## 5 Results and Discussion
### Discussion  
To further evaluate the estimation accuracy of the \(C_{aged}\) of the BA-ELM model, we constructed neural network models of ELM, BP, Elman, RBF, and GRNN. The estimated error of the \(C_{aged}\), RMSE, mean absolute error (MAE), mean absolute percentage error (MAPE), MAX error (MAX), and MIN error (MIN) were used to compare the estimated accuracies of six models. The results are shown in Figures 17-19 and Tables 6 and 7.  
The RMSE and MAE of the BA-ELM model are 0.5354% and 0.4326%, respectively. The RMSE of the other 5 models is between 2% and 4%, and the MAE is between 1% and 4%. The MAPE of the BP model is the largest, reaching 4.06%, and the MAPEs of BA-ELM, ELM, and GRNN are 0.44%, 0.20%, and 0.29%, respectively. The maximum error of the SOH result estimated by RBF is 0.1913 Ah, and the maximum error of the SOH result estimated by BA-ELM is only 0.017 Ah. The minimum error of the result of ELM estimation is 0.1212 Ah, and the minimum error of the result of BA-ELM estimation is only 0.062 Ah. Compared with the other five models, the results of the BA-ELM estimation show a smaller RMSE, MAE, MAX, and MIN.  
To further compare the performance of the six models on various error indicators, we sorted them according to the pros and cons of each indicator. Furthermore, we assigned 6 points for ranking first, 5 points for ranking second, and so on, with 1 point assigned for ranking sixth. The higher the score of a certain model on a certain error index, the better the performance of the algorithm on this error index. The results are shown in Table 8. We drew the radar chart of the 5 error indicators for the 6 models, and the results are shown in Figure 20. The shadow area covered by the BA-ELM model is the largest while the  
Figure 19: Comparison of MAX and MIN.  
Figure 18: Comparison of RMSE MAE and MAPE.  
shadow area covered by BP and RBF is smaller. According to the total score of the five error indicators, the results of all models are as follows:  
BA-ELM (28) > ELM (21) > Elman (15) = GRNN (15) > BP (13) > RBF (12).  
The predictive results of all models are illustrated in Figure 21a-f in the form of scatter plots and determination coefficients. It can be seen from the scatter plots that the BA-ELM provides less scattered estimates. Compared to other models, the fit line equation of BA-ELM is closer to the exact line (\(y=x\)), with a higher determination coefficient. It is followed by the ELM, BP, and Elman models. RBF and GRNN provide poor estimates while these two models also have lower determination coefficients of 0.9819 and 0.9883, respectively.  
Figure 21: Capacity of actual vs predicted in the form of scatter plot and R\({}^{2}\), respectively for: (**a**) BA-ELM; (**b**) ELM; (**c**) BP; (**d**) Elman; (**e**) RBF; (**f**) GRNN.  
In Figure 22, from the violin diagrams, the BA-ELM model demonstrates its advantage by having a distribution that is a more similar distribution to the actual capacity as compared to other models. The data in the BA-ELM model is the most concentrated. A small amount of data in the ELM and RBF models, respectively, is far from the actual value. The median in the BP model is on the high side. A graphical comparison justifies the results obtained using the statistical indicators presented in the previous tables. Based on the above discussion, it can be concluded that compared with the other five models, the BA-ELM model has a better performance and can accurately estimate the actual remaining capacity and SOH of the battery.