Online Estimation of State-of-charge Based on the H ... - Science Direct

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ScienceDirect Energy Procedia 105 (2017) 2791 – 2796

The 8th International Conference on Applied Energy – ICAE2016

Online estimation of state-of-charge based on the H infinity and unscented Kalman filters for lithium ion batteries Quanqing Yu, Rui Xiong*, Cheng Lin National Engineering Laboratory for Electric Vehicles, School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China.

Abstract The state of charge (SOC) is a key indicator for the battery management system (BMS) of electric vehicles. A SOC joint estimation method based on the H infinity filter (HF) and unscented Kalman filter (UKF) algorithms is proposed in this paper, HF based parameters identification can trace the parameters online according to the working conditions while he UKF based state estimation method does not require the jacobian matrix derivation and the linearization for nonlinear model. The HF-UKF SOC joint estimation method has been experimentally validated at different temperatures. The results show that this method is robust to the inaccurate initial SOC value and the different working temperatures. © 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.

Keywords: state of charge; H infinity filter; unscented Kalman filter; lithium-ion battery;

1. Introduction The accurate state of charge (SOC) estimation is not only the main concern in the battery management system (BMS), but also the basis of the efficient, safe and reliable operation of EVs. Due to the strong nonlinear and time-variability characteristics of the power battery, the performance of the battery is easily affected by the ambient environment, aging and other factors. Therefore, the SOC estimation approach based on the identified model is difficult to accurately estimate the SOC of whole working conditions [12]. SOC cannot be directly measured when the battery is working. The general approach is to estimate the SOC through some measured information, such as current and voltage. A number of approaches have been proposed for SOC estimation and each one has its own merits and drawbacks. Ampere-hour counting is the one of the most conventional method, but it suffers from the unknown information of the

* 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.2017.03.600

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initial SOC and the accumulated error over time due to the integration process [3]. The open-circuit voltage (OCV) based estimation is another effective approach, but the drawback is the relationship between the SOC and OCV varies with ambient temperatures and aging [4]. Compared with the offline electrochemical impedance spectroscopy (EIS) method [5], the online observer-based techniques, such as the Kalman filter (KF) [6], extended Kalman filter (EKF) [7], unscented kalman filter (UKF) [8] have been developed and become popular recently to compensate the over-potential dynamics of the battery. In contrast to the KF, the accuracy of H infinity filter (HF) does not depend on the specific noise statistics. Hence, they are more robust to any possible noise statistics pre-setting [9]. Parameter identification is the foundation of SOC estimation, the most commonly used parameter identification methods are genetic algorithm (GA) [10], recursive least squares (RLS) approach [11], GA is offline method without considering the effect of working conditions on the battery performance, the RLS method requires the system model to conform to the least squares model, and cannot accurately represent the internal dynamic characteristics of the system. So in this paper, Considering the effect of ambient temperatures on the battery parameters, we proposed HF-UKF based SOC joint estimation method which can achieve the online of parameter identification and state estimation in the actual working conditions. 2. HF-UKF based SOC joint estimation 2.1. Battery model Thevenin model, as one of the typical equivalent circuit model, with one RC group and resistors, and voltage sources to represent the battery dynamic operations, which shows sufficient precision and simplicity compared with the EIS model and electrochemical model. From the schematic diagram as shown in Fig. 1, we can obtain the electrical behavior which shows in Eq. (1): + Up -

IL Cp

Ro

+

Rp

Ut Uoc -

Fig. 1. Schematic diagram of Thevenin model. 't 't   ­ °U p ,k 1 e C p ,k uRp ,k U p ,k  (1  e C p ,k uRp ,k ) R p I L ,k ® °U t ,k U oc ,k  U p ,k  I L ,k Ro ¯

(1)

Where Uoc means the battery OCV and Ut is the terminal voltage. IL is the battery current, Rp, Cp and Ro are the polarization resistance, polarization capacitance and ohmic resistance, respectively. Up is the voltage across the Cp, The OCV has a relationship with the SOC, the function can be defined as Eq. (2): Uoc (s)

c0  c1s  c2 s 2  c3s3  c4 s 4  c5s5  c6 s 6  c7 s 7  c8s8

(2)

