- Email: [email protected]

ScienceDirect Energy Procedia 105 (2017) 2678 – 2683

The 8th International Conference on Applied Energy – ICAE2016

The Estimation of State of Charge for Power Battery Packs used in Hybrid Electric Vehicle Shanshan Xiea,b, Rui Xionga,b, Yongzhi Zhanga,b, Hongwen Hea,b,* a

National Engineering Laboratory for Electric Vehicles, School of Mechanical Engineering, Beijing Institute of Technology, Beijing, 100081, China b Collaborative Innovation Center of Electric Vehicles in Beijing, Beijing Institute of Technology, Beijing, 100081, China

Abstract The estimation of state of charge (SOC) for power battery packs is significant for a hybrid electric vehicle with providing data supports for the efficient and fine energy managements. In this brief, extended Kalman filter (EKF), multi-model extended Kalman filter (MMEKF) and adaptive fading extended Kalman filter (AFEKF) are used to estimate SOC respectively, then they are combined with a switching strategy to match their own features with that of SOC in different working areas. The estimation error of SOC is below 2.5 percent. And the strategy is verified to have good initialization stabilities and convergence behaviors. © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2016 The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and/or peer-review under responsibility of ICAE Peer-review under responsibility of the scientific committee of the 8th International Conference on Applied Energy. Key words: Power battery pack; State of charge; Extended Kalman filter; Hybrid electric bus

1. Introduction The estimation of state of charge is significant for the control and management of a hybrid electric vehicle. But it is a tacit parameter and cannot be detected directly. Many esitmation approaches based on different battery models and algorithms are proposed [1]. Among them, Plett [2, 3] proposed to implement parameter identification and SOC estimation of power batteries with extended Kalman filter based on simplified equivalent electric battery model. This method is explored and discussed thoroughly in the next several years. In order to improve the model accuracy and model robustness against inaccurate initial values, battery models and filters are both improved [4, 5]. But there are still many problems like great calculation burden, real-time performance to be solved. In this brief, multiple linear regression is used to identify specific parameters of Thevenin battery model offline. Then EKF, MMEKF and AFEKF are used in turn to estimate SOC. According to the results, main error sources for SOC estimation in different SOC ranges are analyzed. Considering the matching specific characters of algorithms and corresponding SOC changing laws can be conducive to

* Corresponding author. Tel.: +86 (10) 6891 4842; fax: +86 (10) 6891 4842. E-mail address: [email protected]

1876-6102 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the 8th International Conference on Applied Energy. doi:10.1016/j.egypro.2017.03.774

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solving the problems, EKF, MMEKF and AFEKF are combined with a transferring strategy to estimate SOC. 2. The Model of Battery The battery sample is composed with 10 individual cells that are connected in series. The total capacity is 36 Ah. And the anode material of the lithium battery is lithium manganite. The primary components of the battery test bench is Arbin BT2000 (Fig.1), a mulita-function battery test system. The machine conduct charge-discharge experiments on batteries according to the programs that are set in advance. Moreover, the corresponding reactions of batteries are shown on the screen by TIP/IP. The results of three experiments, which are open-circuit voltage experiment (showing the function relationship between opencircuit voltage and SOC), improved HPPC [6] and BJDC experiments (the city driving cycle in Beijing) respectively.

Fig. 1. (a)The configuration of the battery test bench; (b) Thevenin model of power battery (symbol, see Table 1)

As for the battery model, Thevenin model belongs to equivalent circuit model and can depict the dynamic external characteristics of batteries effectively [7]. The equations (Eq. (1)) derive from Kirchhoff Voltage Law (KVL) and Kirchhoff Current Law (KCL). After discretizing the equations with Taylor’s formula, combining the experiment data of improved HPPC, the related parameters can be identified offline by Multiple Linear Regression (MLR). Table 1. Equations of Thevenin model for different uses Item

State equation/ Observer equation

Original form

EKF

Up x IL (1) °U p C R p pC p ® °U U U I R oc p L 0 ¯ t

T (2) 0 °ª Soc(k 1)º 1 Soc(k ) q (k ) Q0 T °« ][ ][ ][ I L (k )] [ 1 ] [0 1 » T q2 ( k ) °¬ U p (k 1) ¼ C p R p U p (k ) ® Cp ° SoC ° wU oc 1][ ] [ R0][ I L (k )] [r (k )] °[U L (k )] [ Up wSoC ¯

* U t denotes the terminal voltage,

U oc is

the open-circuit voltage, R0 is the ohm inner resistance,

polarization resistance, C is the equivalent polarization capacitor, U is the voltage of p p

q1 and q 2 are the system noises, T is the sampling period. 3. Estimations of SOC based on EKF, MMEKF and AFEKF 3.1. Introduction of EKF, MMEKF and AFEKF

