# Sizing Methodology of a Fast Charger for Public Service Electric Vehicles Based on Supercapacitors
## 2 Materials and Methods
### Electrical Analysis of an SC Working in a Fast-Charging System
#### 2.1.1 Circuit Current and SC Voltages  
Applying [PERSON]'s voltage law and considering the relationship between the SC voltages and the circuit current, the following second order differential equation can be defined for the calculation of the current, i(t):  
\[\frac{\text{di}(\text{t})^{2}}{\text{dt}^{2}}+\frac{\text{R}_{T}}{\text{L}} \frac{\text{di}(\text{t})}{\text{dt}}+\frac{1}{\text{L}_{\text{Ceq}}}\text{ i}(\text{t})=0, \tag{1}\]  
Figure 2: Wiring diagram of the fast-charging station. \(\mathrm{R_{T}}\) being the total resistance of the circuit,  
\[\mathrm{R_{T}}=\mathrm{R_{1}}+\mathrm{R_{2}}+\mathrm{R_{L}}, \tag{2}\]  
and \(\mathrm{C_{eq}}\) being the equivalent capacitance,  
\[\mathrm{C_{eq}}=\left(\frac{1}{\mathrm{C_{1}}}+\frac{1}{\mathrm{C_{2}}}\right) ^{-1}. \tag{3}\]  
As there are no generators in the circuit, the solution of the homogeneous differential equation and the complete differential equation will be the same. The two solutions of the homogenous characteristic equation are the following:  
\[\mathrm{D_{1-2}}=-\frac{\mathrm{R_{T}}}{2\mathrm{L}}\pm\sqrt{\left(\frac{ \mathrm{R_{T}}}{2\mathrm{L}}\right)^{2}-\frac{1}{\mathrm{L\cdot C_{eq}}}}=- \alpha\pm\sqrt{\alpha^{2}-\omega_{0}^{2}}=-\alpha\pm\beta, \tag{4}\]  
\(\alpha\) being the damping coefficient (s\({}^{-1}\)),  
\[\alpha=\frac{\mathrm{R_{T}}}{2\mathrm{L}}, \tag{5}\]  
and \(\omega_{0}\) being the angular resonance frequency (s\({}^{-1}\)), which only depends on the inductance and the total capacitance:  
\[\omega_{0}=\sqrt{\frac{1}{\mathrm{L\cdot C_{eq}}}}. \tag{6}\]  
\(\beta\)(s\({}^{-1}\)) is defined as follows:  
\[\beta=\sqrt{\alpha^{2}-\omega_{0}^{2}}. \tag{7}\]  
Depending on the type of solution, \(-\alpha\pm\beta\), of the differential equation, the system can be underdamped, critical, or overdamped. In this application, the values of \(\mathrm{R_{T}}\), \(\mathrm{L}\), and \(\mathrm{C_{eq}}\) must be chosen so that the system is always overdamped. In the first case the voltage would suffer oscillations that could exceed the maximum admissible voltage of the SC arrays. The second is the inflection point between the first and third, which in practice cannot occur, so only the strict underdamping situation should be avoided. In addition, if the system were underdamped, there would be time instants where the vehicle's accumulator would inject current into the charging station, which is incompatible with the normal operation of the system. To solve (1), the initial voltages of \(\mathrm{C_{1}}\) and \(\mathrm{C_{2}}\) must be known. By defining u\({}_{1}\) (t = 0) = U\({}_{01}\) and u\({}_{2}\) (t = 0) = U\({}_{02}\), being that U\({}_{01}\) > U\({}_{02}\), and \(\Delta\)U = U\({}_{01}\) - U\({}_{02}\), the following solution for the circuit current expressed as a function of the parameters \(\alpha\) and \(\beta\) is obtained:  
\[\mathrm{i(t)}=\frac{\Delta\mathrm{U}}{2\mathrm{\cdot L\cdot\beta}}.\left( \mathrm{e}^{(-\alpha+\beta)\mathrm{\cdot t}}-\mathrm{e}^{(-\alpha-\beta) \mathrm{\cdot t}}\right). \tag{8}\]  
\(\mathrm{C_{1}}\) and \(\mathrm{C_{2}}\) voltages, u\({}_{1}\)(t) and u\({}_{2}\)(t), are directly calculated from (8):  
\[\mathrm{u_{1}(t)}=\mathrm{U_{01}}-\frac{\Delta\mathrm{U}}{2\mathrm{\cdot L\cdot \beta\cdot C_{1}}}\left(\frac{\mathrm{e}^{(-\alpha+\beta)\mathrm{\cdot t}}}{- \alpha+\beta}+\frac{\mathrm{e}^{(-\alpha-\beta)\mathrm{\cdot t}}}{\alpha+ \beta}\right)-\frac{\mathrm{C_{2}}}{\mathrm{C_{1}}+\mathrm{C_{2}}}.\Delta \mathrm{U}, \tag{9}\]  
Figure 2:0When the circuit reaches the steady state, both SC arrays will have the same voltage value:  
Figure 2:1  
The final voltage, \(U_{final}\), only depends on the capacitances of \(C_{1}\) and \(C_{2}\) and their initial voltages, \(U_{01}\) y \(U_{02}\).