A battery model including hysteresis for State-of-Charge estimation ...

IEEE Vehicle Power and Propulsion Conference (VPPC), September 3-5, 2008, Harbin, ChinaA battery model including hysteresis forState-of-Charge estimation in Ni-MHbatteryFeng Xuyun * and Sun Zechang ***Automotive College Tongji University, Shanghai, China. Email:catia33@126.com**Automotive College Tongji University, Shanghai, China. Email:sunzechang@fcv-sh.comAbstract—According to the measurement of the Ni-MHbattery electrochemical impedance spectroscopy (EIS) andthe relationship of the OCV-SOC , one basicequivalent-circuit model structure including hysteresisvoltage for Ni-MH battery is presented, which can beapplied for the estimation of battery state of charge (SOC).The model approach is evaluated by comparing themeasured and the simulated battery voltage correspondingto several current profiles. Voltage hysteresis is shown toplay a critical role in the Ni-MH battery model. Then therelationships among different physic variables and statesvariables of this model are expressed by the discrete-timestate-space functions. Based on the Extended Kalman Filter,the Stare of Charge (SOC) for the Ni-MH battery model isestimated by using the date such as current and voltagewhich are real-time tested by the battery test instrument.Finally, the behavior of the algorithm in terms of accuracyis examined. Voltage hysteresis is shown to play a criticalrole, and this effect makes determining the SOCparticularly difficult for Ni-MH batteries relative to otherbattery systems.Keywords—Ni-MH Battery; Battery Model; Hysteresis;State-of-Charge (SOC)I. INTRODUCTIONTo cope with the complexity of the future electricpower system (EV&HEV) and to find favorable systemarchitecture or optimal operation strategies,simulation-based design methods are becomingincreasing important. For these simulations, suitablesub-models of all vehicle system components aremandatory. However, compared to the sub-models ofmany electrical or mechanical devices, modeling of thedynamical behavior of electrochemical power sources isan important issue and difficult to obtain.Besides, one suitable and accurate battery model canbe applied to the estimation algorithm of SOC. Until tonow, there were a lot of battery modeling approachesbeen introduced in the references [1]. However, some ofthem didn’t consider the typical hysteresis phenomenonwhich significant influence the Ni-MH batteryequilibrium-potential, or weren’t suitable for beingapplied to estimate the battery SOC. A variety oftechnique has been proposed to measure or monitor theSOC of the battery used in the EV&HEV [2] . Eachmethod has its relative merits. Just as it is important toknow the amount of the fuel remaining, the SOC of thebattery is an essential factor in the EV&HEV operation.The goal of this paper is to provide a powerfulmodeling method for the Ni-MH battery. And then basedon the proposed model, the Extended Kalman Filter isapplied to estimate the battery SOC. The paper will beorganized as follows. In Section II, the dynamic Ni-MHbattery model is presented and evaluated by comparingthe measured and the simulated battery voltagecorresponding to several current profiles. In Section III,the battery SOC estimation algorithm is built on theproposed battery model, and Section IV presents somesimulation and test that confirm the effectiveness of theestimation method. Finally the summary is presented inthe Section V.II.THE Ni-MH BATTERY MODEL INCLUDINGTHE HYSTERESISA. Impedance-based Battery Model [3]The impedance-based modeling approach employselectrochemical impedance spectroscopy (EIS). Themeasurements were carried out with the “EISmeter”which possible measure impedances at frequencies fromdown to frequencies as low as with very high accuracy.Impedance spectra were obtained by applying an acsingle ( Iac) with different defined frequencies andevaluating the system’s response at nearly any workingpoint (temperature, SOC, dc-current). However, in thispaper, we just want to identify the basic model structureof the Ni-MH battery by the analysis of the impedancespectra. There is no need to test the impedance spectra ofevery possible working point.Im(Z)/mOhm151050217Hz214mHz2.1mHz-55 10 15 20Re(Z)/mOhmI_dc=0@T=20 o CFigure1. Impedance spectra with no dc-current of a Ni-MH battery at70% SOC and 27978-1-4244-1849-7/08/$25.00○C 2008 IEEEAuthorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:49 from IEEE Xplore. Restrictions apply.

