## 3 MMC-Based MTDC Power Grid Modelling Overview
### MMC Main Circuit  
Detailed switch model (DSM), average value model (AVM), simplified average value model (SAVM), detailed equivalent model (DEM), and equivalent discharging model (EDM) are the major types of accurate and efficient models to represent the dynamics of the MMC main circuit under different scenarios [19]. The principles and application scenarios of each MMC main circuit model are introduced, as follows. * DSM: in a DSM, a detailed switch is applied to present the operational times of the switching devices and the on/off-state resistances/voltages in the SMs [20]. The detailed representation requires a small simulation time step to accurately track the switching voltage edges that are essential for the preservation of various frequency components in the AC and DC waveforms. Because a large number of electrical nodes is created in DSM, it leads to huge simulation time cost. With the increasing number of SMs in MMCs, the DSM is facing a significant challenge in computation [7]. DSM is not suitable for the dynamic analysis, control design, and fault analysis for large scale DC power systems, and simplifications shall be made. * AVM: the AVMs of MMC has been proposed to approximate system dynamics, where the switching details are not explicitly modeled and the dynamics of MMC is equivalent through using controlled voltage and current sources [18]. The relationship of the arm currents (\(i_{\text{p}}\) for upper bridge arm and \(i_{\text{n}}\) for lower bridge arm) and the capacitor voltages (\(u_{\text{p}}\) for the upper bridge arm and \(u_{\text{n}}\) for lower bridge arm) can be expressed as \[S_{\text{p}}\cdot i_{\text{p}}=C\frac{\text{d}u_{\text{cp}}}{\text{d}t},S_{ \text{n}}\cdot i_{\text{n}}=C\frac{\text{d}u_{\text{cn}}}{\text{d}t}\] (1) where \(C\) is the capacitance of each submodule capacitor; and, \(S_{\text{p}}\) and \(S_{\text{n}}\) are the average switching functions of the upper and lower arms. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**SM Topologies** & **Output Voltage Level** & **DC Short Circuit Fault Ride through Capability** & **Power Losses** \\ \hline HB-SM & \(0,u_{\text{c}}\) & No & Low \\ FB-SM & \(0,+u_{\text{c}}\) & Yes & High \\ CD-SM & \(0,u_{\text{cL}},u_{\text{cL}},u_{\text{cL}}+u_{\text{cL}}\) & Yes & Moderate \\ Three-level FC & \(0,u_{\text{cL}},u_{\text{cL}},u_{\text{cL}}-u_{\text{cL}}\) & No & Low \\ Three-level NPC & \(0,u_{\text{cL}},u_{\text{cL}}+u_{\text{cL}})\) & No & Moderate \\
5 LCC-SM & \(0,u_{\text{cL}},u_{\text{cL}},u_{\text{cL}}+(u_{\text{cL}}+u_{\text{cL}})\) & Yes & Moderate \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of different SM topologies. The upper and lower bridge arm voltages \(u_{\rm p}\) and \(u_{\rm n}\) are  
\[u_{\rm p}=NS_{\rm p}u_{\rm cp},u_{\rm n}=NS_{\rm n}u_{\rm cn} \tag{2}\]  
The average switching function model of the MMC is composed of (1) and (2). Ignoring the differences between the circuit parameters of MMC six bridge arms and high order circulating current components, the arm current contains a DC component that equals \(I_{\rm dc}/3\); a fundamental frequency AC component equaling to \(I_{\rm s}/2\); and, even-order circulating current components with an amplitude of \(I_{\rm cir}\). The upper and lower arm currents are presented as  
\[\left\{ \begin{array}{l}i_{p}=\frac{1}{3}I_{\rm dc}-\frac{1}{2}I_{s}\sin \left(\omega t+\beta_{1}\right)+I_{\rm cir}\sin\left(2\omega t+\beta_{2}\right) \\ i_{n}=\frac{1}{3}I_{\rm dc}+\frac{1}{2}I_{s}\sin\left(\omega t+\beta_{1}\right) +I_{\rm cir}\sin\left(2\omega t+\beta_{2}\right)\end{array} \right. \tag{3}\]  
The real switching function of the bridge arm is a stepped waveform that indicates when SMs are inserted or out of work [21]. However, with the increasing number of SMs, the average switching function \(S_{\rm p}\) and \(S_{\rm n}\) can be represented by continuous waveforms:  
\[\left\{