A Precise FCC Estimation Algorithm Based on Recursive Least-Squares Identification of Li-Ion Batteries with Adaptive Forgetting Factor Tuning
Authors: Ono Hironori , Lin Lei , Fukui Masahiro , Takaba Kiyotsugu 
Published: February 2020

Li-ion batteries are remarkable devices due to their high capability to store energy densely. The FCC (full charge capacity) estimation is essential for capturing the state of secondary battery. In this paper, precise FCC estimation is devised by estimation of internal parameter and SOC (state of charge).
Comparted with the conventional identification techniques [1], the adaptive forgetting factor is more suitable for Li-ion battery. The open circuit voltage, OCV, is affected by SOC value. Therefore, the rate of change of OCV is not linear to time. In order to solve this problem, a heuristic adaptive forgetting factor is introduced based on terminal voltage and terminal current. To apply the system identification approach, the battery equivalent circuit model shown in Fig.1 is used. The subject of the system identification is OCV. OCV is physical quantity depending on SOC, and the relationship between OCV and SOC is given as Fig. 2. OCV-SOC curve has a significant role to achieve SOC estimation.

The difference equation of u
L is obtained by the equivalent circuit. Then, Equation (1) is obtained as observation function using the variables defined in Equation (2), and adding w (k) which is observation noise. T
s is the sampling period. In equation (1), parameterθ regards as a unknown parameter, being estimated by using recursive least-squares identification. The value of R
a, R
b, C
b, u
OCV can be calculated by parameterθ with equation (3). SOC is obtained by referring the OCV-SOC curve. A heuristic forgetting factor λ(k) is defined by Equation (4). In this equation, G is gain, m
uL is average variance of voltage u
L preserving in buffer, and i
avg is average value of current i.

Estimation of OCV is performed using discharge patterns indicated in Fig.3. Table1 shows the average values of SOC estimation error. The average error is the smallest in the case of using the adaptive forgetting factor. Table2 shows the error dependent on temperature in the case of using the adaptive forgetting factor. SOC estimation is successful since the maximum value of average SOC estimation error is less than 0.05.
Then, FCC is estimated with the estimated SOC. FCC is calculated by quotient of the value of current accumulation from time t0 to t1 and the change of estimated SOC value from t0 to t1. Ten percent discharges are executed from the 10 points of SOCs. The starting points of the discharge are from 0.97 to 0.92 by 0.05 intervals. The result of FCC estimation is shown Fig.4. In the case of 0.5A, the maximum value of average SOC estimation error is 0.049. Compared with the conventional techniques [2], the amount of calculation with proposed method is much less. The conventional technique [3] needs huge amount of data before estimation. In contrast, proposed method needs few data.
This FCC estimation system pulls the maximum capability of secondary batteries with precise SOC estimation, and controls the batteries’ state by grasp of FCC. In addition, the amount of calculation and data needed preparing before estimation are less than conventional technique. Therefore, this proposed method can be implemented easily with inexpensive microcomputer. The experimentation has been executed in several conditions, and similar accurate results were obtained.
[1] C. Fleischera, W. Waaga, H. Heyna, and D. U. Sauera, “On-line adaptive battery impedance parameter and state estimation considering physical principles in reduced order equivalent circuit battery models: Part 1. Requirements, critical review of methods and modeling,” J. Power Sources, vol. 260, no. 15, pp. 276-291, 2014.
[2] T. Kim, Y. Wang, Z. Sahinoglu, T. Wada, S. Hara, W. Qiao, “A rayleigh quotient-based recursive total-least-squares online maximum capacity estimation for litium-ion batteries, ”IEEE trans. on energy conversion, vol. 30, no.3, pp. 842-850, 2015.
[3] C. Hu, G. Jain, C. Schmidt, C. Strief, M. Sullivan, “Online estimation of lithium-ion battery capacity using sparse Bayesian learning, ” J. Power Sources, 289, pp. 105-113, 2015

Figure 1