IEEE Vehicle Power and Propulsion Conference (VPPC), September 3-5, 2008, Harbin, ChinaThe impedance spectra (Figure1) were obtained byapplying the “EISmeter” with no dc-current of theNi-MH battery at 70%SOC and temperature. Theequivalent elements included in the battery model can beidentified by evaluation of the impedance spectra [3] : Thespectra show an inductive behavior ( L ) at the highfrequencies; The intersection point with the real axis candetermine the battery pure ohmic resistance R ; Thesemicircle from the high frequencies to the lowerfrequencies can be modeled by a parallel connection of anon-linear resistor R ctand a capacitor Cdl; At thelower frequencies, the battery diffusion behavior of thebattery becomes apparent which is commonly modeledby a so-called Warburg impedance Zw. Hence, based onthe analysis of the impedance spectra, the basic Ni-MHbattery model structure can be developed as the figure2.However, according to the equivalent transform relations,different circuits may processes completely identicalimpedance spectroscopy. The basic model structurepresented in this paper is the combination of theconsideration of the simplicity and sufficient knowledgeinvolved in the impedance spectra.OCV( V)7.17.06.96.86.76.66.56.46.36.26.1Di s c har geChar ge6.00 10 20 30 40 50 60 70 80 90 100SOC( %)Figure 3. Measured Ni-MH battery Open-circuit voltage values underdifferent SOC, the lower curve measured after discharging steps (-1C)with rest periods (3600s) in between, and the upper curve measuredafter charging steps (+1C).Besides the test of completed charge/dischargeSOC-OCV relationship, we also gained the relationshipof OCV-SOC under partial charge/discharge as the figure4.LRC d lZ w7.17.06.9Di s c har geChar geFigure2. The basic equivalent electrical circuit structure of the Ni-MHbattery in frequency domain determined by the impedance spectrawithout modeling the equilibrium potential.R c tOCV( V)6.86.76.6B. Hysteresis Phenomenon in Ni-MH BatteryFor most electrochemical system the equilibriumpotential is unambiguously defined by the state of charge(SOC). However, the Ni-MH battery shows a significanthysteresis of the open circuit potential [4] , with thepotential on charge being higher than on discharge atevery SOC (Figure2). The more detailed explanation ofthe voltage hysteresis characteristics could refer to thepaper [4], [5].The figure 3 is the depiction of theSOC-OCV relationship of one certain Ni-MH batterieswhich is composed of five single cells in series. In thefigure 3, we can obviously recognize the stabledifference of the open-circuit potential undercharge/discharge condition. That means that the potentialdoes not depend solely on the state of charge but also onthe history of charging and discharging of the electrode.6.56.410 20 30 40 50 60 70 80 90 100SOC( %)Figure 4. Measured Ni-MH battery OCV-SOC under partialcharge/dischargeIII. SCHEMATIC PRESENTATION OF THEEQUIVALENT CIRCUIT MODELFinally, based on the impedance-based model structureand Ni-MH battery hysteresis phenomenon testbehaviors, one equivalent circuit model including thehysteresis voltage model is presented as the figure. In thefigure, one hysteresis voltage model Vhwas included inthe part of V oc.The inductive element L hasn’t beenincluded in the model since the Ni-MH batteries are lessused at so high frequencies. Besides, we used one linearresistor paralleled with the capacitor in the model and weneed the further investigation on the non-linear resistorin the future work. Any other else electric elements inthe model are the same as the explanations ofpre-provided impedance-based model.Authorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:49 from IEEE Xplore. Restrictions apply.

