## 4 Short-Circuit Characteristic Calculation Method  
### Node Voltage Equations for the Pre-Fault DC Grid  
In order to describe the calculation method more clearly, the nodes and branches need to be defined firstly. It is assumed that the DC grid has \(2n\) nodes (\(n\) nodes in the positive pole and \(n\) nodes in the negative pole) and \(b\) branches. The nodes of the positive pole and negative pole are denoted as \(n_{i}\) and \(n_{i+n}\) (\(i=1,2\ldots\)\(n\)), respectively. The reference direction for the branch currents of the lines is defined as the direction from node \(n_{i}\) to \(n_{j}\). DC line from node \(n_{i}\) to node \(n_{j}\) is denoted as \(l_{ij}\). The numbering subscript in the voltage, the current, the equivalent admittance and the equivalent current source in a specific branch is in accordance with the numbering subscript of the branch. The MMC bus nodes are defined as the real nodes, and the line connection nodes are defined as the virtual nodes. It is assumed that there is an MMC between the corresponding virtual nodes of the positive pole and the negative pole, and the admittance \(G_{\rm m}\) of the virtual MMC are set to be zero. According to the defined positive directions of currents and voltages, the simplified model between any two adjacent nodes of the positive pole and the negative pole is shown in Figure 6.  
Figure 5: Discrete adjoint model of MMC. Figure 6: Simplified model between any two adjacent nodes. For node \(n_{i}\) and \(n_{j}\), the node voltage equations could be derived as follows:  
\[ \begin{array}{l}\left(G_{mi}+\sum\limits_{q\in i}G_{iq}\right)V_{i}^{k+1}-G_{mi }V_{i+n}^{k+1}-\sum\limits_{q\in i}G_{iq}V_{q}^{k+1}=-J_{mi}^{k}-\sum\limits_{q \in i}J_{iq}^{k}\\ \left(G_{mj}+\sum\limits_{p\in j}G_{pj}\right)V_{j}^{k+1}-G_{mj}V_{j+n}^{k+1}- \sum\limits_{p\in j}G_{pj}V_{p}^{k+1}=-J_{mj}^{k}+\sum\limits_{p\in j}J_{pj}^{ k}\end{array}  \tag{13}\]  
For node \(n_{i+n}\) and \(n_{j+n}\), the node voltage equations could be derived as follows:  
\[ \begin{array}{l}\left(G_{mi}+\sum\limits_{q+n\in i+n}G_{(i+n)(q+n)}\right)V_{ i+n}^{k+1}-G_{mi}V_{i}^{k+1}-\sum\limits_{q+n\in i+n}G_{(i+n)(q+n)}V_{q+n}^{k+1}=J _{mi}^{k}-\sum\limits_{q+n\in i+n}J_{(i+n)(q+n)}^{k}\\ \left(G_{mj}+\sum\limits_{p+n\in j+n}G_{(p+n)(j+n)}\right)V_{j+n}^{k+1}-G_{mj} V_{j}^{k+1}-\sum\limits_{p+n\in j+n}G_{(p+n)(j+n)}V_{p+n}^{k+1}=J_{mj}^{k}+ \sum\limits_{p+n\in j+n}J_{(p+n)(j+n)}^{k}\end{array}  \tag{14}\]  
Here, \(V\) means the node voltage. The symbol \(\in\) means that the two nodes are connected to each other. \(G_{ij}\) and \(J_{ij}\) are the equivalent admittance and the equivalent current source of line \(l_{ij}\). The other symbols have been defined above. The node voltage matrix \(\mathbf{V_{0}}\), the equivalent current source matrix \(\mathbf{J_{0}}\) of the DC lines, and \(\mathbf{J_{c}}\) of the MMCs are defined as Equations (15)-(17). It is assumed that the rows from 1 to \(n\) of these matrices correspond to the equivalent current source of the nodes in the positive pole, and the rows from \(n+1\) to the end correspond to the equivalent current source of the nodes in the negative pole. \[V_{0}=\left[V_{1}V_{2}\cdots V_{n+1}\cdots V_{2n}\right]_{2n}^{T} \tag{15}\]  
\[J_{0}=\left[J_{12}\cdots J_{ij}\cdots J_{(n+1)(n+2)}\cdots J_{(i+n)(j+n)} \cdots\right]_{b}^{T} \tag{16}\]  
\[J_{c}=\left[-J_{c1}\cdots-J_{cn}J_{c1}\cdots J_{cn}\right]_{2n}^{T} \tag{17}\]  
Thus, the relationship between the node voltages and the equivalent current sources can be obtained in Equation (18). \[G_{0}\bullet V_{0}=J_{c}-A_{0}\bullet J_{0} \tag{18}\]  
Here in Equation (18), \(\mathbf{A_{0}}\) is the pre-fault correlation matrix between the nodes and the branches, and its dimension is \(2n\times b\). \(\mathbf{G_{0}}\) is the pre-fault admittance matrix, and its dimension is \(2n\times 2n\). According to the reference direction of the current defined as above, the elements in \(\mathbf{A_{0}}\) can be defined in Equation (19). \[a_{ij}=\left\{