## 5 Control Algorithm and Numerical Analysis
It can be noted that the capacitor voltage, kept practically constant in the steady states conditions, indeed increases during the first transient. This is due to the time response of the PV MPPT algorithm, which induces an average positive voltage derivative in the DC-Link capacitor when the solar radiance rapidly decreases. Finally, in Figure 7c, the space vector module of grid current and common mode converter current is reported, together with the common mode homopolar component. It can be noted that, before the fault occurring event, the common mode currents are kept at very small values (<0.1%). As expected, after the fault event, the module of the space vector associated with the common mode current simply coincides with half of the remaining converter one (see Equation (22)), while the correspondent  
Figure 6: Simulink model of the controlled system. Figure 7: Main electrical quantities during the whole time window: (a) grid and PV Powers and solar irradiation; (b) capacitor voltage and time derivative; (c) space vectors modules of grid current and converter common mode current with homopolar component. homopolar component is null. The module of the line currents space vector is practically constant in the steady state condition, i.e., a low ripple is induced in the line currents by the proposed MPPT algorithm. With reference to the steady state condition at 1000 W/m\({}^{2}\), Figure 8a shows the PV and the grid instantaneous active power together with the grid power, averaged across one modulation period \(\left(\overline{P_{s}}\right)\). The power ripple induced by the MPPT algorithm on the PV power is very small (around 1.1 kW), while the grid power ripple due to the modulation is practically completely filtered out by the LC. The capacitor voltage variation with respect to \(v_{MPP}\) (\(\Delta v_{c}=v_{c}-v_{MPP}\)) is shown in Figure 8b, together with the time derivative of the capacitor voltage. While the voltage ripple is kept really low (<0.3 %), the time derivative is not kept within the upper hysteresis band. This is due to the delay introduced by the digital low pass filter on the capacitor voltage measurement. The overall effect is an asymmetrical hysteresis equivalent controller, which produces a voltage ripple higher than the one obtained in the ideal case. Finally, two line currents are depicted in Figure 8c. As expected, the line currents are sinusoidal and symmetrical; in particular, there is no appreciable oscillation caused by the MPPT. The same quantities of Figure 8 are reported in Figure 9 with reference to the shading condition (250 W/m\({}^{2}\)). The behaviors are time centered on the fault event instant. The PV power ripple in steady state conditions is practically equal to the previous case (around 1.1 kW); this testifies the effectiveness of the proposed power slope adaptive control. As expected, the capacitor voltage ripple is instead increased. Indeed, the reduction of the power slope widens the PV power period of repetition and the capacitor filtering behavior becomes less effective. The line currents appear sinusoidal and symmetrical, even after the fault event. At the time instant of the converter failure, the remaining operating converter is able to quickly compensate for the missing channel. Indeed, the line currents are a little affected (see Figure 9c). Nevertheless, an evident power grid spike is present at the fault instant. In the presence of this perturbation, the MPPT is able to quickly recover the previous operating point (Figure 9a,b). Figure 8: Main electrical quantities during steady state condition at 1000 W/m\({}^{2}\) radiation value: (**a**) grid and PV power; (**b**) capacitor voltage and time derivative; (**c**) line currents. Figure 9: Main electrical quantities during steady state condition at 250 W/m\({}^{2}\) radiation value around the fault instant: (**a**) grid and PV Powers; (**b**) capacitor voltage and time derivative; (**c**) line currents.