## 2 Description of MMC-VSG Structure and Modeling
### MMC-VSG Topology and Control Strategy
#### 2.2.2 VSG Control Strategy  
As a control method, the VSG enables the inverter to mimic the SG's droop-based operating performance more accurately. The damping and inertia come from the rotor of the conventional SG, so the SG's rotor motion equation is as follows  
\[J\omega_{0}\frac{\mathrm{d}\omega}{dt}=P_{m}-P_{e}-D(\omega-\omega_{0}) \tag{9}\]  
where \(\omega\) is output voltage angular frequency for VSG, \(\omega_{0}\) is the rated voltage angular frequency, \(P_{e}\) is the output active power, and \(P_{m}\) is the virtual mechanical power, while \(D\) is the virtual damping, and \(J\) is the virtual inertia. The virtual speed controller adopts proportional control. \[P_{m}=P_{ref}+k(\omega_{0}-\omega) \tag{10}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**T1** & **T2** & **Working State** \\ \hline on & off & on \\ off & on & bypass \\ off & off & locked \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Submodule working status. in which \(P_{ref}\) is the active power command, and \(K\) is the droop coefficient. Therefore, the VSG active power frequency control equation can be obtained by combining Equations (2) and (3)  
\[J\omega_{0}\frac{\mathrm{d}\omega}{dt}=P_{ref}-P_{e}-D_{P}(\omega-\omega_{0}) \tag{11}\]  
where \(D_{P}=D+k\) is the equivalent damping coefficient. Converting Equation (4), we find  
\[\frac{\omega-\omega_{0}}{P_{ref}-P}=\frac{1}{Js\omega_{0}+D_{P}} \tag{12}\]  
\[\omega=\frac{P_{ref}-P}{Js\omega_{0}+D_{P}}+\omega_{0} \tag{13}\]  
It can be seen from Equation (5) that a first-order transfer function makes up the VSG's inertia, and the output power can be obtained from VSG grid-connected circuit, which is  
\[P_{e}=3\frac{E_{g}V_{g}}{X}\sin\delta\approx 3\frac{E_{g}V_{g}}{X}\delta=K_{P}\delta \tag{14}\]  
\[Q_{e}=3\frac{E_{g}\cos\delta-V_{g}}{X}V_{g}\approx 3\frac{E_{g}-V_{g}}{X}V_{g} \tag{15}\]  
where \(V_{g}\) is the grid peak voltage, \(E_{g}\) is the MMC peak output voltage, \(X\) is the equivalent impedance of the line, and \(K_{P}=3\frac{E_{g}V_{g}}{X}\). The VSG control block diagram for active power can be achieved from Equations (4) and (7) as shown in Figure 3.  
Figure 2: The main circuit topology and control framework of MMC-VSG. The MMC equivalent AC output port voltage \(e_{g}\) consists of three parts. The first part is MMC-VSG virtual no-load potential \(e_{0}\), which characterizes the terminal voltage of MMC during no-load operation. The second part \(e_{Q}\) corresponds to the part for the reactive power regulation, which can be defined as  
\[e_{Q}=k_{q}\Big{(}Q_{ref}-Q\Big{)} \tag{16}\]  
where \(k_{q}\) is the reactive adjustment coefficient, \(Q_{ref}\) is reference reactive power command of the MMC-VSG, and \(Q\) is the MMC-VSG instantaneous reactive power. The third part \(e_{v}\) corresponds to the output of the MMC-VSG voltage-regulating unit and, thus, can be expressed as  
\[e_{v}=k_{v}\Big{(}E_{ref}-E_{g}\Big{)} \tag{17}\]  
where \(k_{v}\) is the voltage regulation coefficient, and \(E_{ref}\) is the MMC reference peak output voltage. We consider that the SG has a primary voltage-regulating curve as shown  
\[J\omega_{0}\frac{\mathrm{d}\omega}{dt}=P_{ref}-P_{e}-D_{P}(\omega-\omega_{0}) \tag{11}\]0  
Thus, the MMC-VSG excitation controller mathematical model can be expressed by combining Equations (9) to (11) together  
\[J\omega_{0}\frac{\mathrm{d}\omega}{dt}=P_{ref}-P_{e}-D_{P}(\omega-\omega_{0}) \tag{11}\]1  
where \(k_{q}^{\prime}=k_{q}-\frac{k_{v}}{D_{q}}\) can be expressed as the equivalent reactive adjustment coefficient.