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Hindawi Publishing CorporationAdvances in Mechanical EngineeringVolume 2013,Article ID 154831,7 pageshttp//dx.doi.org/10.1155/2013/154831ResearchArticleA Method of Remaining Capacity Estimation forLithium-Ion BatteryJunfu Li, Lixin Wang, Chao Lyu, Weilin Luo, Kehua Ma, and Liqiang ZhangSchoolof Electrical Engineeringand Automation, Harbin Institute of Technology,Harbin 150001,ChinaCorrespondence should be addressedto Lixin Wang; wlxhit.edu.cnReceived8September 2013;Revised22 October 2013;Accepted 22October 2013Academic Editor Xiaosong HuCopyright 2013Junfu Li et al. This is an open accessarticle distributed under the Creative Commons Attribution License,whichpermits unrestricted use,distribution, and reproduction in any medium, provided the original work is properly cited.Combining particle filter PF with sample entropy feature of discharge voltage, a method of remaining capacity estimation forlithium-ion battery isproposed. The sampleentropy calculatedfrom dischargevoltage curve can serveasan indicator for assessingthe condition of battery. Under a certain working condition, a functional relationship between sample entropy and dischargecapacityis createdand estimations computed from the function aretaken asobservations to propagateparticles in PF.The resultsindicate that the algorithm enhancesthe accuracy.Due to the establishment of functions atdifferent dischargeratesand temperaturemodification, prognostic accuracy of discharge capacity hasbeen improved under multi-operating working conditions.1. IntroductionWith the rapid development of industrial technology, theexploration and utilization of new energy havebeen in urgentneed. Electric vehicle occupies a pivotal position in newenergy automobile. Battery management system BMS isspecially designed to improve efficient utilization, to preventovercharge or overdischarge, to prolong the service life, andto monitor the state of the battery. A more sophisticatedprognostic of battery health state is much needed for highrequirements of reliability, stability, and security of batteries.Consequently, the prediction of remaining battery life isconsidered asone of the promising research fields. Numerouspapers have reported the studies on state of charge SOCand state of health SOH which are the focus of batteryPrognostic and Health Management PHM.Battery discharge capacity reaching its criteria withoutany omen leads to a disastrous failure in some cases.Theaccurate prediction of remaining useful life RUL of batteryis essential for long-time efficient use. The causesof capacityfading are internal factors such as anodic and cathodicactive material changes and SEI membrane incrassation[ 1, 2]. Accurate battery SOC estimation is of great signi-ficance to battery electric vehicles and hybrid electric vehi-cles. SOC estimation aims at the management of energyflows of electric vehicles and avoiding battery overcharge orundercharge. Lee et al. [3] proposed an Extended KalmanFilter EKF method along with a measurement noise modeland data rejection of lithium-ion battery SOC estimation.The proposed algorithm and model approach were verifiedthrough severalexperiments. An adaptive unscented Kalmanfiltering method to estimate SOC of lithium-ion battery waspresented [4]. The proposed SOC estimation method had abetter accuracy compared with previous works. Lee et al. [ 5]estimated the SOC and the capacity of a lithium-ion batterywith a modified OCV-SOC model. The method overcame thevariation in conventional OCV-SOC.Methods of battery capacity estimation are proposedbased