## 4 Analysis and Suppression Method of Short Circuit Current in MMC-Based MTDC Power Grid
### Short Circuit Current Calculation Method
#### 4.1.1 Pole-to-Pole Short Circuit Current Calculation Method  
Once a pole-to-pole DC short circuit fault occurs in MMC-based MTDC power grids, the capacitors discharge rapidly, which leads to massive short circuit current in both transmission lines and each MMC bridge arm. These phenomena usually happen within 5\(\sim\)10 ms after the fault occurs [17; 34]. For an MMC-based MTDC power grid, the dominant component of the short-circuit currents is the discharge currents of SM capacitors. The AC infeed contributions are exclusively present after 10 ms, which can be faithfully ignored [34; 43]. It should be noted that the dynamics of MMCs that are based on various SM topologies under DC faults are developed in the same way before blocking, including HB-SM and other SM topologies with DC fault ride through capability [3; 7; 13]. An equivalent circuit method for calculating the pole-to-pole short circuit fault current in MMC-based MTDC power grids is presented in [13]. In the proposed calculation method, each MMC is represented by (10) and the DC line is equivalent to a series RLC circuit, as shown in Figure 8. The nodes that are connected to a real converter are defended as \"real nodes\" and the intersection nodes of the transmission lines without connection to a real converter are defined as \"virtual node\". When considering a DC power grid with \(b\) branches, \(N\) real nodes, and \(M\) virtual nodes on the positive pole layer, the total number of the nodes is \(n\). The branch currents matrix \(\mathbf{i_{0}}\), node capacitor voltages matrix \(\mathbf{u_{0}}\), and node injection current matrix \(\mathbf{i_{c0}}\) can be expressed by  
\[ \begin{split}\mathbf{i_{0}}&=\left[\begin{array}{c}i_{ 12},\ldots i_{ij},\ldots\end{array}\right]_{b}^{\mathrm{T}}\\ \mathbf{u_{0}}&=\left[\begin{array}{c}u_{\mathrm{c1}},u_{\mathrm{c2}},\ldots u _{\mathrm{cN}},u_{\mathrm{cN}+1},\ldots u_{\mathrm{cN}+M}\end{array}\right]_{ n}^{\mathrm{T}}\\ \mathbf{i_{c0}}&=\left[\begin{array}{c}i_{\mathrm{c1}},i_{\mathrm{c2}},\ldots i _{\mathrm{cn}}\end{array}\right]_{n}^{\mathrm{T}}\end{split}  \tag{19}\]  
Figure 8: Equivalent circuit of a three-terminal MMC-based DC power grid. Branch current equations in matrix form can be expressed as  
\[\mathbf{A}_{0}\cdot\mathbf{u}_{0}=\mathbf{R}_{0}\cdot\mathbf{i}_{0}+\mathbf{L}_{0}\cdot\dot{ \mathbf{i}}_{0} \tag{20}\]  
where \(\mathbf{A}_{0}\) is the incidence matrix with \(n\) nodes and \(b\) branches before fault. \(\mathbf{R}_{0}\) and \(\mathbf{L}_{0}\) is the resistance matrix and inductance matrix before fault, respectively. Assume the fault point that is indicated by 0 is set between node \(i\) and node \(j\) and \(R_{t}\) is the fault resistance. By applying fault branch, the \(R_{ij}\) and \(L_{ij}\) are replaced by \(R_{i0}\), \(R_{j0}\) and \(L_{i0}\), \(L_{j0}\). \(i_{i0}\) and \(i_{j0}\) denote the currents of branch \(i0\) and \(j0\), respectively. The modified matrices are defined as \(\mathbf{R}_{t}\) and \(\mathbf{L}_{t}\), respectively. \(\mathbf{A}_{0}\) is modified to \(\mathbf{A}_{t}\) with the branch \(ij\) that is replaced by branch \(i0\) and \(j0\). Therefore, (20) can be rewritten as  
\[\mathbf{A}_{t}\cdot\mathbf{u}_{0}=\mathbf{R}_{t}\cdot\mathbf{i}+\mathbf{L}_{t}\cdot\dot{\mathbf{i}} \tag{21}\]  
The relationship between the post-fault branch currents \(\mathbf{i}_{\text{c}}\) and post-fault node injection currents \(\mathbf{i}\) can be represented by  
\[\mathbf{i}_{\text{c}}=\mathbf{A}_{t1}^{\text{T}}\cdot\mathbf{i} \tag{22}\]  
where \(\mathbf{A}_{t1}\) is the first to \(N\)th columns in \(\mathbf{A}_{t}\)  
The relationship between the capacitor voltages and currents can be presented by  
\[\dot{\mathbf{u}}=\mathbf{P}\cdot\mathbf{i} \tag{23}\]  
where \(\mathbf{P}\) is given by  
\[\mathbf{P}=-\text{diag}\left[ \begin{array}{cc}1/C_{\text{c1}},1/C_{\text{c2}} \ldots 1/C_{\text{cN}}\end{array}