A-simplified-model-for-determining-capacity-usage-

JournalofPowerSources183(2008)376–380ContentslistsavailableatScienceDirectJournalofPowerSourcesjournalhomepage:fiedmodelfordeterminingcapacityusageandbatterysizeforhybridandplug-inhybridelectricvehiclesPaulAlbertus∗,JohnNewmanDepartmentofChemicalEngineering,UniversityofCalifornia,Berkeley,CA94720,UnitedStatesarticleinfoArticlehistory:Received18March2008Receivedinrevisedform1May2008Accepted1May2008Availableonline7May2008Keywords:HybridelectricvehiclePlug-inhybridelectricvehicleLithium-ionbatteryModelingBatterysizeCapacityusageabstractWedevelopasimplifiedmodeltoexaminetheeffectoftheshapeandmagnitudeofthebatterypulse-powercapabilityoncapacityusageandbatterysize.Thesimplifiedmodelexpressesthecapacityusageandadimensionlessbatteryareaintermsofadimensionlessenergy-to-powerratioandaparameterthatcharacterizestheshapeofthepulse-powercapability.Wealsopresentdimensionalresultsthatshowhowthecapacityusagedependsontheequivalent-electricrangeandseparatorarea,andhowthebatteryareadependsontheequivalent-electricrange.KeyresultsincludethepresenceofaLangmuir-likerelationshipbetweenthecapacityusageandthedimensionlessenergy-to-powerratio,andalinearrelationshipbetweenthedimensionlessenergy-to-powerratioandadimensionlessarea,withaslopeandoffsetthatdependontheshapeofthepulse-powercapability.Wealsofoundthataflatpulse-powercapabilitycurveincreasescapacityusageanddecreasesbatterysize,andthattwoimportantparametersforbatterydesignare(U−Vmin)Vmin/R,whichreflectsthemaximumpowercapability,andQV,whichreflectsthebatteryenergy.Theresultsandanalysiscontainedhereinareusedtohelpinterprettheresultsfromacombinedbatteryandvehiclemodel,presentedinacompanionpaper.©2008ElsevierB.V.Allrightsreserved.1.IntroductionIncreasingattentionisbeinggiventoenergyuseinthetrans-portationsector,andsomeofthisattentionisfocusedonthereductionsinfuelconsumptionthatcanresultfromhybridelec-tricvehicles(HEVs)andplug-inhybridelectricvehicles(PHEVs).HEVscanvaryaccordingtothedegreeofhybridization,butofferimprovementsinefficiencyofuptoapproximately50%[1].AnothermajorjustificationforthedevelopmentofPHEVsistheabilitytoaccomplishfuelswitching,forexamplereplacingoilwithelectric-ity(derivedfromnaturalgas,coal,orrenewables).Anultimategoalistoelectrifythetransportationindustryandderivetheelectricityfromrenewablesources.Thiswouldsimultaneouslyaddresstheissuesof“energysecurity”andclimatechange.TwoofthemainchallengesthathinderthegrowthoftheHEVandPHEVtechnolo-giesarecostandthereducedcargospacethatresultsfromtheinclusionofalargebatteryinavehicle.Itshouldalsobenotedthatcostandsizearelinked;ingeneralalargerbatterycostsmore.Thus,thereissignificantpressuretouseabatterythat,whilemeet-ingperformancerequirements,isassmallaspossible.Onewayto∗Correspondingauthor.E-mailaddresses:[e-mail](P.Albertus),[e-mail](J.Newman).achieveasmallerbatteryistofindawaytoincreasethecapac-ityusage[2].Thiscouldbeachievedwiththeuseofbettercontrolalgorithms,orwithachemistrythatprovidesarelativelyflatpulse-powercapability[3].Herewepresentasimplifiedmodelforcapacityusageandbat-terysizeinaHEVandPHEV.Thegoalsaretoexaminetheunderlyingrelationshipsbetweenkeysystemvariablesinordertoimproveunderstanding,andassistwiththeinterpretationoftheresultsfromacompanionpaperthattreatsaspecificsetofcellchemistrieswithadetailedbatterymodelcombinedwithasimplevehiclemodel[4].Thispaperisdividedintothreesections.Thefirstsectiondis-cussesmaximumpowercapability,thesecondsectiondevelopsasimplifiedmodelforPHEVsandexpressesresultsintermsofadimensionlessenergy-to-powerratio,andthethirdsectionappliesthesimplifiedmodeltopracticalquestionsencounteredbybatterydesigners.2.DefiningthemaximumpowerAssumingalinearsystem(oneforwhichtheresistanceisinde-pendentofappliedcurrent),themaximumpowerperunitareaofabatteryisgivenbyPmax,dischargeA=U24R,(1)0378-7753/$–seefrontmatter©2008ElsevierB.V.Allrightsreserved.doi:10.1016/j.jpowsour.2008.05.004P.Albertus,J.Newman/JournalofPowerSources183(2008)376–380377NomenclatureListofsymbolsAtotalseparatorareaofabattery(m2)Bslopeofthepulse-powercapabilitycurvec1constantcalculatedfromthepositionofVminc2constantcalculatedfromthepositionofVmaxEenergyrequiredatthewheelsduringadrivingcycle(J)Ebatteryenergyforavehicletotravelagivendistance(Jm−1)icurrentdensity(Am−2)Lequivalent-electricdrivingdistance(m)Pmaxmaximumpowerrequiredfromabatteryforagivenconfiguration(W)(P/A)0maximumspecificpulse-powercapabilityforagivenchemistryandcelldesign(Wm−2)Qspecificcoulombiccapacity(Cm−2)Rcellresistance(ohmm2)SODstate-of-dischargeUequilibriumpotential(V)Vinstantaneouscellpotential(V)Vaveragecellpotential(V)Vmaxuppercutoffpotential(V)Vminlowercutoffpotential(V)whereRistheresistanceandUistheequilibriumpotential.ThemaximumpoweroccursatacurrentofU/2R,atwhichpointthecellpotential,V,isequaltoU/2.Thereisanadditionalconstraintonpracticalsystems,namelytheneedtostayabovethelowercut-offpotential,Vmin,andbelowtheuppercutoffpotential,Vmax.IfthemaximumdischargepowergivenbyEq.(1)occursatap