## 3 MMC-Based MTDC Power Grid Modelling Overview
### MMC Control Systems  
Figure 5 shows a typical structure of MMC control systems. The control systems for a single MMC consist of vector current controls (VCCs), phase-lock-loops (PLLs) and CCSCs [18; 24; 25; 37; 38]. The VCCs consist of inner and outer control loops. The control modes of MMCs are determined by the outer loop control objectives that can be chosen among active/reactive power control mode, DC voltage/reactive power control mode, active power/AC voltage control mode, and DC/AC voltage control mode. Figure 4: EDM of MMC under DC fault. Figure 5: Structure diagram of an MMC control system. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Model** & **Single** & **Control** & **Circuit** & **System** & **Control** & **SM** & **DC** \\  & **SM** & **System** & **Current** & **Level** & **Strategy** & **Level** & **Fault** \\  & **Dynamics** & **Dynamics** & **Dynamics** & **Analysis** & **Design** & **Analysis** & **Analysis** \\ \hline DSM & \(\surd\) & \(\surd\) & \(\surd\) & \(\times\) & \(\times\) & \(\surd\) & \(\surd\) \\ AVM & \(\times\) & \(\surd\) & \(\surd\) & \(\surd\) & \(\times\) & \(\surd\) \\ SAVM & \(\times\) & \(\surd\) & \(\times\) & \(\surd\) & \(\times\) & \(\surd\) \\ DEM & \(\surd\) & \(\surd\) & \(\surd\) & \(\times\) & \(\surd\) & \(\times\) \\ EDM & \(\times\) & \(\times\) & \(\times\) & \(\times\) & \(\times\) & \(\times\) & \(\surd\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of different MMC model. The VCC typically adopts d-q decoupling control strategy for power or voltage, with proportion-integral (PI) controllers. as shown in Figure 5. In VCC, the d-q decoupling control strategy is realized through four PI controllers. The outputs of the VCC are the dynamic phasors of the fundamental frequency components. Therefore, a fourth-order model can be formulated in order to illustrate the dynamics of PI controllers. \[ \begin{cases}\dfrac{\mathrm{d}x_{1}}{\mathrm{d}t}=i_{\mathrm{dref}}-i_{ \mathrm{sd}}\\ \dfrac{\mathrm{d}x_{2}}{\mathrm{d}t}=i_{\mathrm{qref}}-i_{\mathrm{sq}}\\ \dfrac{\mathrm{d}x_{3}}{\mathrm{d}t}=P_{\mathrm{ref}}-P\quad\text{or}\quad \dfrac{\mathrm{d}x_{3}}{\mathrm{d}t}=U_{\mathrm{dref}}-U_{\mathrm{dc}}\\ \dfrac{\mathrm{d}x_{4}}{\mathrm{d}t}=Q_{\mathrm{ref}}-Q\quad\text{or}\quad \dfrac{\mathrm{d}x_{4}}{\mathrm{d}t}=U_{\mathrm{acref}}-U_{\mathrm{ac}}\end{cases}  \tag{11}\]  
where \(P_{\mathrm{ref}}\), \(Q_{\mathrm{ref}}\)\(U_{\mathrm{dref}}\) and \(U_{\mathrm{acref}}\) are the reference values of active power \(P\), reactive power \(Q\), DC voltage \(U_{\mathrm{dc}}\), and AC voltage \(U_{\mathrm{ac}}\), respectively. The reference currents \(i_{\mathrm{dref}}\) and \(i_{\mathrm{qref}}\) are given below. \[ \begin{cases}i_{\mathrm{dref}}=k_{\mathrm{pl}}\left(P_{\mathrm{ref}}-P\right) +k_{i1}x_{3}\quad\text{or}\quad k_{\mathrm{pl}}\left(U_{\mathrm{dref}}-U_{ \mathrm{dc}}\right)+k_{i1}x_{3}\\ i_{\mathrm{qref}}=k_{\mathrm{p2}}\left(Q_{\mathrm{ref}}-Q\right)+k_{i2}x_{4} \quad\text{or}\quad k_{\mathrm{p2}}\left(U_{\mathrm{acref}}-U_{\mathrm{ac}} \right)+k_{i2}x_{4}\end{cases}  \tag{12}\]  
where \(k_{\mathrm{p}j}\) and \(k_{ij}\) are the proportional constant and integral constant of the \(j\)th PI controller, repectively. The outputs of the MMC control system denoted by \(U_{\mathrm{cd}}\) and \(U_{\mathrm{cq}}\) are the dynamic phasors of the fundamental frequency components in the d-q frame, as follows  
\[ \begin{cases}u_{\mathrm{cd}}=u_{\mathrm{d}}-\omega Li_{\mathrm{q}}-k_{\mathrm{ p3}}\left(i_{\mathrm{dref}}-i_{\mathrm{d}}\right)-k_{i3}x_{1}\\ u_{\mathrm{cq}}=u_{\mathrm{q}}+\omega Li_{\mathrm{d}}-k_{\mathrm{p4}}\left(i_{ \mathrm{qref}}-i_{\mathrm{q}}\right)-k_{i4}x_{2}\end{cases}  \tag{13}\]  
PLL consists of two PI controllers. The dynamic of the PLL can be described as  
\[ \begin{cases}\dfrac{\mathrm{d}x_{5}}{\mathrm{d}t}=u_{\mathrm{sq}}\\ \dfrac{\mathrm{d}x_{\mathrm{pll}}}{\mathrm{d}t}=-k_{\mathrm{ppll}}U_{\mathrm{sq }}-k_{\mathrm{ipll}}x_{5}\end{cases}