## 3 Detailed Nonlinear Modeling of Grid-Connected VSCs during Asymmetric Faults
### Main Circuit Modeling
Transforming Equations (5) and (6) into the rotating coordinate system \(d^{+}q^{+}\) and \(d^{-}q^{-}\), respectively, by using the coordinate transformation in (4), we obtain the following:  
\[\left\{\begin{array}{l}\frac{d\mathbf{I}_{t1 dq^{+}}}{dt}=\left(\begin{array}{ l}j\omega_{\text{pl}}{}^{+}\mathbf{I}_{t1 dq^{+}}+a_{11}\mathbf{I}_{t1 dq^{+}}e^{-j\pi/6}+a_{12}\mathbf{I}_{ gdq^{+}}+a_{13}(\mathbf{I}_{t1 dq^{-}})^{*}e^{j(\theta_{\text{pl}}{}^{-}+\theta_{ \text{pl}}{}^{+}+\pi/6)}+\\ a_{14}\mathbf{V}_{t1 dq^{+}}e^{-j\pi/6}+a_{15}\mathbf{V}_{g}{}^{+}e^{-j(\theta_{\text{s} }{}^{-}\theta_{\text{pl}}{}^{+})}+a_{16}(\mathbf{V}_{t1 dq^{-}})^{*}e^{j(\theta_{ \text{pl}}{}^{-}+\theta_{\text{pl}}{}^{+}+\pi/6)}+\\ \frac{d\mathbf{I}_{gdd^{+}}}{dt}=\left( \begin{array}{l}j\omega_{\text{pl}}{}^{+} \mathbf{I}_{gdd^{+}}+a_{21}\mathbf{I}_{t1 dq^{+}}e^{-j\pi/6}+a_{22}\mathbf{I}_{gdd^{+}}+a_{ 23}(\mathbf{I}_{t1 dq^{-}})^{*}e^{j(\theta_{\text{pl}}{}^{-}+\theta_{\text{pl}}{}^ {+}+\pi/6)}+\\ a_{24}\mathbf{V}_{tdd^{+}}e^{-j\pi/6}+a_{25}\mathbf{V}_{g}{}^{+}e^{-j(\theta_{\text{s} }{}^{-}\theta_{\text{pl}}{}^{+})}+a_{26}(\mathbf{V}_{t1 dq^{-}})^{*}e^{j(\theta_{ \text{pl}}{}^{-}+\theta_{\text{pl}}{}^{+}+\pi/6)}\\ \frac{d\mathbf{V}_{tdd^{+}}}{dt}=j\omega_{\text{pl}}{}^{+}\mathbf{V}_{tldq^{+}}+(\mathbf{I }_{tdd^{+}}-\mathbf{I}_{t1 dq^{+}})/C_{f}^{+}\\ \frac{d\mathbf{I}_{tdd^{+}}}{dt}=j\omega_{\text{pl}}{}^{+}\mathbf{I}_{tdq^{+}}+(\mathbf{E }_{dq^{+}}-\mathbf{V}_{tdq^{+}})/L_{f}^{+}\\ \end{array} \right. \tag{7}\]  
\[\left\{\begin{array}{l}\frac{d\mathbf{I}_{tdq^{-}}}{dt}=\left(\begin{array}{ l}j\omega_{\text{pl}}{}^{-}\mathbf{I}_{t1 dq^{-}}+a_{31}\left(\mathbf{I}_{t1 dq^{+}}\right)^{*}e^{j( \theta_{g}{}^{+}+\theta_{\text{pl}}{}^{-}+j\pi/6)}+a_{32}\left(\mathbf{I}_{gdq^{+ }}\right)^{*}e^{j(\theta_{g}{}^{+}+\theta_{\text{pl}}{}^{-})}+\\ a_{33}\mathbf{I}_{t1 dq^{-}}e^{-j\pi/6}+a_{34}\left(\mathbf{V}_{tdq^{+}}\right)^{*}e^{j( \theta_{g}{}^{+}+\theta_{\text{pl}}{}^{-}+j\pi/6)}+a_{35}\mathbf{V}_{g}{}^{+}e^{ j(\theta_{g}{}^{+}+\theta_{\text{pl}}{}^{-})}+a_{36}\mathbf{V}_{tdq^{-}}e^{-j\pi/6}\\ \frac{d\mathbf{V}_{tdq^{-}}}{dt}=j\omega_{\text{pl}}{}^{-}\mathbf{V}_{tdq^{-}}+\left(\mathbf{ I}_{tdq^{-}}-\mathbf{I}_{t1 dq^{-}}\  
* DSOGI/PNSC modeling  
From Figure 1, the input of DSOGI is \(\overrightarrow{\mathbf{V}_{ta\beta}}\) and the output is \(\hat{\overrightarrow{\mathbf{V}}_{ta\beta}}\) and its quadrature signal \(\hat{\overleftarrow{\mathbf{U}}_{ta\beta}}\). The control structure based on SOGI-QSG can be obtained as follows:  
\[\left\{ \begin{array}{l}d\hat{\overrightarrow{\mathbf{V}}_{ta\beta}}/dt=\omega_{0 }k_{\text{SOGI}}\left(\overset{\rightarrow}{\mathbf{V}}_{ta\beta}-\hat{ \overleftarrow{\mathbf{V}}_{ta\beta}}\right)-\omega_{0}\hat{\overleftarrow{\mathbf{U }}_{ta\beta}}\\ d\hat{\overleftarrow{\mathbf{U}}_{ta\beta}}/dt=\omega_{0}\hat{\overleftarrow\mathbf{{V}}_{ta}}\end{array} \right.\quad\left\{ \begin{array}{l}d\hat{\overleftarrow {\mathbf{V}}_{t\beta}}/dt=\omega_{0}k_{\text{SOGI}}\left(\overset{\rightarrow}{ \mathbf{V}}_{t\beta}-\hat{\overleftarrow{\mathbf{V}}_{t\beta}}\right)-\omega_{0}\hat{ \overleftarrow{\mathbf{U}}_{t\beta}}\\ d\hat{\overleftarrow{\mathbf{U}}_{t\beta}}/dt=\omega_{0}\hat{\overleftarrow\mathbf{{V}}_{t\beta}}\end{array} \right. \tag{10}\]  
where \(k_{\text{SOGI}}\) is the proportional gain coefficient of SOGI. The mathematical model of DSOGI can be obtained by sorting out Equation (10) as shown in Equation (11). \[\left\{ \begin{array}{l}d\hat{\overleftarrow{\mathbf{V}}_{ta\beta}}/dt=\omega_{0 }k_{\text{SOGI}}\left(\overset{\rightarrow}{\mathbf{V}}_{ta\beta}-\hat{\overleftarrow {\mathbf{V}}_{ta\beta}}\right)-\omega_{0}\hat{\overleftarrow{\mathbf{U}}_{ta\beta}}\\ d\hat{\overleftarrow{\mathbf{U}}_{ta\beta}}/dt=\omega_{0}\hat{\overleftarrow\mathbf{{V}}_{ta\beta}}\end{array} \right. \tag{11}\]  
where \(k_{\text{SOGI}}\) is the proportional gain coefficient of SOGI.