SCAPE:SafeChargingwithAdjustablePowErHaipengDai,YunhuaiLiu,GuihaiChen - PDF document

SCAPE:SafeChargingwithAdjustablePowErHaipengDai,YunhuaiLiu,GuihaiChenz,XiaobingWu,TianHeStateKeyLaboratoryforNovelSoftwareTechnology,NanjingUniversity,Nanjing,Jiangsu210023,CHINAThirdResearchInstituteofMinistryofPublicSecurity,Shanghai,CHINAShanghaiKeyLaboratoryofScalableComputingandSystems,ShanghaiJiaoTongUniversity,Shanghai200240,CHINAComputerScienceandEngineering,UniversityofMinnesota,Minneapolis,MN55455,USAdhpphd2003,yunhuai.liugchen,wuxb@nju.edu.cn,tianhe@cs.umn.eduÐWirelesspowertransfertechnologyisconsideredasoneofthepromisingsolutionstoaddresstheenergylimitationproblemsforend-devices,butitsincurredpotentialriskofelectromagneticradiation(EMR)exposureislargelyoverlookedbymostexistingworks.Inthispaper,weconsidertheSafeChargingwithAdjustablePowEr(SCAPE)problem,namely,howtoadjustthepowerofchargerstomaximizethechargingutilityofdevices,whileassuringthatEMRintensityatanylocationinthe®elddoesnotexceedagiventhreshold.WepresentnoveltechniquestoreformulateSCAPEintoatraditionallinearprogrammingproblem,andthenremoveitsredundantconstraintsasmuchaspossibletoreducecomputationaleffort.Next,weproposeadistributedalgorithmwithprovableapprox-imationratio(1.Throughextensivesimulationandtestbedexperiments,wedemonstratethatour(1-approximationalgorithmoutperformstheSet-Coveralgorithmbyupto23%andhasanaverageperformancegainof411%overtheSCPalgorithmintermsoftheoverallchargingutility.I.INTRODUCTIONInrecentyears,wirelesspowertransfertechnology[1]hasbeenattractinggreatinterestsofindustryandresearchers.Asacommercializedandcontrollabletechnology,itisoneofthepromisingtechnologiestoaddresstheenergylimitationproblemsforend-devicessuchasRFIDs[2],sensors[3],cellphones[4],laptops[5],vehicles[6]andunmannedplanes[7].Thoughtherehasemergedavarietyofworksdedicatedtoenergyef®ciencyissueswithrespecttowirelesspowertransfertechnology[8]±[13],mostofthemoverlookedthepotentialriskofelectromagneticradiation(EMR)broughtbythistechnology.ExposuretohighEMR,however,hasbeenwidelyrecognizedasathreattohumanhealth.Itspotentialrisksincludebutnotlimitedtomentaldiseases[14],tissueimpairment[15]andbraintumor[16].Inaddition,therehasbeensolidevidencethatpregnantwomenandchildrenareevenmorevulnerabletohighEMRexposure[17][18].Forexample,Gandhietal.[18]foundthatchildren'sheadsabsorbovertwotimesofRFthanadults,andtheirabsorptionoftheskull'sbonemarrowcanbetentimesgreaterthanadults.ThesefactssuggesttheneedforconsideringEMRsafetywhenapplyingwirelesspowertransfertechnology.Inthispaper,weattempttoimprovetheoverallchargingperformanceunderEMRsafetyconcern,wherechargerscancontinuouslyadjusttheirpowerlevelwithinanappropriaterange.Basically,ourobjectiveistomaximizetheoverallchargingutilityofdevicesbyadjustingthepowerofchargers,whileassuringthatnolocationhasEMRintensityexceedingagiventhreshold.Intuitively,thisproblemisquitechallengingastheEMRsafetyrequirementisimposedoneverypointinthe®eld,whichcorrespondstoanin®nitenumberofconstraints.Tomaketheproblemtractable,wepresentanapproximationapproachtoreformulatetheproblemasalinearprogrammingproblemwithlimitedconstraints,andalsodeviseanoveldistributedapproachtoreducethecomputationaleffortsoftheproblem.Afterthat,wedevelopa(1distributedalgorithmtodealwiththisproblem.Themaincontributionsofthispaperarelistedasfollows.Tothebestofourknowledge,thisisthe®rstpaperconsideringtheproblemofmaximizingthechargingef®ciencyofthenetworkunderEMRsafetyconcern,byadjustingthepowerofchargers.Weformulatethisprob-lemasSafeChargingwithAdjustablePowEr(SCAPE)Wepresentanareadiscretizationtechniquetohelpre-formulatingtheproblemintoatraditionallinearpro-gramming(LP)problem.Further,weproposeanoveldistributedredundantconstraintreductionschemetocutdownthenumberofconstraints,andthusreducethecomputationaleffortsoftheLPproblem.WedevelopadistributedalgorithmtodealwiththeSCAPEproblem,andprovethatitachieves(1approximationratio.Webuildatestbedtoevaluatetheperformanceofouralgorithms.ExperimentalresultsshowthatouralgorithmsuccessfullycontrolsthemaximalEMRinthe®eldunderagiventhreshold,andhasanaverageperformancegainof411%comparedwiththeSCPalgorithm.Further-more,wealsoconductcomprehensivesimulations.TheresultsshowthatouralgorithmoutperformstheSet-CoveralgorithmbyuptoTheremainderofthepaperisorganizedasfollows.WereviewrelatedworkinSec.II,andformallyde®netheprobleminSec.III.Sec.IVintroducesanovelapproachtoreformulatetheproblem,andSec.Vproposesadistributedmethodtoreduceitscomputationalefforts.Next,wepresentanapprox-imationalgorithminSec.VI.Sec.VIIandSec.VIIIpresentextensivesimulationresultsandtestbedexperimentresultstovalidateourtheoretical®ngings,andSec.IXconcludes. II.RELATEDORKInthissection,webrie¯yreviewrelatedworksstudyingenergyef®ciencyproblemsinwirelessrechargeablesensornetworkswithwirelesspowertransfertechnology,andthatconsideredEMRsafety.First,weconcentrateontheworksabouthowtodeploystaticchargerstomaximizethechargingef®ciencyofsensornodes.Forexample,Heetal.