# Concept for Generating Energy Demand in Electric Vehicles with a Model Based Approach
## 4 Simulation and Model Environment
### Vehicle Model
#### 4.1.2 Supercapacitor Model  
In order to describe the dynamic characteristics of a supercapacitor mathematically, the equivalent circuit shown in Figure 5 was used.  
This model describes the dynamic behaviour of the supercapacitor during charge and discharge cycles, i.e., this model makes it possible to determine the voltage delivered by the supercapacitor as a function of the charge or discharge current.  
The model shown in Figure 5 is a second-order RC model. The order of the model depends on the time step size in which the supercapacitor operates [31]. In [31], a comparison of different models of supercapacitors is presented and their precision is evaluated considering the load change frequency. In electric vehicles, the load (speed) changes every few  
Figure 5: The equivalent circuit the supercapacitor model is based on [30].  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Value** & **Unit** \\ \hline
**Nominal Voltage** & 3.7 & V \\
**Nominal Capacity** & 61 & Ah \\
**Min./Max. Voltage** & 2.70/4.10 & V \\ \hline
**Material Cathode** &
\begin{tabular}{c} NCM \\ (Nickel-Cobalt-Manganese) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 3: Characterized parameters of the lithium-ion cell of Samsung SDI.  
seconds (which can be considered as low frequency), so it is sufficient to use a second-order RC model [32]. A simpler model, i.e., a model with only one RC circuit, is not suitable because the error of the supercapacitor output voltage is significant, as shown in [33].  
The model consists of three parts. The first part includes the resistor \(R_{f}\), which represents the current losses inside the cell (losses) [30]. The part with the resistor \(R_{1}\) and the capacitor \(C_{1}\) (main cell) determines the instantaneous response (seconds) of the supercapacitor to a charge or discharge current [33]. And finally, the part with resistor \(R_{2}\) and capacitor \(C_{2}\) (slow cell) determines the slow response (minutes) of the supercapacitor to a charge or discharge current [33].  
In order to implement the model shown in Figure 5, it is necessary to know the values of all components describing the circuit, i.e., resistors \(R_{f}\), \(R_{1}\), and \(R_{2}\) and capacitors \(C_{1}\) and \(C_{2}\). The values of all components were obtained from [30], where the _Maxwell/BCAP3000_ supercapacitor is simulated. Table 4 shows the characteristic values of the supercapacitor.  
The following Equation (5) is used to calculate the voltage of the supercapacitor:  
\[U_{\text{SC}}=N_{\text{SC}}(V_{1}+R_{1}\frac{I_{\text{SC}}}{N_{P_{\text{SC}}}}) \tag{5}\]  
The calculation of the voltage \(V_{1}\) is from:  
\[V_{1}=\frac{-C_{0}+\sqrt{C_{0}^{2}+2C_{\text{\tiny P}}Q_{1}}}{C_{\text{\tiny P }}} \tag{6}\]  
The calculation of the voltage \(V_{2}\) is from:  
\[V_{2}=\frac{1}{C_{2}}\int\frac{1}{R_{2}}(V_{1}-V_{2})dt \tag{7}\]  
In addition, the voltage of the supercapacitor is calculated based on its state of charge using Equation (8):  
\[SoC_{S}C=\frac{U_{S}C}{NominalVoltage} \tag{8}\]