Predicting State of Charge of Lead-Acid Batteries for HEV by EKF

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Predicting state of charge of lead-acid batteries for hybridelectric vehicles by extended Kalman filter

A. Vasebi *, S.M.T. Bathaee, M. Partovibakhsh

Hybrid Electric Vehicle Research Center, Department of Electrical and Electronic Engineering, K.N. Toosi University of Technology, Tehran, Iran

Received 9 October 2006; accepted 20 May 2007Available online 17 July 2007

Abstract

This paper describes and introduces a new nonlinear predictor and a novel battery model for estimating the state of charge (SoC) oflead-acid batteries for hybrid electric vehicles (HEV). Many problems occur for a traditional SoC indicator, such as offset, drift and longterm state divergence, therefore this paper proposes a technique based on the extended Kalman filter (EKF) in order to overcome theseproblems. The underlying dynamic behavior of each cell is modeled using two capacitors (bulk and surface) and three resistors (terminal,surface and end). The SoC is determined from the voltage present on the bulk capacitor. In this new model, the value of the surfacecapacitor is constant, whereas the value of the bulk capacitor is not. Although the structure of the model, with two constant capacitors,has been previously reported for lithium-ion cells, this model can also be valid and reliable for lead-acid cells when used in conjunctionwith an EKF to estimate SoC (with a little variation). Measurements using real-time road data are used to compare the performance ofconventional internal resistance (Rint) based methods for estimating SoC with those predicted from the proposed state estimationschemes. The results show that the proposed method is superior to the more traditional techniques, with accuracy in estimating theSoC within 3%. 2007 Elsevier Ltd. All rights reserved.

Keywords: Batteries; Extended Kalman filter; Hybrid electric vehicle; State of charge

1. Introduction

Peak power demands of hybrid electric vehicles are sub-ject to large dynamic transients in current and power. Anexample is the Manhattan driving cycle that shows roaddata collected from a Toyota Prius HEV, where therequired maximum charge and discharge current are 10 A

and 25 A, respectively, when subjected to a series of vehicledriving tests [1]. The satisfaction of such operating condi-tions needs a management system that has accurate knowl-edge of the peak power buffers state of charge to facilitatesafe and efficient operation.

Various electric equivalent circuit models have beenapplied to lead-acid batteries to determine the SoC. How-

ever, accurate description of the complex nonlinear electro-chemical processes that occur during power transfer to/from the battery are dynamically difficult. These processesinclude the flow of ions, amount of stored charge, ability todeliver instantaneous power, the effects of temperature,internal pressure etc. [2,3]. A variety of techniques havebeen proposed to measure or monitor the SoC of a cell

or battery, each having its own characteristics, as reviewedby Piller et al. [4]. Coulomb counting or current integrationis the most commonly used technique. It requires dynamicmeasurement of the cell/battery current, and its time inte-gral is used to provide a direct indication of the SoC [5].However, because of the reliance on integration, errors interminal measurements due to noise, resolution and round-ing are cumulative, and large SoC errors can result. A resetor recalibration action is, therefore, required at regularintervals in all electric vehicles (EVs). This may be per-formed during a full charge or conditioning discharge,

0196-8904/$ - see front matter 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enconman.2007.05.017

* Corresponding author. Tel.: +98 2188462459; mobile: +98 9329411278;fax: +98 2188462066.