IEEE Vehicle Power and Propulsion Conference (VPPC), September 3-5, 2008, Harbin, ChinaIU 0=V ocC 0RV h Z wFigure 5. Equivalent circuit model for the Ni-MH battery.According to some former works on the Ni-MHbattery hysteresis phenomenon and the reference [6], thehysteresis voltage model can be described as thefollowing equation:Charge:t−RC h0 h,maxU = U * + U ∗(1 − RC h ) (1)hDischarge:et−RC h0 h,maxU = U * −U∗(1 − RC h ) (2)heAnd the implementation of the Warburg impedanceZ in the Matlab/Simulink will refer to the paper [7].wIV.V tTHE Ni-MH BATTERY SOC ESTIMATIONUSING EKF BASED ON THE PRESENTED MODELIn order to use Kalman-based methods for the Ni-MHbattery SOC estimation, we must first have a cell modelin a discrete-time state-space form. Specifically, weassume the form:x = + 1f( x , u ) + w(3)k k k ky = gx ( , u)+ v(4)k k k kEq.(3) is called the “state equation” or “processequation”, and Eq.(4) is the “output equation”. Then, weconvert the proposed battery model to the state equation.The battery SOC is constrained to be a member of thestate vector xk; the vector ukcontains theinstantaneous cell current ik.the cell loaded terminalvoltage is considered to the output yk. Finally, theoverall state-space equations for the Ni-MH batterymodel are:⎡ 0 sign ( ik) F( ik)⎤⎡Uhk ( + 1) ⎤ ⎡1 −Fi(k) 0⎤⎡Uhk ( ) ⎤ ⎢ ikη t⎥⎡ ⎤⎢ ⎥= ⎢Z 0 1⎥⎢ ⎥+ Δ0⎢U⎥(5)k+1Z ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ k ⎦ h,max⎢ C⎣ ⎦⎣ n ⎥ ⎦U = U ( z ) + i R+ U + U(6)eek ocv k k ct ( k ) h( k )The battery SOC and hysteresis voltage are denotedZU hkask( )( )and respectively. Where andZkare the two members of the state vectorxk . In theUEq.(), Eq.(),k is the battery loaded terminal voltage,−C dl−R ctttU hkikis the discharge/charge current, R is the cell internalUocvand Uctare the open-circuit voltageR C in parallel,resistance,and the voltage of the ct dlUh,maxisthe max value of the hysteresis voltage.Finally the EKF basic principle can be applied toestimate the state of the equation. The main EKFalgorithm equations are summarized in the TableI:TABLE I.SUMMARY OF THE NON-LINEAR EXTENDKALMAN FILTER (EKF) [9]Non-linear state-pace modelxk + 1= f ( xk , uk) + wkyk= g( xk, uk)+ vkDefinitionsˆ ∂ f ( xk, uk) ,Ak − 1=ˆ ∂ g ( xk, uk)Ck=∂ xk+ ∂ xx k = xˆk −k−1x k = xˆkInitializationˆ + + + Tx0 = E ( x0), P ˆ ˆ0= E[( x0 − x0 )( x0 − x0) ]ComputationFor k=1,2,3,…compute− +State estimation time update: xˆ( ˆk= f xk−1, uk−1)−Error covariance time update: P Tk= Ak− 1Pk−1Ak−1+Qk−1− T − T −1Kalman gain matrix: Kk = P C ( )k kCkP Ck k+ RkState estimation measurement update:+ − −xˆ ˆ [ ( ˆk= xk + Kkyk − g xk , uk)]Error covariance measurement update:+ −Pk = ( E − KkCk)PkThen, we should summarize some model function, andaccording to the basic principle of the EKF, someunknown parameters and function in the EKF algorithmcould also be derived from the presented model equation(5),(6), which are summarized in the Table IITABLE II.SUMMARY OF THE PARAMETERS IN THE MODELAND EKF ALGORITHMF ( i ) = exp( −βi Δt)kk⎡sign( ik) F ( ik)Uh,max⎤⎡1 − F( ik) 0⎤ f ( x , u ) = x⎢η t⎥⎢ u0 1⎥ + Δ⎣⎦⎢ ⎥⎢⎣Cn ⎥⎦g ( x , u ) = U ( x [1]) + x [2] + U + RuAˆk − 1k k k kk k ocv k k ct( k )k∂ f ( x , )1 (k) 0ku− F ik⎡⎤= =∂ x⎢k0 1 ⎥⎣ ⎦+x k = xˆk − 1ˆ ∂ g ( x , u ) dUCk= = ( ,1)dSOCk k OCV∂ xk x −k = x ˆ kThe parameter β could be derived from the hysteresisAuthorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:49 from IEEE Xplore. Restrictions apply.

