# Explainable Artificial Intelligence for State of Charge Estimation of Lithium-Ion Batteries
## 4 Experiments and Results
### E3: SoC Estimation Guided by xAI and Importance Estimates
Here, we used the importance estimate method PIEE [26], which is more suitable when considering the time steps of a multivariate time series context because it can examine each time point from a global standpoint. We applied PIEE to the NNs trained with CURRENT and VOLTAGE. The results of the importance  
Figure 8: xAI: InputXGradient’s average explanation over all samples, with red or blue color signifying importance. (**a**) Based on CNN-LSTM. (**b**) Based on GRU. CURRENT and VOLTAGE have high relative importance. estimates are shown in Figure 9, which reinforce the previous observation regarding time steps within the input window. Utilizing the information of the time steps, we retrained CNN-LSTM and GRU with reduced window sizes, which were informed by their corresponding results of PIEE. The results of the retrained NNs with reduced window sizes are available in Table 6, and the visuals of CNN-LSTM and GRU are available in Figure 4c and Figure 5c, respectively. Regarding Table 6, again, we are only concerned with the test sets in order to be representative of the challenges of real application. Here, we note that there are results of SoC estimation that have a minimum difference when compared to their counterparts from Table 5 (up to 0.1 \(\times\) 10\({}^{-4}\) difference in MSE, up to 0.3 \(\times\) 10\({}^{-3}\) difference in MAE); however, there are also results where there is a noticeable difference when compared to their counterparts (5 \(\times\) 10\({}^{-4}\) difference in MSE and 5 \(\times\) 10\({}^{-3}\) difference in MAE). This is due to the open interpretation of PIEE's estimate. From Figure 9, we can observe that the method does not produce a definitive value regarding the smallest effective window size; instead, it produces an estimate of importance for each time step, which can be combined to form a heatmap of importance. Using the heatmap of importance, it becomes possible to make informed choices regarding the approximate effective window size. This is demonstrated by the results of CNN-LSTM with window size 53, shown in Table 6, which is based on Figure 9a. This can also be visually compared in Figure 4c. However the effective window size is not necessarily always clear, and this is evident in the example of Figure 9b, which can lead to a result such as GRU with a window size 45 from Table 6 and Figure 5c. In such cases, a systematic reduction evaluation approach can be used to reach a conclusion. This is also demonstrated in Table 6. An extended version of the table can be found in Appendix A's Table A3. The key findings from the results suggest that the window size can be reduced up to 9-25% and still retain performance. \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{3}{c}{**Train**} & \multicolumn{3}{c}{**Validation**} & \multicolumn{3}{c}{**Test**} \\ \cline{2-7}  & **MSE** & **MAE** & **MSE** & **MAE** & **MSE** & **MAE** \\ \hline
**GRU** & 3.3 \(\times\) 10\({}^{-4}\) & 13.3 \(\times\) 10\({}^{-3}\) & 6 \(\times\) 10\({}^{-4}\) & 16.7 \(\times\) 10\({}^{-3}\) & 5.2 \(\times\) 10\({}^{-4}\) (+1.7 \(\times\) 10\({}^{-4}\)) & 16.2 \(\times\) 10\({}^{-3}\) (+3 \(\times\) 10\({}^{-3}\)) \\
**CNN-LSTM** & 5.6 \(\times\) 10\({}^{-4}\) & 16.5 \(\times\) 10\({}^{-3}\) & 8 \(\times\) 10\({}^{-4}\) & 18.9 \(\times\) 10\({}^{-3}\) & 7.1 \(\times\) 10\({}^{-4}\) (+2.6 \(\times\) 10\({}^{-4}\)) & 18.2 \(\times\) 10\({}^{-3}\) (+2.8 \(\times\) 10\({}^{-3}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: E3’s performance results of NNs for train, validation, and test sets according to the evaluation metrics only using the signals: CURRENT and VOLTAGE. The values in brackets indicate the difference compared to the respective metric from E1’s Table A1. Figure 9: PIEE’s multivariate importance estimate, with red signifying importance. (**a**) Based on CNN-LSTM. (**b**) Based on GRU. This suggests that not all time steps within the input window are important for SoC estimation.