E-mail address: [email protected] (A. Vasebi).

www.elsevier.com/locate/enconman

Available online at www.sciencedirect.com

Energy Conversion and Management 49 (2008) 7582


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but it is not appropriate for standard HEV operationwhere full SoC is rarely achieved. Other factors that ulti-mately influence the accuracy of SoC estimates and causeadditional complications to the traditional integrationbased techniques are the variation of cell capacity with dis-charge rate, temperature and Coulombic efficiency losses

[2,6].When considering flooded lead-acid cells, the specificgravity of the electrolyte is known to be a good measureof SoC [6]. However, the estimation of SoC gets compli-cated when using valve regulated lead-acid (VRLA) cellsdue to the nominal amount of electrolyte being immobi-lized in the glass fiber separator mat or gel. Nevertheless,since the open circuit terminal voltage of a VRLA batteryvaries almost linearly over the majority of the batterysSoC (Fig. 1) [2,3,7], it has been used in many SoC estima-tors. Note that this curve has been individually obtainedfrom a case study battery in room temperature and con-stant discharge rate. To be an effective method, however,

corrections must be made for temperature and electrolyteconcentration gradients (concentration polarization)formed during high rate charges and discharges (long set-tling times may be required to allow such concentrationgradients to disperse prior to making an open circuit volt-age reading [3]).

Another broad category of cell modeling and SoC deter-mination technique involves measuring cell impedancesover wide ranges of AC frequencies at different states ofcharge. Values of the model parameters are found by tak-ing least squares of measured impedance values. Thismethod is not suitable for our application because it needs

to inject signals directly into the cell to measure its imped-ances [6,8].

Other reported methods for estimating SoC have beenbased on artificial neural networks [9] and fuzzy logic [10]principles, although the latter was reported to have rela-tively poor performance. Although such techniques causelarge computation overhead on the battery pack controller,which previously caused problems for online implementa-

tion, the increasing computational power of digital signalprocessing chips and the accompanying reduction in devicecosts may, in the near future, make their application anattractive alternative. Neural networks, in particular, havebeen used to avoid the need of the large number of empir-ically derived parameters required by other methods.

Indeed, for portable equipment application, where thetask of prediction of SoC is less demanding, a neural net-work modeling approach has been shown to give meanerrors of 3% [11]. Also, a neural network model for predict-ing battery power capacity during driving cycles has beenadded to the ADVISOR EV and HEV modeling environ-ments [1].

Here then, model based state estimation techniques areproposed to predict the states of a cell that are normallydifficult or expensive to measure or are subjected to the sig-nificant problems described previously. In this case, theSoC is the key state. Using an error correction mechanism,the observers provide real-time predictions of SoC. Specif-

ically, the well known extended Kalman filter (EKF),developed during the 1960s to provide a recursive solutionto optimal linear filtering for both state observation andprediction problems [12], is used for this study; a uniquefeature of the EKF is that it optimally (minimum variance)estimates states affected by broadband noise containedwithin the system bandwidth, i.e. that cannot otherwisebe filtered out using classical techniques, and enablesempirical tradeoffs between modeling errors and the influ-ence of noise. A KF based method has been used in Ref.[2] with a linear state space battery model for SoC estima-tion, whereas in this paper, we have employed the EKF due

to the nonlinear nature of the battery.

2. Battery model

A dynamic model of the battery, in the form of statevariable equations, is necessary to predict the SoC. Here,a generic model [13,14] consisting of a bulk capacitor Cbulkto characterize the ability of the battery to store charge, acapacitor to model surface capacitance and diffusion effectswithin the cell Csurface, a terminal resistance Rt, surfaceresistance Rs and end resistance Re, is employed as shownby Fig. 2. The voltage across the bulk and surface capaci-tors are denoted Vcb and Vcs, respectively. In this model,

00.10.20.30.40.50.60.70.80.911.7

1.8

1.9

2

2.1

2.2

2.3

2.4

soc

Vt(V)

Fig. 1. Open circuit voltage versus SoC. Fig. 2. RC battery model.

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battery current is positive for the charging mode and neg-ative for the discharging mode.

The initial parameters of the cell are calculated fromexperimental data where open circuit voltage (OCV) testswere performed upon successive discharges of the batteryby application of current pulses. An initial estimate of Cbulk

is obtained by analyzing the amount of stored energy in thecell, while the provisional value of Csurface relies on calcu-lating the time constant of the cell in response to high fre-quency excitation. Complete derivation details, along withthe initial parameters for the cells considered, are given inthe following section for completeness.

