## 4 Experimental Results and Discussion
### Results of the Estimation Obtained at Variable Current without Aging Effect
In the following, n represents the overall sample size, the ith real and predicted values of the target variable are denoted by Y\({}_{i}\) and \(\widehat{\mathrm{Y}}_{i}\), respectively, and \(\widehat{\mathrm{Y}}_{i}\) corresponds to the mean value of the observations. The MSE is expressed by the following formula:  
Figure 10: SoC estimated vs. measured SoC and estimation error on the Panasonic Sanyo UR18650 ZY battery validation dataset. The DNN network consisted of three layers and nine neurons in each layer. \[\text{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(\text{Y}_{i}-\hat{\text{Y}}_{i} \right)^{2}} \tag{16}\]  
The variation in MSE at different epochs for the training, test, and validation datasets is illustrated in Figures 13a and 14a. The best performance of the model is represented by the green circle. Figures 13b and 14b illustrate the error histograms from various datasets. These two figures demonstrate that errors approached zero. These corresponded to the difference between the DNN outputs and targets. The findings presented in Figures 13 and 14 were generated by a four-layer DNN, the first three of which contained nine neurons while the last consisted of one neuron. In that case, the target was the SoC calculated from simulation and experiment, respectively. Figure 11: The SoC predicted by the DNN compared to the real SoC and the error of estimation on the Li-ion 18650 battery validation dataset. The DNN was composed of three layers with nine neurons in each layer. Figure 12: SoC estimated by the DNN compared to the simulated and error of estimation on the Li-ion battery \(LiCoO_{2}\) validation dataset. The DNN was composed of three layers with nine neurons in each layer. Figures 15 and 16 show the SoC estimation results for the two types of experimental and simulation data, respectively. Both approaches were subject to the same current profile mentioned above. The two models were capable of estimating the SoC successfully, but the LSTM offered a slight improvement compared to the GRU. A nonlinear transformation of the input data was performed by two stacked hidden layers, the first of which contained 64 units and the next 32. An activation function, the hyperbolic tangent (Tanh), which converts values varying from \(-1\) to \(1\), was integrated into both layers. The purpose of this was to ensure that the output of the neurons did not become too large, since this would have necessitated more calculation time and memory. The 20% dropout regularization was applied between hidden layers to prevent overfitting and enhance network generalization. The Adam optimizer was used to minimize the loss function [88], and the learning rate was set to \(10^{-3}\). For experimental data, the GRU provided a mean error value under 2.2% and a maximum error value under 4.6%, while for simulation data, it provided a mean error value under 2.4% and a maximum error value under 5.4%. In contrast, the LSTM provided a mean error value under 2.2% and a maximum error value under 5.5% for experimentally acquired data, and a mean error value under 2.4% and a maximum error  
Figure 14: (**a**) Mean square error for the training, testing, and validation datasets for different epochs. (**b**) Error histogram of the difference between the target and the DNN output. Figure 13: (**a**) Mean square error for training, testing, and validation datasets for different epochs. (**b**) Error histogram of the difference between the target and the DNN output. value under 4.4% for simulation-generated data. Table 4 displays the statistical metrics obtained for both models using the two data sources. The DNN model performed better than the LSTM and GRU ones in terms of RMSE, MAE, and MSE values of 0.341%, 0.463%, and \(~{}0.21~{}\times~{}10^{-4}\), for data obtained experimentally, respectively, and for data generated by simulation, the RMSE, MAE, and MSE values were 0.107%, 0.040%, and \(~{}0.011~{}\times~{}10^{-4}\), respectively.