## 3 Detailed Nonlinear Modeling of Grid-Connected VSCs during Asymmetric Faults
### Coordinate System  
The coordinate system used for modeling in this paper is shown in Figure 7, where \(abc\) is the three-phase stationary coordinate system, \(a\beta\) is the two-phase stationary coordinate system, \(d^{+}q^{+}\) is the two-phase positive-sequence (PS) rotating coordinate system, and \(d^{-}q^{-}\) is the two-phase negative-sequence (NS) rotating coordinate system. For Figure 7a-c, the details are as follows:  
As shown in Figure 7a, for the \(d^{+}q^{+}\) coordinate system, the \(q^{+}\)-axis lags the \(d^{+}\)-axis by \(90^{\circ}\), and the \(d^{+}\)-axis is oriented in the direction of the positive-sequence voltage vector of PCC. As shown in Figure 7b, for the \(d^{-}q^{-}\) coordinate system, the \(q^{-}\)-axis exceeds the \(d^{-}\)-axis by \(90^{\circ}\), and the \(d^{+}\)-axis is oriented in the direction of the negative-sequence voltage vector of PCC. As shown in Figure 7c, the synthetic vector of the PS and NS vectors is denoted as \(\overrightarrow{\mathbf{X}}\). The NS vector is usually non-zero under asymmetrical faults, and thus the trajectory of the endpoints of the synthetic vector is an ellipse when the synthetic vector is rotated by one week. As shown in Figure 1, the high-voltage-side zero-sequence circuit is closed through the grounding connection of the neutral point. However, the low-voltage-side zero-sequence circuit is open. Thus, the voltages and currents of the VSC, denoted as \(x_{\text{abc}}\), contain only PS and NS components. The vectors (complex vectors) of the symmetrical PS components and NS components can be denoted by \(\overrightarrow{\mathbf{X}}=\mathbf{X}^{+}e^{i(\omega_{0}t+\varphi^{+})}\) and \(\overrightarrow{\mathbf{X}}=\mathbf{X}^{-}e^{i(-\omega_{0}t+\varphi^{-})}\), respectively, where \(X^{+}\) and \(X^{-}\) are the amplitude of the PS and NS components of \(x_{\text{abc}}\), respectively, while \(\varphi^{+}\) and \(\varphi^{-}\) are the initial phase angle of the PS and NS components of \(x_{\text{abc}}\), respectively. The phasors of \(\overrightarrow{\mathbf{X}}\) and \(\overrightarrow{\mathbf{X}}^{-}\) are defined via the symmetrical components method as \(\dot{\mathbf{X}}^{+}=\mathbf{X}^{+}e^{i\varphi^{+}}\) and \(\dot{\mathbf{X}}^{-}=\mathbf{X}^{-}e^{i\varphi^{-}}\), respectively. The relationship between the vectors and the phasors is then expressed by Equation (1). \[\overrightarrow{\mathbf{X}}^{+}=\dot{\mathbf{X}}^{+}e^{i\omega_{0}t}\overrightarrow{ \mathbf{X}}^{-}=\dot{\mathbf{X}}^{-}e^{-i\omega_{0}t} \tag{1}\]  
The synthetic vector of the PS and NS vectors is denoted as \(\overrightarrow{\mathbf{X}}\), and the specific expression is shown in (2). The NS vector is usually non-zero under asymmetrical faults,  
Figure 7: The coordinate system is used for modeling in this paper. (**a**) PS vector; (**b**) NS vector; (**c**) The synthetic vector of the PS and NS vectors. and thus the trajectory of the endpoints of the synthetic vector is an ellipse when the synthetic vector is rotated by one week, as shown in Figure 7. The projection of the synthetic vector under the \(abc\) coordinate axis corresponds to the \(x_{\text{abc}}\) instantaneous value. \[\overrightarrow{\mathbf{X}}=\overrightarrow{\mathbf{X}}^{+}+\overrightarrow{\mathbf{X}} \tag{2}\]  
As can be seen from Figure 7, the transformation relations of the positive- and negative-sequence vectors \(\overrightarrow{\mathbf{X}}^{+}\) and \(\overrightarrow{\mathbf{X}}^{-}\) in the \(\alpha\beta\) and \(dq\) coordinate systems are shown in Equations (3) and (4), respectively. \[\left\{ \begin{array}{l}\overrightarrow{\mathbf{X}}^{+}=\overrightarrow{\mathbf{X} }^{+}_{\alpha}+\cdot j\overrightarrow{\mathbf{X}}^{+}_{\beta}\triangleq \overrightarrow{\mathbf{X}}^{+}_{\alpha\beta}\\ \overrightarrow{\mathbf{X}}^{-}=\overrightarrow{\mathbf{X}}^{-}_{\alpha}+j \overrightarrow{\mathbf{X}}^{-}_{\beta}\triangleq\overrightarrow{\mathbf{X}}^{-}_{ \alpha\beta}\end{array} \right. \tag{3}\]  
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