# Sizing Methodology of a Fast Charger for Public Service Electric Vehicles Based on Supercapacitors
## 3 Discussion
By taking the solution between the functions \(\beta_{1}(\gamma)\) and \(\beta_{2}(\gamma)\), the peak current will never reach 1 kA, and the charging time will always be less than 30 s. Considering that in  
Figure 8: Graphical representation of the functions \(\beta_{1}(\gamma)\), \(\beta_{2}(\gamma)\), \(\beta_{3}(\gamma)\), and \(\beta_{4}(\gamma)\) and the pair (\(\gamma\), \(\beta\)) = (1.006, 41) chosen to obtain the required inductance. this case \(\mathrm{C_{eq\_max}}\) = 18.229 F (see Table 3), the values of the inductance, L, and the minimum total resistance, \(\mathrm{R_{Tmin}}\), according to expressions (34) and (35), will be L = 2.71 mH and \(\mathrm{R_{Tmin}}\) = 223.66 m\(\Omega\). Since in this case \(\mathrm{R_{1 min}}\) = 3.26 m\(\Omega\) and \(\mathrm{R_{2}}\) = 54 m\(\Omega\), the inductance would have an internal resistance \(\mathrm{R_{L}}\) = 166.4 m\(\Omega\). These values can be achieved by using two inductances parallelly connected, identical to those as used in [20], which withstand 500 A of peak current. The equivalent resistance of this combination would be 77.5 m\(\Omega\), so a series resistor of 88.9 m\(\Omega\) could be added to complete the 166.4 m\(\Omega\) required resistance. The fast-charging station would therefore be completed using commercially available elements. Figure 9 shows the total resistance values of the charging station, \(\mathrm{R_{T}}\) [m\(\Omega\)], as a function of the number of modules connected to the charger and for different cell aging. It has been considered that, at the end of the lifetime of the cells, the internal resistance of the charger and the vehicle accumulator will double. In the case of the smoothing inductor, no variation in its internal resistance over time has been considered. Figure 10 shows the variation of the equivalent capacitance of assembly, \(\mathrm{C_{eq}}\) [F]. It has been considered that at the end of the lifetime of the cells, their capacitance will be reduced by 20% of their initial value. In this case, an aging of 20% means that the cells have lost 4% of their capacitance. The same value is considered for both the vehicle accumulator and the charger cells. Figure 10: Capacitance of the assembly, \(\mathrm{C_{eq}}\) [F], for different number of modules and different degrees of aging of the cells. Figure 9: Resistance of the assembly, \(\mathrm{R_{T}}\) [m\(\Omega\)], for different number of modules and different degrees of aging of the cells. The study will then be completed by simulating and plotting the results, and the validity of the design will be demonstrated in all cases including the aging of the cells in both the charger and the vehicle's accumulator. From the values that both the total resistance and the capacitance of the assembly will take as a consequence of its aging, the charging times can be calculated by means of (12). The results are shown in Figure 11, where it can be seen that in no case are the 30 s of charging time, set in the design, exceeded. Figure 12 presents the values of the maximum current, using (3), (5), (7) and (16) and also considering the aging process. The maximum value will appear at the beginning of the lifetime of the cells and at the minimum voltage of the vehicle accumulator (187.5 V). In this case, the peak current will be 936.48 A, meeting again the selected limit of 1 kA. For higher voltages in the accumulator and higher degree of aging of all cells, the peak current will decrease. Figure 11: Charging time, \(t_{\text{charge}}\) [s], for different number of modules and different degrees of aging of the cells. Figure 12: Peak current, \(I_{\text{max}}\) [A], for different number of modules and different degrees of aging of the cells.