## 3 MMC-Based MTDC Power Grid Modelling Overview
### MMC Main Circuit
\begin{array}{l}S_{p}=\frac{1}{2}-\frac{1}{2}M\sin(\omega t+\alpha)+ \frac{U_{\rm cir}}{U_{\rm dc}}\sin(2\omega t+\varphi)\\ S_{n}=\frac{1}{2}+\frac{1}{2}M\sin(\omega t+\alpha)+\frac{U_{\rm cir}}{U_{\rm dc }}\sin(2\omega t+\varphi)\end{array} \right. \tag{4}\]  
where \(M\) is the modulation index and \(\alpha\) is the phase angle of the fundamental frequency component of the switching function. \(U_{\rm cir}\) and \(\phi\) are the amplitude and phase of a control signal generated by the circulating current suppression controllers (CCSCs). In the AVM, although the individual capacitor voltages are not calculated, the large-scale dynamic behavior can be accurately modeled [22]. Instead, a single DC voltage is calculated, so that the AVM can use larger simulation time steps and it requires less computational resources [23]. It provides faster simulation speed and is more efficient than models that consider the detailed MMC converter topology. It has been proposed for studying the dynamics behavior of both single MMC [24; 25] and DC power grids [26; 27; 28; 29]. * SAVM: the SAVM assumes that all of the internal variables in the MMC are perfectly controlled, all submodule capacitor voltages are perfectly balanced, and second harmonic circulating currents in each phase are suppressed [22; 26]. It is a simplified model that is based on the AVM by ignoring the internal circulating current dynamic process. Therefore, the average switching function can be simplified to \[\left\{ \begin{array}{l}S_{p}=\frac{1}{2}-\frac{1}{2}M\sin(\omega t+\alpha) \\ S_{n}=\frac{1}{2}+\frac{1}{2}M\sin(\omega t+\alpha)\end{array} \right.\] (5) It is suitable for the study of large-scale MMC-based MTDC power grids and system-level controller design, as discussed in [30]. * DEM: based on the equivalent branch model of a bridge arm, a DEM of MMC is proposed in [31; 32]. In the DEM, the dynamics of each SM capacitor is represented by an equivalent controlled voltage source \(u_{\rm c}\) in series with resistance \(R_{\rm c}\)[31]. The values of \(u_{\rm c}\) and \(R_{\rm c}\) are \[\left\{ \begin{array}{l}R_{\rm c}=\frac{\Delta t}{2\zeta}\\ u_{\rm c}(t)=u_{\rm c}(t-\Delta t)+2R_{\rm c}i_{\rm c}(t-\Delta t)\end{array} \right.\] (6) where \(C\) is the capacitance of each SM's capacitor, \(\Delta t\) is the time-step for simulation, and \(i_{\rm c}\) is the SM capacitor current. The DEM has three major assumptions: an off-state switch is equivalent to an open circuit; and on-state that is equivalent to \(R_{\rm on}\); \(i_{\rm c}(t)\) is equal to the corresponding bridge arm current when the SM is inserted. \(i_{\rm c}(t)\) is equal to 0, when the SM is passed by [32]. As a result of the aforementioned assumptions, each MMC arm can be modeled as a series connection of arm inductor \(L_{\text{arm}}\), equivalent voltage source \(U_{\text{armEQ}}\), and equivalent resistance \(R_{\text{armEQ}}\). Figure 3 shows the equivalent circuit of each submodule of the MMC in DEM. The equivalent bridge voltage can be given by \[U_{\text{armEQ}}(t)=\sum_{i=1}^{N}U_{\text{Th}_{i}}(t)\] (7) where \(i\) is the index of SM; \(N\) is the number of SM in each bridge arm; and, \(U_{\text{Th}_{i}}(t)\) satisfies \[U_{\text{Th}_{i}}(t)=\left\{ \begin{array}{l l l l}0&\text{if}&\text{SM}_{i}& \text{is}&\text{off}\\ U_{\text{GEQ}_{i}}(t)&\text{if}&\text{SM}_{i}&\text{is}&\text{on}\end{array} \right.\] (8) Meanwhile, \[R_{\text{armEQ}}(t)=N_{\text{on}}\times R_{\text{c}}+N\times R_{\text{on}}\] (9) where \(N_{\text{on}}\) is the number of SMs in on-state. The DEM provides a detailed representation of SMs switching events and individual capacitor ripples. It is feasible for arbitrary number of submodules. The DEM can be applied in order to design the balancing control of the capacitor voltages in the bridge arm. It can also represent the blocked mode, which is essential for accurately describing the MMC behaviors during startup and fault conditions [31; 33].