## 5 Numerical Simulation
PV array average voltage is still very close to 154 V in a good agreement with the value expected from PV module specifications. From \(t=4.5\) s to \(t=5\) s, sun irradiance is maintained at 500 W/m\({}^{2}\) and MPPT continues to track the maximum power. On the other hand, under irradiance changes, the DC-link PV voltage \(V_{\rm dc}\), undergoes a very small change and remains very close to the MPP voltage and the PI controller is capable of maintaining the voltage \(V_{\rm dc}\) at the desired value after short transient period (0.02 s) due MPPT perturbations. The results clearly demonstrate the capability of the designed system to operate at the MPP regardless of the value of the irradiance and the MPPT control is capable of tracking the MPP when irradiance vary. Figure 11 shows the steady-state waveforms of the PV voltage and its reference, the extracted power, the power \(P\) from the PV source and its maximum power \(P_{\rm max}\), the inductor currents \(i_{1}\) and \(i_{2}\) and the capacitor voltages \(v_{o1}\) and \(v_{o2}\) for irradiance level \(S=\)500 W/m\({}^{2}\) (\(P_{\rm max}\approx 670\) W) and 1000 W/m\({}^{2}\) (\(P_{\rm max}\approx 1.4\) kW). It can be observed that in the vicinity of the MPP, the reference voltage \(V_{\rm mpp}\) dictated by the MPPT controller oscillates according to the values of the MPPT sampling period \(T_{\rm mppt}=0.1\) s and the perturbation amplitude \(\Delta V_{\rm mpp}=4\) V as shown in Figure 11a,b for the two different values of the sun irradiance. The inductor currents and capacitor voltage are well balanced and are phase shifted by 180\({}^{\circ}\) as depicted in Figure 11c,d for the same values of the sun irradiance level. Moreover, the steady-state averaged values of the capacitor voltages and the inductor current are in a close agreement with the theoretical predictions given in (8)-(11) and depicted in Figure 5. Note also that, the DC PV voltage is regulated to the MPP voltage and that it is practically a sinusoidal signal as predicted by (37) with DC component equal to \(V_{\rm mpp}\pm\Delta V_{\rm mpp}\) dictated by the MPPT controller during the MPPT sampling period \(T_{\rm mppt}=0.1\) s and peak-to-peak AC ripple \(\Delta\overline{v}_{\rm dc}\approx 14.65\) V for \(P_{\rm max}=1.4\) kW and it is \(\Delta\overline{v}_{\rm dc}\approx 7.02\) V for \(P_{\rm max}=670\) W in a remarkable agreement with (36). Figure 10: Startup and steady-state behavior and response to changes in the irradiance level of the boost inverter between 500 W/m\({}^{2}\) (\(P_{\rm max}\approx 670\) W) and 1000 W/m\({}^{2}\) (\(P_{\rm max}\approx 1.4\) kW). \(\Theta=25\)\({}^{\circ}\)C.  
Figure 12 shows the steady-state waveforms of the grid current \(i_{g}\) together with the grid voltage \(v_{g}\) for irradiance level \(S=500\) W/m\({}^{2}\) (\(P_{\max}\approx 670\) W) and \(1000\) W/m\({}^{2}\) (\(P_{\max}\approx 1.4\) kW). Observe that grid voltage and current are in phase and have practically the same sinusoidal shape demonstrating a close to unity power factor. Moreover, the steady-state averaged values of the grid current is in a close agreement with the theoretical predictions. Namely, for irradiance level \(S=500\) W/m\({}^{2}\) (\(P_{\max}\approx 670\) W), the RMS value of the grid current \(i_{g}\) is about 2.91 A and for \(S=\)1000 W/m\({}^{2}\) (\(P_{\max}\approx 1.4\) kW), the RMS value of \(i_{g}\) is about 6.1 A. To check the power quality, the THD of the grid current \(i_{g}\) in steady-state was calculated for both values of the irradiance. Namely, for irradiance level \(S=500\) W/m\({}^{2}\) (\(P_{\max}\approx 670\) W), the THD value of the grid current \(i_{g}\) is about 1% and for \(S=\)1000 W/m\({}^{2}\) (\(P_{\max}\approx 1.4\) kW), the THD value of \(i_{g}\) is about 1.2%. Therefore, it can be observed that the grid current \(i_{g}\) exhibits a low THD of about 1% as calculated by PSIM\({}^{\copyright\) software for both values of sun irradiance level.