## 3 Voltage-Sharing Driving Method for RMVSC
### Design for Voltage-Sharing Driving Method  
To minimize the switching frequency of the reinjection circuit and reduce energy loss, each submodule should be connected and bypassed exactly once during each current pulsation period (\(\pi/3\)). By designing the drive pulses to be symmetrical about the half-period, the complexity of the submodule drive pulses can be reduced. Since the drive pulses for the Y reinjection circuit and the D reinjection circuit are complementary, only the drive pulses for the Y reinjection circuit need to be designed.  
Based on Section 2.2, the instantaneous magnitude of the reinjection voltage is determined by the number of reinjection circuits conducting at any given moment. For an m-level reinjection circuit, a simple drive pulse can be designed to generate an ideal reinjection voltage waveform, meeting the required specifications. The switching function of the \(i\)th submodule (\(i=1,2,3\dots\text{m}-1\)) during the \(k\)th reinjection voltage interval (k = 1, 2, 3...) for the Y reinjection circuit can be expressed as  
\[f_{\text{sy}i}=\left\{\begin{array}{l}1\ \frac{(2i-1)\pi}{ 12(m-1)}+\frac{k\pi}{3}+\alpha<\omega t\leq\frac{(25-2i)\pi}{12(m-1)}+\frac{k \pi}{3}+\alpha\\ 0\ \frac{(25-2i)\pi}{12(m-1)}+\frac{k\pi}{3}+\alpha<\omega t<\frac{(2i-3)\pi}{12 (m-1)}+\frac{(k+1)\pi}{3}+\alpha\\ (i=1,2,\cdots m)\ \ (k\in\text{Z})\end{array}\right. \tag{15}\]  
where \(\alpha\) is the trigger angle. The switching function described by Equation (15) is shown in Figure 4.  
Figure 4: The m-level simple drive pulse sequence.  
As indicated by Equations (8) and (14), for the aforementioned driving pulse, the current passing through each submodule is not identical, which prevents voltage balance across the submodules. Therefore, the following principle is proposed: within one current pulsation cycle, if a submodule has a longer on-time, its on-time should be shortened in the next conduction interval, and vice versa. Based on this principle, the driving pulses in Figure 4 can be processed cyclically, meaning that each submodule cyclically adopts the switching functions of all submodules in the reinjection circuit. Consequently, a cyclic triggering method is designed, with its switching function expressed as  
\[\left\{\begin{array}{l}f_{\text{sly}i}(\omega t)=f_{\text{sly}(i-1)}(\omega t \pm\pi/3)\\ f_{\text{sjd}i}(\omega t)=f_{\text{sjd}(i-1)}(\omega t\pm\pi/3)\end{array}\right. \tag{16}\]  
where \(\pi/3\) represents the phase shift interval. An m-level RMVSC requires a phase shift period of \((\text{m}-1)\pi/3\) to complete one cycle of cyclic triggering, during which each submodule adopts the switching functions of all other submodules in the reinjection circuit once. Using this method, a cyclic driving pulse is designed for the 7-level reinjection circuit, with a cycle period of \(2\pi\), and the cyclic driving pulse sequence is illustrated in Figure 5.  
After adopting cyclic driving pulses for the 7-level RMVSC, the submodules adjust the triggering intervals within each current pulsation cycle. This adjustment enables the driving pulse to circulate to be completed within \(2\pi\). Based on Equations (8) and (14), the cyclic driving pulses satisfy the condition for sub-module capacitor voltage balancing, ensuring that each sub-module undergoes an identical charging and discharging process within \(2\pi\), thereby achieving sub-module capacitor balance. Thereby satisfying the condition for submodule capacitor voltage balance. Furthermore, each driving pulse is symmetrical with respect to the half-current pulsation cycle without increasing the complexity of the driving  
Figure 5: Cyclic drive pulse sequence of 7-level RMVSC. (**a**) Number of conduction reinjection switches on Y side. (**b**) Driving pulse waveforms of Y reinjection switches under the phase-shift switching functions.  
sequence. This approach achieves low-frequency operation of the RMVSC switches while maintaining equal voltage of the submodules.