## 2 Theoretical Study of Serial Submodule RMVSC
### Principle of Harmonic Reinjection
During a single cycle, the commutation instances of the switches in the Y and D main bridges of the RMVSC are as follows:  
\[ \begin{split} t_{Y}(\omega t)&=\alpha+\left\{0,\frac{ \pi}{3},\frac{2\pi}{3},\pi,\frac{4\pi}{3},\frac{5\pi}{3}\right\}\\ t_{D}(\omega t)&=\alpha+\left\{\frac{\pi}{6},\frac{\pi}{2},\frac{5 \pi}{6},\frac{7\pi}{6},\frac{3\pi}{2},\frac{11\pi}{6}\right\}\end{split}  \tag{6}\]  
From Equations (2) and (6), it is evident that during the commutation moment of the main bridge, the number of reinjection voltage levels is 1, corresponding to a zero-voltage state, with the zero-voltage interval being \(\pi/6(m-1)\). This allows for ZVS in the RMVSC main bridge devices. Based on the aforementioned analysis of the RMVSC working principle, assuming the reinjection voltage is an ideal multilevel voltage, the dc voltage is the unit voltage, and ignoring device losses, the theoretical waveform of the 7-level RMVSC within one cycle, with a trigger angle of 30\({}^{\circ}\), is shown in Figure 2.  
Figure 2a,b illustrate the trigger pulses of the RMVSC's Y and D main bridges, where each switch operates for 180\({}^{\circ}\) per cycle, with commutation occurring every 60\({}^{\circ}\). As shown in Figure 2c,d, the dc voltage reaches zero at the commutation moment, thereby achieving zero-voltage commutation in the main bridge. For instance, at \(\omega\)t = 0, the Y main bridge  
Figure 2: The theoretical voltage waveforms of 7-level RMVSC. (**a**) The switching signal of the Y main bridge. (**b**) The switching signal of the D main bridge. (**c**) Y reinjection voltage. (**d**) D reinjection voltage. (**e**) The A-phase voltage on the valve side of the Y-winding transformer. (**f**) The A-phase voltage on the valve side of the D-winding transformer. (**g**) The A-phase voltage on the grid side of the RMVSC. (**h**) The spectrum of the A-phase voltage. undergoes commutation. When the drive signal \(f_{\text{sy1}}\) transitions from low to high, the reinjection voltage is zero, allowing switch \(S_{\text{y1}}\) to achieve zero-voltage turn-on. At \(\omega\)t = \(\pi\), the reinjection voltage causes switch \(S_{\text{y1}}\) to achieve zero-voltage turn-off. In each power cycle, the Y bridge and D bridge experience six zero voltage intervals, respectively. The IGRTs in the Y bridge and D bridge undergo commutation during their corresponding six zero voltage intervals. Therefore, the reinjection circuit facilitates the realization of ZVS in the main bridge of the RMVSC by adjusting the access and bypass of a submodule, significantly reducing dynamic losses and the impact on switches while improving device stability. Figure 2c,d illustrate the reinjection voltages \(U_{\text{YY}}\) and \(U_{\text{JD}}\) for the Y bridge and D bridge, respectively. Both voltages are 7-level stepped voltages with a phase difference of 30\({}^{\circ}\), which corresponds to the phase difference in the drive signals of the main bridge. At each moment, the sum of the two reinjection voltages remains a constant dc voltage. According to Equation (5), the A-phase voltage waveforms \(U_{\text{YA}}\) and \(U_{\text{DA}}\), generated on the valve side of the transformer after the main bridge converter processes the reinjection voltages \(U_{\text{JY}}\) and \(U_{\text{JD}}\), are shown in Figure 2e,f. The two waveforms are serially superimposed through the transformer's grid-side windings, resulting in the A-phase voltage waveform, as shown in Figure 2g. The corresponding spectrum is presented in Figure 2h, where the total harmonic distortion (THD) of the voltage \(U_{\text{A}}\) is only 2.36%, which is significantly lower than the 15% THD of a 12-pulse converter. Furthermore, the 11 th and 13 th harmonic content is significantly reduced compared to that of a 12-pulse converter, thereby greatly improving power quality. Consequently, RMVSC can output high-quality power even under low switching frequency conditions.