IEEE Vehicle Power and Propulsion Conference (VPPC), September 3-5, 2008, Harbin, ChinadUOCVvoltage test, andcould be calculateddSOCfrom the relationship curve of the OCV-SOC.V. EVALUATION OF THE MODEL ANDMETHOD OF SOC ESTIMATIONFirst, the presented Ni-MH battery model is evaluatedby comparing the measured and the simulated batteryvoltage corresponding to several current profiles. As thefigure 6, the battery was charged and discharged underpulse current (1C, about 60S). We used two differentmodels that one includes the hysteresis model and theother without. The comparison result indicated that thehysteresis model plays one great role in the Ni-MHbattery model.Voltage(V)1.501.481.461.441.421.401.381.361.341.321.301.28Measured VoltageSimulated Voltage(no Hysteresis Model)Simulated Voltage1.260 200 400 600 800 1000 1200 1400Time(s)Figure6. Comparison of the measured voltage with the model simulatedvoltage, one simulation model including the hysteresis model and theother one without the hysteresis modelIn the figure 7, the measured battery voltage wascompared with the model simulation voltage under onerandom current load.Vol t age( V)46444240383634Mode l Si mul at i on Vol t ageMeasur ed Vol t age0 500 1000 1500 2000 2500 3000 3500 4000Ti me( S)1.3701.3651.3601.355600 640 680Figure7. Comparison of measured and simulated voltage correspondingto one random current profileSecond, the behavior of the battery SOC algorithmwhich is based on the presented Ni-MH battery model isalso evaluated under one random discharge/chargecurrent test. The Figure 8 is the current profile of the testand figure9 is the SOC estimation result of using themethod EKF and Ah-integration respectively.Cur r ent ( A)40200-20-40-60-801000 2000 3000 4000 5000 6000 7000 8000 9000Ti me( S)Figure8. Input random current profiles to evaluate the SOC estimationalgorithmSOC( %)6050403020100-10-20-301000 2000 3000 4000 5000 6000 7000 8000 9000Ti me( S)EKFAh- i nt egr at i onFigure9. Comparison of the SOC estimation using EKF andAh-integration respectively under the current loads as the figure 8In the Figure 9, the SOC Estimation results werecompared with the Ah-integration method which canreach one certain accuracy in one short time. We couldrecognize that there is the obvious deviation between thetwo estimation results. We could see that the estimationresult of the Ah-integration method reach to one negativevalue when the battery was nearly to be fully discharged.Apparently, the EKF estimation method avoided thedeficit that the estimation errors have been integratedgradually during the test in the Ah-integration method,and one more accurate estimation result could be gained.VI. CONCLUTIONIn this paper one equivalent circuit model includingthe hysteresis voltage model for Ni-MH battery has beenpresented. We used the method of EIS to identify theNi-MH battery model structure. Then combined withsome current-voltage response tests, the basic Ni-MHCDABAuthorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:49 from IEEE Xplore. Restrictions apply.

IEEE Vehicle Power and Propulsion Conference (VPPC), September 3-5, 2008, Harbin, Chinabattery model including the hysteresis was established.And then an EKF based algorithm has been developed toestimate SOC. The proposed method can accuratelymodel the dynamic behavior of the battery which makesit appropriate for HEV application. Finally, the capabilityand accuracy of the proposed method in estimation SOCof the Ni-MH batteries is verified by the evaluationworks.VII. ACKNOWLEDGMENTThis work was greatly supported by the ShanghaiScience and Technology Commission of Shanghai andShanghai Fuel Cell Vehicle Powertrain Co., Ltd. Thebattery test laboratory in Automobile College, TongjiUniversity, provided the test facilities used in the wholeperiod of investigation. Many enlightening discussionswith Pro. Sun Zechang, Associate Pro. Wei Xuezhe andPh.D Dai Haifeng are gratefully acknowledged. Besidesmany colleagues such as Yuan Yongjun, Wang Jiayuanare also acknowledged.REFERENCES[1] Chen Quanshi, Lin Chengtao. “Summarization of studies onperformance models of batteries for electric vehicle”. AutomobileTechnology, 2005, 37(3):1-5.[2] S.Piller, M.Perrin, A.Jossen. “Methods for state-of-chargedetermination and their application”. Journal of Power Sources,96: (2001)113-120.[3] Marc Thele, Oliver Bohlen, Dirk Uwe Sauer. “Development of avoltage-behavior model for NiMH batteries using animpedance-based modeling concept”. J. Power Sources, 2008,175 (2008) 635-643.[4] Venkat Srinivasan, John W.Weidner, and John Newman.“Hysteresis during Cycling of Nickel Hydride Active Material”. J.Electrochemical Society, 2001, 148(9):696-980.[5] M.W. Verbrugge, Edwad Tate. “Adaptive state of chargealgorithm for nickel metal hydride battery including hysteresisphenomena”. J of Power Sources,2004,126: 236-249.[6] Feng Xuyun, Wei Xuezhe. “Study on the Ni-MH BatteryEquivalent Circuit model. Battery”, 2007, Vol.37(4):286-288.[7] S.Buller. “Impedence-based simulation models for energy storagedevices in advanced automotive power system”. PhD Thesis,RWTH Aachen Univercity, 2002, ISBN: 3-8322-1225-6.[8] Wu Hongjie, Qi Bojin, Zheng Minxin, Liu Yongzhe. “Ni-MHbattery state-of-charge estimation based on Kalman filter”. J.Beijing University of Aeronautics and Astronautics, 2007,33(8):945-948.[9] Deng zili. Kalman filter and Vina filter [M]. Harbin: HarbinIndustry University Publisher, 2001.Authorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:49 from IEEE Xplore. Restrictions apply.