## 7 Design Example and Simulation Results
### Numerical Analysis and Design  
The parameters of the P&O algorithm, as reported in Table 1, were calculated following the procedures validated in [26] and [27], which are the starting point for the design procedure summarized in Section 6.3. Then, solving the simultaneous equation system formed by (24), (26), (31), (35) and (38) for a settling time lower than \(T_{a}\) (\(t_{s}<1.0\) ms) results in the feasible \(k_{1}\) and \(\Omega\) values reported in Figure 6a, where the maximum \(\Omega\) value is 725 krad/s, while the maximum value of \(k_{1}\) is 11 A/V (out of the figure). The \begin{table}
\begin{tabular}{l c} \hline \hline  & **PV Panel** \\ \hline MPP voltage (\(v_{mpp}\)) & 18.5 V \\ MPP power (\(p_{mpp}\)) & 85.5 W \\ Short-circuit current (\(i_{sc}\)) & 5.0 A \\ Open-circuit voltage (\(v_{oc}\)) & 22.1 V \\ \hline \multicolumn{2}{c}{**Flyback Converter**} \\ \hline Input capacitor (\(C_{pv}\)) & 100 μF \\ Magnetizing inductor (\(L_{m}\)) & 75 μH \\ Leakage inductor (\(L_{k}\)) & 11 μH \\ Turn-ratio (\(n\)) & 8.0 \\ Average output voltage (\(v_{o}\)) & 220 V \\ \hline \multicolumn{2}{c}{**P&\(\&\)O Algorithm** \\ \hline Perturbation amplitude (\(\Delta v_{pv}\)) & 0.5 V \\ Perturbation period (\(T_{a}\)) & 1 ms \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters for the design example. Figure 5: Implementation of the proposed control system. numerical analysis shows that high \(k_{1}\) values reduce the settling time of the PV voltage, while high \(\Omega\) values (i.e., \(\Omega>80\) krad/s) do not affect, significantly, such a settling time. Figure 5(a) reports a first option for the SMC and filter parameters (\(k_{1}\) and \(\Omega\)) to regulate the application example of this section. This first option, named Option 1, is characterized by \(\Omega=100\) krad/s and \(k_{1}=0.49\) A/V, which ensures a settling time \(t_{s}=0.8\) ms \(<1\) ms, where \(t_{s}=0.8\) ms was selected to provide a 20% safe margin under the perturbation period \(T_{a}=1\) ms. The left side of Figure 5(a) shows that low values of \(\Omega\) have an impact on both the settling time and \(k_{1}\) values; hence, Figure 5(b) shows a zoom for 4 krad/s \(\leq\Omega\leq 10\) krad/s, which confirms the impact of \(\Omega\) on those qualities. In fact, such a figure shows that the same settling time is achievable with two couples of \(k_{1}\) and \(\Omega\) values, where increments in \(k_{1}\) allow the reduction in \(\Omega\) for the same \(t_{s}\) value. For example, Figure 5(b) reports a second option for the SMC and filter parameters, named Option 2, which is characterized by \(\Omega=9.6\) krad/s and \(k_{1}=0.61\) A/V, which ensures the same settling time of Option 1 (\(t_{s}=0.8\) ms). It is remarkable that reducing \(\Omega\) by 10.4 times only requires multiplying \(k_{1}\) by 1.24 to obtain the same settling time with global stability. This condition is confirmed by the time responses depicted in Figure 7, where Figure 6(a) shows the time response of Option 1 (high \(\Omega\)) and Figure 6(b) shows the time response of Option 2 (low \(\Omega\)). The time response of Option 1 (high \(\Omega\)) shows that the PV voltage fulfills the desired \(t_{s}=0.8\) ms. However, even though the derivative of \(v_{mppt}\) fulfills the stability restriction, the high-value of \(\Omega\) imposes a fast rise in \(v_{mppt}\), and thus, a high voltage slew-rate (\(SR_{f}=50\) V/ms) is needed. The time response of Option 2 (low \(\Omega\)) shows that the PV voltage also ensures the desired \(t_{s}=0.8\) ms, but the low value of \(\Omega\) imposes a slow rise in \(v_{mppt}\); thus, a low voltage slew-rate (\(SR_{f}=4.8\) V/ms) is needed, and a cheaper operational amplifier can be used to implement the reference filter in comparison to Option 1. Such a slew-rate difference is confirmed in Figure 6(c),d, which shows the maximum slew-rate acceptable and the waveform of \(v_{mppt}\), and in both cases, \(SR_{f}\) is fulfilled, but Option 1 exhibits a much higher voltage derivative.