## 6 Validation Using Circuital Simulations
A second simulation was carried out to test both the stability and dynamic performance of the proposed control system. This new simulation introduces a change of \(\Delta v_{po}\) in the voltage reference to test the controller's ability to ensure the desired settling time \(t_{s}\) defined in Table 3. Figure 17 reports the detailed simulation results, where the reference change is imposed at \(t=4.17\) ms with the derivative defined in Table 3 (dashed black waveform \(v_{r}\)), i.e., \(\max\left(\left|\frac{d\
u_{z}}{dt}\right|\right)=0.061\) V/us. This simulation also includes the verification of the theoretical dynamic behavior of the PV voltage reported in Equation (58), which was evaluated using the parameters of Table 3 to obtain the numerical version \(G_{pv}\) for this particular example:  
\[G_{pv}=\frac{v_{pv}}{v_{r}}=\frac{2.965\cdot s+1.999\times 10^{4}}{1.1\times 10^{ -4}\cdot s^{2}+2.965\cdot s+1.999\times 10^{4}} \tag{67}\]  
The simulation results reported in Figure 17, at the top, show the correct prediction of the PV voltage (\(v_{pv}\) in blue trace) provided by the theoretical transfer function \(G_{pv}\) (red trace), which exhibits an absolute-relative-error (68) of 0.52% that is mainly caused by the switching ripple, thus confirming the correctness of the voltage design dynamics proposed in Section 4. For the calculation of the ARE, \(v_{sim}\) corresponds to the voltage obtained in the PSIM circuital simulation, while \(v_{theo}\) corresponds to the theoretical value. \[ARE=100\times\frac{\sqrt{\sum_{i=1}^{N}(v_{sim}-v_{theo})^{2}}}{\sqrt{\sum_{i= 1}^{N}(v_{theo})^{2}}}\ [\%] \tag{68}\]  
Figure 16: Stable operation of the proposed PV system. The waveforms in Figure 17 also confirm the correct tracking on the reference (\(v_{r}\) in dashed black trace) with the desired settling time \(t_{s}=400\)\(\upmu\)s, which validates the accurate voltage control design proposed in Section 4.1. The compensation of such a reference change is introduced, by the voltage controller, into the SMC by means of the current reference \(i_{r}\), which acts on the switching circuit by means of the switching function \(\Psi\). The simulation results show the waveform of \(i_{r}\), where the voltage controller compensation can be observed. It is worth noting that this accurate regulation of the PV voltage is performed in presence of a large perturbation in the output voltage \(v_{b}\) (i.e., the DC-link). In this example, the perturbation is equal to a 25% peak-to-peak sinusoidal oscillation at double the grid frequency (120 Hz), which corresponds to the perturbation introduced by a grid-connected inverter with a non-electrolytic DC-link. This perturbation is large in comparison with the DC-link oscillations considered for the validation of other first-stage solutions: Table 4 reports some examples of the peak-to-peak oscillation amplitudes adopted in literature to validate the first stage of microinverters, which are between 1.9 and 500 times smaller than the oscillation adopted in this example, thus requiring a much larger DC-link capacitor. Figure 17: Dynamic performance of the proposed control system. This comparison provides a measurement of the reduction in the DC-link capacitance requirements obtained with the solution proposed in this paper. The simulation results given in Figure 17 also report the duty cycle produced by the SMC, where no saturation occurs (\(0<d<1\)), confirming that the equivalent control condition analyzed in Section 3.3 is fulfilled. In addition, Figure 17 reports the output current \(i_{2}\) of the NEC boost converter, which is always continuous as expected, hence, providing a much better power quality in comparison with the classical boost converter. Moreover, it is observed that the SMC modulates into \(i_{2}\) (and \(d\)) the perturbation present on \(v_{b}\), thus avoiding the transmission of such a perturbation into the PV voltage, which ensures stable operation of the PV module even under large output voltage oscillations.