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Adaptive unscented Kalman ﬁltering for state of charge estimation of a lithium-ion battery for electric vehicles Fengchun Sun, Xiaosong Hu*, Yuan Zou, Siguang Li National Engineering Laboratory for Electric Vehicles, Beijing Institute of Technology, Beijing 100081, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 3 October 2010 Received in revised form 22 March 2011 Accepted 22 March 2011 Available online 22 April 2011

An accurate battery State of Charge estimation is of great signiﬁcance for battery electric vehicles and hybrid electric vehicles. This paper presents an adaptive unscented Kalman ﬁltering method to estimate State of Charge of a lithium-ion battery for battery electric vehicles. The adaptive adjustment of the noise covariances in the State of Charge estimation process is implemented by an idea of covariance matching in the unscented Kalman ﬁlter context. Experimental results indicate that the adaptive unscented Kalman ﬁlter-based algorithm has a good performance in estimating the battery State of Charge. A comparison with the adaptive extended Kalman ﬁlter, extended Kalman ﬁlter, and unscented Kalman ﬁlter-based algorithms shows that the proposed State of Charge estimation method has a better accuracy. Ó 2011 Elsevier Ltd. All rights reserved.

Keywords: Battery management system Electric vehicle Adaptive unscented Kalman ﬁlter State of charge Lithium-ion battery

1. Introduction Battery Electric vehicles (BEVs) and hybrid electric vehicles (HEVs) potentially can take advantage of renewable electricity sources, reduce reliance on fossil fuels, and are widely viewed as an important transitional technology towards sustainable transportation [1,2]. For instance, Kühne [3] exempliﬁed that electric bus is an energy efﬁcient urban transportation means. Traction battery packs, critical sub-systems of BEVs/HEVs, are currently both performance and cost bottlenecks of BEVs/HEVs [4]. Due to the transient and demanding vehicle operations in daily driving, a battery management system (BMS) is required to ensure safe and reliable battery operations. The BMS needs to provide accurate knowledge of the states of the traction battery pack to operate the battery reliably and efﬁciently. A critical variable that must be estimated (because no direct measurement is available) is the battery State of Charge (SOC). Failure to estimate SOC precisely can easily cause under- or over-charging situations which result in a decrease of the power-output capability and the lifetime of the traction battery pack. Moreover, the efﬁciency of the whole vehicle energy management might be lowered severely. Therefore an accurate SOC estimation is of great signiﬁcance for BEVs/HEVs.

* Corresponding author. Tel./fax: þ86 10 6891 4625. E-mail address: [email protected] (X. Hu). 0360-5442/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2011.03.059

A number of methods to estimate the battery SOC were proposed, each with its own advantages and disadvantages [5]. Depending on the choices of battery models, some commonly used methods can approximately be categorized into three types. The ﬁrst type is the nonmodel-based Coulomb counting method which was applied to many battery management systems of BEVs/HEVs and battery storage simulations. This approach measures the current constantly and computes the accumulated charge to estimate the SOC. It is simple and online. For example, Aylor et al. [6] implemented a SOC monitoring technique combining the open circuit voltage and coulometric measurements on a microcomputer-based circuit for lead-acid batteries used in electric wheelchairs. An open circuit voltage reading was taken to recalibrate the accumulated error caused by the Coulomb counting method [6]. An SOC estimator based on the Coulomb counting method has been designed in [7] for battery charging control and management. A combination of the actual battery discharge time versus discharge rate data and coulometric measurements has also been investigated in [8] to estimate the SOC of sealed lead-acid battery blocks used as telecommunication back-up power supplies. In addition, the Coulomb counting method was used to calculate the battery remaining capacity for optimization of photovoltaic systems with battery storage [9,10]. Banerjee et al. [11] adopted the Coulomb counting method to compute the battery capacity when implementing optimum sizing of a battery-integrated diesel generator through a design-space methodology. The Coulomb counting-

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F. Sun et al. / Energy 36 (2011) 3531e3540

