State of charge estimation of lithium-ion batterie_动视

State of charge estimation of lithium-ion batterie

2025-09-26 10:40:54 责编:小OO

State of charge estimation of lithium-ion batteries using the open-circuit voltage at various ambient

temperatures

Yinjiao Xing a ,⇑,Wei He b ,Michael Pecht b ,Kwok Leung Tsui a

a Department of Systems Engineering and Engineering Management,City University of Hong Kong,83Tat Chee Avenue,Kowloon,Hong Kong b

Center for Advanced Life Cycle Engineering (CALCE),University of Maryland,College Park,MD 20740,USA

h i g h l i g h t s

g r a p h i c a l a b s t r a c t

10002000

3000400050006000

020

6080

Time(s)

S O C (%)

True & Estimated SOC [ambient temperature:40o

True SOC Initial Guess

Estimated SOC based on OCV-SOC

Estimated SOC based on OCV-SOC-40°C

a r t i c l e i n f o Article history:

Received 2April 2013

Received in revised form 28June 2013Accepted 3July 2013

Available online 7August 2013Keywords:

Electric vehicles

Lithium-ion batteries SOC estimation

Open-circuit voltage

Temperature-based model Unscented Kalman ﬁltering

a b s t r a c t

Ambient temperature is a signiﬁcant factor that inﬂuences the accuracy of battery SOC estimation,which is critical for remaining driving range prediction of electric vehicles (EVs)and optimal charge/discharge control of batteries.A widely used method to estimate SOC is based on an online inference of open-circuit voltage (OCV).However,the fact that the OCV–SOC is dependent on ambient temperature can result in errors in battery SOC estimation.To address this problem,this paper presents an SOC estimation approach based on a temperature-based model incorporated with an OCV–SOC–temperature table.The unscented Kalman ﬁltering (UKF)was applied to tune the model parameters at each sampling step to cope with various uncertainties arising from the operation environment,cell-to-cell variation,and mod-eling inaccuracy.Two dynamic tests,the dynamic stress test (DST)and the federal urban driving schedule (FUDS),were used to test batteries at different temperatures.Then,DST was used to identify the model parameters while FUDS was used to validate the performance of the SOC estimation.The estimation was made covering the major working range from 25%to 85%SOC.The results indicated that our method can provide accurate SOC estimation with smaller root mean squared errors than the method that does not take into account ambient temperature.Thus,our approach is effective and accurate when battery oper-ates at different ambient temperatures.Since the developed method takes into account the temperature factor as well as the complexity of the model,it could be effectively applied in battery management sys-tems for EVs.

1.Introduction

Electric vehicles (EVs)are bringing new life to the automobile industry as an alternative way to reduce consumption of fossil fuels.As one of the critical components in EVs,battery perfor-mance determines the safety,reliability,and operating efﬁciency of the vehicle system.Accurate and instantaneous information on the state of the battery,such as state of charge (SOC)and state of health (SOH),should be provided to the drivers by a battery man-agement system (BMS)to

guarantee safe and reliable battery oper-ation.[1–4].The SOC quantiﬁes the usable energy at the present

⇑Corresponding author.Tel.:+852********;fax:+852********.

E-mail addresses:

yxing3@student.cityu.edu.hk (Y.Xing),weihe@calce.umd.edu (W.He),pecht@calce.umd.edu (M.Pecht),kltsui@cityu.edu.hk (K.L.Tsui).

There are three main types of methods for SOC estimation:cou-lomb counting,machine learning methods,and their combination using a model-based estimation approach.These three types of methods are described below.

Coulomb counting is a straightforward method for estimating SOC that accumulates the net charge at the last time period in units of ampere-hours(Ah).Its performance is highly reliant on the pre-cision of current sensors and the accurate estimation of the initial SOC[3,13].However,coulomb counting is an open-loop estimator that does not eliminate the accumulation of measurement errors and uncertain disturbances.In addition,it is not able to determine the initial SOC,and address the variation of the initial SOC caused by self-discharging.Without the knowledge of the initial SOC,this method will cause accumulating errors on SOC estimation.Taking into account these factors,regular recalibration is recommended and widely used by methods such as fully discharging the battery, or referring to other measurements such as open-circuit voltage (OCV),as suggested in[3,6,7,14].

