# Three-Phase PV CHB Inverter for a Distributed Power Generation System
## 3 Numerical Analysis
As can be seen, the voltages are synthesized by means of seven discrete levels, and the currents are symmetric and balanced; in this case (uniform illumination), the homopolar component \(v_{o}\) (see Equation 4) is set to zero. The observed time interval of 100 ms corresponds to a MPPT period, while the switching frequency was \(f=5\) kHz. Moreover, the THD values of the phase currents and voltages have been calculated up to the 40 th harmonic, as defined by the standard rules. The numerical results for the three-phase currents are 0.23%, 0.28%, and 0.27%, while for the modulated voltages they are 5.76%, 5.77%, and 6.40%. The results gained when solar panels were not uniformly irradiated are reported in Figures 8-10. In these experiments, the nine solar panels were subjected to the irradiance levels reported in Table 3, so that each of the three legs supplied a different power. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Phase** & **1** & **2** & **3** \\ \hline cell 1 & 625 W/m\({}^{2}\) & 625 W/m\({}^{2}\) & 425 W/m\({}^{2}\) \\ cell 2 & 300 W/m\({}^{2}\) & 425 W/m\({}^{2}\) & 425 W/m\({}^{2}\) \\ cell 3 & 625 W/m\({}^{2}\) & 625 W/m\({}^{2}\) & 425 W/m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Non-uniform case: PV panel irradiance values. Figure 7: Sunny case: comparison among (**a**) phase currents \(i_{grid,\,p^{2}}\) and (**b**) phase modulated voltages \(v_{inv,p}\). Figure 8: Partial shading: the operating point position corresponding to three panels in the \(p\)-th phase leg (namely, panel 1, \(p\) (pink squares), panel 2, \(p\) (orange circles), and panel 3, \(p\) (green triangles)) is highlighted with respect to the individual panel P–V curves. The solid-line P–V curves correspond to the panels under sunny conditions, while the dashed lines describe panels 2, 1 and 2, 2 under partial shading conditions. (**a**) Phase leg 1; (**b**) phase leg 2; (**c**) phase leg 3. Figure 9: Partial shading: time behavior of the output currents compared to the grid voltages related to (**a**) phase 1; (**b**) phase 2; and (**c**) phase 3. It can be observed from Figure 8 that--also in the case of partial shading--all solar panels operate at their respective MPP. What is now more challenging for the control system is to guarantee balanced grid currents, even if the supplied powers on the three legs were different from each other. The consequence of a power imbalance is that the outputs \(v_{imv,p}\) of the multilevel inverters are also unbalanced. As explained in the previous section, the equilibrium of the three grid currents is achieved thanks to the addition of the homopolar voltage component \(v_{o}\). The effectiveness of this strategy is clearly visible in Figure 9, showing the superposition of the grid currents with the grid voltages, and Figure 10 showing the phase relation among the grid currents and the modulated voltages corresponding to three phase legs. In particular, from Figure 10b it can be seen that the zero-sequence injection provides no more balanced modulated voltages while assuring balanced currents. The THD values of the phase currents and voltages have been calculated up to the 40 th harmonic, as for the uniform case. The numerical results are 0.60%, 0.68%, and 0.47% for the three-phase currents, while for the modulated voltages they are 5.60%, 6.00%, and 6.20%. The last experiment was performed by linearly varying the irradiance impinging on the first solar panel of the first leg from 625 W/m\({}^{2}\) to 400 W/m\({}^{2}\) in a time interval starting at 0.4 s and ending at 0.8 s, so that the MPP of that panel changed from 73 W to 40 W. From Figure 11, we observe that the MPP of the shaded panel is correctly tracked during time; at the same time, the power factor is unity in all three legs (Figure 12). As evidenced in Figure 13, the homopolar correction (4) works as well; indeed, the three grid currents are well balanced (Figure 13a), while the phase relation among the voltages (Figure 13b) changes during time.