on the following two ideas. One method is feature-based.In one sense,asvariations of voltage, current, and tem-perature characteristic curves could reflect the battery agingprocesses or internal resistance variations, some charactersare often extracted from them. Salkind et al. [6] proposed apractical method that resistancesobtained by electrochemicalimpedance spectroscopy EIS measurement and coulombcounting techniques were employed in predicting SOC andSOH. The advantage of the work was that there was no needto know previous discharge or cycling history. Gomez et al.[ 7] made a detailed analysis on EIS and pointed out thataging information could be extracted from the parametersof EIS equivalent circuit model. Pincus [ 8] firstly introducedthe concept of approximate entropy mainly to compute the2 AdvancesinMechanicalEngineeringcomplexityoftimeseries.Widodoetal.[9] tooksampleentropyfeaturesobtainedfromdischargevoltagecurvesasinputsofsupportvectormachineSVMandrelevancevectormachineRVMforSOHprediction.Theresultsshowedthatthemethodproposedwasplausible.Theotherismodel-based.Generally,faultfeatureiscloselyrelatedtotheparametersofthemodel.Correctionandadjustmentofmodelparameterscanenhancethepre-dictionaccuracy.Themodel-basedtechniquescontributetoanin-depthunderstandingofthemechanismandhavetheadvantageofreal-timefaultprediction.Amodelofbatterysystemstateisestablishedtodescribethedischargebehaviororbatteryhealthstate.Abbasetal.[10]introducedanintegratedmethodologybasedonbothphysicsoffailuremodelsandBayesianestimationmethodsforprognosisofelectricalcomponents.Anempiricalformulawasproposedtodepictdischargingbehavioroflithium-ionbatteries[11–13]. SimulationresultsindicatedthatPFalgorithmwasappropriateforthepredictionofbatteryhealthstate.Sahaetal.[14] presentedseveralalgorithmsincludingARIMA,RVM,EKF,andPF.ARVM-PFframeworkhadsignificantadvantagesovertheconventionalmethodsofRULestimationlikeARIMAandEKF.Someresearchershavealsoestablishedelectrochemicalnumericalmodelandthermalmodelforthestudyonbatteryinternalcharacteristics.PorouselectrodemodelwithliquidelectrolytewasproposedbyWestetal.[15].Thatelectrolytedepletionwastheprimarylimitingfactorofcapacitywasdemonstrated.Parketal.[16]presentedanelectrochemicalheatconductionphenomenalmodel.Abetterunderstandingofconductionphenomenaoflithium-ionbatterieswaspre-sented.Kimetal.[17] extendedone-dimensionalmodelingapproachtothreedimensionstocapturegeometricalfeaturessuchasshapesanddimensionsofcellcomponents,tosimulateoventestsandtodeterminehowalocalhotspotcanpropagatethroughthecell.Thoughsomekeybehaviorsofbatterycellscanbecapturedinthesemodels,itiscomplextodeployalargenumberofunknownparametersduetothememoryandcomputation.Lumpedbatterymodelsarelikelytobethepreferredchoicewitharelativelyfewerparameters.Asystematiccomparativestudyoftwelvelumpedbatterymodelswasconducted[18].ThedevelopedcellvoltagemodelscouldbeusedinSOCestimationinBMS.Thisworkisconductedbythecombinationofthetwoideasmentionedabove.Inthefollowingsection,wefirstlyintroducethetheoryaboutsampleentropyandbasicuti-lizationofparticlefilterintermsofprognosticsoflithium-ionbatteryRUL.Then,wepresentthedetailedpredictionprocedure.2. Theory and Intelligent Prognostic Method2.1.SampleEntropy.SampleentropyisdefinedasgenerationrateofnewinformationbyRichmanandMoorman[19]forthecalculationofcomplexityoftimeseries.ItcanbeexpressedasSampEn,,, whereisagiventotalnumberofdata,isthetoleranceforacceptingmatrices,andisthedimensionofvectors.Thespecificalgorithmofsampleentropyisasfollows.Foragivenseries{},weform- 1vectorsas[,1,.,- 1],for1to- 1. 