## 3 Results
### Energy Storage System: SC Array Dimensioning and Operation  
The array of capacitors was dimensioned and simulated. The dimensioning procedure was the result of previous studies in this field done by our research group [35; 36; 37]. A thorough Simulink(tm) model was developed, and the cells were selected to obtain the calculated energy with the minimum weight. The SC array was calculated to allow the engine to accelerate from 3500 RMP to 7500 RPM without any fluctuation in the DC bus. Although the acceleration process took around 1.5 s, the storage energy system was oversized to maintain a constant voltage for 3 s. The SC array needed to provide 3750 W. The following allows the calculation of the discharge time when the SCs are operated at a constant power:  
\[t=\left[(U_{0}^{2}-U_{min}^{2})+\left(U_{0}\cdot\sqrt{U_{0}^{2}-4\cdot P\cdot R _{eq}}\right)-\left(U_{min}\cdot\sqrt{U_{min}^{2}-4\cdot P\cdot R_{eq}}\right)- 4\cdot P\cdot R_{eq}\cdot L_{m}\left(\frac{U_{0}+\sqrt{U_{min}^{2}-4\cdot P \cdot R_{eq}}}{U_{min}+\sqrt{U_{min}^{2}-4\cdot P\cdot R_{eq}}}\right)\right] \tag{1}\]  
where \(P\) is the discharged power, \(U_{0}\) is the initial internal voltage of SC array (50 V), \(R_{eq}\) and \(C_{eq}\) are the equivalent resistance and capacitance, respectively, and \(U_{min}\) is the minimum voltage the SC array can reach:  
\[U_{min}=\sqrt{4\cdot R_{eq}\cdot P} \tag{2}\]  
With this method, all the possible combinations were studied; furthermore, the weight and volume of all of the combinations were also calculated. The result led to the selection of the used SC cell: a Maxwell(tm) 150F in an arrangement of two parallel branches of 18 series-connected cells. Table 7 shows the result of the study and the different options that were analysed. Figure 20 shows a diagram of the SC array as it was built. The 18 series-connected cells as well as the parallel branches are represented. Its dimensions were very small, allowing it to be attached to the bottom of the bench where the whole system was installed. The weight of the array, already included in the dry weight of the system provided in Table 3, was 1044 g. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Branches** & **Cell (F)** & **R (f)** & **Cell (g)** & **D (mm)** & **L (mm)** & \
\begin{tabular}{c} **Req** \\ **(f)** \\ \end{tabular} & **Ceq (F)** &
\begin{tabular}{c} **Array** \\ **(g)** \\ \end{tabular} \\ \hline
1 & 325 & 0.0019 & 65.3 & 33 & 62.5 & 0.0342 & 18.05 & 1175.4 \\ \hline
**2** & **150** & **0.0065** & **29** & **25** & **50** & **0.0585** & **16.66** & **1044** \\ \hline
4 & 100 & 0.012 & 21.1 & 22 & 46 & 0.054 & 22.22 & 1519.2 \\ \hline
6 & 50 & 0.016 & 12.2 & 18 & 41 & 0.048 & 16.66 & 1317.6 \\ \hline
11 & 25 & 0.025 & 6.7 & 16 & 25.5 & 0.0409 & 15.27 & 1326.6 \\ \hline
24 & 10 & 0.03 & 3.1 & 10 & 30.5 & 0.0225 & 13.33 & 1339.2 \\ \hline
46 & 5 & 0.045 & 2.1 & 10 & 20.5 & 0.0176 & 12.77 & 1738.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Selection of the optimal cell. Array used highlighted in bold. Figure 20: SC array: distribution and dimensions. Costs were not included in the study, although SCs are relatively expensive. The selected cell was the one that formed the array with the minimum weight and volume. Prices change with the number of ordered cells. In our case, where the order was of 50 units, extra cells were used compared with previous research, and the price of the SC array was EUR 864. As with all the other parts of the prototype, the SC array and the two-way converter, which managed the stored energy under the control of the FPGA, were simulated using Simulink(tm). The simulation model presented different levels for electrical and thermal analysis, as validated in [35; 36; 37]. The SCs' internal resistance and temperature were inputted into the model in order to analyse the behaviour of different types of SCs with different specifications. The model of the SCs was placed inside a very simple bi-directional electronic converter.