[8]consideredthedeploymentproblemofchargerssuchthatstaticormobilerechargeabletagscanreceivesuf®cientpowertokeepcontinuousoperation,whiletherequirednumberofchargerscanbeminimized.Daietal.[9][13]furtherimprovedthesolutionbytakingintoconsiderationpracticalissuessuchasbatteryconstraintsoftags.In[10],Chiuetal.studiedtheproblemofmaximizingthesurvivalrateofend-deviceswithpriorknowledgeofthemobilitymodelofsensornodes.Liaoetal.[11]adoptedamorepracticalchargingmodelbyassumingthatthecoverageareaofachargerisacone,andconsideredtheplacementprob-lemofchargersinthree-dimensionalspace.Theirobjectiveistominimizethenumberofdeployedchargerswhileassuringthatallsensornodesarecovered.AlltheaboveschemesdidnotconsiderEMRsafetyduringchargingprocess.[19]isthe®rstandonlywork,asfarasweknow,tostudytheenergyef®ciencyproblemunderconcernofEMRsafety.Weemphasizethatthisworkisfundamentallydifferentfromthatof[19]inthefollowingaspects.Firstofall,[19]consideredasimpli®edchargerschedulingmodelinwhichchargerscanbeonlyineitheroftheon/offstates,whileweassumethatthepowerofchargersisadjustableinthispaper.Second,theproposedalgorithmin[19]isessentiallyacentralizedalgorithm.Incontrast,thealgorithmpresentedinthispaperisadistributedone.Third,thoughthealgorithmin[19]hasbeenprovedtooutperformtheoptimalsolutionfortheproblemwitharelaxedEMRthreshold(1,itis,however,notanapproximationalgorithmsinceitrelaxestheEMRconstraints.Conversely,ourdistributedalgorithmprovablyachievesanapproximationratioof(1III.PROBLEMTATEMENTA.PreliminariesSupposethatthereisasetofidenticalstationarywirelesspowerchargers;s;:::;srechargeablede-;o;:::;odistributedonatwo-dimensionalplane.Thedevicescanharvestwirelesspoweroriginatedfromthechargersandthusmaintainnormalworking.Weassumethatallthechargerscancontinuouslyadjustitspowerlevelfrom0toamaximumpower.Whenachargerworksatthemaximumpower,thereceivedpowerbyadevicewithadistancefromthechargercanbequanti®edbyanempiricalmodel[8],i.e.,)= ;d;d�Dareknownconstantsdeterminedbythehardwareofthechargerandthereceiver,aswellastheenvironment.Becauseofthehardwareconstraint,thereceivedpowerfromthechargerdecreasesdramaticallyasthedistanceincreases,andtheenergy®eldfarawayfromthechargerwillbetoosmalltobereceivedbyanode.Wecharacterizethispropertybyusingtodenotethefarthestdistanceachargercanreach,asEq.1illustrates.Wede®neadjustingfactor(0;i=1;:::;nastheratioofthecurrentadjustedpowertothemaximumallowedpowerforthecharger.Therefore,thepoweradevicereceivedfromachargerwithdistanceandadjustingfactorcanbeexpressedas.Besides,weassumethewirelesspoweroriginatingfrommultiplechargersreceivedbyareceiverisadditive[8].Weassumethateachchargerisawareofitslocation.Twochargersareneighborstoeachotherifandonlyiftheircoverageareasintersect.Formally,wedenotebythesetofneighborsofthecharger.Eachchargercansimultaneouslycommunicatewiththeirneighborswirelesslyduringchargingprocess[20][21],whichimpliesthatthewirelesscommunicationrangeisatleasttwicethechargingrange,i.e.,.Thisassumptionispracticalsincetheeffectivechargingdistanceformostoff-the-shelfproductsisusuallyshort,e.g.,lessthan10forTX91501powertransmittersproducedbyPowercast[3],whilethewirelesscommunicationrangeforchargersistypicallylargerthan20Forthechargingutilitymodel,wede®nethechargingutilitytobeproportionaltothechargingpower,namely)==1;o));oisthedistancefromthechargertothedevice,andisapredeterminedconstant.WeadopttheEMRmodelwhichisproposedandexper-imentallyveri®edby[19].Thatis,theintensityofEMRisproportionaltothereceivedpowerthere,i.e.,)=isthedistanceandistheconstanttocapturethelinearrelation.AssumingEMRisalsoadditive,theaccumulatedEMRatalocationisthus)=;p))=;p))AsummaryofthenotationsinthispaperisgiveninTableI.B.ProblemDescriptionWiththeaforementionedmodels,wedescribeandmathe-maticallyformulateourprobleminthissubsection.InordertocontroltheEMRleveloverthe®eld,weestablishanappropriateEMRthresholdandrequirethatEMRatanypointinthe®eldshouldnotexceed.ByEq.3,thisrequirementcanbeformallyexpressedas;C=1;p)) TABLEIOTATIONS Symbol Meaning ;S Charger,chargerset;O Device,deviceset Receivedpowerfromdistance Farthestdistanceachargercanreach;o Distancefromchargertodevice;p Distancefromchargertopoint Chargingutilityofdevice;e EMRfromdistance,EMRatpoint Adjustingfactorofcharger HardthresholdofEMRsafety Neighborsetofcharger Twochargersareneighborstoeachotherifandonlyiftheircoverageareasintersect. Area Discretization and Problem ReformulationSCAPE Distributed Redundant Constraint ReductionSec. IVSec. V Sec. VI -Approximation Distributed Algorithm