Machine learning approaches,including artiﬁcial neural net-works,fuzzy logic–based models,and support vector machines, have been used to estimate SOC online.Li et al.[15]designed a 12-input-2-level merged fuzzy neural network(FNN)that was fused with a reduced-form generic algorithm(RGA)to estimate SOC.Bo et al.[16]developed parallel chaos immune evolutionary programming(PICEP)to train a neural network model in which ﬁve input variables were selected.This approach was used to esti-mate the SOC of nickel–metal hydride(Ni/MH)batteries.The per-formance of the kind of black-box models is reliant on the reliability of the training data,i.e.whether it is sufﬁcient to cover the entire loading conditions.Once the battery operated at the un-known loading conditions,the robustness of these models was subject to challenge.Wang[17]employed a support vector ma-chine to model the dynamic behavior of a Ni/MH battery under dy-namic current loading.However,model training is time consuming and requires a large amount of data.Also,the estimation based on this model causes a large prediction error due to the uncertainty of the new data set.

A model-basedﬁltering estimation approach is being widely ap-plied due to its close-loop nature and concerning various uncer-tainties.Both electrochemical models and equivalent circuit models aim to capture the dynamic behavior of the battery.The former are usually presented in the form of partial differential equations with many unknown parameters.They are accurate but not desirable in practice because of a high requirement for memory and computation.To guarantee the accuracy of the model and the feasibility,equivalent circuit models have been imple-mented in BMSs such as the enhanced self-correcting(ESC)model and the hysteresis model,as found in[10,18,19],and one or two-order resistance–capacitance(RC)network models[1,2,10,11,20]. OCV is a vital element in the above-mentioned battery equivalent models and is a function of SOC in nature.The premise of utilizing OCV–SOC is that the battery needs to rest a long time and terminal voltage approaches the OCV.However,in real life,a long resting time may not be possible.To make up for theﬂaws of OCV meth-ods,nonlinearﬁltering techniques based on state-space models have been developed to enhance SOC estimation through combin-ing coulomb counting and OCV[7].Plett applied extended Kalman ﬁlters(EKF)into BMS to implement SOC estimation of a lithium polymer battery(LiPB)using different battery models in [10,21,22].Plett later proposed the use of two sigma-point Kalman ﬁltering(SPKF)estimators,including the unscented Kalmanﬁlter (UKF)and central difference Kalmanﬁlter(CDKF),in[18,23].Sub-sequently,adaptive EKF[7,20],dual EKF[11],and adaptive UKF[3] were developed to improve the accuracy of the SOC estimation based on their own sample sets and some common equivalent cir-cuit models.Charkhgard and Farrokhi[13]also proposed the com-bination of NN and EKF to estimate SOC.NN was employed to train a lithium-ion battery model using some charging data from the battery.The effectiveness of this method was not veriﬁed under the dynamic discharging data,which would lead to a larger uncer-tainty on estimating SOC.

However,several existing issues are seldom addressed in the literature.Firstly,the temperature dependence of the OCV–SOC lookup table is seldom discussed with regard to battery SOC esti-mation.Instead,a single OCV–SOC table constructed at a certain temperature(e.g.,room temperature)is widely employed.It will cause a large error in inferring SOC when the battery is operating at other ambient temperatures(not room temperature) [1,8,10,11].Secondly,lithium-ion batteries have a relativelyﬂat OCV curve over the SOC,especially for lithium iron phosphate (LiFePO4)batteries,which are widely used in the electric vehicle market[24].That means a small error on the inferred OCV will produce a larger deviation in SOC.Thirdly,different models were adopted by individuals based on their own experimental data. Although a sophisticated model with more parameters might be able to provide a smaller modeling error,it would run the risk of adding more uncertainties,such as over-ﬁtting problems and the introduction of unnecessary noises,especially concerning temperature.Therefore,it makes more sense to investigate a gen-eric but accurate temperature-based model with fewer parame-ters for real-time applications.Kim et al.[25]considered temperature as an input variable into aﬁrst-order RC circuit model.However,the effect of temperature on the OCV–SOC was ignored due to a slight difference between OCV curves from 30%SOC to80%at different temperatures.Moreover,their samples have an obvious linear slope of OCV–SOC that is prone to infer SOC accurately.Nevertheless,for a relativelyﬂat OCV curve dependent on ambient temperature,it is signiﬁcant not only to develop an accurate and generic model considering ambient temperature,but also to enhance the capability of online estimation due to the uncertainty,including unit-to-unit varia-tion,measurement noise,operational uncertainties,and model inaccuracy[26].

