## 4 SOC Estimation Method
### EKF Method
Based on the above four rules, 15 control rules for linguistic variables were obtained, shown in Table 4. The fuzzy linguistic output is an unavailable signal for the SOC estimation system. Therefore, the defuzzification is necessary, which is the process of relating the output membership function to a value. The centroid defuzzification method was utilized in this paper, and it can be expressed as follows:

\[k_{FLC}=\frac{\int z\mu(z)dz}{\int\mu(z)dz} \tag{15}\]

Figure 10: Membership function. (**a**) \(\Delta K_{\text{SOC}}(k)\); (**b**) \(\Delta K_{\text{SOC}}{{}^{\prime}}(k)\); and (**c**) \(k_{FLC}\). Figure 9: Block diagram of the fuzzy logic control. The proposed fuzzy fusion algorithm uses the incremental features of EKF and AH to design fuzzy characteristics, which effectively avoids the shortcomings of the two algorithms and takes advantage of their advantages. ## 5 Results and Discussion

### Estimation Results Based on EKF

The SOC estimation error based on EKF is shown in \[\Delta\text{SOC}_{AH}(k)=\text{SOC}_{AH}(k)-\text{SOC}_{AH}(k-1) \tag{11}\]0 It can be seen that the SOC accuracy based on the PNGV model in the low SOC area was higher than that based on the 2 RC model, and that SOC estimation accuracy based on off-line identification was higher than that based on on-line identification. The results indicate that the proposed MM-MA method can improve the accuracy of the model and the SOC estimation significantly. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(k_{FLC}\)} & \multicolumn{6}{c}{\(\Delta K_{SOC}\)(\(k\))} \\ \cline{2-7}  & & **VS** & **S** & **M** & **L** & **VL** \\ \hline \multirow{3}{*}{\(\Delta K_{SOC}^{\prime}(k)\)} & N & VL & L & M & S & VS \\ \cline{2-7}  & Z & L & M & S & VS & VS \\ \cline{2-7}  & P & M & S & S & VS & VS \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Rules of the fuzzy logic control for linguistic variable. Figure 11: SOC estimation results using EKF method. (**a**) Comparison between the PNGV model and the 2 RC model; and (**b**) SOC error based on MM-MA. The reason why the accuracy of off-line identification was higher than that of on-line identification is that the more complex model and more suitable identification algorithm (4 RC plus MFO) were used for off-line identification. However, greater computing burden and more identification time were needed for off-line identification, which is shown in Figure 6.

### Case Studies for the Fuzzy Fusion Algorithm

To verify and evaluate the effectiveness of the proposed fuzzy fusion algorithm, six simulated cases of real LIBs, considering sensor and model errors, were set, as shown in Table 5. The fuzzy fusion algorithm was carried out based on these cases. In addition, the AH and EKF methods were applied to the same cases for comparison. In Table 5, \(e_{SOC0}\) is the initial SOC error, ECM\({}_{drift}\) is the ECM error, \(I_{drift}\) is the drift current of the current sensor, \(U_{drift}\) is the drift voltage of the voltage sensor, and \(e_{OCV}\) is the SOC-OCV curve error. In this study, SOH is expressed as Equation (16), where \(SOH_{i}\) means the SOH value in the \(i\)th cycle, \(C_{i}\) represents the capacity at the \(i\)th cycle, \(C_{0}\) represents the initial capacity. With aging of the cell, the cell capacity shows a downward trend. \[SOH_{i}=\frac{C_{i}}{C_{0}}\times 100\% \tag{16}\]

To evaluate the performance of different SOC estimation methods, the root-mean-square errors (RMSE) of SOC were calculated as follows:

\[\text{RMSE}=\sqrt{\frac{1}{m}\sum_{k=1}^{m}\left(SOC_{rel}(k)-SOC_{Estimated }(k)\right)^{2}} \tag{17}\]

where \(SOC_{rel}\) and \(SOC_{Estimated}\) denote the real and estimated values of the SOC, respectively. Figure 12 shows the SOC estimation results for Case A. It indicated the influence of initial SOC error (\(e_{SOC0}\)) on the AH, EKF, and fuzzy fusion algorithm (fuzzy). It was clear that \(e_{SOC0}\) had no effect on the EKF method, but significantly affected the AH and the fuzzy methods.