Fig.1.IllustrationofSCAPEwork¯owOntheotherhand,ourobjectiveistomaximizetheoverallchargingutilityfromalldevices,namely,=1.ByEq.2,wehave=1)==1=1;o)))Tosumup,theSafeChargingwithAdjustablePowErproblem(SCAPE)canthusbede®nedasfollowsmax=1=1;o)));C=1;p))1(=1;:::;nItisverychallengingtosolveSCAPEseeingfromtheaboveformulation.TheconstraintinSCAPEisimposedoneverypointontheplane,whichmeansthatthereisindeedanin®nitenumberofconstraints.Wewillintroducetheoverviewofoursolutiontoaddressthisproblembelow.C.OverviewofOurSolutionToconveythemainideaoftheschemeinthispaper,we®rstpresentanoverviewofoursolution.Thework¯owofoursolutionisillustratedinFig.1.Firstofall,inviewofthechallengecausedbythein®nitenumberofconstraintsinproblemP1,weproposeanovelareadiscretiza-tionmethodtopartitionthe®eldintoalimitednumberofsub-areas.ByexpressingtheEMRsafetyrequirementineachofthesesub-areasasaconstraint,wereformulatetheoriginalintractableproblemP1intoaclassicallinearprogrammingproblem(Sec.IV).Next,wediscusshowtoimplementtheareadiscretizationmethodinadistributedmanner.Asthenumberofobtainedlinearconstraintsexplodeswiththenetworkscaleandthegranularityofareadiscretizationandleadstohighcomputationalefforts,wedevelopadistributedalgorithmtoidentifyandreducetheredundantconstraintsatthesecondstep(Sec.V).Finally,weproposeadistributedalgorithmtoaddresstheproblem,whichprovablyachieves(1ratio(Sec.VI). (1)(2) F Fig.2.IllustrationofplanediscretizationIV.AREAISCRETIZATIONANDROBLEMEFORMULATIONInthissection,we®rstintroduceanareadiscretizationmethodtoreducethenumberofconstraintsfromin®niteto®nite,andtherebyreformulateourproblemintoatraditionallinearprogrammingproblem.Finally,wediscusshowtoimplementthismethodinadistributedmanner.A.AreaDiscretizationInthissubsection,wewilldemonstratehowtodiscretizethe2DareabasedonapiecewiseconstantapproximationofTobeginwith,weusemultiplepiecewiseconstantsegmentstoapproximatetheEMRfunction.Weintelligentlycontrolthenumberofsegmentsandtheirendpoints(1);:::;`)((0)=D;`)=suchthattheap-proximationerrorislessthanagivensmallnumber.Afterthat,wedrawconcentriccircleswithradius(1);:::;`foreachcharger,respectively.TheapproximatedEMRfromthechargeratanypointbetweenadjacentcirclesshouldbeuniform.Finally,thewholenetworkplaneispartitionedintomultiplesub-areafaces,whichareshapedbythesecircles.TakingFig.2asanexample.TheEMRfunctionisap-proximatedby2piecewiseconstantsegmentswithendpoints(1)(2).Then,given3chargersontheplane,wedraw2concentriccirclesforeachofthem,andtherefore,thenetworkplaneispartitionedinto13sub-areafaces,includingtheouterfacewithnoEMR.Foreachsub-areaface,suchasinFig.2,theapproximatedEMRatanypointwithinitfromeachchargershouldbeconstant.Wede®nethefollowingapproximationofEMRandbounditsapproximationerror.(0)=0)=,and)=((1+k=1)=1;:::;K1),thepiecewiseconstantcanbede®nedas)=(0));d(0)1));`1)d)(=1;:::K;d&#x-278;&#x.802;D:Notethatisagivenerrorthreshold.With,theapproximationerrorofEMRissubjectto 1+; WeproceedtoboundtheapproximationerrorofEMRineachsub-areaface,aswellasthenumberofsub-areafaces.betheapproximatedEMRofanypositionintheface,namely,)==1qi,whereqiisaconstantthatdenotestheapproximatedEMRstemmingfromthechargerintheface.Then,wehavethefollowingTheapproximationerrorofanypositionintheface 1+;Proof:Clearly,wehave =1qi =1;p)).Com-biningEq.6,theresultfollows. Thenumberofsub-areafacesProof:ByDef.4.1,wecanderivethatthenumberofdividedsub-areasisgivenbyln(0)=e)) ln(1+,which.Besides,itisclearthatthenumberofallconcentriccirclesisNK.Basedontheclassicalresultsof[22],thenumberofsub-areafacesformedbyNKNKNK+2.Theresultfollows. B.ProblemReformulationAftertheapproximationproceduresintroducedabove,wereducethenumberofconstraintsinP1fromin®niteto®nite.Asaresult,SCAPEisreformulatedasmax=1=1;o)))=1qi=1;:::;Z=1;:::;nThisisatypicallinearprogrammingproblem.Mathemati-cally,itcanbesolvedbyintegerprogrammingsolverssuchasCPLEX[23].However,thenumberofthefaceswillexplodewithalargenetworksizeandasmallvalueofTotacklethischallenge,itisdesirabletodevelopdistributedalgorithmstoaddresstheproblem.Beforegoingintothedetailsofdistributedalgorithms,weneedtoimplementtheareadiscretizationmethodinadistributedmanner.C.DistributedImplementationofAreaDiscretizationRecallthatchargersareassumedtoknowitslocationapriori.Eachchargercanexchangethelocationinformationwithitsneighbors,andtherebydiscretizeitscovereddiskareaaccordingly.Afterthat,eachchargerwillobtainalistoflinearconstraints,andneighboringchargersmustshareatleastoneconstraintastheyhaveintersectionsub-areas.V.DISTRIBUTEDEDUNDANTONSTRAINTEDUCTIONInthissection,weinvestigatehowtoidentifyandremoveredundantlinearconstraintsinadistributedwaytoreducethecomputationaleffort.Inparticular,weproposeatwo-stepalgorithmtodealwiththisproblem.Wenotethatthis 1 F 2 F 121:252:253:454:455:425xx d d 3 F 4 F 5 F 6 F 7 F 8 F 9 F 121212126:2457:2258:2259:445xxxxxxxxdddd