ðiÞ

Nomenclature

S

S

l a

battery State of Charge Coulombic efﬁciency battery nominal capacity Cn Dt sampling interval I battery current battery discharging current Iþ battery charging current I u process noise k time step index kjk 1 time instant just before the measurement at time step k is given U battery output voltage R battery internal resistance battery discharging resistance Rþ battery charging resistance R H half the hysteresis between charging and discharging voltages minus the voltage loss caused by the battery internal resistance v measurement noise h function of the sign of the battery current e small positive constant K0, K1, K2, K3 and K4 parameters of the battery open circuit voltage c battery model parameter vector U battery output voltage sequence M regressor matrix N the number of the sample data ^ State of Charge estimate at time step k Sk ^ predicted State of Charge value at time instant kjk 1 Skjk1 a posteriori State of Charge estimation error covariance Pk priori State of Charge estimation error covariance Pkjk1 Q process noise covariance V measurement noise covariance L ðiÞ window size for covariance matching ! sigma points S

h

based SOC estimation method was also integrated into an energy control algorithm for a hydraulic/electric synergy system for heavy hybrid vehicles [12]. However, the performance of the Coulomb counting method is highly dependent on the precision of current sensors. In practical electric vehicle operations, the open-loop algorithm can easily lead to accumulated measurement errors due to uncertain disturbances. A recalibration is, therefore, required at regular intervals. Nevertheless, the recalibration cannot easily be implemented in highly dynamic operations of BEVs/HEVs. Additionally, the determination of the initial SOC value is quite difﬁcult and error-prone. The second type is based on the black-box battery models that describe the non-linear relationship between the battery SOC and its inﬂuencing factors. The black-box models were often established by computational intelligenceebased approaches. Given an appropriate training data set, this kind of methods can provide good SOC estimates, due to the powerful capability of computational intelligence to approximate non-linear function surfaces. For example, an artiﬁcial neural network (ANN)-based available capacity computation model for lead-acid batteries in electric vehicles was built in [13]. The ANN model was demonstrated to be more accurate than the models based on the Peukert equation [13]. Shen et al. [14] established an ANN model to describe the nonlinear relationship between the SOC, the discharging current and the temperature of a lead-acid battery. The ANN method has also

ðiÞ

Wm ðiÞ Wc ðiÞ U kjk1 ^ U kjk1 Dk Ek G

m F Zmae,k Jk

predicted values for the sigma points scaling parameter small positive value mean weights covariance weights predicted output voltages for the sigma points at time instant kjk 1 predicted output voltage of the battery model at time instant kjk 1 ﬁlter covariance of the predicted output voltage at time instant kjk 1 ﬁlter cross-covariance between the predictions of the output voltage and State of Charge at time instant kjk 1 Kalman gain output voltage residual of the battery model approximation to the covariance of the voltage residual mean absolute error for the State of Charge estimates up to and including time step k derivative of the model output equation with respect to State of Charge at time instant kjk 1

List of abbreviations AEKF Adaptive Extended Kalman Filter ANN Artiﬁcial Neural Network AUKF Adaptive Unscented Kalman Filter BEVs Battery Electric Vehicles BMS Battery Management System BTS Battery Testing System CAN Controller Area Network CDKF Central Difference Kalman Filter EKF Extended Kalman Filter FUDS Federal Urban Driving Schedules HEVs Hybrid Electric Vehicles MAE Mean Absolute Error SOC State of Charge SPKF Sigma-Point Kalman Filter UKF Unscented Kalman Filter

been used in [15,16] for depicting the available capacity as a function of the discharged/regenerative capacity distribution and the temperature for lead-acid and nickel-metal hydride batteries. An SOC estimation model based on a radial basis function neural network has been studied in [17] for the SOC indication of lead-acid batteries. This model had good indication accuracies for different sized batteries and various degradation states, since the battery degradation degree was used as one of the model input signals [17]. In order to overcome the disadvantages of the ANN backpropagation training algorithm, Cheng et al. [18,19] applied immune evolutionary programming to train the ANN-based SOC estimation models for nickel-metal hydride batteries. In addition to the ANN-based methods, the fuzzy logic methodology has been studied for estimating the battery SOC. Salkind et al. [20] utilized the fuzzy logic methodology to implement the SOC estimation for lithium-ion and nickel-metal hydride batteries, on the basis of the training datasets obtained by impedance spectroscopy and coulomb counting techniques. Adaptive neuro-fuzzy inference systems have been investigated in [21,22] for describing the available capacity as a non-linear function of the discharged capacity distribution and the temperature for nickel-metal hydride and lithium-ion batteries. Singh et al. [23] built a fuzzy logic-based SOC meter for lithium-ion batteries for portable deﬁbrillators. The impedance and voltage recovery measurements were used as the input parameters for the fuzzy logic-based SOC meter [23]. In order