In this paper,a temperature-based internal resistance(R int)bat-tery model combined with a nonlinearﬁltering method was put forward to improve the SOC estimation of lithium-ion batteries un-der dynamic loading conditions at different ambient temperatures. The research proceeds as follows.Three tests at different tempera-tures are introduced in Section2.The dynamic stress test(DST)and federal urban driving schedule(FUDS)are two kinds of dynamic loading conditions tested at different temperatures to identify the model parameters and verify the estimated performance, respectively.The purpose of the OCV–SOC–temperature(OCV–SOC–T)test is to extend the OCV–SOC behavior to temperature ﬁeld.Due to various uncertainties of the system,UKF-based SOC estimator is proposed due to its superiority of reaching to the 3rd order of any nonlinearity over the EKF.The implemented procedure for our battery study is followed by Section3.The experimental results are presented in Section4to compare our developed method based on OCV–SOC–T with the original

Y.Xing et al./Applied Energy113(2014)106–115107

estimation using a single OCV–SOC table.The robustness is vali-dated and compared under the different initial true values and dif-ferent initial guesses of SOC.2.Experiments

The experiment setup is shown in Fig.1.It consisted of (1)lith-ium-ion cells (LiFePO4)of the 18650cylindrical type (the key spec-iﬁcations are shown in Table 1);(2)Vötsch temperature test chamber (The cells were placed in cell holders in the chamber);(3)a current and voltage sampling cable for loading and sampling;(4)a battery test system (Arbin BT2000tester);(5)a PC with Ar-bins’Mits Pro Software (v4.27)for battery charging/discharging control;(6)Matlab R2009b for data analysis.During battery oper-ation,the sampling time of current,voltage was 1s.Three separate test schedules were conducted on the battery test bench for model identiﬁcation,OCV–SOC–T table construction,and method valida-tion,respectively.

2.1.Model identiﬁcation test

For model identiﬁcation,the dynamic stress test (DST)was run from 0°C to 50°C at an interval of 10°C.DST is employed to investigate the dynamic electric behavior of the battery.It is designed by US Advanced Battery Consortium (USABC)to simu-late a variable-power discharge regime that represents the expected demands of an EV battery [27].A completed DST cycle is 360s long and can be scaled down to any desired maximum demand regarding the speciﬁcations of the test samples.There-fore,in our study,DST was run continuously from 100%SOC at 3.6V to empty at 2V over several cycles in a discharge process.The positive current responds to discharging while the negative denotes charging.The measured current and voltage proﬁle at 20°C is shown in Fig.2.

2.2.The OCV–SOC–T test

OCV is a function of SOC for the cells.If the cell is able to rest for a long period until the terminal voltage approached the true OCV,OCV can be used to infer SOC accurately.However,this method is not practical for dynamic SOC estimation.To address this issue,the SOC can be estimated by combining the online identiﬁcation of the OCV with the predetermined ofﬂine OCV–SOC lookup table.Taking into account the temperature dependence of the OCV–SOC table,the OCV–SOC test was conducted from 0°C to 50°C at an interval of 10°C.The test procedure at each temperature is the same as fol-lows.Firstly,the cell was fully charged using a constant current of 1C-rate (1C-rate means that a full discharge of the battery takes approximately 1h)until the voltage reached to the cut-off voltage of 3.6V and the current was 0.01C.Secondly,the cell was fully dis-charged at a constant rate of C/20until the voltage reached 2.0V,which corresponds to 0%SOC.Finally,the cell was fully charged at a constant rate of C/20to 3.6V,which corresponds to 100%SOC.The terminal voltage of the cell is considered as a close approximation to the real equilibrium potential [6,10].As

shown

Fig.1.Schematic of the battery test bench.

Table 1

The key speciﬁcations of the test samples.Type Nominal voltage Nominal capacity Upper/lower cut-off voltage Maximum continuous discharge current LiFePO 4

3.3V

1.1Ah

3.6V/2.0V

30A

010002000300040005000-2

024

Time (s)

C u r r e n t (A )

2.3.Method validation test

A validation test with a more sophisticated dynamic current proﬁle,the federal urban driving schedule(FUDS),was conducted to verify the estimation algorithm based on the developed model. FUDS is a dynamic electric vehicle performance test based on a time–velocity proﬁle from an automobile industry standard vehi-cle[27].In the laboratory test,a dynamic current sequence was transferred from the time–velocity proﬁle,programmed to charge or discharge the battery and applied to battery performance test [3,17,20,29].Similar to the DST test,the current sequence is scaled toﬁt the speciﬁcation of the test battery and the limitation of the testing system of Fig.1.The current proﬁle of FUDS causes varia-tion of the SOC from fully charged at3.6V to empty at2V.The FUDS test was also run form0°C to50°C at an interval of10°C. The measured current,voltage proﬁle,and the cumulative SOC at 20°C are shown in Fig.4.A completed FUDS current proﬁle over 1372s is emphasized in another graph in Fig.4(a).