Fig.3.Illustrationofanexampleofconstraintreductionalgorithmisonlyperformedonceafterthedeploymentofchargers,anditdoesn'tneedtheknowledgeofpositionorchargingpowerofdevices.Therefore,itscomputationaltimeisamortizedovertimeandthusisnegligible.Byredundantconstraints,wemeanthoseconstraintswhosepresencedoesnotaltertheoptimumsolution.Wereferthereaderto[24]fortheformalde®nitionofredundantcon-straints.Astheexistingalgorithmsforredundantconstraintreduction,suchas[25]and[26],arecentralizedandthuscannotbedirectlyappliedtoourscenarios,wedevelopourowndistributedalgorithminspiredby[25][26].Inoursettings,eachchargermaintainsalistoflinearconstraintswhichcorrespondstothesub-areafacesitcov-ers,andneighboringchargersmustshareatleastonelinearconstraint.Ingeneral,ourgoalistocollaborativelyreducetheredundantconstraintsamongchargerssuchthatthenumberoftheaggregatedlinearconstraints(whichmergestheidenticallinearconstraintsbetweenneighbors)isminimized.Duetospacelimit,wejustsketchourDistributedRedun-dantConstraintReduction(DRR)algorithm.Basically,thedistributedalgorithmconsistsoftwostages.The®rststageistolocallyremovetrivialconstraintswhichcanbealwayssatis®ed.Then,thechargerselectivelycollectsneighboringlinearconstraintsfromneighborswhichdon'tinvolveitself(achargerissaidtobeinvolvedinalinearconstraintifandonlyifthecoef®cientofinthisconstraintispositive),andemployslinearprogrammingmethodtofurtherremoveitsredundantconstraints.Inparticular,toidentifytheredundantconstraints,theleft-handsideofeachconstraintisoptimizedsubjecttotheremainingconstraints.Theoptimalvalueisthencomparedwiththeright-handsidevalueofthecorrespondingconstrainttodecidewhetheritisredundantornot.TakingFig.3asanexample.Thewholeareaispartitionedsub-areafacesafterareadiscretizationfortwochargers.SupposetheEMRthreshold=5,theapprox-imatedchargingpowerininnerandoutersub-areasisrespectively.Wethusobtain9linearconstraintsaslistedinFig.3.Atthe®rststageofDRR,theconstraintareidenti®edastrivialconstraintsandremovedatbothchargersastheycanalwaysbesatis®ed.Next,bothchargersbroadcasttheirconstraintstoeachother.Thecharger,forexample,®ndsthatallthereceivedconstraintsinvolveitself,andthereforeitneglectsallofthem.Next,fortherestconstraints,suchastheconstraint,it+2,theleftsideofthisconstraint,subjecttotheconstraints.Theoptimalvalueis,whichissmallerthan.Therefore,theconstraintcanberemoved.Sodoestheconstraint.Finally,onlytheconstraint 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1000 2000 3000 4000 5000 Threshold Aggregate Number of Constraints Original Reduced (centralized) Reduced (distributed) (a)ConstraintNumbervs. 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 5 10 15 20 25 30 35 40 Threshold Indivisual Number of Constraints Original Reduced (distributed) (b)ConstraintNumbervs. 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0 200 400 600 800 1000 1200 1400 1600 1800 EMR Threshold R Aggregate Number of Constraints Original Reduced (centralized) Reduced (distributed) (c)ConstraintNumbervs.Fig.4.IllustrationofRedundantConstraintReduction 2 D 2 D  Fig.5.IllustrationofthecoverageWedisclosetheperformanceofthealgorithminthefol-lowingtheorembycomparingittoitscentralizedversion,whichconductsthetwo-stepschemebasedonthecompleteknowledgeofallconstraintscomingfromallchargers.5.1:TheDRRalgorithmachievesthesameper-formanceasthecorrespondingcentralizedversioninthattheaggregationofitsobtainedlinearconstraintsisidenticaltothatofthecentralizedalgorithm.Proof:Weomitthedetailsoftheprooftosavespace. Fig.4showstheperformanceofourDRRalgorithmintermsofbyadoptingthesameparametersettingasthatinSectionVII-C1.InFig.4(a),theoriginalaggregatednumberofconstraintsgenerallyincreaseswithadecreasingandrisesto4629when=0.Incontrast,thenumberofreducedconstraintsofDRRalgorithmissubstantiallysmallerthantheoriginalnumberofconstraints,e.g.,itisonly348=0,nearlyoftheoriginalone.Moreover,itisexactlyequaltothatproducedbythecorrespondingcentralizedalgorithm.Further,Fig.4(b)showsthattheaveragenumberofconstraintspossessedbyindividualchargersisnearlyproportionaltotheaggregatednumberofconstraints.Besides,Fig.4(c)showsthatthenumberofreducedcon-straintsofDRRalgorithmorcentralizedalgorithmdecreasessteadilywhenbecomeslarger,anddropsto0when=0isevenbiggerthanthemaximumpossibleEMRovertheplane.Incontrast,theoriginalnumberofconstraintsremainsunchanged.VI.(1-APPROXIMATIONISTRIBUTEDLGORITHMInthissection,wediscusshowtodevelopa(1approximationalgorithm.Tobeginwith,wedividethewholeareaintouniformsquareswithsize,whereisthediskradiusofchargers'coveragearea.Sinceeachchargerisawareofitslocation,itcaneasilyclassifyitselfintoaspeci®cregularsquaregivenaglobalreferenceanchorpoint.Apparently,byapplyingthispartitionmethod,thechargersinnon-adjacentsquareswillnothavetheircoverageareasintersected.Byºadjacentº,wemeantwosquaresshareatleastoneendpoint.Fig.5showsanexamplewherefoursquaresareadjacenttoeachother.AsshowninFig.6,wegroupsquaresintoalargergrid,whichwecallgridforshort.Forsimplicity,supposethatthewholeareacanbedividedintoanintegralnumberofgrid(ifitisnotthecase,wecanaddphantomsquaresto mDmD 2D