Where sk is the SOC at the kth sampling time, the ci (i=0,1…,8) is the polynomial coefficients to fit the OCVs and SOCs. 2.2. HF-UKF based SOC joint estimation method

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Parameter identification is the foundation of SOC estimation, and if we want to get the accurate parameters of the battery model in real-time for the more accurate SOC estimation, the online parameter identification method is needed. In this paper, we employ the HF algorithm to identify the parameters online. For the system equation as shown in Eq. (3), the state vector xh, system input uˈmeasurement vector y, optimal vector z and the coefficient matrices F, H and L can be defined in Eq. (4). ­ xh ,k 1 f h ( xk , uk )  wh ,k | Fk xh ,k  wh ,kˈwh ,k ~(0, Q h ) ° ® yk hh ( xk , uk )  vh,k | H k xh,k  vh,kˈvh ,k ~(0, Rh ) ° ¯ zk Lk xh ,k

­ xh ªU oc U t U p C p R p Ro º ¬ ¼ ° ªU oc º ° ­ «U  U  I R » wf ° oc p L o °F = « » ° wx t t ' ' ° «  »  ° C p uR p C uR ° U p ,k  (1  e p p ) R p I L » ° f h ( x, u )= «e ° « » ° ® ° H = >0 «C p » ° ° «R » ° ° « p » ® ° «¬ Ro »¼ ° ª0 ° ˈ ° «0 ° y Ut ° L= « ° T ° «0 ª º z U C R R ° p p o¼ ¬ t ¯ ° « ° ¬0 ° ¯

x

(3)

xˆ

1 0 0 0 0@

(4) 1 0 0 0

0 0 0 0

0 1 0 0

0 0 1 0

0º 0 »» 0» » 1¼

Table1 The general process of the HF algorithm for parameters identification Step 1: Initialization: for k

0 , set xˆh,0 , Ph,0 , Sh,0 ,Th, , Qh,0 , Rh,0

Step 2: Time update: For k 1, 2,..., calculate  Prior estimate of state: xˆh,k

Fk 1xˆh,k 1

(5)

 h,k

 T k 1 h,k 1 k 1

Prior estimate of error covariance: P

F  Qh,k 1

F P

(6)

T k

Symmetric positive definite matrix update: Sk

L Sk Lk

(7)

Step 3: Measurement update: Innovation update: eh,k

yk  Hk xˆh,k

Kalman gain matrix update: Kh,k

(8)  k h ,k

FP

 I T

 h ,k

 h , k h ,k

1 h ,k



 1 k h ,k

S P H R H P T k

T k

1 h ,k

(9)

H R

 h ,k

xˆ  Kh,k eh,k

Measurement update of state estimate: xˆ

 h ,k

Measurement update of error covariance: P

 h ,k

P

 I T

(10)

 h , k h ,k

1 h ,k



 1 k h ,k

S P H R H P T k

(11)

Step 4: Time update: return to Step 2.

With HF algorithm (as shown in Table 1), the online parameters identification could be conducted. then we could use the UKF algorithm to estimate the state. UKF does not require linearized by the first order Taylor expansion and shows excellent robustness. suppose n is the dimension of the state vector x, IL is the system input vector u and Ut is the measurement vector y. The system equation is listed in Eq. (12), The UKF algorithm can be seen in Table 2. 't ­  ª  C 'utR º C uR ° «e p ,k p ,k U p ,k  (1  e p ,k p ,k ) R p I L ,k » ªU p ,k 1 º ° » ° xk +1 « » ˈf ( xk +1 , uk ) « « soc  Ki I L 't » ® ¬ sock 1 ¼ «¬ k »¼ ° Cn ° ° ¯ yk U t ,k U oc ,k  U p ,k  I L ,k Ro

(12)

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Where Δt is the sample intervals, Ki means the charge or discharge efficiency, Cn is the maximum available capacity of the battery. Table 2 UKF approach for SOC estimation Step 1: Initialization: for k

