## 5 Numerical Results  
The proposed control strategy has been tested in the Matlab/Simulink(r) environment; the simulation parameters are shown in Table 1.  
In order to test the effectiveness of the used approach, the system undergoes two strong perturbations. At 0.5 s, starting from a steady state no load condition, the power absorbed by the LV DC-Link varies instantaneously from 0 to 100 kW. At 1.5 s, the power absorbed by the LV DC-Link varies instantaneously from 100 kW to \(-\)100 kW. Figure 7 shows the whole simulation results. It can be noted that both the grid and the transformer powers accurately follow their reference. The ripple in the transformer power, averaged over the sampling time, is linked to the MMC homopolar differential voltage output, which is characterized by unwanted oscillations due to the differential currents control. The effective tracking of the reference powers allows a good dynamic response of the DC-Link controllers, as synthetically shown in the correspondence of the power step variation in correspondence of the second perturbation by the DC-Link overshot, contained under 4% of its rated value. The balancing of the DC-Link capacitors operates with satisfying results: the maximum voltage difference per leg is kept around 1% of the cells' rated voltage during the whole simulation, except for the first step variation, when the maximum difference is characterized by a strong overshoot of about 8%, quickly recovering under 0.1 s. Finally, \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Symbol** & **Quantity** & **Value** & **Symbol** & **Quantity** & **Value** \\ \hline \(T_{s}\) & Sample Time Interval & 100 μs & \(f_{m}\) & Modulation Frequency & 5 kHz \\ \(f_{R}\) & Transformer Frequency & 5 kHz & \(P_{n}\) & System Rated Power & 100 kVA \\ \(f_{n}\) & Grid Rated Frequency & 50 Hz & \(V_{n}\) & Grid Rated Voltage & 20 kV \\ \(N_{leg}\) & Modules per Leg & 9 & \(C_{MV}\) & MV DC-Links Capacitance & 16 μs \\ \(L_{s}\) & Grid Boost Inductance & 0.5 H & \(L_{\text{int}}\) & MMC Inductance & 0.12 H \\ \(L_{aq}\) & Power Transfer Inductance & 0.19 H & \(C_{LV}\) & LV DC-Link Capacitance & 10 mF \\ \(k_{b}\) & Primary Boost Factor & 1.4 & \(k_{R}\) & Transformer Ratio & 50 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. Figure 6: Converter output voltages and transformer current for positive (**a**) and negative (**b**) power. the RMS value of the transformer current is close enough to the ideal value (1 p.u.) obtained with an ideal square wave switching with no phase shift delay. Figures 8 and 9 show the behavior of the homopolar differential voltage and of the transformer currents in correspondence of the two power step variations. The voltages are expressed in p.u. of their rated values, whereas the transformer current is normalized with respect to the magnitude value obtainable with an ideal square-wave current in phase with the MMC ideal output voltage which would produce a power flow of \(P_{n}=100\) kVA. It can be noted that the output voltage \(v_{\Delta 0}\) is characterized by square shaped oscillations around its mean value \(v_{\Delta 0,id}\) (i.e., the ideal square-wave voltage of the PSC strategy) due to the asymmetrical switching of the upper and lower converters. This unwanted ripple forces the transformer current to deviate from its ideal shape (see Figure 3). The resulting interaction between \(v_{\Delta 0}\) and \(i_{R}\) introduces the observed ripple in the transformer power. Finally, it can be noted that the phase shift between \(v_{\Delta 0}\) and \(v^{\prime}_{LV}\) quickly changes from a small value to a positive value in correspondence of the first step variation (see middle diagram in Figure 8) and from a positive value to a negative value in correspondence to the second step variation (see middle diagram in Figure 9).