## 5 Simulation Results  
In this section, the PEMFCs including the fuel cell, the boost, and the designed MPPT based on the PI and backstepping algorithm are implemented in a Matlab-SimulinkTM environment. Comparison results between the two controllers are also analyzed and discussed in this section. The parameters values of the boost converter components and the controllers are listed in Table 3.  
In order to verify the performances of the proposed MPPT method, variations in temperature, hydrogen, and load resistance are applied at different times. Thus, as shown in Figure 13, load resistance variations are performed at \(t=2\) and \(t=4\) s, from \(20\) to \(100\,\Omega\), and from \(100\) to \(10\,\Omega\), respectively, temperature variations are performed at \(t=1.5\) and \(t=2.5\) s, from \(25\) to \(35^{\circ}\), and from \(35\) to \(50^{\circ}\), respectively, and hydrogen variations are performed at \(t=1\) and \(t=3\) s, from \(0.01\) to \(0.1\) bar, and from \(0.1\) to \(0.6\) bar, respectively. Figure 12: Synoptic diagram of the backstepping algorithm. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Ideal DC–DC Boost Converter**} & \multicolumn{2}{c}{**PI Controller**} & \multicolumn{2}{c}{**Backstepping**} \\ \hline
**Parameter** & \(R\) & \(L\) & \(C\) & \(K_{p}\) & \(K_{i}\) & \(b_{1}\) & \(b_{2}\) \\
**Value** & \(20\,\Omega\) & \(69\times 10^{-3}\,\mathrm{H}\) & \(C=1500\times 10^{-6}\,\mathrm{F}\) & \(0.05\) & \(10.7\) & \(9\) & \(220\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Boost converter components and controller parameters. Figures 14-16 show, respectively, the waveforms of the PEMFC current, voltage, and power. They illustrate the behaviour of the MPPT method based on PI and the backstepping algorithm to track the MPP under the variation of temperature, hydrogen, and load resistance. These figures confirm that the proposed MPPT shows satisfactory results for maintaining the PEMFC at high-performance operation. Hence, the proposed method manifest a gradual and smooth increase to the MPP value. It is clearly presented that, in the presence of hydrogen variations (\(t=1\) and \(t=3\) s), the MPP is obtained quickly with fantastic rigor and global stability of the closed-loop system. Thus, these figures show the validity of the proposed MPPT method to keep the PEMFC generating an efficient power response. In addition, good performance such as high tracking accuracy is achieved even for large system parameter variations. On the other hand, Figures 14-16 show that the proposed backstepping algorithm provides better results compared to the conventional PI controller for tracking the reference current (\(I_{mpp}\)). Moreover, these figures show the excellent recovering features of the backstepping algorithm against load variation. The fuel cell current and voltage signals controlled by backstepping and the PI controller are respectively shown in Figures 14 and 15. According to these figures, it is clearly presented that the proposed backstepping algorithm offers gradual and smooth escalations to the reference current value. It offers a fast start-up with a response time equal to 320 ms, while the PI controller takes approximately 450 ms. Furthermore, these figures illustrate the PI and backstepping behavior when facing hydrogen variations at \(t=1\) and \(t=3\) s, and load resistance variations at \(t=2\) and \(t=4\) s. It is noticeable that the backstepping technique shows better tracking performance compared to the conventional PI controller. Thus, at \(t=1\) s, the PI shows an overshoot current of 80 mA and an undershoot voltage of 45 mV, while the backstepping shows an undershoot current of 50 mA and an undershoot voltage of 30 mV. At \(t=3\) s, the PI shows an overshoot current of 50 mA and an undershoot voltage of 30 mV, while the backstepping shows an undershoot current of 40 mA and an overshoot voltage of 25 mV.