# Sizing Methodology of a Fast Charger for Public Service Electric Vehicles Based on Supercapacitors
## 2 Materials and Methods
### Calculation of the Value of the Variables Involved in the Charging Process
#### 2.2.1 Selection of the Charger Capacitance and Its Initial Voltage  
Commercial SCs can endure a voltage slightly above the maximum for short periods of time. For example, a typical 2.7 V cell can be used with a voltage of 2.85 V for a restricted time. This limiting value shall be designated as U\({}_{2\rm Nmax}\). The initial voltage of the charger shall be taken equal to this limit value, U\({}_{01}\) = U\({}_{2\rm Nmax}\), or slightly lower. As there is a voltage drop during the charging process, the final value of the voltage of the vehicle's accumulator (U\({}_{\rm final}\)) will always be below U\({}_{2\rm Nmax}\), thus maintaining the safety of the cells as well as the maximum charging voltage. Once the value of U\({}_{01}\) has been set, the following relationship between the capacitances must exist for the voltages of the SCs to converge to a final value, U\({}_{\rm final}\), according to (11):  
\[\rm C_{1}=\left(\frac{U_{final}-U_{02}}{U_{01}-U_{final}}\right)\cdot C_{2}. \tag{24}\]  
Considering that U\({}_{01}\) is a low value, the capacitance of the charger will be considerably higher than that of the vehicle's SC bank. It can be concluded from (24) that it will be necessary to modify the value of C\({}_{1}\) according to the initial voltage of the vehicle's accumulator, U\({}_{02}\). This will be done by adding or removing modules of SCs in a discrete way (Figure 3). As stated above, the minimum voltage of the SC bank will be 50% of its rated voltage, U\({}_{02\rm min}\) = 0.5U\({}_{2\rm N}\). For U\({}_{02\rm min}\), a first value for the final voltage, U\({}_{0}\), will be set. At that point, the capacitance of the charger will be the maximum, C\({}_{1\rm max}\). If the value of C\({}_{1\rm max}\) remains constant and no SC module of the charger is disconnected, the higher the initial voltage of the vehicle's accumulator, U\({}_{02}\), the higher the final voltage of the assembly, U\({}_{\rm final}\). When U\({}_{\rm final}\) = U\({}_{2\rm N}\), the initial voltage of the accumulator will present the value U\({}_{02}\) = U\({}_{02\rm min}\) + \(\Delta\)V. As soon as this value is surpassed, a module of the charger's SCs, with \(\Delta\)C capacitance, will be removed, changing the capacitance of the charger to a new value, C\({}_{1}\) = C\({}_{1\rm max}\) - \(\Delta\)C. For this new value of capacitance and with this new initial voltage in the vehicle's accumulator, the final voltage of the set would be U\({}_{\rm f1}\). For the new charger capacitance, C\({}_{1}\) = C\({}_{1\rm max}\) - \(\Delta\)C, by increasing the value of U\({}_{02}\), the final voltage would increase again so that for U\({}_{02}\) = U\({}_{02\rm min}\) + 2\(\Delta\)V, the final voltage U\({}_{\rm final}\) = U\({}_{2\rm N}\) would be reached again. This process is repeated iteratively. The final voltage can be expressed as a function of C\({}_{1}\), C\({}_{2}\), U\({}_{01}\), and U\({}_{02}\) as follows:  
\[\rm U_{final}=\left(\frac{C_{1}}{C_{1}+C_{2}}\right)\cdot U_{01}+\left(\frac {C_{2}}{C_{1}+C_{2}}\right)\cdot U_{02}. \tag{25}\]  
As can be reasoned from (25), the final voltage is a linear function, in which decreasing C\({}_{1}\) decreases the ordinate at the origin and increases the slope of the line. If the first value of the final voltage, U\({}_{0\rm f0}\), is very close to the value U\({}_{2\rm N}\), \(\Delta\)V will have a lower value and, therefore, more steps in the capacitance of the charger will be necessary. This situation  
Figure 3: Final voltage, U\({}_{\rm final}\), for each initial voltage at the load, U\({}_{02}\), as a function of the values taken by the capacitance C\({}_{1}\). implies a higher cost of the installation. However, the values of all final voltages (U\({}_{\rm f0}\), U\({}_{\rm f1}\),..., U\({}_{\rm fn}\)) will be closer to the rated value, U\({}_{\rm 2N}\), so that the vehicle accumulator will always be closer to its rated voltage and will therefore have more energy stored for the next trip.