## 2 Materials and Methods
### Laboratory Tests--The Proposed Model for EDLC  
Our five laboratory EDLCs (90 Wh, 16 V) have undergone testing. Each device was subjected to a duty cycle of six constant-current charge and discharge cycles at 5, 10, 15, 20, 25, and 33 A (see Figure 8a). This testing approach reflects locomotive start-up demands, as simulations indicate that OESD supplies power for 2-4 min, limiting the tested EDLC current to approximately 30-40 A. According to tests and literature, the ideal model does not work and gives significant and cumulative errors of up to 20% (Figure 8b). If the rated capacity is C\({}_{0}\), the voltage decreases from the ideal voltage line defined by slope Ig\({}_{0}\)/C\({}_{0}\), and it decreases more and more with the rise of voltage (see Figure 8b). This fact can be \"explained\" by several RC networks that suit the voltage trend according to their time constants, but it is stated in the literature that capacitors are not linear, and many electrolytes are not linear, too, which changes charge and discharge periods. Consequently, a non-linear model seems the best option, but it is usually considered more complex to solve it from a mathematical point of view, but this is not right, as will be stated later in this sub-section. Our proposal in this paper is to implement a simple model with good accuracy that reflects the physical base of EDLC and that enables engineers to solve and implement it easily. Such a model can help to size and test the performance of the OESD in railway units. Specifically, the model is presented in Figure 9, connected to an ideal current source (power source and electrical load units in mode CC presented in Figure 5). \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**SoC**} & \multicolumn{2}{c}{**Open Voltage**} & **Internal Impedance** \\ \cline{2-3}  & **Value (V)** & **Value (m\(\Omega\))** \\ \hline
0.9 & 13.41 & 24.2 \\
0.8 & 13.36 & 24.3 \\
0.6 & 13.25 & 24.3 \\
0.4 & 13.20 & 24.6 \\
0.2 & 13.05 & 25.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of 1.28 kWh Li-ion battery (for SoC foreseen operation). Figure 8: Tests for EDLC: (**a**) An example of laboratory tests; (**b**) Real measurements (charging at 5 A) and the response of the ideal/linear capacitor model. Parameters in Figure 9 are the following:  
\(R_{P}\): is the parallel resistance of the capacitor (\(\Omega\)). \(C_{0}\): is the constant capacity of the capacitor, which changes depending on charge and discharge duties due to the consideration of hysteresis on the electrolyte (F). \(K_{V}\): is the coefficient of the non-linear term (quadratic) of the capacitor (F/V\({}^{2}\)). \(R_{S}\): is the series resistance (ESR) given by the manufacturer (\(\Omega\)). \(R_{L}\): the resistance of the test measurement circuit (\(\Omega\)). \(Ig_{0}\): current in A of Power Source Unit (PSU) or Electrical Load (EL). \(u(t)\): is the output voltage in V of EDLC. \(u_{C}(t)\): is the voltage in V of the ideal capacitor. [PERSON]'s relations should consider that capacitance is variable, and the current is conducted by the change of charge q(t), \(\frac{dq(t)}{dt}\). In this way, the mathematical model is the initial value problem (IVP) involving a non-linear first-order equation given by  
\[\frac{d\left[\left(K_{V}u_{C}^{2}(t)\right)\left(u_{C}(t)\right)\right]}{dt}+ \frac{d\left[\left(C_{0}\right)\left(u_{C}(t)\right)\right]}{dt}+\frac{u_{C}(t )}{R_{P}}=-Ig_{0} \tag{8}\]  
or  
\[3K_{V}u_{C}^{2}(t)\frac{du_{C}(t)}{dt}+C_{0}\frac{du_{C}(t)}{dt}+\frac{u_{C}(t )}{R_{P}}=-Ig_{0}, \tag{9}\]  
With the initial condition \(u_{C}(t)=u_{0}\), \(u_{0}\) is a real constant.