0 , set xˆ0 , P0 , Q0 , R0

Step 2: Time update: For k 1, 2,..., calculate Generate the sigma points˖ xˆki 1 , i 1: 2n  1

xˆk01 m 0

w

xˆk+1  n  O

xˆk+1ˈxˆki 1

O nO

c 0

ˈw

O nO





xˆk+1  n  O

Pk 1 , i 1, 2,..., nˈxˆki 1 i

 1  E  D ˈw 2

m i



Pk 1 , i i

n +1,...2n

(13)

1 , i 1, 2,...2n 1(n  O )

c i

w

xˆki

Propagate the sigma points: xˆki , i 0,1, 2,..., 2n



f ( xˆki 1 , uk ), i 0,1, 2,..., 2n 2n

¦w

xˆk

Calculate the mean and covariance of the state: xˆk , Pk

xˆ ˈPk

m i, f i k

i 0

(14)

2n

¦ w ( xˆ c i

i k

 xˆk )( xˆki  xˆk )T

(15)

i 0

Step 3: Measurement update: Propagate the sigma points: yˆki , i

yˆki

0,1, 2,..., 2n

¦w

yˆ k

Calculate the mean of the measurement: yˆ k

h( xˆki uk ), i 0,1, 2,..., 2n

2n

m i

yˆ ki

(16) (17)

i 0

2n

Pkh

Calculate the mean of the measurement: Pkh

¦ w ( yˆ c i

i k

 yˆ k )( yˆ ki  yˆ k )T

(18)

i 0

2n

Calculate the mean of the state and measurement: Pk fh Kalman gain matrix update: K k

Pk fh  Pkh

Pk fh

xˆ  Kk ( yk  yˆk )  k

Measurement update of error covariance: P

i k

 xˆk )( yˆ ki  yˆ k )T

(19) (20)

 k

Measurement update of state estimate: xˆ

c i

i 0

1

 k

¦ w ( yˆ

 k

h k k

P +K P K

T k

(21) (22)

Step 4: Time update: return to Step 2.

3. Experiments and Verifications The test bench in [5] is applied in this paper to carry out the experiments. The experimental data were acquired on the LiNiMnCoO2 (NMC) lithium-ion cells, each cell has a nominal output voltage of 3.65V, the normal capacity is 2.1Ah, the upper and lower cut-off voltage are 4.2V and 2.5V respectively. The dynamic stress test (DST) loading profile is applied to verify the performance of HF-UKF SOC joint estimation method at 25ºC. On top of that, Considering the effect of temperature on the parameters of the battery, the DST tests are conducted at different temperatures to evaluate its performances. Fig. 3(a) is the comparative profiles between the simulated OCV and reference OCV (obtained via Eq. (2)). Fig. 3(b) is the comparative profiles between the simulated voltage and reference voltage from the sensors, which indicates the estimated voltage is accordance with the reference voltage except at the beginning of the discharge. Fig. 3(c) is the voltage across the polarization capacitance. Fig. 3(d)-Fig. 3(f) are the battery parameters Rp, Cp and Ro respectively; Considering the difficulty to obtain accurately initial SOC value in real conditions, we assume that the initial guess value of SOC is the 80% of the reference SOC. It is noted that the reference SOC is calculated through Coulomb counting method via the high precision current sensors. Moreover, we define the converge time of the estimation as the first time that the estimated SOC approach the 5% error bound compared with the reference SOC.

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With the parameters identified through HF algorithm, we could execute the UKF online state estimation, Fig. 4 shows the results of the HF-UKF based SOC joint estimation method under the DST test at 25ºC. Fig. 4 (a) is the comparative profiles between the simulated voltage and reference voltage from the sensor, which indicates the estimated voltage error is less than 20 mV except at the beginning discharge. Fig. 4(b) is the comparative profiles between the estimated SOC and reference SOC, the estimated SOC converges to the reference SOC within 3s under the condition of SOC in the inaccurate initial value. (a)

(b)

(c)

(d)

(f)

(e)

Fig. 3 Parameters and states identified by HF algorithm at 25ºC

(a)

