# Parameter Identification Method of a Double-Layer Supercapacitor by Using a Real Voltage Source
## 4 Results and Discussion
Table 4 shows the value of the Normalized Initial Capacitance, k\({}_{0}\), the estimated rated capacitance, C\({}_{\text{N-est}}\), calculated by expression (2), and the circuit time constant, \(\tau\), considered for the variable capacitance model, as defined in [31], whose theoretical expression is as follows:  
\[\tau=\text{R}_{\text{total}}\cdot\bigg{(}\text{C}_{0}+2\cdot\text{k}_{\text{c }}\cdot\bigg{(}\text{U}_{0}+\frac{\text{E}-\text{U}_{0}}{\text{e}}\bigg{)} \bigg{)}. \tag{39}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\text{C}_{\text{N}}\) [F] & **Method** & \(\text{C}_{\text{N-est}}\) [F] & \(\text{k}_{0}\) & \(\tau\) [s] \\ \hline \multirow{2}{*}{150} & 3 PM & 165.544 & 0.613 & 88.89 \\  & LSM & 165.771 & 0.611 & 88.92 \\ \hline \multirow{2}{*}{310} & 3 PM & 362.228 & 0.729 & 206.76 \\  & LSM & 365.773 & 0.717 & 208.14 \\ \hline \multirow{2}{*}{400} & 3 PM & 445.953 & 0.604 & 248.61 \\  & LSM & 443.909 & 0.604 & 247.23 \\ \hline \multirow{2}{*}{600} & 3 PM & 575.288 & 0.794 & 336.23 \\  & LSM & 560.387 & 0.822 & 330.36 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Rated and normalized capacitances (\(\text{C}_{\text{N}}\) and \(\text{k}_{0}\)) and the time constant of the circuit. Figure 8: Measurement of the charging current, i(t), for the 150 F (red), 310 F (black), 400 F (magenta), and 600 F (blue) cells. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\text{C}_{\text{N}}\) [F] & **Method** & \(\text{R}_{0}\) [m\(\tau\)] & \(\text{C}_{0}\) [F] & \(\text{k}_{0}\) [F/V] \\ \hline \multirow{2}{*}{150} & 3 PM & 4.268 & 101.575 & 21.323 \\  & LSM & 4.195 & 101.343 & 21.476 \\ \hline \multirow{2}{*}{310} & 3 PM & 1.498 & 263.974 & 36.390 \\  & LSM & 1.714 & 262.158 & 38.376 \\ \hline \multirow{2}{*}{400} & 3 PM & 2.173 & 269.304 & 65.425 \\  & LSM & 1.458 & 268.362 & 65.017 \\ \hline \multirow{2}{*}{600} & 3 PM & 4.231 & 456.807 & 43.882 \\  & LSM & 4.712 & 460.918 & 36.840 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Main branch values for all the SCs tested. In (39), \(R_{total}\) is the total resistance of the circuit (series connection of all that are present) and e is the Euler number. The theoretical external voltage, \(u_{co}(t)\), can be obtained from the estimated model parameters (Table 3) using Equations (4)-(7) and (10). It is also possible to analyze theoretically the behavior of the SC (one branch model with variable capacitance) considering the relationship between the external voltage, \(u_{co}(t)\), and the current absorbed, i(t), according to the passive sign convention, as follows:  
\[u_{co}(t)=R\cdot i(t)+u(t)=R\cdot i(t)+U_{0}+\int_{0}^{t}\frac{i(t)\cdot dt}{C_{0 }+2\cdot k_{c}\cdot u(t)}. \tag{40}\]  
In both ways, the external voltage can be calculated. The second option can be easily modeled with a simulation program. Figure 9 shows the block diagram created with SimulinkTM to reproduce the behavior of an SC considering only the main branch of variable capacitance, which will need the three parameters of the model (R, \(C_{0}\), and \(k_{c}\)) and the value of the initial internal voltage, \(U_{0}\). Connecting this block to a real voltage source (E, \(R_{c}\)) identical to the one used in the experimental tests, the theoretical value of the external voltage, \(u_{co}(t)\), will be obtained. The theoretical external voltage, \(u_{co}(t)\), was obtained using SimulinkTM software version R2021b, with the block diagram described in Figure 9 and graphically represented for the four cells tested alongside the actual recorded measurements, during the first 80 s, as shown in Figure 10. It can be observed that the theoretical and measured values of the external voltage are in perfect agreement. The model parameters chosen were those corresponding to the LSM method (Table 3). Table 5 shows the values of the absolute error (minimum, maximum, and mean), calculated as the absolute value of the difference between the measured external voltage and the one estimated with the theoretical formulation.