3. Calculation of initial parameters

The initial parameters required for the battery model aredetermined from experimental data. In this paper, we haveused a 6 Ah, 2 V sealed lead-acid cell manufactured by theSABA Battery Co. Iran, where OCV tests are performedupon successive discharges of the battery by injection ofcurrent pulses.

3.1. Capacitor Cbulk

The capacitance is determined by analyzing the amountof stored energy. Fig. 3 shows the OCV when dischargecurrent pulses of 1.53 A are applied for 3600 s at 5400 sintervals [2]. The energy stored in Cbulk is determined fromthe OCV at 0% SOC and 100% SOC, using the followingexpression:

ECbulk 12CbulkV2 1

2CbulkV2100% SOC V20% SOC 1

ECbulk is equivalent to the rated Amp-sec capacity of thebattery, giving:

Cbulk-initial RatedAmp-sec V100% SOC

12

V2100% SOC V20% SOC

2

V100%SOC and V0%SOC have been presented in Fig. 3.

3.2. Capacitor Csurface

The initial value of Csurface relies on the results of highfrequency excitation of the cell to determine the time con-stant given by the surface capacitor and its associated resis-tance. As before, OCV tests are performed. Dischargepulses of 10 A are applied at 500 ms intervals, thereby iso-lating the results from the effects of Cbulk. From Fig. 4, it isseen that

V1 2:168; V2 2:102

V3 2:157; V4 2:1645

Dt 0:5 s

The time constant is approximated using the followingrelationship:

Vno-load V1 V3 V4 V3I ets 3

and solving for s gives:

s Dtln 1 V4 V3V1 V3

0:58 s 4

The time constant is described by

s Rs ReCsurface 5

Hence, the initial estimate of the surface capacitor is deter-mined as

Csurface-initial s

Re Rs6

3.3. Battery resistance

The internal resistance of the battery is measured as5.6 mX. It is assumed that Rs and Re are equivalent andaccount for 80% of the total resistance. Hence, Rt is

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

-1.5

-1

-0.5

0

time(sec)

Current(A)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

1.7

1.8

1.9

2

2.1

2.2

time(sec)

Vt(v)

Vt soc100%

Vt soc 0%

Fig. 3. Cell terminal voltage when discharge current pulses of 1.53 A are

applied.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 22.08

2.09

2.1

2.11

2.12

2.13

2.14

2.15

2.16

2.17

2.18

time(sec)

Voltage(v)

V1

V2

V3

V4

Fig. 4. Cell terminal voltage when a discharge current of 10 A pulse is

applied at 500 ms intervals.

A. Vasebi et al. / Energy Conversion and Management 49 (2008) 7582 77


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5/8

Ak dfx; u

dxjxk;uk

Bk dfx; u

dujxk;uk

Ck dCx

dxjxk;uk C

22

Now, we calculate these matrices for our system:

Ak

a11 a12 0 a14

a21 a22 0 0

a31 0 a33 a34

0 0 0 0

26664

37775

a11 a12 x4k

2Re; a14

x1k x2k IkRe2Re

a21 a22 1

2ReCsurface; a31

x4k

2Re

1

2ReCsurface

a33 x4k

2Re

1

2ReC

surface

; a34 x1k

2Re

x3k

2Re

IkRt

2Re

23

and

Bk x4k

21

2Csurfaceb13 0

h iT

b13 1

2CsurfaceRtx4k

2Re

Rt

2ReCsurface

24

ultimately,

C 0 0 1 0 25

Assuming the applied input u is constant during each sam-pling interval, a discrete time equivalent model of the sys-tem is given by:

xk1 Adxk Bduk

yk1 Hxk126

where

Adk % I AkDT; Bdk BkDT

H C27

and DTis the sampling period. The system is now assumedto be corrupted by stationary Gaussian white noise via theadditive vectors rk and lk. The former vector is used torepresent system disturbances and model inaccuracies,and the latter represents the effects of measurement noise.Both rk and lk are considered to have a zero mean value,

for all k, with the following covariance matrices (E denot-ing the expectation operator):

ErkrTk Q for all k

ElklTk R for all k

28

The resulting system is, therefore, described by

xk1 Adkxk Bdkuk rk

zk1 Hxk1 lk129

where z is the vector of measured outputs after being cor-rupted by noise.

