## 3 Results and Discussion
### SOC Estimation Results
At 25 \({}^{\circ}\)C, the range becomes narrower, between 0.0207 and 0.0371, indicating more stable behaviour at moderate temperatures. At 40 \({}^{\circ}\)C, a moderate increase in variability is noted, with values ranging from 0.0231 to 0.0611, where once again, Case 3 registers the highest measurement. Figure 15 presents the SOC estimation performance under optimal input conditions at three distinct ambient temperatures. Across all three temperature conditions, the SOC estimation follows the general trend of the measured SOC with good alignment. The SOC estimation model performs optimally at 40 \({}^{\circ}\)C, with strong alignment between the estimated and measured SOC. This suggests that the model functions well under conditions where the battery experiences higher levels of electrochemical activity. Figure 14: SOC estimation results for different cases at ambient temperature of 40 \({}^{\circ}\)C.  
The RMSE values corresponding to the three temperature levels (\(-10\)\({}^{\circ}\)C, \(25\)\({}^{\circ}\)C, and \(40\)\({}^{\circ}\)C) are presented in Table 6. The data reveal that the error is lower at \(-10\)\({}^{\circ}\)C (\(0.0144\)) and \(40\)\({}^{\circ}\)C (\(0.0105\)), with a moderate increase observed at \(25\)\({}^{\circ}\)C (\(0.0197\)). In comparison with the results shown in Table 5, there is a significant reduction in RMSE, indicating improved performance under optimized conditions. This reduction in error underscores the enhanced reliability of the system across varying temperature conditions following optimization. These findings highlight the critical role of optimization in minimizing fluctuations and improving the overall stability of the system. The SOC estimation model developed for a single lithium-ion cell can be scaled to battery packs, which are composed of many cells connected in series and parallel configurations. However, larger battery packs introduce non-uniformities between individual cells, which can affect SOC accuracy. These non-uniformities arise due to differences in cell aging, temperature gradients, and manufacturing tolerances, leading to deviations in the performance of individual cells within the pack. If the SOC estimation model is applied without accounting for cell balancing and cell-to-cell variability, the SOC predictions at the pack level may be inaccurate. For example, under dynamic loading conditions or extreme temperatures, certain cells within the pack may experience higher internal resistance or capacity fade than others, causing the pack-level SOC estimation to deviate from the true state of the individual cells. Temperature gradients are more pronounced in larger battery packs because cells in the centre of the pack may heat up more quickly due to limited \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(-\mathbf{10}\)\({}^{\circ}\)C & \(\mathbf{25}\)\({}^{\circ}\)C & \(\mathbf{40}\)\({}^{\circ}\)C \\ \hline
**RMSE** & 0.0144 & 0.0197 & 0.0105 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: RMSE values of SoC estimation using EKF for each case under different temperature conditions. Figure 15: Estimation results under optimal input conditions for ambient temperatures of \(-10\)\({}^{\circ}\)C (left), \(25\)\({}^{\circ}\)C (middle), \(40\)\({}^{\circ}\)C (right). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Cases** & \(-\mathbf{10}\)\({}^{\circ}\)C & \(\mathbf{25}\)\({}^{\circ}\)C & \(\mathbf{40}\)\({}^{\circ}\)C \\ \hline
1 & 0.0202 & 0.0273 & 0.024  
cooling, while cells near the edges may stay cooler. As temperature significantly affects the internal resistance, polarization, and overall electrochemical behaviour of lithium-ion cells, this introduces additional complexities in maintaining accurate SOC estimation across the pack. Also, cell balancing becomes more crucial in large battery packs to maintain consistent SOC across all cells. Passive and active balancing strategies can be employed to equalize the SOC and voltage drift between cells.