Fig.6.Illustrationofoverallpartitionachievethisgoal).InFig.6,thereare6gridsenclosedbybluedottedboundariesaftergrouping.TodecomposeP2intominoronesandsolvetheminadistributedmanner,wecanselectivelyturnoffsomechargerssuchthattheentireareacanbeseparatedintoseveralsub-areas.Fig.6showsanexamplethatthechargerslocatedinthosewhitestripsareswitchedoff,thenthewholeareaisre-partitionedinto12sub-areas,eachofwhichcontainsatmostsquares.Speci®cally,werequirethateachgridthesameselect-and-turn-offpolicy,namely,turningoffallthechargerslocatedatthe-throwandthe-thcolumnofthegrid.Weuseatwo-tuplei;jtodenotesuchpolicy.Fig.7demonstratestwodifferentpolicies,.Bythepolicy,forexample,allthechargersatthe-throwand-thcolumnshouldbeturnedoff.Fig.6illustratestheultimateresultwheneachgridadoptsthepolicyConsequently,ineachpartitionedsub-area,suchasthose12sub-areasinFig.6,wecanapplyalocallinearpro-grammingmethodtodeterminethepowersofthechargersinsideindependentlyofothersub-areas.Thisisbecausethenearestdistancebetweenanysub-areasisatleast,whichissuf®cienttoavoidthein¯uenceofEMRfromchargersinothersub-areas.Ifnoconfusionarises,wecallthesenewlyformedsub-areasasnewgridsforsimplicity.Sofar,ourproblemhasbeenboileddowntotwosub-problems,namely,howtodeterminethesizeofangrid)andhowtodeterminetheselect-and-turn-offpolicyadoptedineachgrid.Ingeneral,forthe®rstsubproblem,wewillprovethatonlyrelatestotheerrorthresholdandhasnothingtodowiththepresentdevicedistribution.Incontrast,thesecondsubproblemshouldbeaddressedbasedontheknowledgeofdevicelocationsandtheirchargingutility. Algorithm1TheNearOptimalAlgorithmatSinkNode TheerrorthresholdTheselect-and-turn-offpolicy.InitializationPhase1:2(2+ =2) =,broadcastittoallchargerstobuildgrids.WorkingPhase1:Requireallsquareheadsofsquarestocomputeandreportitslocalchargingutility,thencollectallthechargingutilityfromallsquares.2:Findtheselect-and-turn-offpolicywiththeleastoverallperfor-manceloss,andsendthepolicytoallsquareheads. mDmD ˜ 2D 4,4¢²6,6 ¢ ² Fig.7.IllustrationofgridInparticular,weestablishasinknodetocollectthechargingutilityofallsquares,®ndtheselect-and-turn-offpolicywiththeleastoverallperformanceloss,anddisseminatethe®nalsolutiontoallsquares.Byintelligentlychoosingthepolicywiththeleastperformanceloss,wecanachieveafactorof(1oftheoptimum.Wepresentthedetailsofthenearoptimalalgorithmsperformedatthesinknodeandatthesquareheads,whichservesasanagentbetweenthesinknodeandthedevicesintheirsquare,inAlg.1andAlg.2,respectively.Ingeneral,intheinitializationphase,thesinknodedecidesthesizeofgrids,namely2(2+ =2) =,andbroadcaststhemtoallchargers,whichconstructgridsaccordingly.Wewillprovethatsuchavalueofcontributestoachieving(1approximationratio.Notethatlinearconstraintextractionandredundantconstraintreduction,togetherwithsquarepartition,havealreadybeendonebeforethisprocedureatsquareheadsinallsquares.Next,theworkingphasemainlyincludestwostages.Atthe®rststage,thesinknodecollectsallnecessaryinformationfromallsquareheads,andtherebydeterminestheselect-and-turn-offpolicywiththeleastoverallperformancelossandsendsitout.Forthesecondstage,afterreceivingthepolicy,eachsquareheadreassignsitselftoanewgrid.Andineachnewgrid,aheadiselectedtofacilitatethelocalcomputationofthelinearprogrammingproblem.Notethatthedistributedalgorithmconductedinchargersthatarenotsquarenodesissimplytoreportinformationtosquareheads,receivecommandandadjustpoweraccordingly.Theorem6.1revealstheperformanceofthedistributed6.1:Thedistributedalgorithmachieves(1approximationratio. Algorithm2TheNearOptimalAlgorithmatSquareHeads Thelocation,themaximumreceivedpowersofitscovereddevices.Theobjectivepower.InitializationPhase1:Useareadiscretizationtechniquewitherrorthreshold=toextractlinearconstraints,thenapplyDRRalgorithmtoreduceredundantconstraints.2:Classifyitselfintoaspeci®csquare,andelectalocalsquare3:Receivetheparameterfromthesinknode,andclassifyitselfintoagridWorkingPhase1:Whenreceivingtherequestfromthesinknodetoreportlocalchargingutility,disseminatetherequesttootherchargersinthissquare,andcollectalltheinformationfromthem.2:Computethelocaloptimalsolutioninthissquare,sendittothesinknode.3:Receivethecommandfromthesinknode.Ifthecommandistoturnoffthechargersinthissquare,disseminatethecommandtoeachchargerinthissquare,turnitselfoff.Otherwise,assignitselftoanewgridaccordingtothereceivedcommand,andelectalocalheadofthegrid4:ifItisthegrid5:Disseminatetherequestforinformationtoothersquareheadsinthisgrid,andcollectalltheinformationfromthem.6:Computethelocaloptimalsolutioninthisgrid,sendittosquareheadsinthisgrid.Adjustitspoweraccordingtotheoptimalsolution.7:8:Whenreceivingtherequestfromthegridheadtoreporttherelatedinformation,sendittothegrid9:Receivethecommandfromthegridhead,disseminateittoallchargersinthesquare,andadjustitspoweraccordingtothecommand.endif Proof:WeprovethistheoremintheAppendixforabetter¯owofthepaper. Weremarkthat,theelectionofgridheadsatStep3inAlg.2canbedoneintheinitializationphasetosavetherealtimecomputationalcostandreducedelay.Thiscallsforenumerationforallpossiblenumberofformationofnewgridsinadvance,andthespacetostorecomputedresults.Moreover,Alg.2canbeadaptedtobefullydistributedbyremovingthenotionofsinknode,squareheadandgridhead,andlettingchargersexchangeinformationwiththeirneighborsroundbyrounduntilalltherequiredinformationiscollected,andthencomputingtheoptimalsolution.Suchanadaptedalgorithmis,apparently,morerobust,butismorecomputationallydemanding.Supposethechargerdeploymentdensityinthe®eldisbounded,thenfortypicalchargernetworkswehavethefollowingtheorem.Notethatisthenumberofchargers.6.2:Thenumberofexecutionroundsofthenearoptimalalgorithmis(logVII.SIMULATIONESULTSInthissection,we®rstpresentsimulationresultstoevaluateourproposedalgorithmsintermsoferrorthreshold,chargernumber,EMRthresholdanddelay.Afterthat,wegive 