJ /F7 11.955 Tf 11.95 0 Td[(1))]TJ /F17 11.955 Tf 5.48 -9.69 Td[(EUtiIfQ=ig(t))]TJ /F3 11.955 Tf 11.96 0 Td[(mi+EUtQ=EUtQ, (C) where( C )follows NXi=1(i)]TJ /F7 11.955 Tf 11.95 0 Td[(1))]TJ /F17 11.955 Tf 5.48 -9.68 Td[(E[UtiIfQ=ig(t)])]TJ /F3 11.955 Tf 11.95 0 Td[(mi=0, sincewehave i8><>:=1,ifE[UtiIfQ=ig(t)]1,ifE[UtiIfQ=ig(t)]=mi. Inaddition,theconstraintE[UtwIfQ=wg(t)]mwalwaysholds. 114

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Hence,with( 5 )andN=2asaspecialcaseof( C ),theoptimalityofQisproved.Nowweprovethattheconstraint( 5 )ontheintegrationpercentageofwindenergyisequivalentto( C ). Substituting( 5 )intoQ=argmini(iUti),weobtain Q=argminifcpt,w(qtt+kt)g(C) whichisindependentoftheforecastwindenergygenerationW(t)attimet.Then( C )canbere-writtenas EptW(t)IfQ=cg(t)=E[W(t)]EptIfQ=cg(t) (C) mc, (C) where( C )followsthatW(t)isindependentofQandpt.Inaddition,Qsatises( 5 )andwehave EW(t)IfQ=cg(t)=E[W(t)]E[IfQ=cg(t)]E[W(t)])]TJ /F3 11.955 Tf 11.96 0 Td[(rE[D(t)] (C) Comparing( C )and( C ),weneedthefollowingtohold E[W(t)])]TJ /F3 11.955 Tf 11.96 0 Td[(rE[D(t)] E[IfQ=cg(t)]=mc E[ptIfQ=cg(t)]. (C) Since IfQ=cg=8><>:1,ifw(qtt+kt)cpt0,otherwise, 115

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wehavethefollowing EptIfQ=cg(t)=PfwqtcptgE[(qtt+kt)jIfQ=cg(t)=1] (C) whichisaconstantandcanbecalculatedfromthedistributionofpt,qt,tandkt.RememberthatE[IfQ=cg(t)]=Pfw(qtt+kt)cptg,andwehavemc=E[ptjIfQ=cg(t)=1](E[W(t)])]TJ /F3 11.955 Tf 11.96 0 Td[(rE[D(t)]),whichcompletestheproofofTheorem 5.1 116

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BIOGRAPHICALSKETCH ZongruiDingwasborninTianjin,Chinain1983.ShereceivedaB.E.degreeincommunicationengineeringin2006fromBeijingUniversityofPostsandTelecommunications,China.In2008,shebecameaPhDstudentintheDepartmentofElectricalandComputerEngineeringatUniversityofFlorida.HerresearchfocusesonstochasticoptimizationinsystemslikewirelessnetworksandtheSmartGrid. 123