## 4 SOC Estimation Method
### EKF Method
Then, EKF and AH methods can be fused through the following incremental form:

\[SOC_{f}(k)=SOC_{f}(k-1)+k_{FLC}\Delta SOC_{EKF}(k)+(1-k_{FLC})\Delta SOC_{AH}(k) \tag{13}\]

where \(SOC_{f}(k)\) is the SOC value determined by the fusion algorithm, \(k_{FLC}\) is ratio coefficient (\(0\leq k_{FLC}\leq 1\)). The most critical step of the fusion algorithm is the determination of the more credible SOC increments between \(\Delta\text{SOC}_{AH}(k)\) and \(\Delta\text{SOC}_{EKF}(k)\). In other words, the key to SOC estimation in (13) is to select suitable \(k_{FLC}\) based on the relationship between \(\Delta\text{SOC}_{AH}(k)\) and \(\Delta\text{SOC}_{EKF}(k)\). We assume that the incremental ratio of SOC estimated by the two methods at time k is \(\Delta K_{SOC}(k)\), and it can be expressed as follows:

\[\Delta K_{SOC}(k)=\frac{|\Delta\text{SOC}_{EKF}(k)|}{|\Delta\text{SOC}_{AH }(k)|} \tag{14}\]

In this study, the fuzzy logic control was used to determine the appropriate \(k_{FLC}\) based on \(\Delta K_{SOC}(k)\). The range of \(\Delta K_{SOC}(k)\) was set as 0 to 3 in this paper. In the fuzzy logic control, the input and output were divided into fuzzy subsections and expressed by a linguistic variable. As shown in Figure 9, we chose \(\Delta K_{SOC}(k)\) and its derivative \(\Delta K_{SOC}{}^{\prime}(k)\) as input variables and \(k_{FLC}\) as output variables. The fuzzy variable of \(\Delta K_{SOC}(k)\) was divided into very large (VL), large (L), medium (M), small (S), and very small (VS). \(\Delta K_{SOC}{}^{\prime}(k)\) was divided into positive (P), zero (Z), and negative (N). The fuzzy variable of output with respect to \(k_{FLC}\) was divided into VL, L, M, S, and VS. The triangular fuzzy membership function was chosen in this paper because the triangular membership function has the characteristics of simplified calculation and good control performance. The membership function of FLC is shown in Figure 10.

Considering the sensor and model errors, the AH method may not be very accurate, but it had higher reliability increments due to less affected parameters. The EKF method had good accuracy, but it may not be stable. Therefore, we can choose the SOC increment with high reliability and accuracy based on the relative relationship between \(\Delta\text{SOC}_{AH}(k)\) and \(\Delta\text{SOC}_{EKF}(k)\) to calculate SOC according to the value of \(\Delta K_{\text{SOC}}(k)\). Based on the above characteristics, the following rules were proposed for determining the higher credibility SOC increment between the two methods: (1) If \(\Delta K_{\text{SOC}}(k)\leq 1\), then \(\Delta\text{SOC}_{EKF}(k)\) is considered to be more credible (large \(k_{FLC}\) should be chosen); (2) if \(\Delta K_{\text{SOC}}(k)>1\), then \(\Delta\text{SOC}_{AH}(k)\) is considered to be more credible (small \(k_{FLC}\) should be chosen); and (3) if \(\Delta K^{\prime}_{\text{SOC}}(k)<0\), it means that \(\Delta K_{\text{SOC}}(k)\) has a decreasing trend, otherwise there is an increasing trend. Therefore, the fuzzy rule of the fuzzy fusion algorithm can be expressed as follows:

* When \(\Delta K_{\text{SOC}}(k)\) is relatively small and \(\Delta K_{\text{SOC}}{{}^{\prime}}(k)\) is negative, very large \(k_{FLC}\) should be chosen to ensure that \(\Delta\text{SOC}_{EKF}(k)\) is more credible in the fuzzy fusion algorithm. * When \(\Delta K_{\text{SOC}}(k)\) is relatively large and \(\Delta K_{\text{SOC}}{{}^{\prime}}(k)\) is positive, very small \(k_{FLC}\) should be chosen to ensure that \(\Delta\text{SOC}_{AH}(k)\) is more credible in the fuzzy fusion algorithm. * When \(\Delta K_{\text{SOC}}(k)\) is relatively large and \(\Delta K_{\text{SOC}}{{}^{\prime}}(k)\) is negative, small \(k_{FLC}\) should be chosen. * When \(\Delta K_{\text{SOC}}(k)\) is relatively small and \(\Delta K_{\text{SOC}}{{}^{\prime}}(k)\) is positive, medium \(k_{FLC}\) should be chosen to improve the stability of the control system.