0.2 0.3 0.4 0.5 0.6 0.7 0.8 5 10 15 20 25 30 35 40 Threshold Overal Utility Optimal SolutionNear Optimal Algorithm6HW&RYHU$OJRULWKPLightweight Algorithm Fig.8.OverallUtilityvs.Threshold 20 40 60 80 100 120 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Number of Chargers Overal Utility Optimal SolutionNear Optimal AlgorithmSetCover Algorithm/LJKWZHLJKW$OJRULWKP Fig.9.OverallUtilityvs.Charger 0.01 0.015 0.02 0.025 0.03 0.035 0.04 5 10 15 20 25 30 35 40 EMR Threshold R Overal Utility Optimal SolutionNear Optimal Algorithm6HW&RYHU$OJRULWKPLightweight Algorithm Fig.10.OverallUtilityvs.EMR 10 10 0 5 10 15 20 25 30 35 40 Network Size Delay Near Optimal Algorithm SetCover Algorithm Lightweight Algorithm Fig.11.Delayvs.NetworkSizeinsightsofhowthedistributionofchargersaffectstheoverallchargingutility.A.EvaluationSetupUnlessotherwisespeci®ed,weusethefollowingevaluationsettings.Assumetherearechargersand10devicesuniformlydistributedovera2Dsquarearea.Moreover,weset=100=100=20forthechargingmodel,=1fortheEMRmodeland=0fortheutilitymodel.Theerrorthresholdis=0,andtheEMRthresholdis=0.Moreover,everypointonthesimulationcurvesstandsfortheaveragevalueof100instanceswithdifferentrandomseedsanddevicedeployments.B.BaselineSetupAscurrentlythereisnoalgorithmavailableforsafecharg-ingwithadjustablepower,weintroducetwoalgorithmsforcomparison.The®rstalgorithmborrowstheideaofSet-Cover.Aftercollectingtheinformationfromallchargers,thisalgorithmgreedilypicksthechargerwhichcoversthemaximumnumberofdevices,andadjustsitspoweraslargeaspossiblewhileassuringtheEMRsafety,untilnofurtheradditionofchargersispossibleconsideringtheEMRsafety.Then,itbroadcaststheresultstoallchargers.Thesecondalgorithmisalightweightalgorithm.Inthisalgorithm,eachsquareheadsimplygatherstheinformationlocallyinthesquare,computestheoptimalsolution,andthencutsdownthepowerofeachchargerinthesolutionto.TheobtainedsolutionmustsatisfytheEMRsafetyrequirementbysimilaranalysisintheprooftoTheorem6.1.Inaddition,weapproximatetheoptimalsolutionbyparti-tioningtheareaina®ne-grainedwayandsolvingtheobtainedlinearprogrammingprobleminacentralizedway.C.PerformanceComparison1)ImpactofThresholdWe®rstinvestigatethein¯uenceoftheerrorthresholdontheoverallutility.AsdepictedinFig.8,theoverallutilityoftheoptimalsolutionremainsunchangedandisequalto35asgrows,whilethatoftheotherthreealgorithmsdecreasessteadily.Speci®cally,ournearoptimalsolutionisalwayslargerthan(1oftheoptimalvalue,andoutperformsthatoftheSet-Coveralgorithmby4%onaverage.Suchsmallperformancegapbetweenthesetwoalgorithmsisduetothefactthatthechargerdistributioninthiscaseisrelativelysparseandmostchargersareisolatedfromothers,andtherefore,theoutputsofthetwoalgorithmsareverysimilar.ThelightweightalgorithmperformstheworstbecauseofthehugepotentialutilitylosswhenitcutsdownthecomputedpowertoFurther,weobservethattheachievedutilityofthenearoptimalsolutionexperiencesadropwhen=0.Thisisbecausethespeci®edsizeofgridisbiggerthanthenetworksizewhen,andthusthereisnoneedforgridpartition,andnofurtherperformancelosswilloccur.Thecaseistheoppositefor2)ImpactofChargerNumber:Weproceedtoexaminethein¯uenceofthechargernumberontheoverallutility.Toreducethecomputationaltime,wereducethe®eldsizetoandthenumberofdevicesto,and=0.InFig.9,itcanbeseenthattheperformancegapbetweentheoptimalalgorithmandthenearoptimalsolution,aswellastheSet-Coveralgorithm,isprettysmallwhenthenumberofchargersissmall.Thisisbecausewithasparsedistribution,mostofthechargerscanbesettoitsmaximumpowerwithouthurttheEMRsafetyforallthesethreealgorithms.Nevertheless,thegapsbetweenthethreealgorithmsbecomeconspicuouswithmorechargersdeployed.Thenearoptimalsolutionhasanperformancegainofupto230%overtheSet-Coveralgorithm,anditsperformancelosscomparedwiththeoptimalsolutionisnomorethan135%muchlessthanallowedbyTheorem6.1.Again,thelightweightalgorithmhastheworstperformance.3)ImpactofEMRThresholdWearealsointerestedintheimpactoftheEMRthresholdontheoverallutility.AsillustratedinFig.10,notsurprisingly,theoverallutilityofallalgorithmsgrowswithanincreasing.Notethatwe=0.ThenearoptimalalgorithmalsooutperformstheSet-Coveralgorithm.Overall,theoptimalsolutionisnearly7%higherthanthatofournearoptimalalgorithm,whichinturnenjoysaperformancegainofupto2%overtheSet-Coveralgorithm.Furthermore,whenexceedsallthechargercanbetunedtoitsmaximumpowerwhileguaranteeingtheEMRsafety,andtherebytheoverallutilityoftheoptimal,nearoptimalandSet-Coveralgorithmsreachthemaximumvalue38.Similarly,thelightweightalgorithmkeepstheutilityvalueofexceeds4)ImpactofNetworkSizeonDelay:Westudytheimpactofnetworksizeontheoveralldelayinthispart.We®xthedensityofchargerstobe,set=0,andletthecommunicationradiusbetwicethechargingradius,i.e.,40.InFig.11,wecanseethatthedelayofthelightweightalgorithm 