# 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
#### 2.2.2 Selection of the Properties of the Inductor: Inductance and Internal Resistance
However, if the vehicle arrives at the charging station with a voltage higher than U\({}_{\rm{02 min}}+\Delta\)V, it will be necessary to disconnect SC modules of capacitance \(\Delta\)C, which will decrease the value of C\({}_{\rm{eq\_max}}\) and increase the value of the total resistance. If for a given capacity C\({}_{\rm{eq\_max}}\) a point (\(\gamma\), \(\beta\)) with a very low initial beta value is selected, it is possible that, as the charger capacity, C\({}_{\rm{1}}\), is reduced, which leads to a decrease in C\({}_{\rm{eq}}\), \(\beta\) decreases so much so that the system becomes underdamped. As stated above, this type of transient regime is not admissible due to the oscillations that can appear in the voltages of both banks of SCs, which could exceed their maximum admissible voltage. The value of \(\beta\) must be such that, with the minimum equivalent capacity (when the maximum number of \(\Delta\)C modules have been disconnected), the system never goes into an underdamped regime. This condition shall be satisfied if \(\beta>0\) whatever the value C\({}_{\rm{eq}}\) and R\({}_{\rm{T}}\), which is the same as imposing that the following inequality is satisfied  
\[\rm{\left(\frac{R_{Tmax}}{2L}\right)^{2}\geq\frac{1}{L\cdot C_{eq\_min}}}. \tag{36}\]  
In the calculation of R\({}_{\rm{Tmax}}\) and C\({}_{\rm{eq\_min}}\), it is necessary to consider the change that aging causes in the properties of the charger's and vehicle accumulator's SCs. Data provided by the manufacturers indicate that at the end of their lifetime, the internal resistance doubles and the capacitance is reduced by 20%. If the system is to be operated for any level of cell aging both in the vehicle and in the charger, the maximum of total resistance, R\({}_{\rm{Tmax}}\), and the lowest value of the equivalent capacitance, C\({}_{\rm{eq\_min}}\), must be considered in (36). These two values can be calculated from the initially chosen pair (\(\gamma\), \(\beta\)) (at the beginning of the lifetime and with minimum U\({}_{\rm{02}}\)). Since at this point C\({}_{\rm{1 min}}\), R\({}_{\rm{1 max}}\) (charger capacitance and resistance at the beginning of the lifetime with all capacitance steps \(\Delta\)C disconnected), C\({}_{\rm{2}}\), and R\({}_{\rm{2}}\) (vehicle accumulator capacitance and resistance at the beginning of their lifetime) are known, and considering that the charger and the vehicle cells are equally aged, that is, at the end of their lifetime, their resistances double and their capacitances decrease by 20%; therefore, the values of R\({}_{\rm{Tmax}}\) and C\({}_{\rm{eq\_min}}\) can be calculated as follows:  
\[\rm{R_{Tmax}}=2\cdot R_{\rm{1 max}}-R_{\rm{1 min}}+R_{\rm{2}}+\frac{2\cdot\gamma} {\beta\cdot C_{eq_{max}}\cdot(\gamma^{2}-1)}=\Delta R_{\rm{max}}+\frac{2\cdot \gamma}{\beta\cdot C_{eq_{max}}\cdot(\gamma^{2}-1)}, \tag{37}\]  
\[\rm{C_{eq\_min}}=0.8\cdot\left(\frac{C_{\rm{1\_min}}\cdot C_{\rm{2}}}{C_{\rm{ 1\_min}}+C_{\rm{2}}}\right). \tag{38}\]  
By defining \(\Delta\)R\({}_{\rm{max}}\) as the largest increase that the total resistance of the circuit will suffer, considering both the minimum capacity of the charger and the maximum aging in the SC banks of the charger and vehicle, through the substitution of the values of \(\mathrm{R_{Tmax}}\) and \(\mathrm{C_{eq\_min}}\) given in (37) and (38), respectively, into (36), the following curve \(\beta_{3}(\gamma)\) is obtained:  
\[\beta^{2}=\alpha^{2}-\frac{1}{\text{L}\cdot\text{C}_{\text{eq}}}, \tag{31}\]0  
\(\beta_{3}(\gamma)\) cuts the abscissa axis at the point where \(\gamma\) takes the following value:  
\[\beta^{2}=\alpha^{2}-\frac{1}{\text{L}\cdot\text{C}_{\text{eq}}}, \tag{31}\]1  
Any pair (\(\gamma\), \(\beta\)) which is a valid solution, chosen at the beginning of the lifetime of the cells, and which is above the curve \(\beta_{3}(\gamma)\) will ensure that, with the minimum system capacity and maximum resistance, the transient period stays overdamped.