F. Sun et al. / Energy 36 (2011) 3531e3540

to take the time-dependent characteristics of a lead-acid battery into account, a simple fuzzy logic-based learning system has been developed in [24] for the battery SOC estimation. Due to a better capability to approximate non-linear functions than those of the ANN and fuzzy logic, support vector regression has also been used to estimate the battery SOC. For example, a support vector based battery SOC estimator has been developed in [25] for a large-scale lithium-ion-polymer battery pack. Shi et al. [26] established a v-support vector regression-based SOC estimation model which was demonstrated to perform better than the backpropagation ANNbased model. Hu et al. [27] used a fuzzy clustering based support vector regression algorithm to build a SOC estimator for lithium-ion batteries for electric vehicles and the SOC estimator could provide better SOC estimates in comparison to the standard support vector regression-based estimator. Despite good accuracies of these blackbox SOC estimation models, most of them had to be ofﬂine established in that the training processes were quite computationally heavy. Furthermore, the performance of the black-box modeling method is strongly reliant on the amount and reliability of the training data. Therefore, a poor robustness to varied battery operating conditions sometimes may occur. The third type is based on the Kalman ﬁltering approaches by means of state-space battery models. Despite a higher computational cost than the Coulomb counting-based methods, this type of method has the advantages of being closed-loop (self-corrected) and online as well as the availability of the dynamic SOC estimation error range. Thus, the Kalman ﬁltering methods are increasingly popular and more suitable for real-time battery management and control for BEVs/HEVs than the foregoing two types of SOC estimation methods. For instance, a Kalman ﬁlter based on a linear resistance-capacitance battery model has been designed in [30] for estimating the SOC of a lead-acid battery. Extended Kalman ﬁlters (EKF) based on the non-linear state-space battery models have been studied in [31,32] for the SOC estimation of batteries. However, in practice, the use of EKF has some shortcomings. For example, linearization would cause highly unstable ﬁlters if the assumptions of local linearity were violated [33]. Additionally, the derivation of the Jacobian matrices is nontrivial and error-prone in many applications [33,34]. As an alternative approach to state estimation for non-linear systems, sigma-point Kalman ﬁlter (SPKF) has a higher order of accuracy in estimating the mean and the error covariance of the state vector than EKF [33]. Additionally, SPKF does not require the calculation of the Jacobian matrices and the computational complexity is comparable to EKF [35]. The two most common SPKF-based methods are central difference Kalman ﬁlter (CDKF) and unscented Kalman ﬁlter (UKF). A CDKF based on a non-linear enhanced self-correcting battery model has been studied in [36] for the SOC estimation of a lithium-ion battery, demonstrating a better accuracy than EKF. A UKF based on a nonlinear electrochemical battery model has also been investigated in [37] for estimating the SOC of a lithium-ion cell. On the other hand, in the framework of Kalman ﬁlters, the process and measurement noise covariances are critical for the ﬁltering performance and stability. In EKF and SPKF, constant values of the process and measurement noise covariances often need to be prespeciﬁed by the trial-and-error method which is very timeconsuming, laborious and error-prone. If the process noise covariance and/or the measurement noise covariance were too small at the beginning of the estimation process, the uncertainty tube around the true value would tighten and a biased solution would result [38]. If the process noise covariance and/or the measurement noise covariance were too large, ﬁlter divergence would occur [38]. For the battery SOC estimation, the error would be large or even diverge if inappropriate values of the noise covariances were used. Therefore, adaptive extended Kalman ﬁlters (AEKF) have been

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studied in [39,40] so as to improve the accuracy of the EKF-based SOC estimation by adaptively updating the process and measurement noise covariances. In this paper, based on the zero-state hysteresis battery model [31,39], an adaptive unscented Kalman ﬁlter (AUKF) is developed to online estimate the SOC of a lithium-ion battery for battery electric vehicles. This method not only has the advantages of UKF over EKF, but also adaptively adjusts the values of the process and measurement noise covariances in the SOC estimation process, on the basis of the output voltage residual sequence of the battery model. Due to a simple structure of the zero-state hysteresis battery model, the AUKF algorithm has a low computational load so that the recursive SOC estimation can easily be implemented. Experimental results show that the AUKF-based SOC estimation algorithm has a good performance. A comparison with the AEKF, EKF, and UKF-based approaches indicates that the proposed method can estimate the SOC of the lithium-ion battery more accurately. The remainder of this paper is organized as follows. A description of the structure and parameterization of the zero-state hysteresis battery model is given in Section 2. The AUKF-based SOC estimation algorithm is depicted in Section 3. The estimation results of the proposed algorithm are discussed in Section 4. A comparison with the AEKF, EKF and UKF-based SOC estimation approaches is illustrated in Section 5. Finally, conclusions are drawn in Section 6. 2. Battery modeling 2.1. Model structure The zero-state hysteresis battery model structure is used to describe the battery voltage behavior, which was often applied to the Kalman ﬁlter-based SOC estimation [31,39,40]. The discretetime state and output equations of the battery model are shown as follows:

Skþ1 ¼ Sk

Uk ¼ K0

hk Dt Cn

Ik þ uk

K1 K2 Sk þK3 lnðSk ÞþK4 lnð1Sk ÞRIk hk H þvk Sk

(1)

(2)

where S is the battery SOC, h is the Coulombic efﬁciency and Cn is the nominal capacity of the battery. Dt and the subscript k are the sampling interval and the time step index, respectively. U and I represent the battery output voltage and the current, respectively. R denotes the internal resistance. u and v are the process and measurement noises. H is half the hysteresis between charging and

Fig. 1. Schematic diagram of the battery test bench.

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F. Sun et al. / Energy 36 (2011) 3531e3540

Fig. 3. Experimental SOC proﬁle.

Fig. 2. Current and voltage proﬁles.

discharging voltages minus the RI loss. h is a function of the sign of the current described as follows:

8 < 1; hk ¼ 1; : hk1 ;

Ik > e Ik < e jIk j e

(3)

where e is a small positive number. K0, K1, K2, K3 and K4 are ﬁve unknown parameters to depict the battery open circuit voltage. 2.2. Model parameter calibration 2.2.1. Battery test bench As shown in Fig. 1, the experimental setup consists of a Digatron Battery Testing System (BTS-600), a battery management module, a controller area network (CAN) communication unit and a Labview-based virtual measurement unit. The Digatron Battery Testing System is responsible for loading the battery based on the designed program with maximum voltage of 500 V and maximum charging/discharging current of 500 A. The recorded signals include load current, terminal voltage, temperature, accumulative Amphour (Ah) and Watt-hour (Wh). The battery management module can collect the voltage and temperature of each cell in the battery. The measured load current by a Hall current sensor is transmitted to the battery management module through CAN bus driven by the Labview program and CAN communication unit. The errors of the Hall current and voltage sensors are less than 0.2% and 0.5%, respectively. Both the Labview-based measurement unit and the management module have a low-pass ﬁltering function incorporated to implement large noise cancellation. Given the sampled current and voltage, the battery SOC estimates can be attained by the designed algorithm in the battery management module. The measured voltage, temperature and estimated SOC are then transmitted through CAN bus to the Labview for real-time display.

2.2.2. Estimation data A lithium-ion battery module composed of sixteen cells in series was used in experimentation. Each healthy cell has a nominal output voltage of 3.6 V and a nominal capacity of 100 Ah. The actual capacity of the module was 86.5 Ah, less than the nominal capacity of a healthy cell due to deviant behaviors of cells in the module. The lithium-ion battery was developed for heavy battery electric vehicles. Since the Federal Urban Driving Schedules (FUDS) testing cycle was often applied to validate diverse battery SOC estimation algorithms for battery electric vehicles [15,16,18,19,22,25,40], a FUDS test for the lithium-ion battery was conducted. An actual peak power value for the battery was speciﬁed in the FUDS test. In order to obtain a typical battery SOC range in the daily driving of heavy BEVs, the initial SOC of the battery was chosen to be 93%. The experimental SOC values were computed by the Digatron Battery Testing System (BTS-600) which can offer sufﬁciently accurate accumulative Amp-hours ﬂowing out of the battery. The sampling interval was half a second. The sampled current and voltage during FUDS cycles are shown in Fig. 2. The discharging current is positive and the charging current is negative. That means that both discharging and charging processes existed in the test. However, the general descending trend of the battery voltage indicates that the test mainly discharged the battery. Fig. 3 shows the experimental SOC values. The SOC seems to keep falling. The magniﬁed view, however, can show more information on the rise and fall of the battery SOC. 2.2.3. Calibration algorithm and result Given the sampled current, voltage and experimental SOC values of the battery, the parameter vector c of the battery model can be solved using the least squares algorithm as follows:

h iT 1 c ¼ K0 ; K1 ; K2 ; K3 ; K4 ; Rþ ; R ; H ¼ M T M MT U

(4)

where U ¼ [U1,U2,.,UN]T is the output voltage sequence, M ¼ [M1,M2,.,MN]T is the regressor matrix in which M k ¼ ½1; ð1=Sk Þ; Sk ; lnðSk Þ; lnð1 Sk Þ; Ikþ ; Ik ; hk .As shown Table 1 Result of the model parameter calibration. K0

K1

K2

K3

K4

Rþ (U)

R (U)

H

64.172

0.179

0.168

2.817

0.247

0.0184

0.0167

0.003

F. Sun et al. / Energy 36 (2011) 3531e3540

Fig. 4. Measured and estimated battery voltage responses.

in [31] and [39], different values of the internal resistance R should be used for the battery discharging and charging processes so as to accurately describe the battery voltage behavior. Here, Rþ and R are used to represent the discharging and charging resistances, respectively. Ikþ and Ik accordingly denote the discharging and charging currents, respectively. During the discharging process (Ik 0), Ikþ ¼ Ik and Ik ¼ 0; during the charging process (Ik < 0), Ikþ ¼ 0 and Ik ¼ Ik . The subscript N denotes the number of the sample data. The result of the parameter calibration is shown in Table 1. Fig. 4 shows the measured and estimated voltage responses of the battery. To be more readable, those for the ﬁrst FUDS cycle are shown in Fig. 5. A magniﬁed portion of the ﬁrst FUDS cycle is shown in Fig. 6 to show more details. The model prediction error is shown in Fig. 7. The maximum and mean relative errors are 2.012% and 0.178%, respectively. The battery model can essentially simulate the battery voltage response.

3. AUKF algorithm based on the battery model UKF consists of the prediction and update processes, as illustrated in detail in [34,35,41]. In the two processes of UKF, constant

Fig. 5. Measured and estimated voltages in the ﬁrst FUDS cycle.

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Fig. 6. A magniﬁed portion of the measured and estimated voltages in the ﬁrst FUDS cycle.

covariance values of the process and measurement noise are used. In order to achieve an AUKF algorithm, adaptively adjusting covariance values of the process and measurement noise should be employed. Herein, the idea of covariance matching on the basis of the model output residual sequence proposed by Mohamed et al. [38] is extended to the UKF setting. The covariance matching in the UKF setting is used to realize the adaptive adjustment of the noise covariances so that a residual-based AUKF algorithm can be developed. The steps of the AUKF-based SOC estimation algorithm based on the battery model are illustrated as follows. (1) Initialization Initial SOC estimate: ^ S0 Initial a posteriori error covariance: P0 Initial process noise covariance: Q0 Initial measurement noise covariance: V0 Window size for covariance matching: L (2) Prediction (a) Generate sigma points at time step k 1

!ð0Þ S k1 ¼ ^ Sk1

(5)

Fig. 7. Relative voltage error.

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F. Sun et al. / Energy 36 (2011) 3531e3540

!ð1Þ S k1 ¼ ^ Sk1 þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 þ lÞPk1

(6)

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ !ð1Þ S kjk1 ¼ ^ Skjk1 þ ð1 þ lÞPkjk1

(16)

!ð2Þ Sk1 S k1 ¼ ^

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 þ lÞPk1

(7)

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ !ð2Þ Skjk1 ð1 þ lÞPkjk1 S kjk1 ¼ ^

(17)

!ð0Þ !ð1Þ !ð2Þ where S k1 , S k1 and S k1 are the three sigma points at time step k 1, l ¼ 3a2 1 is a scaling parameter. a determines the spread of the sigma points around ^ Sk1 and is set to a small positive value. (b) Compute the predicted SOC ^ Skjk1 and a priori error covariance Pkjk1

!ðiÞ h Dt ðiÞ Skjk1 ¼ S k1 k1 Ik1 ði ¼ 0; 1; 2Þ Cn 2 X

^ Skjk1 ¼

2 X

þ Hhk ði ¼ 0; 1; 2Þ

ðiÞ ðiÞ Wm Skjk1

(9)