0 100 200 0 50 100 150 200 x (m) y (m)(1) 2D Gaussian Distribution 0 100 200 0 50 100 150 200 x (m) y (m)(2) 1D Gaussian Distribution (a) 0.05 0.1 0.15 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 EMR Threshold R Overal Utility Uniform Distribution1D Gaussian Distribution'*DXVVLDQ'LVWULEXWLRQ Fig.12.Illustrationofnon-uniformdistributionsandresultskeepsconstant,sinceitonlyrequireslocalcommunicationwithinasquare.Incontrast,thedelayoftheSet-Coveralgorithmincreasesnearlyproportionaltothelogarithmofthenetworksize,whichismainlyduetothealgorithm'sdatacollectionandsolutiondisseminationprocesses.Forthenearoptimalalgorithm,theslopeofthecurveistwicethatoftheSet-Coveralgorithmwhenthenumberofchargersissmallerthan,andisthesameotherwise.Thisisbecausethesizeofgridsisprecisely.Therefore,ifthenumberofchargersexceeds,sincethenumberofrequiredcomputationstepsinagridisconstant,theincrementofdelayonlydependsonthe®rststageofdatacollectionanddisseminationprocess.D.InsightsInthissection,weprovideinsightsofhowthedistributionofchargersaffectstheoverallchargingutility.Supposethereare50chargersdeployedina®eld,anddevicesuniformlydistributedinthesame®eld.Weconsiderthreekindsofdistributions,i.e.,uniformdistribution,2DGaussiandistributionand1DGaussiandistribution.By2DGaussiandistribution,bothofthex-coordinateandy-coordinateofachargerarerandomlyselectedfromaGaussiandistribution=100=100.Comparedwith2DGaussiandistribution,theonlydifferenceof1DGaussiandistributionisthatitsx-coordinateisuniformlyselectedamong[0200].Fig.12(a)illustratesexamplesofthem.AsshowninFig.12(b),alltheutilityofthreedistributions®rstincreasesrapidlyandthensmoothly,and®nallybecomesconstantsinceallthechargersareadjustedtotheirmaximumpower.Aninterestingobservationisthattheutilityachievedfortheuniformdistributionis®rstsuperiortothatcorrespond-ingtotheothertwodistributions,andsoonbecomesinferiortothem.Thisisbecausewhenissmall,thechargerswithuniformdistributionhavelargeraveragedistancebetweeneachother,andthusareallowedtotunetheirpowertoagreatervaluewhileassuringEMRsafety,whichresultsinahigherutility.Incontrast,withalarge,thechargersareabletoadoptalargerpower,eventhemaximumpower,withoutviolatingtheEMRsafety,andtherefore,thenumberofcovereddevicesbecomesthedominantfactoraffectingtheoverallutility.Notethat,thereareanumberofchargerslocatedneartheboundariesofthe®eldintheuniformdistributioncase,whosechargingareainevitablycoverstheregionoutsidethe®eld.Thiscoveragewastethusleadstoalossoftheoverallutility.Incontrast,thechargersfortheothertwodistributionsaremoreconcentrated,andthusresultsinasmallercoveragewasteaswellasalargeroverallutility.Furthermore,forthesamereasons,theperformanceforthe1DGaussiandistributioncaseisbelowthatofthe2DGaussiandistributiongivenalargeVIII.FIELDXPERIMENTSInthissection,webuildtestbedandconduct®eldexperi-mentstoverifyourtheoretical®ndings.A.TestbedGenerally,weusethesametestbedasthatin[19].Thetestbedconsistsof8TX91501powertransmittersproducedbyPowercast[3][27],and2rechargeablesensornodes.Weplacethechargersontheverticesandmiddlepointsofedgesofasquarearea,andonewirelessrechargeablesensornodeatthecenterofthesquarearea,andtheothertotherightsideofthe®rstonewithdistance,asillustratedinFig.13.Alsoshowninthis®gurearealaptopandanAPconnectingtoit.TheAPisresponsibletocollecttheinformationofreceivedpowerfromthesensornodes,andthensendittothelaptop.NotethatTX91501powertransmittersareactuallydirec-tional,whosechargingregioncanbemodeledasasectorwith60andradius4.Wetunetheorientationsofcharger1to8as56564426564463445656,respectively.Apartfromthis,asthepowerofTX91501powertransmittersisnotadjustable,weputapieceofcopperfoiltapewithproperlengthandwidthinfrontofchargers,andadjustitspositionandbendinganglesuchthatthereceivedpowerandEMRatlocationsfurtherthanthetapearenearlyuniformlycutdowntoadesiredlevel.B.AdaptedAlgorithmandComparisonAlgorithmInconsiderationofthechargingfeaturesoftheTX91501powertransmitters,wemakenecessarymodi®cationstoouralgorithmstoaccommodatethiscase.Weomitthedetailstosavespace.Second,weletthetwosensorssamplethechargingpowerfromeachchargeratthebeginningofthealgorithmtoreducethemodelerrorandmakeouralgorithmrobusttoenvironmentalvariation.Wethenperformouralgorithmsbasedonthesampledvalues.WecompareouralgorithmtotheSCPalgorithm,acen-tralizedalgorithmproposedin[19]thataddressesthesimilarproblem,exceptthatthepowerofchargersisassumedtobenotadjustable.C.ExperimentalResultsAsshowninFig.14,wecomparetheutilitycomputedbasedonsamplingvaluewithrealutilityunderthreedifferentvaluesofforbothSCAPEandSCPalgorithms.Notethatthechargingutilityisthesummationofthatofthetwosensornodes.Itcanbeobservedthatthecomputedutility