## 3 Detailed Nonlinear Modeling of Grid-Connected VSCs during Asymmetric Faults
### Main Circuit Modeling
Based on the control structure of the PNSC, the positive-sequence and negative-sequence components \(\hat{\overleftarrow{\mathbf{V}}_{ta\beta}}^{+}\) and \(\hat{\overleftarrow{\mathbf{V}}_{ta\beta}}\) and \(\hat{\overleftarrow{\mathbf{V}}_{ta\beta}}\) can be expressed, respectively, as follows:  
\[\left\{ \begin{array}{l}\hat{\overleftarrow{\mathbf{V}}_{ta\beta}}^{+}=\hat{ \overleftarrow{\mathbf{V}}_{ta}}^{+}+j\hat{\overleftarrow{\mathbf{V}}_{t\beta}}^{+}= \frac{1}{2}\left(\hat{\overleftarrow{\mathbf{V}}_{ta}}+j\hat{\overleftarrow{\mathbf{V }}_{t\beta}}-\hat{\overleftarrow{\mathbf{U}}_{t\beta}}+j\hat{\overleftarrow{\mathbf{U} }_{ta\beta}}\right)=\frac{1}{2}\left(\hat{\overleftarrow{\mathbf{V}}_{ta\beta}}+j \hat{\overleftarrow{\mathbf{U}}_{t\beta}}\right)\\ \hat{\overleftarrow{\mathbf{V}}_{ta\beta}}^{-}=\hat{\overleftarrow{\mathbf{V}}_{ta}}^{-} +j\hat{\overleftarrow{\mathbf{V}}_{t\beta}}^{-}=\frac{1}{2}\left(\hat{\overleftarrow {\mathbf{V}}_{ta}}+\hat{\overleftarrow{\mathbf{U}}_{t\beta}}+j\hat{\overleftarrow{\mathbf{V }}_{t\beta}}-j\hat{\overleftarrow{\mathbf{U}}_{ta\beta}}\right)=\frac{1}{2}\left(\hat{ \overleftarrow{\mathbf{V}}_{ta\beta}}-j\hat{\overleftarrow{\mathbf{U}}_{ta\beta}} \right)\end{array} \right. \tag{12}\]  
Combining (12) and (13) and transforming it into the rotating coordinate system \(d^{+}q^{+}\) and \(d^{-}q^{-}\), the mathematical model of DSOGI/PNSC can be obtained as shown below:  
\[\left\{ \begin{array}{l}d\hat{v}_{tdq}^{+}=j\omega_{\text{pll}}{}^{+}\hat{ \overleftarrow{\mathbf{V}}_{tdq}}+\omega_{0}k_{\text{SOGI}}\left({\mathbf{V}_{tdq}}^{+ }-\hat{\mathbf{V}}_{tdq}^{+}\right)-\omega_{0}\hat{\mathbf{U}}_{tdq}^{+}\\ d\hat{u}_{tdq}^{+}=j\omega_{\text{pll}}{}^{+}\hat{\overleftarrow{\mathbf{U}}_{tdq} ^{+}}+\omega_{0}\hat{\mathbf{V}}_{tdq}^{+}\end{array} \right. \tag{13}\]  
\[\left\{ \begin{array}{l}d\hat{v}_{tdq}^{-}=j\omega_{\text{pll}}{}^{-}\hat{ \overleftarrow{\mathbf{V}}_{tdq}}+\omega_{0}k_{\text{SOGI}}\left({\mathbf{V}_{tdq}}^{- }-\hat{\mathbf{V}}_{tdq}^{-}\right)-\omega_{0}\hat{\mathbf{U}}_{tdq}^{-}\\ d\hat{u}_{tdq}^{-}=j\omega_{\text{pll}}{}^{-}\hat{\mathbf{U}}_{tdq}^{-}+\omega_{0} \hat{\mathbf{V}}_{tdq}^{-}\end{array} \right. \tag{14}\]  
Equations (13) and (14) are the mathematical model of the DSOGI/PNSC, denoted as \(\hat{\mathbf{X}}_{\text{D/P}}=\)\(f_{\text{D/P}}(\mathbf{X}_{\text{D/P}},\mathbf{Y}_{\text{D/P}})\), where \(\mathbf{X}_{\text{D/P}}\) is the state variable vectors and \(\mathbf{Y}_{\text{D/P}}\) is the input variable vectors. The specific expressions for \(\mathbf{X}_{\text{D/P}}\) and \(\mathbf{Y}_{\text{D/P}}\) are given below:  
\[\left\{ \begin{array}{l}X_{\text{D/P}}=\left[\hat{V}_{td}^{+},\hat{V}_{tq}^{+}, \hat{\mathbf{U}}_{td}^{+},\hat{\mathbf{U}}_{tdq}^{+},\hat{V}_{td}^{-},\hat{V}_{tq}^{-}, \hat{\mathbf{U}}_{td}^{-},\hat{\mathbf{U}}_{td}^{-},\hat{\mathbf{U}}_{td}^{-}\right]^{T}\\ Y_{\text{D/P}}=\left[V_{td}{}^{+},V_{tq}{}^{+},V_{td}{}^{-},V_{tq}{}^{-} \right]^{T}\end{array} \right. \tag{15}\]  
* \(\text{PLL}^{+}\) and \(\text{PLL}^{-}\) modeling  
The output phase angles \(\theta_{\text{pll}}{}^{+}\) and \(\theta_{\text{pll}}{}^{-}\) of \(\text{PLL}^{+}\) and \(\text{PLL}^{-}\) can be expressed as Equations (16) and (17), respectively. \[\left\{ \begin{array}{l}\frac{d\theta_{\text{pll}}{}^{+}}{dt}=-k_{p}\hat{V}_{ tq}^{+}-k_{i}x_{\text{pll}}{}^{+}+\omega_{0}\triangleq\omega_{\text{pll}}{}^{+}\\ \frac{dx_{\text{pll}}{}^{+}}{dt}=\hat{V}_{tq}^{+}\end{array} \right. \tag{16}\]\[\left\{ \begin{array}{l}\frac{d{\theta_{\rm pl}}^{-}}{dt}=-k_{p}\hat{V}_{ tq}^{-}-k_{i}{x_{\rm pl}}^{-}-\omega_{0}\triangleq\omega_{\rm pl}^{-}\\ \frac{d{x_{\rm pl}}^{-}}{dt}=\hat{V}_{ tq}^{-}\end{array} \right. \tag{17}\]  
where \(k_{p}\) and \(k_{i}\) are the proportional and integral coefficients of the PLL, respectively.