## 4 SOC Estimation Method  
### EKF Method  
For the second-order RC model, the state variables (**x**) can be written as:  
\[\textbf{x}=\left[\text{SOC}_{EKF},\;U_{1},\;U_{2}\right]^{T} \tag{7}\]  
Figure 8: Identification results of the multi-model and multi-algorithm combination (MM-MA) method. Figure 7: Schematic diagram of multi-model and multi-algorithm combinations. From Table 2, the following expressions can be obtained:  
\[f(x_{k},u_{k})=\left[ \begin{array}{ccc}1&0&0\\ 0&\exp(-\Delta t/\tau_{1})&0\\ 0&0&\exp(-\Delta t/\tau_{2})\end{array} \right]\left[ \begin{array}{c}\text{ SOC}_{EKF,k}\\ U_{1,k}\\ U_{2,k}\end{array} \right] \tag{8}\] \[+\left[ \begin{array}{c}-\eta\Delta t/C_{n}\\ R_{1}(1-\exp(-\Delta t/\tau_{1}))\\ R_{2}(1-\exp(-\Delta t/\tau_{2}))\end{array} \right]I_{k}\] \[g(x_{k},u_{k})=U_{oc}(\text{SOC}_{EKF,k})-I_{k}R_{0}-U_{1,k}-U_{2,k} \tag{9}\]  
where \(\Delta t\) is the sample period. The form of the state-space expression of the fourth-order RC and PNGV models was similar to (7)-(9), which is not given here for brevity. According to the reference [27], the standard EKF equations for the battery system are listed in Algorithm 1. Many studies have confirmed that the EKF method produces a good convergence and satisfactory estimation accuracy for small model and sensor errors [14; 16; 28; 29; 30]. However, in actual EV operation, the ECM model error increased with battery ageing, temperature, and other adverse factors. Moreover, the sensor measurement error existed objectively, and the random error of the statistical characteristics of the noise is unknown. Under these circumstances, the accuracy of the SOC estimation error determined by the EKF method would decrease. Therefore, model and sensor errors must be considered for designing SOC estimation algorithm to improve the robustness. ``` The nonlinear state-space model: \(\left\{ \begin{array}{l}\mathbf{x}_{k+1}=f(\mathbf{x}_{k},u_{k})+\mathbf{w}_ {k}\\ y_{k}=f(\mathbf{x}_{k},u_{k})+\mathbf{v}_{k}\end{array} \right.\) where the first equation is the state equation, the second one is the output equation. \(f(x_{k},u_{



### Ampere-Hour Counting Method

The AH method can be expressed as follows [31]:

\[\text{SOC}_{AH}(t)=\text{SOC}(t_{0})+\frac{\eta_{c}\int_{t_{0}}^{t}i(\xi)d\xi}{C _{N}} \tag{10}\]

where SOC\({}_{AH}(t)\) is the SOC value at time \(t\), SOC\((t_{0})\) is the SOC value at the initial time \(t_{0}\), \(C_{N}\) is the nominal capacity, \(i(\xi)\) denotes the current at time \(\xi\) (positive for charging and negative for discharging), and \(\eta_{c}\) denotes the columbic efficiency. Although the AH method is very simple and quite useful, it is an open-loop prediction method, thus it suffers from accumulated errors caused by initial SOC value errors, and noise and measurement errors. Moreover, the battery capacity might change in applications, which can lead to SOC error. ### Fuzzy Fusion Algorithm

From the above analysis, we can see that the EKF method had high accuracy, but it was easy to be influenced by ECM model parameters and voltage sensor errors. The AH method had relatively low accuracy, but the estimation results were relatively stable because relatively few parameters were affected. Based on the characteristics of the above two algorithms, a fuzzy fusion method, combining AH and EKF, was developed to calculate SOC in this paper. In our algorithm, the SOC was estimated by the AH and EKF methods, respectively. Then, the SOC increments were calculated using the following equations:

\[\Delta\text{SOC}_{AH}(k)=\text{SOC}_{AH}(k)-\text{SOC}_{AH}(k-1) \tag{11}\]

\[\Delta\text{SOC}_{EKF}(k)=\text{SOC}_{EKF}(k)-\text{SOC}_{EKF}(k-1) \tag{12}\]

where \(\Delta\text{SOC}_{AH}(k)\) and \(\Delta\text{SOC}_{EKF}(k)\) are the SOC increments for the AH and EKF methods, respectively.