# Sizing Methodology of a Fast Charger for Public Service Electric Vehicles Based on Supercapacitors
## 3 Discussion
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Capacitance** & **Resistance** & **Total Capacitance** & **Total Resistance** \\
**of One Module,** & **of One Module,** & **Vehicle Accumulator,** & **Vehicle Accumulator,** \\
**[F]** & **[m\(\Omega\)]** & **C\({}_{2}\) [F]** & **R\({}_{2}\) [m\(\Omega\)]** \\ \hline
63 & 18 & 21 & 54 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Data of the vehicle accumulator’s SCs. calculated. As a practical example, cells from Maxwell Technologies(tm), one of the best-known manufacturers, have been selected. By connecting 152 cells of 3000 F and 0.15 m\(\Omega\) with a rated voltage of 2.7 V in series, modules with a capacitance C\({}_{\text{m}}\) = \(\Delta\)C = 19.737 F and an equivalent resistance R\({}_{\text{m}}\) = 22.8 m\(\Omega\) will be obtained. The maximum voltage the modules could withstand is 410.4 V, although in the design, the maximum voltage will be set at 400 V in order to improve efficiency. Figure 6 shows a wiring diagram of the charging station connected to the vehicle's accumulator. Depending on the number of modules being used by the charger, connected via contacts S\({}_{\text{i}}\) (i = 1, 2, 3, 4), the equivalent resistance and capacitance values, R\({}_{\text{1}}\) and C\({}_{\text{1}}\), will be those in Table 3.  
The final voltage that the assembly will reach, U\({}_{\text{final}}\), can be defined in sections, corresponding to the variation of C\({}_{\text{1}}\). Each of them will be a straight line, expressed in the form; U\({}_{\text{final}}\) = a + b\(\cdot\)U\({}_{02}\), where both the ordinate at the origin, a, and the slope, b, will change, as shown in (25). With the values of the case study, the results are presented in Table 4.  
Figure 7 shows the values that the final voltage, U\({}_{\text{final}}\) [V], the capacity of the charger, C\({}_{\text{1}}\) [F] (at the beginning of its useful life), and the efficiency of the charging process, \(\varepsilon\) (%), will take as a function of the different values of the vehicle accumulator's initial voltage, \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**U\({}_{\text{final}}\) = a + b \(\cdot\)U\({}_{02}\)**} \\ \hline
**C\({}_{\text{1}}\) [F]** & **C\({}_{\text{eq}}\) [F]** & **U\({}_{\text{02 min}}\) [V]** & **U\({}_{\text{02 max}}\) [V]** & **a [V]** & **b [V\({}^{-1}\)]** \\ \hline
138.158 & 18.229 & 187.5 & 210.94 & 347.222 & 0.1319 \\
118.422 & 17.837 & 210.94 & 234.38 & 339.751 & 0.1506 \\
98.685 & 17.315 & 234.38 & 257.8 & 329.816 & 0.1755 \\
78.948 & 16.588 & 257.8 & 281.25 & 315.956 & 0.2101 \\
59.211 & 15.502 & 281.25 & 304.69 & 295.276 & 0.2618 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Final voltage, U\({}_{\text{final}}\), as a function of the initial voltage of the vehicle, U\({}_{\text{02}}\), and the capacitance of the charger, C\({}_{\text{1}}\). Figure 6: Wiring diagram of the fast-charging station connected to the vehicle accumulator. U\({}_{02}\) [V]. The efficiency presents values between 72.47% and 87.7%, assuming that the initial voltage of the load, U\({}_{02}\), is in the range [187.5, 304.69]. Once the charger design has been completed, it is necessary to calculate the values of the inductance and the internal resistance of the smoothing inductor. Figure 8 shows the four curves, \(\beta_{1}(\gamma)\), \(\beta_{2}(\gamma)\), \(\beta_{3}(\gamma)\), and \(\beta_{4}(\gamma)\), for the worst case (U\({}_{02\min}\) = 187.5 V). In this case, it would be sufficient to choose a combination (\(\gamma\), \(\beta\)) that lies between the curves \(\beta_{1}(\gamma)\) and \(\beta_{2}(\gamma)\), since it will always be below \(\beta_{4}(\gamma)\) and above \(\beta_{3}(\gamma)\). One of the infinite possible combinations has also been pointed out, where the pair of values \(\gamma=1.006\) and \(\beta=41\) has been taken as a valid solution. A high value for \(\beta\) has been chosen so that a low value of inductance is obtained to make it possible to find a commercial product.