## 3 Results and Discussion
### Model Validation and ANOVA Results
The RMSE values of different SOC values (SOC90, SOC50, and SOC30) at an ambient temperature of 25 \({}^{\circ}\)C are shown in Figure 10. The regression models display positive corre  
Figure 8: Optimal experimental parameter results for ambient temperature of \(-\)10 \({}^{\circ}\)C (**left**), 25 \({}^{\circ}\)C (**middle**) and 40 \({}^{\circ}\)C (**right**) operating conditions. lations between the predicted and observed RMSE in all three plots, reflecting consistent model performance across different SOC levels. However, as seen in the \(-10\)\({}^{\circ}\)C analysis, variability increases in the later stages of the discharge process, indicated by the widening of confidence and prediction intervals. In summary, the SOC estimation model demonstrates high accuracy at the beginning of the discharge process at 25\({}^{\circ}\)C, but its performance declines in the later stages, particularly as the battery approaches deeper discharge. The RMSE values for various SOC values at an ambient temperature of 40 \({}^{\circ}\)C are given in Figure 11. Across all stages (SOC90, SOC50, and SOC30), the data points generally follow a positive correlation, with some scatter. The 95% confidence and prediction intervals show that the accuracy of the SOC estimation decreases slightly as the battery approaches lower charge levels. However, the overall model performance remains nearly stable. Briefly, at \(-10\)\({}^{\circ}\)C, the RMSE is higher due to the increased internal resistance and more pronounced voltage drops. At 40 \({}^{\circ}\)C, the RMSE remains relatively low in the early stages but increases as the battery heats up, highlighting the need to account for thermal effects in the model. At lower SOC levels, this resistance increases, causing larger voltage deviations and higher RMSE, which emphasizes the importance of accurate SOC estimation models at different SOC levels. Figure 10: Regression model results for different SOC values at the ambient temperature of 25 \({}^{\circ}\)C, (a) SOC 90, (b) SOC 50, (c) SOC 30. Figure 9: Regression model results for different SOC values at the ambient temperature of \(-10\)\({}^{\circ}\)C, (a) SOC 90, (b) SOC 50, (c) SOC 30. Table 4 illustrates the ANOVA results for RMSE across different stages of battery discharge (SOC90, SOC50, SOC30) under three distinct ambient temperatures: \(-10\)\({}^{\circ}\)C, \(25\)\({}^{\circ}\)C, and \(40\)\({}^{\circ}\)C. Each SOC value is analysed with respect to multiple sources of variation, including regression, pulse time gap, discharge pulse time, C-rate, rest time, and other parameters. The table indicates the percentage contribution of each source to the total variance in RMSE. The regression model consistently contributes the highest percentage to the variance in RMSE across all ambient temperatures and SOC levels, ranging from 90.42% to 99.79%, indicating that the model is the primary driver of SOC estimation accuracy. As the ambient temperature increases from \(-10\)\({}^{\circ}\)C to \(40\)\({}^{\circ}\)C, the regression contribution becomes slightly higher in most cases, reflecting better model performance under higher temperatures. This suggests that the estimation model is more stable and accurate at warmer temperatures, particularly at \(40\)\({}^{\circ}\)C. The results also showed that pulse time gap and discharge pulse time significantly affect the RMSE, especially at low and high ambient temperatures (\(-10\)\({}^{\circ}\)C and \(40\)\({}^{\circ}\)C). The contribution of these parameters (pulse time gap and discharge pulse time) also varies based on the battery's SOC levels. At higher SOC levels (SOC 90%), the battery voltage is closer to its maximum, and the electrochemical reactions are more stable. In this state, the battery is less sensitive to changes in pulse time gap and discharge pulse time. However, at extreme temperatures, especially at \(-10\)\({}^{\circ}\)C, even at high SOC, increased internal resistance causes the battery to struggle with large pulse durations, resulting in higher RMSE values.