A Practical and Accurate SOC Estimation System for Lithium Ion Batteries
Authors: Lin Lei , Kawarabayashi Naoki , Fukui Masahiro , Tsukiyama Shuji , Shirakawa Isao 
Published: February 2020

With the advent of big-scale popularization of secondary batteries, the accuracy as well as the inexpensiveness should be indispensable for measurement systems. Particularly, the SOC (state of charge) estimation is essential for capturing the state of secondary batteries. In this paper, a precise SOC estimation system is devised by means of EKF (Extended Kalman Filter) run on a microcomputer.
Compared with the conventional techniques, such as OCV, internal resistance method, current accumulation method, etc., EKF attains higher accuracy [1, 2]. In fact, EKF is a statistical method which minimizes over time the gap between the prediction and the observation, on the basis of the information acquired from observation of the physical variable with SOC dependence, such as OCV and internal resistance.
To apply EKF to the SOC estimation, the battery equivalent circuit model of Fig.1 is used. Equations (1)-(5) of Fig.2 show differential equations in the circuit. The OCV is given as a function of SOC. It is given, in advance, by calibration of charge and discharge with small current of 0.02C. Note that getting OCV is hard when the battery is in use. Thus, the battery internal impedance is characterized using the least squares method. Battery’s discrete-time system is formulated as Equations (6) and (7) of Fig.2, where equations (6) and (7) are the state equation and the observation equation of discrete-time system, respectively. State vector x(k) denotes the state of the battery at sample time k. The ω(k) and v(k) are the process noise and observation noise, respectively. From Equations (3)-(7), numerical solution of differential equation is obtained as Equations (8) and (9), where Δt is the sampling interval. Since OCV (SOC) is nonlinear, it is linearized as Equations (10) and (11) by the partial differential. The EKF flowchart is given as Fig. 2. It estimates the initial SOC using the initial OCV. The P(0) is the initial error covariance matrix, in this case, P(0) is 4×4 zero matrix. In addition, x^- is the predicted estimate vector, x^ is the filtered estimate vector, P^-is the predicted covariance matrix, and P is the filtered error covariance matrix.
EKF formulates the change of the observed value with a liner function, and defines a differential equation. If the time step is big, the error becomes large at the place where the differentiation by SOC of OCV varies greatly (see circled points in Fig. 3). To solve the problem, we have introduced a new fitting technique which minimizes the amount of differentiation change of OCV. The obtained fitting curve is much smoother than that by least-squares as shown in Fig. 4. Figs. 5 and 6 show errors of SOC estimation obtained by the general fitting method and the new fitting method, respectively. Compared with a general fitting method by least-squares, the maximum errors in SOC of 20% to 80% (area in the rectangle of dashed line) were reduced from 4% to 2%. Moreover, the width of the vibration of the SOC estimation error of Fig. 6 is smaller than that of Fig. 5. It stabilizes the EKF operation and makes convergence easier.
The new fitting technique pulls essential power of EKF and implements more accurate and stable estimation. This means that the SOC estimation system can be implemented with an inexpensive microcomputer with larger time step, and open a new vista as battery sensor system. The experimentation has been executed in several conditions, and similar accurate results were obtained.
[1] C. Liao, Z. Tang, and L. Wang, “SOC estimation of LiFeO4 battery energy storage system,” Proc. APPEEC,pp.1-4, 2010.
[2] M.S. Grewal and A.P. Andrews, Kalman Filtering, WILEY, 2008.