1Thedistancebetweenvectorsandisdefinedas[,]max- ,for,1to- 1,0to- 1.2Foragiven,calculatethenumberwhen[,],for,anddefinethefunction 1- num{[,]}. 3Then,taketheaverageof.Theresultisexpressedas 1- 1-1∑1. 4Similarly,replacewith1andrepeatthestepsfromthebeginning.Afterwards,wecandeterminethetwovaluesand1.Asthesamplelengthisalwayslimited,thesampleentropyisestimatedbySampEn,,- ln[ 1]. 5ThevalueofSampEn,,iscloselycorrelatedwith,,and.Thus,theproperselectedparameterscouldresultinmorereasonablestatisticalproperties.2.2.ParticleFilter.PFisaBayesianlearningtechniqueusingMonteCarlosimulations.TheideaistodescribethesystemstateasaprobabilitydensityfunctionPDFapproximatedbyparticlesthataregeneratedfromaprioridistributionandupdatedfromobservationsthroughameasurementmodel.Modelparametersareincludedasapartofthestatevectortobetracked[11].PFframeworkcanbeappliedtoRULpredictionofbatteryduetoitsgoodstatetrackingperformance.Actualdischargecapacityisassociatedwithmanyfactors.Itisobviousthatchargingdirectlydeterminesthedischargecapacityinonecycle.Besides,reactionproductsforminguparoundtheelectrodeswilldecomposeduringrestorrelaxationperiod,whichleadtotheincreaseofavailablecapacityinnextcycle.Primarily,consideringthemaininflu-encefactorsofbatterycapacity,thefollowingstateequationsarecasttodescribethemodelasfollows112 exp 3Δ , 61V, 1,2,3, 7AdvancesinMechanicalEngineering 3whereiscycleindex,denotesthechargecapacity,Δ istherelaxationperiodbetweenthetwoadjacentcycles,1isthedischargecapacity,1,2,and3 areparametersofthestateequation,andV1,V2,andV3 areindependentzero-meanGaussiannoiseterms.SahaandGoebel[11]establishedameasurementmodelandregardedchargingcapacityastheobservationtoprop-agateparticles.Areasonableobservationformeasuringtheweightsofparticlesandselectivelypropagatingthemplaysanimportantroleinpredictionaccuracy.Inthecaseofourapplication,viathefittingmethod,afunctionalrelationshipofsampleentropyanddischargecapacityisestablishedtoobtainanappropriateobservation.Particularly,sampleentropyiscalculatedfromthedischargevoltagecurveofthecyclenumber.Thecorrespondingoutputofthefunctionisusedastheobservationincycle1.Itisworthmentioningthatthereisnoneedtotakeotherexperimentstoobtainsuchfeatures,forthedischargevoltagecurvescanbeeasilyobtainedduringthemonitoringineachcycle.2.3.IntelligentPrognosticMethod.Theprocedurecomprisesthefollowing.1Datacollectionisasfollows.aExtractbatterydischargevoltagecurvesfromtrainingdataandtheselectedparametersandare2and0.1,respectively.Thefunctionalrelationshipofdischargecapacityandsampleentropyiscreatedunderthecurrentoperatingcondition.bGaindischargecurrentcurves,chargingcapac-ity,andrelaxationtimeofadjacentdischargecyclesfromvalidationtestdata.Inaddition,somehistoricalcapacitydataarealsoneeded.2Particlefilterinitializationisasfollows.aSetthestartingpredictionpointinproportiontothenumberofhistoricalcapacitydata.bObtaininitialparameters 1,2,3 viafitting.c500initialparticlesaregeneratedwithvaluesobtainedin2-bandthevariancesofnoisetermV 1,2,3 areabout10,000timessmallerthan.3Predictionisasfollows.aParticles{}1areupdatedby7 andtheprioridischargecapacityvaluesincycle1arecalculatedthroughthoseupdatedparticles{1}1.bTakesampleentropyfeatureastheinputofthefunctionandcomputetheweightofeachparti-cleperdeviationbetweenthecalculatedobser-vationandpreviousdischargevoltagevalue.Normalizetheveryparticlesusingthefollowingformula11 11∑ 111. 