## 4 Validation of the Real-Time Optimization Based Strategy  
### Simulation Results  
The studied vehicle is a real two-seat electric vehicle, the Tazzari Zero (Imola, Italia) [41], with a mass of 640 kg and an induction machine of 15 kW (Table 1). This kind of vehicle is most of the time used in urban area. The tests have hence been achieved on the low-speed phase (less than 50 km/h) of the standard driving cycle WLTC class 2 (Figure 9) [42]. The battery, SC, and PEM-FC have been designed to comply with energy, power, voltage, mass, and volume. The LFP battery is constituted of 24 A123 cells of 20 Ah connected in series. The SC bank is based on 15 branches connected in parallel. Each branch contains three Maxwell BPAK0350-15 EA modules (San Diego, CA, USA) connected in series. The minimal voltage of the SC has been set to 65% to guarantee a high efficiency [43]. Furthermore, the SoC range of the battery has been defined in a narrow range. In literature, it is stated that the Li-ion battery lifetime is improved at high SoC levels. [PERSON] et al., for instance, achieved ageing tests on LFP batteries from 100% SoC to 30% SoC [44]. The best results were obtained for a SoC range between 80% and 100%. In this study, the minimal and maximal SoC have been set to 90% and 100%, respectively. Figure 8: \(\lambda\)_-control_ strategy scheme. A key issue is the determination of the EMS parameters with the weighting factor \(A_{b}\) and the initial values of the Lagrange multipliers \(\lambda_{sc\text{-}\textit{rsf}}\) and \(\lambda_{b\text{-}\textit{rsf}}\). The initial values of \(\lambda_{sc\text{-}\textit{rsf}}\) and \(\lambda_{b\text{-}\textit{rsf}}\) are determined offline by simulation on the standard driving cycle WLTC class 2. They are computed iteratively to comply with the charge sustaining conditions (12) and (13). The weighting factor \(A_{b}\) is determined by a Pareto optimal front [45] in Figure 10, which shows the compromises between both objectives: fuel cell hydrogen consumption and RMS value of the battery current. The value of \(3.3\times 10^{-7}\) has been chosen for \(A_{b}\) as compromise to try satisfying both objectives. The initial Lagrange multipliers values are \(1.352\times 10^{-5}\) and \(1.354\times 10^{-5}\) for \(\lambda_{sc\text{-}\textit{rsf}}\) and \(\lambda_{b\text{-}\textit{rsf}}\), respectively. Based on these values, we can see in Figure 11b that the FC current is strongly reduced in comparison with a FC-based electric vehicle without any secondary source (FC-only). The limitations and constraints of the SC (Figure 11c) and battery (Figure 11d) are respected. As expected, the solicitations of the battery are weak. The SC allows the current peaks to be actually absorbed along the driving cycle. \begin{table}
\begin{tabular}{c c} \hline \hline \multirow{2}{*}{**Fuel Cell Stack**} & **Type: PEMFC** \\  & **Maximal Power: 20 kW** \\  & **Voltage Range: 50–80 V** \\ \hline Vehicle & 640 kg \\ Electric drive & Rated power: 15 kW \\  & \(u_{sc\text{-}\text{M}}=45\) V \\ Supercapacitor & \(C_{sc}=290\) F \(|\)\(t_{sc}=3.8\) m\(\Omega\) \\ bank & \(u_{sc\text{-}\text{m}}=0.65\)\(u_{sc\text{-}\text{M}}\) \\  & \(u_{sc\text{-}\text{0}}=0.9\)\(u_{sc\text{-}\text{M}}\) \\  & 24 LFP cells (3.3 V/20 Ah) \\ Battery & SoC\({}_{\text{b-}\text{M}}=100\%\) \\  & SoC\({}_{\text{b-}\text{m}}=90\%\) \\  & SoC\({}_{\text{b-}\text{0}}=95\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: System parameters. Figure 10: Corresponding Pareto optimal front to define the weighting factor \(A_{b}\). Figure 9: Standard driving cycle worldwide harmonized light-duty vehicles test cycle (WLTC) class 2 low-speed phase. In order to quantify the benefit of the SC on the battery lifetime, an FC-BAT-based electric vehicle has also been simulated with the _\(\lambda\)-control_. The latter contains the same battery as the FC-BAT-SC-based electric vehicle. The performance index is the minimization of the hydrogen consumption.