(b)

Fig. 4 The results of the state estimation through HF-UKF based SOC joint estimation method Table 3. Statistical results of SOC estimation errors at different temperatures (after the 60s of the discharge) Maximum error

Root mean square error

Time to converge

40ºC

1.01%

0.81%

3s

25ºC

1.04%

0.71%

3s

10ºC

1.07%

0.83%

3s

The Table 3 show the statistical results of SOC estimation errors at different temperatures (after the 60s of the discharge), we could observe that small values for the maximum error and root mean square error for the SOC estimation confirms the need for updating the battery parameters in real-time at different

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working conditions. HF-UKF based SOC joint estimation method can converge to the reference SOC within 3s when the initial value of SOC is inaccurate. 4. Conclusion In order to estimate the SOC more accurately, HF-UKF based SOC joint estimation method is proposed in this paper, it has dual observers which can update the parameters of the battery model in realtime through HF algorithm and robust to the uncertainty initial value through UKF algorithm. additionally, UKF algorithm is quickly to converge for the merits of unscented transform. The proposed method is validated under DST tests at different temperatures, the results indicate that the HF-UKF based SOC joint estimation method can converge to the reference SOC within 3s in the case of the inaccurate initial SOC value. Moreover, the maximum error and root mean square error confirms that updating parameters of the battery model during SOC estimation is key to increase the accuracy of the estimation. References [1] A. Szumanowski and Y. Chang, Battery Management System Based on Battery Nonlinear Dynamics Modeling. IEEE Trans. Veh. Technol., vol. 57, no. 3, pp. 1425-1432, 2008. [2] J. K. Barillas, J. Li, C. Gunther and M. A. Danzer. A comparative study and validation of state estimation algorithms for Liion batteries in battery management systems. Appl. Energy, vol. 155, pp. 455-462, 2015. [3] Y. Xing, W. He, M. Pecht and K. L. Tsui. State of charge estimation of lithium-ion batteries using the open-circuit voltage at various ambient temperatures. Appl. Energy, vol. 113, pp. 106-115, 2014. [4] M. Mastali, J. Vazquez-Arenas, R. Fraser, M. Fowler, S. Afshar and M. Stevens. Battery state of the charge estimation using Kalman filtering. J. Power Sources, vol. 239, pp. 294-307, 2013. [5] H. He, R. Xiong and H. Guo. Online estimation of model parameters and state-of-charge of LiFePO4 batteries in electric vehicles. Appl. Energy, vol. 89, pp. 413-420, 2012. [6] Y. Tian, B. Xia, W. Sun, Z. Xu, and W. Zheng. A modified model based state of charge estimation of power lithium-ion batteries using unscented Kalman filter. J. Power Sources, vol. 270, pp. 619-626, 2014. [7] F. Zhang, G. Liu, L. Fang and H. Wang. Estimation of Battery State of Charge With H∞ Observer: Applied to a Robot for Inspecting Power Transmission Lines. IEEE Trans. Ind. Electron., vol. 59, no. 2, pp. 1086-1094, 2012. [8] L. Zhang, L. Wang, G. Hinds and C. Lyu, J. Zheng and J. Li. Multi-objective optimization of lithium-ion battery model using genetic algorithm approach. J. Power Sources, vol. 270, pp. 367-378, 2014. [9] K. Wang, J. Chiasson. M. Bodson and L. M. Tolbert. A Nonlinear Least-Squares Approach for Identification of the Induction Motor Parameters. IEEE Trans. Autom. Control., vol. 50, no. 10, pp. 1622-1628, 2005. [10] L. Zhang, L. Wang, G. Hinds and C. Lyu, J. Zheng and J. Li. Multi-objective optimization of lithium-ion battery model using genetic algorithm approach. J. Power Sources, vol. 270, pp. 367-378, 2014.

Rui Xiong received the M.E. and Ph. D degrees in vehicle engineering during 2008 to 2014 from the Beijing Institute of Technology, Beijing, China. He is currently working as an associate professor in Beijing Institute of Technology. His research interests include battery management systems and vehicular hybrid power systems.