ðiÞ

Wc

ðiÞ Skjk1 ^ Skjk1

ðiÞ Skjk1 ^ Skjk1

T

þQk1

(10)

ðiÞ

ð0Þ

¼

Wc

ðjÞ

Wm ¼ ðjÞ

Wc

¼

l

(11)

lþ1 l

lþ1

ð18Þ

ðiÞ

ðiÞ

Wm U kjk1

(19)

i¼0 ðiÞ

where Skjk1 are the predictions of the sigma points at time instant ðiÞ ðiÞ kjk 1, Wm and Wc are mean and covariance weights which can be determined by the following equations:

¼

2 X

^ U kjk1 ¼

where U kjk1 are the predicted output voltages for the sigma points at time instant kjk 1. (c) Compute Kalman gain Gk

i¼0

ð0Þ Wm

!ðiÞ !ðiÞ K1 K2 S kjk1 þ K3 ln S kjk1 !ðiÞ S kjk1 !ðiÞ þ K4 ln 1 S kjk1 Rþ Ikþ R Ik

ðiÞ

U kjk1 ¼ K0

(8)

i¼0

Pkjk1 ¼

^ (b) Compute the predicted output voltage U kjk1

þ 1 a2 þ b

(12)

Dk ¼

2 X

ðiÞ

Wc

ðiÞ

^ U kjk1 U kjk1

ðiÞ

^ U kjk1 U kjk1

T

þVk1

(20)

i¼0

Ek ¼

2 X

ðiÞ

Wc

!ðiÞ ðiÞ T ^ S kjk1 ^ Skjk1 U kjk1 U kjk1

(21)

i¼0

Gk ¼ Ek D1 k

(22)

1 ðj ¼ 1; 2Þ 2ð1 þ lÞ

(13)

1 ðj ¼ 1; 2Þ 2ð1 þ lÞ

where Dk is the ﬁlter covariance of the predicted output voltage at time instant kjk 1, Ek is the ﬁlter cross-covariance between the predictions of the output voltage and SOC at time instant kjk 1. (d) Achieve the estimated SOC ^ Sk and a posteriori error covariance Pk at time step k

(14)

^ ^ Skjk1 þ Gk Uk U Sk ¼ ^ kjk1

(23)

Pk ¼ Pkjk1 Gk Dk GTk

(24)

where b ¼ 2 is used to incorporate prior knowledge of the distribution of the SOC estimates. (3) Update (a) Generate sigma points at time instant kjk 1

!ð0Þ S kjk1 ¼ ^ Skjk1

(15)

where Uk is the measured battery voltage at time step k. (4) Adjustment of Qk1 and Vk1

K

mk ¼ Uk K0 1 K2 ^Sk þ K3 lnð^Sk Þ ^ Sk

þK4 lnð1 ^ Sk Þ Rþ Ikþ R Ik Hhk

(25)

Pk m mT Fk z n ¼ kLþ1 n n L

Vk ¼ Fk þ

2 X

ðiÞ

Wc

(26)

ðiÞ

U kjk1 Uk þ mk

ðiÞ

U kjk1 Uk þ mk

T

(27)

i¼0

Table 2 Parameters of the AUKF algorithm.

Fig. 8. Schematic of AUKF-based SOC estimation algorithm.

P0

Q0

V0

L

0.01

0.01

0.5

10

F. Sun et al. / Energy 36 (2011) 3531e3540

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Fig. 9. SOC estimation results of AUKF in Case 1 and Case 2.

Qk ¼ Gk Fk GTk

(28)

Fig. 11. SOC estimation errors of AUKF in Case 1 and Case2 after compensating the initial errors.

where mk is the voltage residual of the battery model at time step k and Fk is an approximation to the covariance of the voltage residual at time step k. The schematic diagram of the proposed AUKF-based SOC estimation algorithm is shown in Fig. 8. After initialization, the estimated SOC ^ S1 and the estimation error covariance P1 can be achieved through the prediction and update processes. The voltage residual m1 is subsequently calculated on the basis of the battery model. Then, the voltage residual-based covariance matching is used to obtain the process noise covariance Q1 and the measurement noise covariance S1 , P1, Q1 and V1 are then used for the next prediction and update V1. ^ processes. Repeat the procedure to be able to recursively estimate the battery SOC. The load voltage and current of the battery are merely needed for the algorithm in the SOC estimation process.