3.Battery modeling

For lithium-ion batteries,the internal resistance(R int)model is generic and straightforward to characterize a battery’s dynamics with one estimated parameter.Although a sophisticated model with more parameters would possibly show a well-ﬁtting result, such as an equivalent circuit model with several amounts of paral-lel resistance–capacitance(RC)networks,it would also pose a risk of over-ﬁtting and introducing more uncertainties for online esti-mation at the same time.Especially taking into account tempera-ture factor,more complexity should be imposed on battery modeling.Therefore,we would prefer a simple model to a sophis-ticated model if the former had generalization ability and provided sufﬁciently good results.In this paper,model modiﬁcation based on the original R int model is proposed to balance the model com-plexity and the accuracy of battery SOC estimation.The schematic of the original R int model is shown in Fig.5.

U term;k¼U OCVÀI kÂRð1Þ

U OCV/fðSOC kÞð2ÞIn Eqs.(1)and(2),U term,k is the measured terminal voltage of the battery under a normal dynamic current load at time k,and I k is the dynamic current at the same time.The positive current re-sponds to discharging while the negative value means charging.R is the simpliﬁed total internal resistance of the battery.U OCV is a function of SOC of the battery that should be tested following the procedure as presented in Section2.2.The battery model Eq.(1) can be used to infer OCV directly according to the measured termi-

Schematic of the internal resistance(R int)model

Y.Xing et al./Applied Energy113(2014)106–1151093.1.Model parameter identiﬁcation

The DST was run on the LiFePO4batteries to identify the model parameter R in Eq.(1).Taking the current and voltage proﬁle of DST at20°C as an example,the voltage and current are measured and recorded from fully charged to empty with a sampling period of1s based on our battery test bench.The accumulative charge(exper-imental SOC)is calculated synchronously from100%SOC.Thus, the parameter R can beﬁtted using a sequence of the current,volt-age,and the ofﬂine OCV–SOC by the least square algorithm.In terms of theﬁtted R value,the model performance can be evalu-ated based on the measured terminal voltage(U term,k)and the esti-mated voltage(U term;k).Fig.6shows the measured and the estimated voltage response on the DST proﬁle at20°C based on the original model.

In statistics,the mean absolute error(MAE)and the root mean squared(RMS)error can be used together to evaluate the good-ness-of-ﬁt of the model.These two indicators are given by the fol-lowing equations respectively.

MAE¼

1X n

k¼1

j e k jð3Þ

RMS error¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

X n

k¼1

ðe kÞ2

ð4Þ

Here,e k is the modeling error(U term,kÀU term;k)at time k.The MAE measures how close forecasts are to the corresponding out-comes without considering the direction.The RMS error is more sensitive to large errors than the MAE.It is able to characterize the variation in errors.The statistics list of the model is shown in Table2.

According to the small graph in Fig.3,the OCV slope,namely, dOCV/dSOC is approximately equal to0.0014between25%and 80%SOC.This means that the deviation on the OCV inference will cause an estimated deviation of SOC up to21%when there is a MAE of0.0288V based on the current model.Additionally,a large mean error was plotted over time in Fig.6.Thus,the residuals should be reduced to improve the model adequacy with smaller MAE and RMS error values.3.2.Model improvement and validation

3.2.1.The OCV–SOC–T table for model improvement

According to the test in Section2.2,six OCV curves were ob-tained from0°C to50°C at an interval of10°C.Fig.7(a)empha-sizes the differences of OCV–SOC curves between30%and80% SOC at different temperatures.It can be seen that SOC0°C is much larger than other SOC values at higher temperatures when the OCV inference is the same,i.e.,3.3V.It makes sense that the releas-able capability of the charge is reduced at low temperatures. Fig.7(b)shows the SOC values if the OCV inference was equal to the speciﬁc values from3.28V to3.32V at intervals of0.01V at three temperatures:0°C,20°C,40°C.