R p denotes the equivalent

R p and C p , r is the measurement noise,

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Kalman filter is a real-time recursion algorithm for linear system. When it comes to nonlinear systems, the system should be linearized firstly by neglecting the high order terms of Tylor series before using KF. The whole process can be called EKF. The common numeric problems of EKF are linearization errors and filter divergences that are caused by the estimation of Jacobian determinant [8]. MMEKF are composed of several independent EKFs with different initial values for system noises and measurement noises, so a better cover of real models can be achieved than EKF does. A residual analysis between real values and every filter result is conducted to measure the reliability of the filters and by weighting all the results, the final estimation can be gotten [9]. In certain occasions, filters are required to perform well in dynamic tracing and AFEKF is advisable. The principle is to introduce a forgetting factor to restrict the memory length of EKF so it can make use of current measurement data fully. By this way, the filter can accommodate to mutations of state variables and the tracing ability is improved [10]. Assuming the state-space equation of a discrete system is Eq. (3), the procedures of calculating the forgetting factor in AFEKF can be seen in Table 2. x(k ) A(k - 1) x(k - 1) B(k - 1)u (k - 1) q(k 1) (3) ® ¯ z (k ) C (k ) x(k ) D(k )u (k ) r (k )

Fig.2. Logical structure of MMEKF Table 2. Algorithm of forgetting factor O in AFEKF Step

Procedure

Operation O (1) , d 0 , K , D , D *

1

Initialization

2

Weighted factor

3

Error variance

4

Regulating variable

5

——

M (k ) C (k ) A(k ) P(k ) P T (k ) AT (k ) **

——

N (k )

6 7

Optimum forgetting factor

°d , e(k )eT (k ) t D d (k ) ® 0 °¯1, e(k )eT (k ) D

(4)

T °O (k 1)e(k )e (k ) /[1 O (k 1)], k ! 2 C0 (k ) ® T ° ¯e(1)e (1) / 2, k 1 ˆ E [ X (k ) Xˆ (k 1)]2K

(6) (7)

d (k )C0 (k ) C (k )QCT (k ) R E (k )

O (k 1) max^1, D tr[ N (k )] / tr[M (k )]`

(5)

***

****

(8) (9)

* K and D are the adjustment coefficients, and D ! 1 . D is the proper limit of estimation error. Other parameters keep consistent with Table 4. **P denotes Error covariance for state forecast . ***Q denotes system noise while R is measurement noise. **** tr[] means Matrix Trace.

3.2. Estimation results of SOC based on EKF, MMEKF and AFEKF When EKF is used to estimate SOC, the noises are considered as white noise, which means the average values are 0. The other initial values can refer to Table 5 except the parameters in Table 3. In order to test the sensitivities of filters on initial values, SOC is set as 1 and 0.8 in the beginning

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respectively. The SOC estimation traces of the method based on EKF are shown in Fig.3 and that based on MMEKF and AFEKF are similar to Fig.3 (a) which are omitted. All the estimation errors are within 8 percent. Table 4 verifies that the three approaches are in the similar level. Table 3. A part of initial values of the three filters State

EKF ª10 0 º « 0 10» ¼ ¬

P0

MMEKF ª1 0 º «0 1 » ¼ ¬ 1000

100

R

AFEKF ª1 0 º «0 1 » ¼ ¬ 1000

Fig. 3. (a) SOC estimation results based on EKF when SOC0 is 1; (b) SOC estimation results based on EKF when SOC 0 is 0.8 Table 4. Statistical information of SOC estimation errors of the three filters Group

x0

Mean

Standard Deviation

Minimum

Median

Maximum

[1 1]

-0.00719

0.02864

-0.08992

-0.00993

0.06761

>0.8 [email protected]

-0.00717

0.02869

-0.08992

-0.00993

0.015

[1 1]T

-0.00707

0.02851

-0.08878

-0.01009

0.05779

>0.8 [email protected]

-0.00733

0.02869

-0.1923

-0.01016

0.05779

-0.00656

0.02811

-0.08091

-0.00991

0.05273

-0.00686

0.02827

-0.19233

-0.00998

0.05273

T

EKF

MMEKF

T

AFEKF

[1 1]

>0.8 [email protected]

T

4. Estimation of SOC based on the Switching Strategy 4.1. The Principle of Switching Strategy MMEKF can convergent swiftly and cover a wide range of noises, thus have a good adaptability for estimation models; with an accurate initial values, EKF is able to achieve a pretty ideal results; according to the principles, AFEKF can improve estimation results when the state variables change suddenly or the models distort. Therefore, at the beginning of the estimation, use MMEKF to enhance the sensitivity on initial values; when convergence comes to an end, turn to EKF to reduce calculation burden when the accuracy is maintained; because battery is a highly non-linear system, when SOC is too low the battery model is easy to distort, so AFEKF is adopted. One key point of this strategy is the switching basis. For the first switching point, namely from

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MMEKF to EKF, there are two methods which refer to convergence rate and time respectively. Here the latter one is adopted. After the estimation has been started for 't , change filter from MMEKF to EKF, where 't is set according to experiences. Then the estimated SOC is used to complete the second alteration, for example, when SOC is lower than 0.45. Although the estimation value may not be precise, it can meet the requirement because the changing point do not need to be too exact. Table 5. Initialization of every filter Group

