# Enhancing Supercapacitor Simulation Accuracy Through a Novel Hybrid Modeling Approach
## 4 Results and Discussions
### Results Analysis and Comparison  
Analyzing the obtained data in the first experiment, it can be observed that in both scenarios, the models with Zubieta and HyMASC parameters follow the SC's electrical behavior experimental curve for both SCs. However, the HyMASC method demonstrates better estimation and higher accuracy, as it follows more closely the real behavior of the SC. This increased accuracy is validated by comparing the predicted voltage-current characteristics during charge and relaxation cycles with experimental data, which shows that the HyMASC method more accurately reflects the dynamics of the SC. The increased performance of HyMASC method is also evident when analyzing the error variation in Figures 10b and 11b. Based on the simulation results, it was observed that the MRE improved from 2.52 to 0.47 in the case of the 300 SC, and from 2.17 to 0.85 in the case of the 400 SC. Using HyMASC, the Max AE and MAE also decreased in both cases. The MRE, MAE, and maximum absolute error (MaxAE) for each simulation result in correlation with the experimental data is shown in Table 2.  
In the second and third experiments, the aim was to evaluate the model behavior for scenarios close to specific usage of the SC. When using the obtained models to simulate full charge and discharge, the variation of the SC voltage in each of these charge/discharge series was compared, and the results were identical. This ensures good repeatability of the obtained results for identical inputs. For the random pulse discharge experiment, the short-term exponential self-discharge in the SC after each discharged current that is applied can be observed. The higher the current that is applied, the higher the voltage drop is, ensuring specific behavior of an SC. The obtained results are further compared to other research results that present modeling methods for SC from the literature. It was observed that: \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & **MRE [\%]** & **Max AE [V]** & **MAE [V]** \\ \hline \multirow{3}{*}{300 SC} & HyMASC method & 0.47 & 0.0145 & 0.004 \\ \cline{2-5}  & Zubieta method & 2.52 & 0.0878 & 0.0492 \\ \hline \multirow{3}{*}{400 SC} & HyMASC method & 0.85 & 0.0192 & 0.0108 \\ \cline{2-5}  & Zubieta method & 2.17 & 0.0378 & 0.0744 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The MRE, MAE, and MaxAE obtained errors. Figure 13: Pulse load test, simulation results for 300 F SC. [PERSON] [30] used a two-branch equivalent circuit to model an SC and a new simplified method to determine its parameters. The method consists of measuring data for only two points at a certain time stamp among the charging curve of the supercapacitor. Using that method, they obtained an MRE error of 4-4.5%. These values are higher compared to the HyMASC method results of 0.47%--0.85 MRE. [PERSON] [8] used two methods for modeling the SC, a two-branch and a multi-branch model, and determined its parameters using the [PERSON]'s method with implementation in MATLAB/Simulink. The MAE for their results is 0.09 V for the two-branch model and 0.08 V for the multi-branch model, which is notably elevated than the results using the HyMASC method, which achieved a lower MAE of 0.004 V and 0.0108 V.  
[PERSON] [28] proposed a new parameter identification method for a two-branch SC model using the synergistic utilization of fundamental information obtained through a single electrical test, coupled with a specialized analytical procedure for parameter identification. Their methodology closely resembles the HyMASC method. However, they employed the newly determined parameters as an initial estimate for implementing the final LSM (least-square method) to optimize the model parameters. As a result, they achieved an MRE of 3.98%, a max AE of 0.17 V, and an MAE of 0.077 V. If these values are compared with the results obtained with the HyMASC method, small improvements in the MRE and MAE errors can be observed.