One issue of interest can be seen in Fig.7(b).That is,the same OCV inference at different temperatures corresponds to different SOC values.For example,the SOC difference between0°C and 40°C reaches approximately22%at an OCV of3.30V.Therefore, we propose adding the OCV–SOC–T to the battery model to im-prove the model accuracy.The improved battery model is as follows:

U term;k¼U OCVðSOC k;TÞÀI kÂRðTÞþCðTÞð5Þwhere U OCV is a function of SOC and ambient temperature(T).C(T)is a function of temperature that facilitates the reduction of the offset due to model inaccuracy and environmental conditions.Fig.8 shows the measured and the estimated voltage response on the DST proﬁle at20°C based on the proposed model.It can be found that the mean error of the new model is reduced with small varia-tions as compared to the original model in Fig.6.

Another issue of interest in Fig.7(b)is that a small deviation of 0.01V in OCV inference will lead to a large difference in SOC at the same temperature condition.It is the same issue as shown in Fig.3. Therefore,if the SOC estimation were directly inferred from a bat-tery model,it would have a high requirement on the model and measurement accuracy.To address this issue and improve the accuracy of the SOC estimation,the model-based unscented Kal-manﬁltering approach was employed and introduced in Section4.

3.2.2.The validation of the proposed model

Based on the developed model in Eq.(5),it is noted that the spe-ciﬁc OCV–SOC look-up table should be selected in terms of the ambient temperature(here it is viewed as an average value).Least squareﬁtting was also used to identify model parameters,R and C. Theﬁtted model parameter list and the statistics list of the pro-posed model are shown in Table3.

In comparison to theﬁtted results at20°C of Table2,here the MAE is one order of magnitude smaller than that of the original model and the RMS modeling error is also reduced.In addition, the correlation coefﬁcient(Corrcoef)was calculated for residual analysis.The Corrcoef(e k,I k)values close to zero indicate that the residuals and the input variable hardly have linear relationship. Thus,the corrected model can be betterﬁtted on the dynamic cur-rent load.Oneﬁnding of interest is C values that can beﬁtted over the ambient temperature(T)using a regression curve,as Fig.9 shows.Referring to the paper[30],the exponential function can be selected toﬁt C values over T because the internal elements of the battery,i.e.,battery resistance follow the Arrhenius equation, which has exponential dependency on the temperature.In our study,ﬁve C values at0°C,10°C,20°C,25°C,30°C,40°C were

Table2

Model parameter and statistics list of theﬁtted error of the model.

R(X)Mean absolute error RMS modeling error

0.24450.0288V0.0301

110Y.Xing et al./Applied Energy113(2014)106–115used for curveﬁtting while C(50°C)was used to test theﬁtted per-formance of this exponential function.The95%prediction bounds are shown in Fig.9based on C values andﬁtted curve.Apparently, C(50°C)drops within the95%prediction bounds.It can be seen that the function of C(T)in Fig.9can be used to estimate C when the corresponding temperature test has not been run.4.Algorithm implementation for online estimation

The online SOC estimation has strong nonlinearity.This point can be seen from any battery model in which OCV has a nonlinear relationship with SOC.Additionally,the uncertainties due to the model inaccuracy,measurement noise,and operating conditions will cause a large variation in the estimation.The model-based nonlinearﬁltering approach has been developed to implement dy-namic SOC estimation.The objective is to estimate the hidden sys-tem state,estimate the model parameters for system identiﬁcation, or both.Thus,an error-feedback-based unscented Kalmanﬁltering approach is proposed by shifting the system noise to improve the accuracy of the estimation.

4.1.Unscented Kalmanﬁltering

The extended Kalmanﬁltering(EKF)technique has become a popular technique for addressing the issue of state or parameter estimation for nonlinear systems.The rationale behind EKF is still the KF approach based on state space modeling.It aims to utilize the error between the current measurement and the model output to adjust the model state by virtue of a Kalman gain.Its principle and implementation can be found in[31].Since KF is only available for linear systems,extended Kalmanﬁltering(EKF)used a lineari-zation process at each time step to approximate a nonlinear system through theﬁrst-order Taylor series expansion[31,32].However, theﬁrst order approximation will probably lead to large errors in

Table3

Fitted model parameter list and statistics list of modelﬁtting.

T(°C)R(X)C Mean absolute

errors(V)RMS

modeling

errors

Corrcoef

(e k,I k)

00.2780À0.05520.01530.0188 1.36eÀ13

100.2396À0.04360.01120.01348.45eÀ14

200.2249À0.03600.00870.0105 1.09eÀ13

250.2020À0.03260.00800.0095 1.02eÀ13

300.1838À0.020.00730.0085À7.62eÀ13

400.1565À0.02370.00600.0071 2.85eÀ13

500.1816À0.02010.00990.0131 3.15eÀ14

Y.Xing et al./Applied Energy113(2014)106–115111

the true posterior mean and covariance of the noise and could even result in divergence of theﬁlter.Under this situation,unscented Kalmanﬁltering(UKF)based on unscented transformation was suggested to avoid the weakness that comes from using Taylor ser-ies expansion.

