## 3 Detailed Nonlinear Modeling of Grid-Connected VSCs during Asymmetric Faults
### Main Circuit Modeling  
Under different types of faults, the boundary conditions of the fault points are different. Based on the boundary conditions under different fault types, the interconnection of the sequence network under single line-to-ground (SLG), double line-to-ground (DLG), and line-to-line (LL) can be represented, as shown in Figure 8, where \(R_{\text{F}}\) represents the ground resistance. Note that \(\mathbf{Z}_{i}^{+}\), \(\mathbf{Z}_{i}^{-}\), and \(\mathbf{Z}_{i}^{0}\) (\(i=L\), \(g\)) represent positive, negative, and zero sequence impedances, respectively. For Figure 8a-c, the details are as follows:  
Figure 8a presents the composite sequence network of the system under single line-to-ground faults. Figure 8: The boundary conditions of the fault points under different types of faults. (**a**) Single line-to-ground (SLG); (**b**) double line-to-ground (DLG); (**c**) line-to-line (LL) faults. \({}^{***}\) represents the complex conjugate operation. Figure 8b presents the composite sequence network of the system under double line-to-ground faults. Figure 8c presents the composite sequence network of the system under line-to-line faults. Based on the circuit shown in Figure 8 and combined with the circuit graph theory, \(\overrightarrow{\mathbf{I}_{I}}^{+}_{t}\), \(\overrightarrow{\mathbf{I}_{I}}^{+}_{t1}\), \(\overrightarrow{\mathbf{I}_{I}}^{+}_{t1}\), \(\overrightarrow{\mathbf{I}_{I}}^{-}_{t}\), \(\overrightarrow{\mathbf{I}_{I}}^{+}_{t}\), \(\overrightarrow{\mathbf{V}_{t}}^{+}_{t}\) and \(\overrightarrow{\mathbf{I}_{g}}^{+}_{t}\) can be chosen as the state variables. Then, the state equation of the main circuit can be obtained based on Kirchhoff current law (KCL) and Kirchhoff voltage law (KVL), as shown in Equations (5) and (6). \[\left\{ \begin{array}{l}\frac{\overrightarrow{\mathbf{I}_{I}}^{+}_{t1}}{dt}=a_{ 11}\overrightarrow{\mathbf{I}_{I}}^{+}_{t1}e^{j\pi/6}+a_{12}\overrightarrow{\mathbf{I }_{g}}^{+}+a_{13}(\overrightarrow{\mathbf{I}_{I}}^{-}_{t1})^{*}e^{-j\pi/6}+a_{14} \overrightarrow{\mathbf{V}_{t}}^{+}e^{j\pi/6}+a_{15}\overrightarrow{\mathbf{V}_{g}}^{+ }+a_{16}(\overrightarrow{\mathbf{V}_{t}}^{-}_{t})^{*}e^{-j\pi/6}\\ \frac{\overrightarrow{\mathbf{I}_{g}}^{+}_{t1}}{dt}=a_{21}\overrightarrow{\mathbf{I}_ {I}}^{+}_{t1}e^{j\pi/6}+a_{22}\overrightarrow{\mathbf{I}_{g}}^{+}+a_{23}( \overrightarrow{\mathbf{I}_{I}}^{-}_{t1})^{*}e^{-j\pi/6}+a_{24}\overrightarrow\mathbf{{V}_{t}}^{+}e^{j\pi/6}+a_{25}\overrightarrow{\mathbf{V}_{g}}^{+}+a_{26}( \overrightarrow{\mathbf{V}_{t}}^{-}_{t})^{*}e^{-j\pi/6}\\ \frac{\overrightarrow{\mathbf{V}_{t}}^{+}_{t}}{dt}=(\overrightarrow{\mathbf{I}_{I}}^{ +}_{t1}-\overrightarrow{\mathbf{I}_{I}}^{+}_{t1})/C_{f}^{+}\\ \frac{\overrightarrow{\mathbf{I}_{g}}^{+}_{t}}{dt}=(\overrightarrow{\mathbf{E}}^{-+} -\overrightarrow{\mathbf{V}_{t}}^{+})/L_{f}^{+}\\ \end{array} \right. \tag{5}\]  
\[\left\{ \begin{array}{l}\frac{d(\overrightarrow{\mathbf{I}_{I}}^{-}_{t1})^{-}} {dt}=a_{31}\overrightarrow{\mathbf{I}_{I}}^{+}_{t1}e^{j\pi/6}+a_{32}\overrightarrow {\mathbf{I}_{g}}^{+}+a_{33}(\overrightarrow{\mathbf{I}_{I}}^{-}_{t1})^{*}e^{-j\pi/6}+ a_{34}\overrightarrow{\mathbf{V}_{t}}^{+}e^{j\pi/6}+a_{35}\overrightarrow{\mathbf{V}_{g}}^{+ }+a_{36}(\overrightarrow{\mathbf{V}_{t}}^{-}_{t})^{*}e^{-j\pi/6}\\ \frac{d(\overrightarrow{\mathbf{V}_{t}}^{-}_{t})^{*}}{dt}=\left((\overrightarrow{ \mathbf{I}_{I}}^{-}_{t})^{*}-(\overrightarrow{\mathbf{I}_{I}}^{-}_{t1})^{*}\right)/C _{f}^{-}\\ \frac{d(\overrightarrow{\mathbf{I}_{I}}^{-}_{t})^{*}}{dt}=\left((\overrightarrow{ \mathbf{E}}^{-})^{*}-(\overrightarrow{\mathbf{V}_{t}}^{-}_{t})^{*}\right)/L_{f}^{-} \\ \end{array} \right. \tag{6}\]  
where \(a_{ij}\) (\(i=1,2,3,j=1,\ldots,6\)) is related to the type of fault. Taking the DLG fault as an example, the expressions for \(a_{ij}\) are provided in Appendix B.