## 5 Results and Discussion  
The estimation performance of the three Kalman observers was compared. Since the true value of SOC in the validation test was calculated by the accumulative Ampere-hours flowing in and out of the battery, the test cells were fully charged by the CC-CV method and rested for 1 h at each test temperature to guarantee the initial SOC\({}_{0}\) to be 100%. The block diagram of our proposed SOC observers is illustrated in Figure 11. The validation test profile was loaded into the cells and Matlab/Simulink model, simultaneously. The sampling time was 1 s, and the model parameters were updated according to the current SOC, I, and T. The parameters of each algorithm were configured as follows:  
\[Q_{k}=[0.1\ 0;0\ 0.00001];\ R_{k}=0.001;\ \alpha=0.002;\ \beta=2;\ \kappa=1.\]  
Figure 10: The flow chart of strong tracking filter (STF) algorithms. The prediction accuracy of the three observers under each specific temperature was discussed. The comparative results between the experimental data obtained by the test equipment and the simulation data calculated by the three aforementioned algorithms under 0 \({}^{\circ}\)C are shown in Figure 12a. Figure 12b is the zoomed figure for Figure 12a from 4000 s to 6500 s. As a result, it can be seen that the performance of STF was better than UKF, and UKF was better than EKF. As shown in Figure 5a, both charging and discharging processes existed in the FUDS cycles. However, the voltage presented a general descending trend which was corroborated by the SOC result as shown in Figure 12b. Hence, it can be said that EKF has the weakest capacity against the undulation which was caused by the charging and discharging conversion. The weakness in voltage tracking makes EKF curve the most fluctuant one in Figure 12b. Meanwhile, the estimation errors were also demonstrated in Figure 12c, and EKF has the lowest robustness to SOC fluctuation. The max data jitter reached 8% when the battery is working at low temperature. In addition, the MAE and RMSE were employed here again to evaluate the accuracy of EKF, UKF, and STF. Figure 13 shows the MAE and RMSE of the three algorithms at each temperature. For EKF and UKF, the MAE and RMSE decreased as the temperature increased. However, there was an anomaly that occurs in the case when the STF was at 0 \({}^{\circ}\)C. We believe that the reason for the anomaly was that when cells were loading at 0 \({}^{\circ}\)C, the external performance deteriorated and the voltage fluctuation became more severe when the current direction changed. Consequently, the strong tracking factor \(\mu_{k}\) began to step in and corrected the estimation results by adjusting the a prior covariance \(P_{k|k-1}\). The statistical data of the MAE and RMSE are summarized in Table 4.  
Figure 11: Block diagram of the proposed battery SOC observers. Figure 12: Comparative results of EKF, UKF, and STF for SOC estimation: (**a**) SOC estimation results; (**b**) the enlarged window from 4000 to 6500 s; (**c**) SOC estimation error. In real applications, it is impossible to obtain the true initial SOC\({}_{0}\) before use. Therefore, the robustness against the unknown initial SOC\({}_{0}\) was also a crucial indicator of the estimation algorithms. Taking the experimental data from 0 \({}^{\circ}\)C as examples, Figure 14 shows the comparative results of the three algorithms with a SOC guess of 80% at 0 \({}^{\circ}\)C. From Figure 14a, all three algorithms could trace the true trajectory accurately and converge to the true value quickly even with large initial errors. From the enlarged window of 0 to 1500 s as shown in Figure 14b, it was clear that STF and UKF converged faster than EKF. Figure 14c demonstrates the estimation errors; the RMSEs of STF and UKF were 1.98% and 2.26%, respectively, which were 48.28% and 29.91% smaller than those of the EKF. The simulation results were consistent with the fact that UT transformation, which is utilized by UKF and STF, will produce less truncation error than first-order Taylor expansion, which is employed by EKF.