The core idea of UKF is easier to approximate the state distribu-tion that is represented by a minimal set of chosen sample points called sigma points,which can capture the mean and covariance of Gaussian random variable when propagated through a nonlinear system.The state-space model of a nonlinear system is represented as follows:

x k¼fðx kÀ1;u kÀ1Þþw kÀ1

¼hðx k;u kÞþm k

ð6Þ

where x k is the system state vector and y k is the measurement vec-tor at time k.Correspondingly,f(Á)and h(Á)are the state function and the measurement function,respectively;u k is the known input vector;w k$Nð0;

Þis the Gaussian process noise;and v k$Nð0;P

Þis the Gaussian measurement noise.Assume the state x has mean x and covariance P x.Using the unscented transformation (UT),the state will be transformed as a matrix of2n+1sigma vec-tors v i with corresponding weights w i.These sigma points are shown in the following equation:

kÀ1

¼½ xð0Þ

kÀ1

; xð1:nÞ

kÀ1

; xðnþ1:2nÞ

kÀ1

¼½ x kÀ1; x kÀ1þ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðnþkÞP kÀ1

; x kÀ1À

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðnþkÞP kÀ1

ð7Þwhere n is the dimension of the state and k is a scaling parameter. These sigma points are propagated through a nonlinear function,re-estimated,and then they are used to capture the posterior mean and covariance to the3rd Taylor expansion,as shown in Eqs.(9) and(10).

¼hðv iÞ;i¼0;...;2nð8Þ y%

X2n

i¼0

wðiÞ

ð9ÞP y%

X2n

i¼0

wðiÞ

f y iÀ y gf y iÀ y

g Tð10Þ

where wðiÞm and wðiÞc are the weights of the corresponding sigma points that can be calculated in[33–35]for details.

4.2.SOC estimation based on proposed method

In our battery study,the state vector is x=[SOC,R]T.Theﬁrst state equation in Eq.(11)follows the coulomb counting method mentioned in Section1.Peukert effect and capacity aging could be partially compensated when introducing the process noise x1,kÀ1.A random walk is applied to the model parameter R regard-ing the cell-to-cell variation and operation uncertainties.Tuning the R will also be able to compensate for the variation of the C in our proposed model.The terminal voltage of the battery is the measured vector y=U term,that is,the proposed battery model as shown in Eq.(5).

State function:

SOC k¼SOC kÀ1ÀI kÀ1ÂD t=C nþx1;kÀ1

R k¼R kÀ1þx2;kÀ1ð11ÞMeasurement function:

U term;k¼U OCVðSOC k;TÞÀI kÂR kðTÞþCðTÞþm kð12Þwhere I k is the current as the input(u k in Eq.(6))at time k;D t is the sampling interval,which is1s according to the sampling rate;and C n is the rated capacity.The rated capacity of the test samples is1.1 Ah.And x1,k,x2,k and m k are zero-mean white stochastic processes with covariances

and

,respectively.According to the proposed state-space model in Eqs.(11)and(12),the procedure of SOC estimation based on UKF is summarized in Table4.

Table4

Summary of the UKF approach for SOC estimation.

Initialize:

–Measure ambient temperature,prepare U OCV(SOC,T)and R0,C0–Initial guess:S0,

–Covariance matrix:P o

–Process and measurement noise covariance:P

;

Generate sigma points at time kÀ1,(k2½l;...;1 ):

kÀ1¼

S kÀ1

R kÀ1

¼½ v kÀ1; v kÀ1þ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðnþkÞP kÀ1

; v kÀ1À

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðnþkÞP kÀ1

Predict the prior state mean and covariance

–Calculate sigma points through state function:

v i

k j kÀ1¼

S i k j kÀ1

R i

k j kÀ1

S i kÀ1ÀI kÀ1ÂD t

C n

R i kÀ1

;i¼1; (2)

–Calculate the prior mean and covariance:

^xÀk ¼

P2n

i¼0

w i

v i

k j kÀ1

;PÀ

P2n

i¼0

w i

v i

k j kÀ1

À^xÀ

v i

k j kÀ1

À^xÀ

i T

Update using the measurement function

–Calculate sigma points y k j kÀ1¼U OCVðS k j kÀ1;TÞÀI kÂRðTÞk j kÀ1þCðTÞ

–Calculate the propagated mean:^yÀ

k ¼

P2n

i¼0

w i m y i

k j kÀ1

–Calculate the covariance of the measurement:

P yÀ

k ;yÀ

P2n

i¼0

w j c½y i

k j kÀ1

À^yÀ

Tþ

–Calculate the cross-covariance and the state and measurement:

of battery SOC estimation by UKF5.Experimental results for SOC estimation

Based on the temperature-based model,the developed method was validated using the FUDS test data,whose input and output are shown in Section2.3.FUDS tests were run at different ambient temperatures to emulate the operation conditions.As analyzed in Section3.2.1,we propose applying an OCV–SOC–T table instead of the conventional OCV–SOC lookup table,which was often estab-lished at room temperature,such as25°C.According to the mea-sured ambient temperature(T),the corresponding OCV–SOC–T is employed as well as the updated model parameters,R(T)and C(T). Aﬂow chart based on the OCV–SOC–T table is shown as Fig.10.Mod-el parameters will follow Table3according to the corresponding OCV–SOC table.A comparison will be made by using two lookup ta-bles and their corresponding models.Initialized parameters,includ-ing the initial guess of S0,measurement covariance,and process noise,were set the same for these two estimators.

Figs.11and12compare the results of the estimated SOC from two different lookup tables using UKF methods.Assuming that the original OCV–SOC table was tested at25°C,Fig.11shows the esti-mated results when a dynamic FUDS test was run at40°C. Fig.11(a)shows the errors between the estimated terminal voltage (U term)and the measured terminal voltage(U term)based on OCV–SOC and OCV–SOC–40°C,respectively.Fig.11(b)shows the esti-mated SOC under these two OCV–SOC tables.Similarly,Fig.12pre-sents similar estimated results when the FUDS proﬁle was performed at10°C while referring to OCV–SOC and OCV–SOC–10°C,respectively.For both of these twoﬁgures,the initial guess of SOC was set at50%,while the true initial SOC was80%.

As shown in Figs.11and12(b),when the selected OCV–SOC–T table is consistent with the ambient temperature,the esti-mated value does capture the true SOC and converge fast.While using the original OCV–SOC,a large deviation from the true SOC will appear.Therefore,we recommend employing the OCV–SOC–T table according to a measured environmental temperature.Be-sides,the error of the estimated SOC at10°C is a bit larger than that at40°C.One reason is that the accuracy of the model at 10°C is less than that at a higher temperature as shown in Ta-ble3.The larger modeling error at low temperature is partially caused by the inaccuracy of the OCV–SOC at low temperature. The OCV hysteresis is a bit larger at low temperature.In addi-tion,operation environment uncertainties would cause less improvement in battery SOC estimation at low temperature.The poor performance of the battery chemistry would present with a less releasable maximum capacity.It would make the battery model underperform,especially when the battery oper-ates under high dynamic loading conditions.

For the untested temperature in OCV–SOC–T,i.e.,35°C,the OCV–SOC–35°C table can be constructed through linear interpo-lating between OCV–SOC–30°C and OCV–SOC–40°C.Although the errors do exist,the approximation can still reduce the effect of temperature on SOC estimation,which is necessary to save the testing time when it covers all the temperature range.

Tables5and6present a comparison of the estimation when the initial guess was at30%,50%and70%SOC,while the true initial value covered its major working range from25%to85% SOC.The aim is to compare the effectiveness and robustness of the method in conjunction with the temperature-based model within the working range.The RMS estimated errors were calcu-lated to assess and compare the estimated performance based on the original model and our developed model.Two points can be drawn from these tables.Firstly,under the conditions of differ-ent initial SOC values,the estimated results based on the tem-perature-based model will provide more accurate estimated results than the estimation without regard to temperature.The former have smaller RMS estimated errors,which are less than 4.7%at both10°C and40°C.Secondly,the estimated errors would be relatively larger when the true initial SOC is between 45%and65%.The reason is that the OCV curve isﬂat in this phase,as shown in Fig.3.Thus,a small deviation from the OCV inference will causeﬂuctuation of the estimated OCV.Esti-mating from this phase will take a longer time to converge to the true SOC.