8cThroughthemethodofrandomsampling,eachparticle{1}1iscopiedorabandonedselec-tivelyaccordingtoitsweightandthennewsample{1}1isobtained.dTheaverageofthesample{1}1representstheprobabilitydensitydistributionexpectationofeachparameterin6.Then,thefinalestima-tion1canbeeasilyfiguredupby6.eRepeatthestepfrom3-ato3-duntilthecapacityreachesitscriterionwhichisa30fadingofratedcapacity.3. Experiment DataThefullsetofagingdatacollectedfromcommerciallyavailable18650-sizelithium-ioncellsprovidedbyNASAAmesPrognosticsCenterofExcellencewastakenasobjectofstudy.BatteryanodeandcathodematerialsaremostlyLiNi0.8Co0.15Al0.05O2 andMAG-10graphite,respectively.Theelectrolyteis1.2MLiPF6 inECEMC37wtandtheseparatoris25mthickPE.Alltestingbatterieswererunthroughdifferentworkingprofilescharge,discharge,andimpedance.BatteriesNo.6andNo.18weretestedbythefollowingsteps1chargingwascarriedoutinaconstantcurrentmodeat1.5Auntilthebatteryvoltagereached4.2V,2aconstantvoltagemodewastheninoperationuntilthechargecurrentdroppedto20mA,3batterieswereputasideforaperiodoftime,4impedancemeasurementwasimplementedwithanelectrochemicalimpedancespectroscopyfrequencysweepfrom0.1Hzto5kHz,5at24° C,dischargingwascarriedoutataconstantcurrentlevelof2Auntilthebatteryvoltagefellto2.5V, 6thesamestepas3,and7thesamestepas4.Repeatedcharginganddischargingresultedinanacceleratedagingprocess.Theexperimentswerestoppedwhenthebatteriesreachedtheend-of-lifecriteriawhichwasa30fadinginratedcapacityfrom2Ahrto1.4Ahr.4. Results and Discussion4.1.SingleWorkingCondition.Figure1depictsthedischargevoltagecurvesindifferentcycles.Ataconstantcurrentof2A,thevoltagedropsfrom4.2Vto2.6V.Obviously,thecurvesvaryfromcycletocycleintheagingprocesses.ItcanbeseenfromFigure1thatthelowestvoltagepointbouncesbackinstantlyattheendofdischargeandsubsequentlyrisesslowlyuntilitcomestoastop.Thetwoarrowspointouttheprocessesmentionedabove.Observingthedefinitionofsampleentropy,wecanfindthatwhenthemaximumdistancecomputedfromtheadjacentvectorsconstitutedbythesequentialsamplesisgreaterthan,thecomplexitynumberofthecorrespondingvectorin3willnotchange4 Advances in Mechanical EngineeringVoltageV4.23.83.432.60 20 40 60 80 100 120 140 160 180 200First cycleSecond cycleThird cycleTimeFourth cycle12Figure 1 Battery voltage curves in different cycles and the twovoltage variation processeswere pointed out by the arrows.2.11.91.71.51.30 20 40 60 80 100 120 140X108Y1.405Actual discharge capacityEstimated value with observationobtained from sample entropyCapacityAhrCycle Figure 2 Prediction of battery No. 6.Error0.080.060.040.02020 40 60 80 100 120 140Cycle Figure 3 Relativeerrors.2.11.91.71.51.31.10 20 40 60 80 100 120 140Actual discharge capacityEstimated value with observationobtained from sample entropyCapacityAhr1 23 456 7Cycle Estimated value with observationobtained from charging capacityFigure 4 Comparative simulation results through different meth-ods.in statistical calculations. Otherwise, if the noise signal isadded to the sampleswith larger amplitude, it will be ignoredby detection, for the distance between the disturbed vectorsis longer than others. In that sense, sample entropy couldcapture the features of voltage variance in a constant currentmode.As battery is aging gradually during the usageperiod, wefind an interesting connection between the sample entropyfeature and the discharge capacity. In consequence, sampleentropy could serveasan indicator for assessingthe conditionof battery. With training data of battery No. 18, a cubicpolynomial fitting is introduced to find out the functionalrelationship between them. When the parameter and aredeployed to 2 and 0.1,respectively, a better fitting effect isobtained with a reasonable statistical result.The starting point and predicting length are 25 and 115.Figures 2 and 3 show the prediction result of battery No. 6and its errors. From the actual disch
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