R

Q1

Q2

P0

MMEKF

1

ª0.01 0 º « 0 0.01»¼ ¬

0 º ª0.001 « 0 0.001»¼ ¬

0 º ª0.01 « 0 0.05»¼ ¬

EKF

900000

üü

0 º ª0.001 « 0 0.001»¼ ¬

ª0.01 0 º « 0 0.05»¼ ¬

AFEKF

900000

üü

0 º ª0.001 « 0 0.001»¼ ¬

ª0.01 0 º « 0 0.05»¼ ¬

Other ptr

ª0.75 0.25º «0.25 0.75» muk ¼ ¬

ª 0 .5 º « 0 .5 » ¬ ¼

üü

O 1 d0

(a)

(b)

(c)

(d)

5D

11 K

0

Fig. 4. (a) SOC estimation trace when declining from1 to 0.23; (b) SOC estimation error when declining from1 to 0.23 (c) SOC estimation trace when declining from0.77 to 0.3; (d) SOC estimation error when declining from0.77 to 0.3 Table 6. Statistical information of SOC estimation errors of switching strategy SOC range

Mean

Standard Deviation

Minimum

Median

Maximum

1~0.23

-0.000635

0.01097

-0.01987

-0.000705

0.05203

0.77~0.3

-0.00806

0.01534

-0.03332

-0.00878

0.2368

Fig. 3. (a) The distribution of working modes in control group; (b) the proportions of different working modes (c) Working points of engine; (d) the distribution of working points of engine, motor and ISG motor

In Fig.4, it is noticeable that no matter the start point of SOC is 1 or 0.7, the initial value is set as 1 and the algorithm can always convergent rapidly. And take SOC range 1~0.23 as an example, the mean of

Shanshan Xie et al. / Energy Procedia 105 (2017) 2678 – 2683

estimation errors of switching strategy (Table 6) are an order lower than that of EKF, MMEKF or AFEKF individually (Table 4) and the corresponding standard deviations half. The fluctuation margins are also improved significantly, comparing 0.07 with around 0.15. Besides, the estimation error is controlled with 2.5 percent, which is much better than 8 percent, the general estimation errors of strategies based on the other three methods. 5. Conclusions x Three SOC estimation approaches are conducted based on EKF, MMEKF and AFEKF in a row and the estimation errors are within 8 percent. x Through analyzing the features of every filter and SOC changing rules in different working areas, these characteristics are matched properly to compose a switching strategy; the bases of switching points are running time and SOC estimated value respectively. x The estimation result of switching strategy is more ideal than that of other three individual filters with an error below 2.5 percent; besides, this method has a good stability on initialization and the start point of SOC. x At one time, only one filter is used in the switching strategy, so the calculation burden is acceptable and real-time performance can be maintained. x The switching strategy is a common idea; the specific components can be altered flexibly; this idea highlights the importance of analyzing the features or the sources of problems. References [1] Moura SJ, Chaturvedi NA, Krstić M. Adaptive Partial Differential Equation Observer for Battery State-of-Charge/State-ofHealth Estimation Via an Electrochemical Model. ASME. J. Dyn. Sys., Meas., Control. 2013;136(1):011015-011015-11. [2]Gregory L Plett.Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 3. State and parameter estimation[J].Journal of Power Sources, 2004, 134(2): 277-292. [3]Gregory L Plett.Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 2. Modeling and identification[J].Journal of Power Sources, 2004, 134(2): 262-276. [4]Santhanagopalan Shriram, White Ralph E. State of charge estimation using an unscented filter for high power lithium ioncells[J].International Journal of Energy Research, 2010, 34(2): 152-163. [6]Freedom Car Battery Test Manual For Power-Assist Hybrid Electric Vehicles. DOE /ID-11069[R]. Washington, DC: US Department of Energy, 2003. [7]Hongwen He, Xiaowei Zhang, Rui Xiong, Yongli Xu, Hongqiang Guo. Online model-based estimation of state-of-charge and open-circuit voltage of lithium-ion batteries in electric vehicles [J]. Energy, 2012, 39: 310-318. [8] Mengyin Fu, Zhihong Deng, Liping Yan. Kalman filter theory and the application in navigation system [M]. Beijing: Science Press. (in Chinese) [9] Xin Liu, Zhihong Deng, Bo Wang, Mengyin Fu. Transfer alignment method based on multi-model filter with uncertain flexural deflection disturbance [J]. Systems Engineering and Electronics, 2013, 35(10): 2145-2151. (in Chinese) [10] Zhong Wang, Zhisheng You, Zhuanli Du, Hui Wang. GPS/INS dynamic Kalman filter optimization algorithms [J]. Journal of Sichuan University (engineering science), 2006, 38(4):141-149. (in Chinese)

Biography Hongwen He is currently a Professor of the National Engineering Laboratory for Electric Vehicles, Beijing Institute of Technology and a researcher with the Beijing Co-innovation Center for Electric Vehicles. His research interests include power battery modeling and simulation on electric vehicles, and control theory of the hybrid power train.

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