Table7presents the estimated results at0°C as similar as Ta-bles5and6show.Our method still showed an improvement on the SOC estimation.However,the maximum RMS estimated errors are up to16.4%.The possible reason is that the battery perfor-mance deteriorates as temperature decreases due to the reducing rate of the internal chemical reaction.This is why preheating is necessary for a battery cranking at low temperature,as in EV.Its purpose is to make the battery operate up to a more efﬁcient ambi-ent temperature.Thus,our battery model performed well in most situations,except at a lower temperature condition,which may require a sophisticated model and a more accurate OCV–SOC curve considered.The investigation of the sophisticated model goes beyond the scope of this article.

6.Conclusions

Ambient temperature affects the relationship between open-circuit voltage(OCV)and SOC and,accordingly,inﬂuences model-based battery SOC estimation.In this paper,we developed a tem-perature-based battery model to address the temperature depen-dence of battery modeling.Firstly,our model is a simple and effective model that improves the accuracy of the estimated SOC

Table5

RMSE(%)of SOC estimation when FUDS was run at40°C.

True Initial SOC(%)OCV–SOC OCV–SOC–40°C

Initial guess of SOC(%)Initial guess of SOC(%)

305070305070

25 1.9266 1.9923 2.4692 1.1516 1.2221 1.9845

35 2.7525 2.5160 2.22 1.1794 1.2200 1.6639

4516.956516.420215.5435 2.7057 2.86 2.85 5514.635014.532814.4129 4.5924 4.5315 4.6877 6511.726511.558011.6536 4.5101 4.2905 4.4098 7512.025511.991911.9558 1.5259 1.25800.9578 8510.287110.254410.4074 2.8114 2.6363 2.6249

Table6

RMSE(%)of SOC estimation when FUDS was run at10°C.

True Initial SOC(%)OCV–SOC OCV–SOC–10°C

Initial Guess of SOC(%)Initial Guess of SOC(%)

305070305070

25 4.45 4.3387 5.1118 2.3135 2.2842 2.6574

35 4.0763 4.02 4.0150 1.7378 1.6058 1.9947

45 6.7356 6.7051 6.7183 3.9229 3.42 3.9552

5510.281610.314810.2542 3.1982 3.4937 3.3147 6513.1013.110313.1261 4.0770 3.9304 3.8824

75 5.72 5.4158 5.0041 2.6257 2.4144 2.2732

85 6.0865 6.1225 6.1734 2.2941 2.3679 2.1650

Table7

RMSE(%)of SOC estimation when FUDS was run at0°C.

True Initial SOC(%)OCV–SOC OCV–SOC(0°C)

Initial guess of SOC(%)Initial guess of SOC(%)

305070305070

25 4.9334 5.0495 6.6315 2.0551 2.2850 3.3146

3511.3011.733611.6550 5.9619 6.1624 6.0038 4510.876210.938610.36 4.1374 4.1375 4.0302 5516.708716.725016.766111.918712.205412.0170 6518.509718.531718.5516 4.2343 4.1238 4.1266 7524.634624.616124.562216.283016.392416.3719 8512.383312.298612.3669 2.3437 2.2317 2.2520for lithium-ion batteries at different ambient temperatures.With few model parameters estimated,this model is easier and more computational efﬁcient for on-board application.Secondly,an OCV–SOC–temperature table was applied in our model based on the OCV test with a temperature interval of10°C from0°C to 50°C.The temperature dependence of the OCV–SOC table was con-cerned to improve the accuracy of the model.Finally,dynamic loading tests were run on the battery at different temperatures to assess the SOC estimation performance using the unscented Kal-manﬁltering approach.A comparison was made between our bat-tery model and the original model without regard to the temperature factor.The estimation covered the working range from25%to85%SOC.The results indicated that the estimation based on the developed battery model provided more accurate SOC values with smaller RMS estimated errors at different temper-atures.The robustness of this method was veriﬁed under the con-ditions of three different initial SOC values.Thus,this approach could be successfully applied in BMSs for electric vehicles.

Two issues about the developed method are worthwhile to be mentioned here for an optimized online application.One is that the OCV–SOC–temperature table can be reﬁned to save memory space in the online system by normalizing the temperature depen-dence of the OCV–SOC.The other is that the estimation based on our developed model provided a sufﬁciently accurate result with RMS estimated errors of less than5%within the major working range.If there were a higher requirement on the estimated accu-racy at temperature lower than even0°C,the SOC estimation based on a more sophisticated model would possibly make more sense.

Acknowledgements

The work presented in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Admin-istrative Region,China(CityU8/CRF/09).The authors would like to thank the members of the Center for Advanced Life Cycle Engineer-ing at the University of Maryland for their support of this work. References

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