Fig.13.Illustrationof®eldexperiment 105 115 125 0 20 40 60 80 100 EMR Threshold RUtility Computed Utility of SC3Real Utility of SC3Computed Utility of SCReal Utility of SC Fig.14.Utilityvs.EMRThreshold 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 x (m) y (m) 30 40 50 60 70 80 90 100 110 120 Fig.15.AnexampleofEMRdistributionofbothalgorithmsisalwayslargerthantherealutility,buttheperformancegapisnomorethan7%.Besides,whendecreasesfromto,theachievedrealutilityoftheSCPalgorithmdropsby339%,whilethatoftheSCAPEalgorithmreducesonly0%.Onaverage,theSCAPEalgorithmis411%betterthantheSCPalgorithm.WesetasW=cm,computetheadjustingfactorsforthecharger1to8,i.e.,60727875andputpiecesofcopperfoiltapeinfrontofthecharger2,4,5and7correspondingly.WemeasuretheEMRvaluesatgridpointsofthesquareregionexceptthelocationsofcharger2,4,5and7sincethepowerthereisnotproperlyadjusted.WeplottheresultsinFig.15,andobservethattheEMRpeaksatthelocationofcharger1andisequaltoW=cmwhichislessthan.ThisfactvalidatesthecorrectnessofourSCAPEalgorithm.Notethat,thoughtheEMRvalueatthelocationsofthecharger2,4,5ismissed,itcanbededucedthattheywon'texceedtoobythesurroundingEMRvalues.IX.CONCLUSIONInthispaper,wehavestudiedtheproblemofhowtoadjustthepowerofchargerstomaximizetheoverallchargingutilitywhileguaranteeingtheEMRsafety.Weemployednoveltechniquestoreformulatetheproblemandfurtherreduceitscomputationalefforts.Wethenpresentedadistributedalgorithmwithapproximationratio(1toaddresstheproblem.Finally,weconductedbothextensivesimulationsand®eldexperimentstovalidateourtheoretical®ndings.Duetoitsdistributednatureandconsiderationofcompu-tationaleffortthroughoutthedesignprocess,ourproposedschemecouldbeeasilyincorporatedintorealsystems.Inourfuturework,wewilltakeotherpracticalconcernsintoconsideration,suchasfairnessofcharging.CKNOWLEDGMENTThisworkissupportedbyNational973projectofChinaunderGrantsNo.2012CB316201andNo.2014CB340303,Na-tionalNaturalScienceFoundationofChinaunderGrantsNo.61133006,No.61321491,No.61373130,No.61170247andNo.613230428,NationalKeyProjectofChinaunderGrantNo.2013ZX01033002-003,ResearchandInnovationProjectforCollegeGraduateStudentsofJiangsuProvincein2012underGrantNo.CXZZ12 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PPENDIXA.ProofofTheorem6.1Proof:Denotebyijthesquarelocatedatthe-throw-thcolumnofthegrid.AtStep2ofWorkingPhaseinAlg.2,eachsquareijcomputestheoptimallocalchargingutility,whichwedenotebyij.LetbetheoptimaloverallchargingutilityfortheproblemP2,itiseasytoseeijObviously,therearedifferentselect-and-turn-offpoli-ciesforallgrids.Foragivenpolicyi;j,theoverallperformancelossisijijTherefore,theoverallperformancelossofthenetworkisijijijNext,weinvestigatethesummationoftheperformancelossforallpossibleselect-and-turn-offstrategies,whichcanbeformallyexpressedasijij))=(21)ijSubsequently,weintendtodisclosetherelationshipbetweenij.We®rstpresentthefollowingsimpleyetimportantobservation.A.1:Anypointinthe®eldcanonlybecoveredbychargersfromatmost4squares.Fig.5illustratesanexampleofthisobservation.Whenapoint,markedinayellowstarinthe®gure,locatedintheup-leftcornerofasquare,itcanonlybecoveredbythechargersfromthissquare,alongwiththosefromtheadjacentthreesquaresasdepicted.BasedonObservationA.1,wesupposetheEMRatapointconsistsoffourparts,i.e.,fromchargersin4adjacentsquares(=0ifthereisnoEMRfromthatsquare).Obviously,wehave.Asaresult,ifwereducethepowerofeachchargerinthe®eldto,thecorrespondingaggregateEMRat)=1,whichmeansthereducedpowerisafeasiblesolution.Consequently,theoverallchargingutilitybecomesijpowerreduction.Sinceisoptimal,wehaveijCombiningEq.13andEq.14givesij4(21)Finally,accordingtothepigeonholeprinciple,weclaimthattheremustexistapolicyi;jthathastheperformancelossij4(21) Inpractice,wecancomputeallthepossiblevalueofijandpickouttheleastone,thenitmustconformtoInequality16.ThisisdoneatStep2ofWorkingPhaseinAlg.1.Afterobtainingtheleastijandtherebydeterminingthepolicyi;j,thesinksendsthepolicytothewholenetwork.ChargersactingassquareheadsreceivedthecommandeitherturnsoffitselfasshownatStep3ofWorkingPhaseinAlg.2forsquareheads,orparticipatesinelectinggridandperformingalocallinearprogrammingalgorithmwithinanewgrid.Supposethesummationoftheobtainedchargingutilityofallnewlyformedgridsis.ItisobviousthatijCombiningEq.16andEq.17,wehave(14(21) Givenanarbitrarilysmallvalueof,weset2(2+ =2) =asatStep1ofInitializationPhaseinAlg.1,wethenhave(1=2)Further,letbetheoptimaloverallchargingutilityoftheoriginalproblemP1.Supposethattheadjustingfactorforthechargerisintheoptimalsolution.Wethenhave=1;p))foranypointSupposethepointisinsidetheface.ByEq.7,theapproximatedEMRatafterareadisretizationsatis®es)==1qi(1+=2)=1;p))Notethatwesettheerrorthresholdforareadiscretizationto=.Itimmediatelyfollows=1qi (1+=2)=1;p))whichimplies (1+isafeasiblesolutionfortheproblemP2istheoptimalsolutionforP2,wethushave=1=1;o))) (1+=2) (1+=2)=1=1;o))) (1+=2)(1=2)CombiningEqs.20and21,wehave(1=2)(1=2)(1Thiscompletestheproof.