## 2 Theoretical Study of Serial Submodule RMVSC
### Principle of Harmonic Reinjection  
Increasing the number of levels in the reinjection circuit reduces the harmonic content in the RMVSC ac voltage. However, an excessive increase in levels in the number of levels leads to higher equipment costs and volume while complicating the switches and their control systems. Increasing from an 11-level to a 101-level configuration reduces the ac output voltage THD from 2% to 1.1% [15]. However, the quantity of switches has increased tens of times, rendering this harmonic reduction approach impractical. Generally, a 7-level RMVSC is adequate to meet system requirements in voltage classes of 35 kV and below. The reinjection circuit periodically supplies a specific level of dc voltage to the main bridge, thereby eliminating harmonics. The switching states of the main bridge circuit and the reinjection circuit are synchronized. When a switching state in the main bridge circuit changes, the reinjection voltage increases or decreases periodically in equal-width steps, with the Y and D reinjection circuits showing entirely complementary behaviors. When the triggering angle is \(\alpha\), the 1 m level increment interval of the Y reinjection circuit is  
\[\omega t\in\left\{\alpha-\frac{\pi}{12(m-1)},\ \alpha-\frac{\pi}{12(m-1)}+ \frac{\pi}{6}\right\} \tag{1}\]  
Figure 1: Serial RMVSC topology submodule. where \(\alpha\) represents the triggering angle of the RMVSC. This interval also corresponds to the decreasing interval of the D reinjection circuit. The interval in which the Y reinjection voltage at the \(i\)th level (\(i=1,2,\ldots m\)) is situated can be defined as  
\[\omega t\in\left\{\alpha+\frac{(2i-3)\pi}{12(m-1)},\ \alpha+\frac{(2i-1)\pi}{12(m-1)}\right\} \tag{2}\]  
The height of this level step is  
\[H_{i}=\frac{i-1}{m-1}U_{dc} \tag{3}\]  
where \(U_{\text{dc}}\) is the dc voltage. The instantaneous value of the dc voltage reinjected into the main bridge circuit can be regulated by adjusting the number of submodules connected at any given moment in the serial submodule reinjection circuit, as illustrated in Figure 1. Assuming each submodule has equal capacity and that the submodule capacitors balance their charge and discharge over one cycle, thereby maintaining a constant average voltage. In an m-level reinjection circuit, (m \(-\) 1) submodules are connected at any moment. If the number of submodules connected to the Y-bridge is n, then the remaining (m \(-\) 1 \(-\) n) submodules will be connected to the D-bridge. The voltage of the reinjection circuit is given by  
\[\left\{ \begin{array}{l}U_{jY}=\frac{n}{m-1}U_{dc}\\ U_{jD}=\frac{n-1}{m-1}U_{dc}\end{array} \right. \tag{4}\]  
Simultaneously, the main bridge operates in a 180\({}^{\circ}\) conduction mode, with a 30\({}^{\circ}\) phase angle difference between the Y and D bridges. Consequently, during one cycle of the RMVSC, the ac voltages of the Y and D bridges are as follows:  
\[U_{\text{DA}}= \begin{cases}0&\alpha+0<\omega t\leq\frac{\pi}{6}+\alpha\\ U_{\text{JD}}(\omega t)&\alpha+\frac{\pi}{6}<\omega t\leq\frac{5\pi}{6}+\alpha \\ 0&\alpha+\frac{5\pi}{6}<\omega t\leq\frac{7\pi}{6}+\alpha\\ -U_{\text{JD}}(\omega t)&\alpha+\frac{7\pi}{6}<\omega t\leq\frac{11\pi}{6}+ \alpha\\ 0&\alpha+\frac{11\pi}{6}<\omega t\leq 2\pi+\alpha\end{cases} U_{\text{YA}}= \begin{cases}U_{\text{JY}}\ (\omega t)/3&\alpha+0<\omega t\leq\frac{\pi}{3}+\alpha\\ 2U_{\text{JY}}\ (\omega t)/3&\alpha+\frac{\pi}{3}<\omega t\leq\frac{2\pi}{3}+ \alpha\\ U_{\text{JY}}\ (\omega t)/3&\alpha+\frac{2\pi}{3}<\omega t\leq\pi+\alpha\\ -U_{\text{JY}}\ (\omega t)/3&\alpha+\pi<\omega t\leq\frac{4\pi}{3}+\alpha\\ -2U_{\text{JY}}\ (\omega t)/3&\alpha+\frac{4\pi}{3}<\omega t\leq\frac{5\pi}{3}+ \alpha\\ -U_{\text{JY}}\ (\omega t)/3&\alpha+\frac{5\pi}{3}<\omega t\leq 2\pi+\alpha\end{cases}  \tag{5}\]  
In addition, owing to the 180\({}^{\circ}\) conduction method, three IGBTs conduct in both the Y and D main bridges at any given moment. Commutation occurs every 60\({}^{\circ}\), with a consistent 30\({}^{\circ}\) phase angle difference between the two bridges.