better evaluate the insensitivity of the proposed algorithm to the initial SOC estimate. The estimation results are shown in Fig. 9. It can be found that the algorithm can quickly compensate the initial SOC errors and accurately track the experimental SOC values in both Case 1 and Case 2. The difference between the two results after correcting the initial errors is virtually indiscernible. Fig. 10 shows the estimation errors. In order to show more details, the estimation errors after compensating the initial errors are shown in Fig. 11. It is clear that the errors in both Case 1 and Case 2 can converge into a 1.7% error band. The mean absolute error (MAE) of the SOC estimates was calculated using the following equation:

4. Experimental results for the SOC estimation

Pk Zmae;k ¼

j¼0

jSj ^ Sj j

kþ1

(29)

The parameters of the AUKF-based SOC estimation algorithm were speciﬁed as shown in Table 2. The initial SOC estimate ^ S0 was set to 70% and 50% in Case 1 and Case 2, respectively, in order to

where Zmae,k is the MAE for the SOC estimates up to and including time step k and Sj is the experimental SOC at time step j. Fig. 12 shows the MAE results. The MAE values in both cases can be reduced to less than 1%.

Fig. 10. SOC estimation errors of AUKF in Case 1 and Case 2.

Fig. 12. MAE results of AUKF in Case 1 and Case 2.

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F. Sun et al. / Energy 36 (2011) 3531e3540

Table 3 Parameters of EKF and UKF. P0

Q0

V0

0.01

0.000001

1

5. Comparison with AEKF, EKF and UKF In order to further evaluate the performance of the AUKF-based SOC estimation algorithm, a comparison with the AEKF, EKF and UKF-based estimation methods was made. The adaptive adjustment of the process and measurement noise covariances in the AEKF algorithm was implemented by the voltage residual-based covariance matching in the EKF context. The equations of the AEKF algorithm based on the battery model are described as follows: Prediction

h Dt ^ Sk1 k1 Ik1 Skjk1 ¼ ^ Cn

(30)

Pkjk1 ¼ Pk1 þ Qk1

(31)

Update

Jk ¼

K1 2 b S kjk1

Fig. 14. Estimation errors of AUKF, AEKF, UKF and EKF in Case 1.

5.1. Adjustment of Qk1 and Vk1

K2 þ

K3 K4 ^ Skjk1 1^ S

(32)

kjk1

Gk ¼ Pkjk1 JkT Jk Pkjk1 JkT þ Vk1

1

(33)

K1 ^ K2 ^ U Skjk1 þ K3 ln ^ Skjk1 kjk1 ¼ K0 ^ Skjk1 Skjk1 Rþ Ikþ R Ik þ Hhk þK4 ln 1 ^

(34)

^ ^ Skjk1 þ Gk Uk U Sk ¼ ^ kjk1

(35)

Pk ¼ Pkjk1 Gk Jk Pkjk1 JkT þ Vk1 GTk

(36)

Fig. 13. SOC estimation results of AUKF, AEKF, UKF and EKF in Case 1.

Vk ¼ Fk þ Jk Pk JkT

(37)

Qk ¼ Gk Fk GTk

(38)

where the approximation to the voltage residual covariance Fk is computed by Eqs. (25) and (26), given ^ Sk determined by Eq. (35). The EKF-based SOC estimation algorithm can be obtained using constant values of Qk and Vk in the prediction and update processes shown in Eqs. (30)e(36) instead of the adjustment in Eqs. (37) and (38). Similarly, the UKF algorithm can be determined by removing the adaptive adjustment of the process and measurement noise covariances from the AUKF algorithm. The window size L for covariance matching in the AEKF was set to 100, which was

Fig. 15. MAE results of AUKF, AEKF, UKF and EKF in Case 1.

