## 4 Comparison Results
### Accuracy Comparison  
The overview of the proposed analyzing method is illustrated in Figure 9. The AE and SE were computed using the voltage datasets and the first-order polynomial fitting was chosen to explore the entropy-capacity mapping [23]. By comparing the accuracy of both estimators, the best suitable TC can be selected. In order to select the best suitable scale, three MSE-based estimators (scale 1, 2, 3) were constructed based on the selected TC. The datasets were divided into the validating group data and training group data, as shown in Figure 10, and the training group data were used to establish the relationship between the battery capacity and entropy. The accuracy of the estimators was verified by the validating group data. Based on the optimal selection strategy of the minimization of the maximum sample entropy relative error, the parameters of \(m\), \(r\), \(N\) of the entropy algorithm were taken as 3, 0.04, and 30, respectively. The accuracy of the capacity estimators was evaluated using the mean percentage error (\(MPE\)) and the root-mean-squared percentage error (\(RMSPE\)), and their expressions are  
\[MPE=\frac{1}{N_{T}}{\sum_{i=1}^{N_{T}}{PE_{i}}} \tag{20}\]  
\[RMSPE=\sqrt{\frac{1}{N_{T}}{\sum_{i=1}^{N_{T}}{({PE_{i}})}^{2}}} \tag{21}\]  
\[PE_{i}=\frac{\left|\hat{C}_{i}-C_{i}\right|}{C_{i}}\times 100\% \tag{22}\]  
where \(C_{i}\) is the reference capacity measured after each round of aging; \(\hat{C}_{i}\) is the estimated capacity; \(N_{T}\) is the total number of test points; and \(PE_{i}\) is the percentage error of the \(ith\) test point. It can be seen from Figure 11 that the AE feature and SE feature under TC5 changed more obviously than that under other TCs. As the value of SOC under TC5 will enter the polarization zone in the future, quick changes of the voltage response at high SOC level will lead to large differences in entropy. In contrast, the SOC will become smaller for TC6 and it is away from the depleted state. Hence, TC6 cannot be used to estimate the capacity, and the results agreed with the previous analysis of polarization characteristics in Section 4.1. The relationships between the AE/SE feature and the capacity were  
Figure 10: Schematic diagram of the validating dataset and training dataset. Figure 9: The overview of the proposed analysis method. acquired based on the first-order polynomial fitting. As shown in Figure 12, the variation of entropy over the capacity was flat under other TCs (except TC5), which will lead to a large estimation error. Consequently, as shown in Figure 13, the percentage error under TC5 was lower than that under other TCs in most ranges of the cycling time. The three-dimensional diagram of MPE and RMSPE with respect to two impact factors is shown in Figure 14. When the SOC was 0.2 and 0.5, the estimation errors were relatively high regardless of what the current direction was because the battery state was in the flat zone. The best accuracy for the AE-based and SE-based estimators was achieved under TC5 as a result of polarization. Similarly, a low SOC level (i.e., smaller than 0.1) can also lead to high accuracy as a result of the polarization; however, this case was not considered in the experiment. In addition, both AE-based and SE-based methods showed very similar estimation errors. Based on these results, it can be concluded that the combined action of the SOC level and the current direction has the main effect on the capacity estimation accuracy. Only if the battery state enters into the polarization zone will high accuracy be achieved. Figure 11: Curve of the calculated entropy during the cycling of the tested battery cell. (**a**) Approximate entropy-based method; (**b**) Sample entropy-based method. Figure 12: First-order curving fitting result. (**a**) Approximate entropy-based method; (**b**) Sample entropy-based method. As the SE-based method in TC5 showed the best performance in terms of MPE and RMSPE, the MSE-based method was selected as a case study to analyze the effect of scale on the capacity estimation accuracy.