## 3 Voltage-Sharing Driving Method for RMVSC  
### Design Principles for Voltage-Sharing Driving Method  
Based on theoretical analysis, the focus of RMVSC in eliminating harmonics and operating at low frequencies is to control the reinjection circuit to generate high-quality multilevel reinjection voltage waveforms. Therefore, the driving method of the reinjection circuit is critical to the RMVSC. It governs not only the balance of capacitor voltages across each submodule but also the generation of a linear multilevel stepped voltage by the reinjection circuit, thereby influencing the quality of the ac output power. This factor is essential for maintaining the stable operation of the system. Each submodule maintains voltage stability by balancing the charge exchange between its capacitor and the main bridge circuit over a single power cycle. For the m-level reinjection circuit, the capacitor voltage of the \(i\)th submodule (\(i\) = 1, 2,..., m \(-\) 1) is given by  
\[\left\{ \begin{array}{l}U_{\text{SMy}i}=\dfrac{1}{j\omega C_{\text{SMy}i}} \int_{-\infty}^{t}I_{\text{SMy}i}dt\\ U_{\text{SMdi}i}=\dfrac{1}{j\omega C_{\text{SMdi}i}}\int_{-\infty}^{t}I_{\text {SMdi}i}dt\end{array} \right. \tag{7}\]  
where \(I_{\text{SMy}i}\) and \(I_{\text{SMdi}i}\) are the \(i\)th submodule currents of the Y bridge and D bridge, respectively. \(C_{\text{SMy}i}\) and \(C_{\text{SMdi}i}\) are \(i\)th submodule capacitors of the Y bridge and D bridge, respectively. Applying the Laplace transform to this gives  
\[\left\{ \begin{array}{l}U_{\text{SMy}i}(s)=I_{\text{SMy}i}(s)/sC_{\text{SMy}i }\\ U_{\text{SMdi}i}(s)=I_{\text{SMdi}i}(s)/sC_{\text{SMdi}i}\end{array} \right. \tag{8}\]  
From Equation (8), the condition for voltage equalization across the submodules is that the capacitor values and the current flowing through each submodule over a complete cycle must be identical. The shorter the period, the more effective the capacitor balance. Given that the capacitor values of the submodules are identical, only the current needs to be considered. The fundamental voltage of the power grid is  
\[\left[ \begin{array}{c}U_{\rm A}(\omega t)\\ U_{B}(\omega t)\\ U_{\rm C}(\omega t)\end{array} \right]=\left[ \begin{array}{c}\sqrt{2}U_{\rm RMS }\sin(\omega t)\\ \sqrt{2}U_{\rm RMS}\sin(\omega t-\frac{2\pi}{3})\\ \sqrt{2}U_{\rm RMS}\sin(\omega t+\frac{2\pi}{3})\end{array} \right] \tag{9}\]  
where \(U_{\rm RMS}\) is the RMS value of the phase voltage. The ac current of the Y main bridge and D main bridge are  
\[\left[ \begin{array}{c}I_{\rm VA}(\omega t)\\ I_{\rm VB}(\omega t)\\ I_{\rm VC}(\omega t)\end{array} \right]=\left[ \begin{array}{c}\sqrt{2}I_{\rm RMS }\sin(\omega t+\varphi)\\ \sqrt{2}I_{\rm RMS}\sin(\omega t-\frac{2\pi}{3}+\varphi)\\ \sqrt{2}I_{\rm RMS}\sin(\omega t+\frac{2\pi}{3}+\varphi)\end{array} \right] \tag{10}\]  
\[\left[ \begin{array}{c}I_{\rm DA}(\omega t)\\ I_{\rm DB}(\omega t)\\ I_{\rm DC}(\omega t)\end{array} \right]=\left[ \begin{array}{c}\sqrt{2}I_{\rm RMS }\sin(\omega t+\frac{\pi}{6}+\varphi)\\ \sqrt{2}I_{\rm RMS}\sin(\omega t-\frac{2}{3}+\varphi)\\ \sqrt{2}I_{\rm RMS}\sin(\omega t+\frac{5\pi}{6}+\varphi)\end{array} \right] \tag{11}\]  
where \(\varphi\) is the displacement factor angle. When the converter operates in a state where the current leads the voltage, \(\theta\) is positive. \(I_{\rm RMS}\) is the RMS value of the phase current. The \(I_{\rm dcY}\) and the \(I_{\rm dcD}\) exhibit a periodicity of \(\pi/3\). Over one period, they can be expressed as follows:  
\[I_{\rm dcY}=-\sqrt{2}k_{\rm n}I_{\rm RMS}\sin(\omega t+\varphi-2\pi/3)\ \ \ \ \ 0< \omega t<\frac{\pi}{3} \tag{12}\]  
\[I_{\rm dcD}=\sqrt{2}k_{\rm n}I_{\rm RMS}\times\left\{ \begin{array}{cc}-\sin (\omega t+\varphi-\pi/2)&0<\omega t\leq\frac{\pi}{3}\\ \sin(\omega t+\varphi+\pi/6)&\frac{\pi}{6}<\omega t<\frac{\pi}{3}\end{array} \right. \tag{13}\]  
where \(K_{\rm n}\) is the transformation ratio of the Y/Y side transformer.