F. Sun et al. / Energy 36 (2011) 3531e3540

3539

MAE results for the AUKF, AEKF, UKF and EKF-based algorithms in Case 2 (^ S0 ¼ 50%) are shown in Fig. 16 and Fig. 17. Likewise, it can be seen that the AUKF-based algorithm has a better accuracy in estimating the battery SOC. 6. Conclusions

Fig. 16. Estimation errors of AUKF, AEKF, UKF and EKF in Case 2.

testiﬁed to be better than 10 used by the AUKF algorithm. The other parameters of the AEKF algorithm were the same as those of the AUKF algorithm. The trial-and-error approach was applied to determine the parameters of EKF and UKF. As shown in Table 3, the identical parameters were speciﬁed for EKF and UKF. The comparison result for the SOC estimation in Case 1 (^ S0 ¼ 70%) is shown in Fig. 13. The associated estimation errors are shown in Fig. 14. In order to be more readable, the MAE results are shown in Fig. 15. It is clear that the AUKF-based algorithm can estimate the battery SOC more accurately, compared to the other three algorithms. Additionally, it can be seen that the precisions of the AUKF and AEKFbased estimation algorithms are higher than those of the EKF and UKF algorithms. This result indicates that adaptive adjusting covariance values of the process and measurement noise are highly beneﬁcial to the enhancement of the SOC estimation accuracy. A better performance for the UKF algorithm can also be observed in comparison to the EKF algorithm. The estimation errors and the

Fig. 17. MAE results of AUKF, AEKF, UKF and EKF in Case 2.

An adaptive unscented Kalman ﬁlter (AUKF) has been developed to online estimate the SOC of a lithium-ion battery for electric vehicles. This ﬁlter not only has the advantages of UKF over EKF, but also adaptively adjusts the values of the process and measurement noise covariances in the SOC estimation process. The idea of covariance matching on the basis of the model output residual sequence is extended to the UKF setting for implementing the adaptive adjustment. The proposed SOC estimation algorithm is based on the zero-state hysteresis battery model. Due to a simple structure of the battery model, the AUKF algorithm has a low computational load which beneﬁts the recursive SOC estimation. Experimental results show that the AUKF-based algorithm can estimate the battery SOC accurately. A comparison with the AEKF, EKF, and UKF-based algorithms indicates that the proposed method has a superior performance. Since the zero-state hysteresis battery model cannot depict the battery relaxation effect, future work could be focused on building a more accurate battery model which can describe the relaxation effect with low computational complexity. Then, based on the more accurate battery model, a better performance of the AUKF-based SOC estimation algorithm could be anticipated. Acknowledgments The authors would like to express deep gratitude to Professor Huei Peng in University of Michigan for many helpful suggestions, as well as to acknowledge the ﬁnancial support from China Scholarship Council. This work was supported by the National Natural Science Foundation of China (50905015) and the National High Technology Research and Development Program of China (2003AA501800). References [1] Johansson B, Mårtensson A. Energy and environmental costs for electric vehicles using CO2-neutral electricity in Sweden. Energy 2000;25:777e92. [2] Smith WJ. Can EV (electric vehicles) address Ireland’s CO2 emissions from transport? Energy 2010;35:4514e21. [3] Kühne R. Electric buses e an energy efﬁcient urban transportation means. Energy 2010;35:4510e3. [4] Åhman M. Primary energy efﬁciency of alternative powertrains in vehicles. Energy 2001;26:973e89. [5] Piller S, Perrin M, Jossen A. Methods for state-of-charge determination and their applications. Journal of Power Sources 2001;96:113e20. [6] Aylor JH, Johnson BW. A battery state-of-charge indicator for electric wheelchairs. IEEE Transactions on Industrial Electronics 1992;39:398e409. [7] Liu TH, Chen DF, Fang CC. Design and implementation of a battery charger with a state-of-charge estimator. International Journal of Electronics 2000;87: 211e26. [8] Çadırcı Y, Özkazanç Y. Microcontroller-based on-line state-of-charge estimator for sealed lead-acid batteries. Journal of Power Sources 2004;129: 330e42. [9] Prasad AR, Natarajan E. Optimization of integrated photovoltaicewind power generation systems with battery storage. Energy 2006;31:1943e54. [10] Avril S, Arnaud G, Florentin A, Vinard M. Multi-objective optimization of batteries and hydrogen storage technologies for remote photovoltaic systems. Energy 2010;35:5300e8. [11] Arun P, Banerjee R, Bandyopadhyay S. Optimum sizing of battery-integrated diesel generator for remote electriﬁcation through design-space approach. Energy 2008;31:1155e68. [12] Sun H, Lifu Y, Jing JQ. Hydraulic/electric synergy system (HESS) design for heavy hybrid vehicles. Energy 2010;35:5328e35. [13] Chan CC, Lo EWC, Shen W. The available capacity computation model based on artiﬁcial neural network for lead-acid batteries in electrical vehicles. Journal of Power Sources 2000;87:201e4.

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