1. Introduction
Developing electric vehicles (EVs) is an important measure to build a modern energy system which is clean, safe, and efficient [
1]. To solve the universal cruising range, safety, battery energy density and battery life problems of EVs, an efficient and reliable battery management system (BMS) is essential. Accurate estimation of state of charge (SOC) is the basis for efficient operation of other functions of BMS [
2].
The SOC estimation methods mainly include five categories [
3,
4]: conventional methods, adaptive filtering algorithms, learning algorithms, nonlinear observers and hybrid methods. The adaptive filtering algorithms, nonlinear observers and hybrid methods derived from them, which provides satisfactory performance in SOC estimation, all rely on an accurate battery model to describe the operating characteristics of batteries. Commonly used battery models can be divided into two major categories [
5]: the electrochemical models and the equivalent circuit models (ECMs). Compared with electrochemical models which involve a number of partial differential equations and unknown parameters [
6,
7], ECMs are more suitable for engineering practice. They have fewer parameters and are convenient to derive empirical results from evaluations and data analyses, and are more likely to achieve higher precision in practices due to its simpler parameter identification [
8,
9]. These inherent merits make ECMS a preferred choice for BMS [
10].
Typical ECMs for lithium-ion battery include an ideal voltage source to describe the open circuit voltage (OCV). The OCV is a very important parameter for lithium-ion battery. Since it has a very close relationship with SOC [
11,
12,
13], it not only affects the identification results of other model parameters, but also affects the accuracy of the ECM and SOC estimation. In order to obtain more accurate SOC estimations, scholars around the world have proposed many advanced algorithms over the years [
14,
15,
16,
17]. Benefiting from these improvements in algorithms, the accuracy and robustness of SOC estimation have been improved, however, the description of the relationship between OCV and SOC, which also affects the estimation results, is often overlooked. In related researches, the OCV is described as a function of SOC by a power function polynomial. Different researchers have selected model equations with different orders in their studies [
18,
19,
20,
21,
22,
23]. It is generally believed that higher polynomial orders lead to more accurate SOC estimation results, and lower orders lead to more convenient calculations. To make the estimation results sufficiently accurate, high-order power function polynomials are usually selected. However, a too high polynomial order will increase the amount of calculation and badly distort the OCV curve (detailed in
Section 3.2), and even cause oscillations due to the Runge’s phenomenon, all of which will reduce the estimation accuracy. In other studies, the OCV equation is expressed by polynomials consisting of power function and logarithmic function [
24,
25]. Because of their low orders, these hybrid polynomials can effectively reduce the computational complexity and do not cause distortion of the OCVs curve. Their disadvantage is that due to the use of a logarithmic function, the error increases when the SOC approaches 0% or 100%.
Establishing a model equation with sufficient accuracy and appropriate complexity is important for SOC estimation. In this paper, a new model equation based on a Gaussian function are proposed. It has higher accuracy than other model equations over the entire range of SOC from 0% to 100%, and is convenient to calculate. The excellent performance of this model equation is demonstrated by applying it to different filtering algorithms. In addition, this paper also studies the model equations based on power function polynomials. The results show that polynomials with high order not only increase the amount of calculation, but also have a negative impact on accuracy. In contrast, the 6th-order power function polynomial has better performance than polynomial with other orders.
The remainder of this paper is organized as follows:
Section 2 introduces our model of a lithium-ion battery and the filtering algorithms for SOC estimation used in this paper.
Section 3 analyzes the shortcomings of the traditional model equations based on power function polynomials, and proposes a model equation based on a Gaussian function trinomial.
Section 4 introduces the battery test bench and experimental projects. In
Section 5, a new typical driving cycle is conducted on the lithium-ion batteries to compare the accuracy of different model equations, then these model equations are applied to different algorithms for SOC estimation, respectively. The experimental results show that the proposed GFT-based model equation can greatly improve the accuracy of SOC estimation. Finally, the conclusions of this work are given in
Section 6.
6. Conclusions
In this paper, a new model equation based on a Gaussian function has been proposed to improve the accuracy of terminal voltage prediction and SOC estimation of lithium-ion battery. By conducting the WLTC test and applying it to different filtering algorithms, the performance of the GFT-based model equation was verified. The results indicated that without improving the filtering algorithm, the GFT-based model equation can reduce the RMSE of the SOC estimation by 0.106% to 1.162%. This progress is even greater than that brought by the improvement of the algorithm. Moreover, unlike the improvement of the algorithm, the proposed model equation does not increase the computational burden. Compared with the most excellent polynomials based on power functions, it has more advantages in computational efficiency. Besides, by studying the model equations based on power function polynomial, we found that the accuracy of SOC estimation does not necessarily increase as the polynomial order of the model equation increases. In contrast, the 6th-order power function polynomial has better performance in SOC estimation than polynomials with other orders. For practical applications, complex calculation often results in an increase in hardware costs. In other words, using complex calculation is not necessarily the best option, especially when it does not lead to improvements in performance. The GFT-based model equation proposed in this paper ensures the estimation accuracy while reducing the computational burden, and thus can be easily applied to new BMS with online SOC estimation. In future work, the proposed GFT-based model equation can be applied to more advanced algorithms to achieve better performance in SOC estimation.