## 3 Results and Discussion
### SOC Estimation Results  
Figures 12-14 show the SOC estimation compared against the measured SOC values for different test cases under ambient temperatures of \(-10\)\({}^{\circ}\)C, \(25\)\({}^{\circ}\)C, and \(40\)\({}^{\circ}\)C, respectively. At an ambient temperature of \(-10\)\({}^{\circ}\)C (Figure 12), across all cases, both the measured and estimated SOC curves display a downward trend as time progresses, indicating that the SOC decreases steadily, consistent with battery discharge behaviour under cold conditions. The estimated SOC generally tracks the measured SOC, though there are deviations in some cases that need closer inspection. The SOC estimation methodology performs reasonably well for the majority of cases, with minimal deviations between the estimated and measured values in many instances, especially at the beginning and middle stages of the discharge process. However, cases, such as Case 2, Case 3, Case 5, and Case 7, exhibit more significant deviations, particularly in the middle and later stages of the discharge cycle, highlighting potential weaknesses in the parameter identification or the effects of extreme temperatures (\(-10\)\({}^{\circ}\)C) on battery performance. The recurring deviations at the end of the discharge cycles across several cases suggest that the model's accuracy could be improved with better handling of SOC dynamics during the final phase of the battery's discharge. The estimated SOC against the measured SOC for different cases at an ambient temperature of \(25\)\({}^{\circ}\)C is given in Figure 13. This represents a more moderate, favourable operating temperature for lithium-ion batteries compared to the previous analysis at \(-10\)\({}^{\circ}\)C and, as a result, the SOC estimation performance under these conditions is expected to improve due to more stable electrochemical activity in the cells. The estimated SOC curves in most cases are more closely aligned with the measured SOC values, indicating that the parameter identification process is more reliable under these moderate-temperature conditions. However, deviations still exist in certain cases, particularly in dynamic operating conditions. Case 1, Case 4, and Case 5 exhibit nearly perfect SOC tracking, suggesting that the model performs optimally in these scenarios. Case 2 presents an exception, with significant noise or instability in the middle of the discharge process, which may indicate an issue with the sensor data or a model mismatch under certain dynamic operating conditions. For the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Ambient Temperature** & **Source** & **SOC90** & **SOC50** & **SOC30** \\ \hline \multirow{8}{*}{\(-10\)\({}^{\circ}\)C} & Regression & 98.72\% & 90.42\% & 98.85\% \\ \cline{2-5}  & Pulse time gap (s) & 40.73\% & 39.83\% & 66.35\% \\ \cline{2-5}  & Discharge pulse time (s) & 37.05\% & 17.59\% & 3.00\% \\ \cline{2-5}  & Discharge pulse C-rate & 1.02\% & 7.49\% & 8.49\% \\ \cline{2-5}  & Rest time & 0.01\% & 0.17\% & 2.16\% \\ \cline{2-5}  & Other parameters & 19.91\% & 25.34\% & 18.85\% \\ \hline \multirow{8}{*}{\(25\)\({}^{\circ}\)C} & Regression & 99.91\% & 99.16\% & 97.66\% \\ \cline{2-5}  & Pulse time gap (s) & 49.48\% & 71.71\% & 39.80\% \\ \cline{2-5}  & Discharge pulse time (s) & 10.49\% & 3.70\% & 21.43\% \\ \cline{2-5}  & Discharge pulse C-rate & 12.91\% & 7.05\% & 6.73\% \\ \cline{2-5}  & Rest time & 1.22\% & 1.69\% & 8.07\% \\ \cline{2-5}  & Other parameters & 25.81\% & 15.01\% & 21.63\% \\ \hline \multirow{8}{*}{\(40\)\({}^{\circ}\)C} & Regression & 99.79\% & 91.74\% & 99.27\% \\ \cline{2-5}  & Pulse time gap (s) & 55.50\% & 31.55\% & 56.91\% \\ \cline{2-5}  & Discharge pulse time (s) & 18.70\% & 36.55\% & 0.06\% \\ \cline{2-5}  & Discharge pulse C-rate & 11.69\% & 15.57\% & 7.41\% \\ \cline{2-5}  & Rest time & 0.41\% & 5.83\% & 0.01\% \\ \cline{2-5}  & Other parameters & 13.49\% & 2.24\% & 34.88\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: ANOVA results of RMSE for the different ambient temperatures.