# State-of-Charge Prediction Model for Ni-Cd Batteries Considering Temperature and Noise
## 4 Analysis of SOC Prediction Results of Ni-Cd Battery Using APSO-GRNN
### SOC Prediction Algorithm Results and Discussion
#### 4.2.3 Model Robustness Test Results for Temperature  
The temperature greatly influenced the available capacity and discharge time of the battery [32]. To study the prediction effect of the model at different temperatures, pulse-discharge experiments of the battery at different temperatures were carried out. The experimental results are shown in Figure 9.  
The results in Figure 9 indicate that an increase in temperature led to an increase in both the available capacity and discharge time of the battery, whereas a decrease in the temperature resulted in a lower available capacity and shorter discharge time. Thus, temperature is a critical parameter in the SOC prediction model. Since the model training set only included battery discharge data at 20 \({}^{\circ}\)C and 40 \({}^{\circ}\)C, this study used 10 \({}^{\circ}\)C pulse-discharge data to verify the advantages of the APCSO-GRNN model over the other models in terms of prediction accuracy and robustness. Figure 10a,b display the experimental outcomes, and the data are presented in Table 5.  
Figure 9: Pulse discharge curves at different temperatures. An implicit relationship between the battery discharge data at different temperatures can be established by introducing a pre-training method. The experimental results demonstrated that the prediction errors of the SVR, XGBoost, and GRNN models increased significantly and fluctuated considerably when experiments were conducted at different temperatures, especially in the early and late stages of discharge, where different temperatures led to a substantial variation in the voltage distribution, thereby amplifying the prediction errors. The APSO-GRNN model fine-tuned the model based on the previous training results using the pre-training model and adaptive adjustment mechanism, acquiring a mapping of the implicit relationship between the different temperatures, which improved the model's robustness. Although the prediction error of the APSO-GRNN model also increased, the error fluctuated considerably only in the early stage of discharge, as seen in Figure 10b. As the battery SOC decreased, the voltage distribution difference gradually decreased, thereby reducing the prediction error of the APSO-GRNN model. Figure 9 shows a clear degradation in the battery capacity when operating below 0 \({}^{\circ}\)C, accompanied by a voltage distribution that significantly deviated from the discharge data observed at room temperature. To investigate the prediction accuracy of the model at different temperatures, prediction experiments were conducted at different temperature nodes. The corresponding experimental outcomes are presented in Figure 11.  
The results presented in Figure 11 indicate that the model exhibited its highest prediction accuracy at approximately 30 \({}^{\circ}\)C, yielding an average relative error of 2.51%. Conversely, the lowest prediction accuracy was observed at \(-\)20 \({}^{\circ}\)C, with an average relative error of 4.11%. As the temperature decreased, both the battery's discharge capacity and the magnitude of the prediction error increased. Notably, the prediction error in the early stage was large, and the stability of the model decreased. This phenomenon can be attributed to the reduced available capacity of the battery and the substantial voltage rebound experienced under low ambient temperatures. When the battery was discharged rapidly, its terminal voltage declined sharply. Compared to normal temperature conditions, the rate of the voltage decline exhibited a significant increase, which resulted in a substantial disparity in the voltage distribution, leading to a drop in the accuracy of the model predictions. Figure 10: Model prediction results (a), model prediction error (b). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\delta\) & \(\delta_{\text{RMSE}}\) & \(\delta_{p}\) \\ \hline SVR & 6.86 & 0.0378 & 0.9421 \\ XGBoost & 5.09 & 0.0326 & 0.9561 \\ GRNN & 7.11 & 0.0418 & 0.9452 \\ APSO-GRNN & 2.56 & 0.0176 & 0.9746 \\ \hline \hline \end{tabular}
\end{table}