For notational purposes, we define ^xk (note the super

minus) to be our a priori state estimate at step kgivenknowledge of the process prior to step k, and ^xk to beour a posteriori state estimate at step kgiven measurementzk. We can then define a priori and a posteriori estimateerrors as

ek xk ^xk

ek xk ^xk30

The a priori estimate error covariance and posteriori esti-mate error covariance are then

Pk Eek e

k

T; Pk EekeTk 31

A property of the EKF is that the estimated state vector ^xkof the system, at time k, minimizes the sum of squared er-rors between the actual and estimated states.

minfPkg minfExk ^xkxk ^xkTg 32

For recursive implementation, the EKF estimate ^xk1 is cal-culated from the previous state estimate ^xk, the input u and

Fig. 5. Recursive EKF algorithm.



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the measurement signals z. The available input/output dataat each sample step is, therefore, considered to be u0, u1,u2 . . . uk, uk+1 and z0, z1, z2 . . . zk, zk+1. The recursiveEKF algorithm is obtained with the predictor/correctorstages being explicitly identified in Fig. 5 [12].

5. Implementation of EKF

The stochastic principles underpinning the EKF areappealing for this investigation, since it is recognized thatthe presence of disturbances stemming from sensor noiseon the cell terminal measurements and the use of non-idealdynamic models make it impossible to predict the states ofthe system over prolonged time periods with certainty. Astatistical predictor/corrector formulation thereby providesobvious advantages.

Since only terminal quantities of the battery can bemeasured, the input is defined as u = I and the measuredoutput is y = Vt. Although no formal stability and tuningmethods are available for initializing the EKF andrecourse to empirical tuning is normally required, its use

is nevertheless widespread. Information about the systemnoise contribution is contained in matrices Q and R and,in essence, the selection of Q and R determines the accu-racy of the filters performance, since they mutually deter-mine the action of the EKF gain matrix Kk+1 andestimation error covariance matrix Pk+1. The covariance

matrix representing measurement noise R can be estimatedfrom knowledge of the battery terminal voltage. The vari-ance is obtained from the square of the root mean square(rms) of noise on each cell and is assumed to be Gaussiandistributed and independent.

Initialization of the covariance matrix describing the dis-turbances on the plant Q is complicated while knowledge ofthe model inaccuracies and system disturbances is limited,particularly as each cell has different characteristics [12]. Ajudicious choice ofQ is, therefore, obtained from experimen-tal studies under the simplifying assumption that there is nocorrelation between the elements ofrkand the noise presenton each cells voltage transducer, thereby leading to a diago-

nal Q. The initial covariance matrix P0 together with Q andR, for our case, are ultimately chosen to be:

Fig. 6. Manhattan driving cycle current.

Fig. 7. Measurement terminal voltage.

Fig. 8. Estimated terminal voltage.

Fig. 9. Terminal voltage error.



7/8

P0

1 0 0 0

0 1 0 0

0 0 1 00 0 0 1

2

6664

3

7775; R 10

Q

0:005 0 0 0

0 0:07 0 0

0 0 0:9 0

0 0 0 0:0001

26664

37775

33

6. Simulation results

The EKF was applied to the real-time estimation of the

SoC of a single cell that was subjected to a Manhattandriving cycle [1]. Fig. 6 shows the cell terminal currentfor this driving cycle. The initial cell SoC was set to 1.0,Note that SoC = 1 is a normalized value used to define afully charged cell. The measured and estimated terminalvoltages for the Manhattan driving cycle are illustratedin Figs. 7 and 8, respectively, and their errors are shownin Fig. 9.

