## 3 Results
### Evaluation of Parameters for the EDLC Model (PBLM-EDLC)  
The coefficients involved in the analytical solution of Equation (10) depend on three physical parameters: \(K_{P}\) (recall that \(K_{P}=3K_{V}\)), \(C_{0}\), and \(R_{P}\). The optimization of those parameters was carried out considering a grid and focusing on minimizing the error of the fitted model. Below, we describe the process in detail. Firstly, a grid was established for the three physical parameters for optimizing (\(K_{P}\), \(C_{0}\), and \(R_{P}\)); see Table 5.  
Secondly, for each combination of parameters in the grid, several goodness-of-fit measurements were computed, such as the MAPE (Mean Absolute Percentage Error), RMSE (Root Mean Squared Error), and RRMSE (Relative Root Mean Squared Error). The errors always refer to the difference between the real voltage of the trials, \(\mathrm{u_{c}(t)}\), and the estimated voltage by the DTM solution given in Equations (18) and (19). The measurements MAPE and RMSE are well known, but it is worth noting that RRMSE can be defined as a percentage dividing the RMSE by the absolute value of the arithmetic mean:  
\[RRMSE=100\,\frac{RMSE}{|\overline{y}|}=100\,\frac{\sqrt{\frac{1}{N}\sum_{t=1 }^{t=N}\left(y_{t}-\hat{y}_{t}\right)^{2}}}{\left|\frac{1}{N}\sum_{t=1}^{t=N}y _{t}\right|} \tag{20}\]  
Here, \(y_{t}\) is the real value, \(\hat{y}_{t}\) is the point forecast, \(\overline{y}\) is the mean of the real data, and N is the length of the dataset under evaluation. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **From** & **To** & **Step** \\ \hline \(K_{P}\) (F/V\({}^{2}\)) & 0.2 & 1.5 & 0.1 \\ \(C_{0}\) (F) & 400 & 600 & 10 \\ \(R_{P}\) (\(\Omega\)) & 20 & 40 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Grid of parameters to optimize the EDLC model. On the other hand, note that the trials have different lengths; therefore, the global measurements (named by global MAPE, global RMSE, and global RRMSE) will give more weight to those trials with a higher amount of data. For this reason, we have also considered the average of the goodness-of-fit measurements in each trial (named by average MAPE, average RMSE, and average RRMSE). The second stage was developed in two different scenarios. Scenario_1: considering all the trials. Scenario_2: splitting trials of charges and discharges. Note that the dielectric behavior can be different for charges and discharges, so Scenario_2 makes sense. Results showed some of the following interesting conclusions:  
1. Behavior of the error measurements is similar for different values of parameter \(R_{P}\), and that is true for Scenarios 1 and 2 as well (see Figure 10 for Scenario_1). Therefore, the accuracy of the fitted model seems to depend on the combination of parameters \(K_{P}\) and \(C_{0}\). 2. Checking the results, lower error measurements were found for specific combinations of parameters (\(K_{P}\) and \(C_{0}\)). In particular, the plot of the MAPEs (RMSEs and RRMSEs) against the quantity \(M=100\cdot K_{P}+C_{0}\) reveals a clear quadratic pattern that allows us to determine the optimum combinations (see Figure 11 for Scenario_1). Similar conclusions were obtained for Scenario_2. 3. It is preferred that the quantity \(M=100\cdot K_{P}+C_{0}\) takes a value in the interval [510; 520] (see Figure 11 and comments in the next section). Indeed, for Scenario_1, the optimal  
Figure 11: Plot of the average MAPE (brown) and global MAPE (blue) against the quantity \(M=100\cdot K_{P}+C_{0}\) in Scenario_1. Figure 10: Plot of the MAPEs against the parameter \(\mathrm{R_{p}}\) in Scenario_1: (**a**) Average MAPE; (**b**) Global MAPE. parameters that minimize the average MAPE are \((K_{P}=0.7\), \(C_{0}=440\), \(R_{P}=40)\) with average_MAPE = 1.93% (note that the optimal parameters can be different depending on what error measurement is minimized). Anyway, if the combination of parameters verifies that quantity \(M\) is in [510; 520], we can ensure that MAPEs and RRMSEs are low enough.