Fig. 9 shows that the maximum terminal voltage error isless than 0.1 V, i.e. less than 4%. Ultimately, we show theresults of the SoC and open circuit voltage (Vcb) estimationby the EKF and compare these results with the Advisor

SoC estimation (Rint based method) in Fig. 10.

7. Conclusion

This paper presented an alternative approach to esti-mate the SoC of a cell pack by the application of anEKF. It was shown that when using a generic model todescribe the dynamic behavior of lead-acid cells, largestate errors can develop over time. In particular, a com-parison between SoC estimation based on the EKF tech-nique and the more conventional methods based oninternal resistance (that was used in Advisor for indicationof the SoC for the Toyota Prius HEV) shows 3% differ-ence in results. We do not use the word error becauseRint based methods are static and are not suitable formodeling dynamic systems, therefore we can not use itas a reference method. This method is only used for com-

parison of the static method against the dynamic method.The results demonstrate that the proposed technique andnew battery model are very suitable for presentation ofthe batterys dynamic behavior and indication of the bat-terys SoC.

References

[1] NREL. Advisor software. http://www.NREL.org.[2] Bhangu BS, Bently P, Stone DA, Bingham CM. Nonlinear observers

for predicting SoC and SoH of lead-acid batteries for hevs. IEEETrans Vehicular Technol 2005;54(3).

[3] Vincent CA, Scrosati B. Modern batteries. 2nd ed. New York: John

Wiley & Sons; 1997.

Fig. 10. (a) Open circuit voltage (Vcb). (b) EKF SoC and Advisor SoC. (c) Difference between EKF SoC and Advisor SoC.



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[4] Piller S, Perrin M, Jossen A. Methods for state-of-charge determi-nation and their applications. J Power Sources 2001;96:11320.

[5] Caumont O, Le Moigne Ph, Lenain P, Rombaut C. An optimizedstate of charge algorithm for lead-acid batteries in electric vehicles. In:Electric vehicle symposium, vol. EVS-15, Brussels, Belgium, Septem-berOctober 303, 1998. Proceedings on CD-ROM.

[6] Plett GL. EKF for battery management system of LiPB-based HEVbattery packs. J Power Sources 2004;134:26276.

[7] Pang S, Farrell J, Du J, Barth M. Battery state-of-charge estimation.American control conference, Arlington, VA June 2527, 2001.

[8] Huet F. A review of impedance measurements for determination ofthe state-of-charge or state of health of secondary batteries. J PowerSources 1998;70:5969.

[9] Chan CC, Lo EWC, Weixiang S. The available capacity computationmodel based on artificial neural network for lead acid batteries inelectric vehicles. J Power Sources 2000;87:2014.

[10] Singh P, Fennie C, Reisner DE, Salkind A. A fuzzy logic approach tostate-of-charge determination in high performance batteries withapplications to electric vehicles. In: Electric vehicle symposium, vol.EVS-15, Brussels, Belgium, SeptemberOctober 303, 1998. Proceed-ings on CD-ROM.

[11] Patillon GO, dAlche-Buc JN. Neural network adaptive modeling ofbattery discharge behavior. In: Artificial neural networks ICANN97,7th international conference, vol. 1327. Lecture notes in computerscience. Berlin, Germany; 1997. p. 1095100.

[12] Welch G, Bishop G. An introduction to the Kalman filter. Universityof North Carolina. .

[13] Vairamohan B. State of charge estimation for batteries. Msc thesis,University of Tennessee, 2002.

[14] Johnson VH, Pesaran AA, Sack T. Temperature-dependent batterymodels for high-power lithium-ion batteries. In: Presented at theproceedings of EVS 17, Montreal, PQ, Canada. 2000.


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Predicting State of Charge of Lead-Acid Batteries for HEV by EKF