## 3 Sequence Admittance Modeling of Grid-Connected Inverter
### Sequential Admittance Modeling of GCI by PR  
According to the phase-domain PR-controlled GCI shown in Figure 1b, the reference value of the grid-connected current under the \(dq\) axis is defined as follows:  
\[\left\{ \begin{array}{l}i_{\rm dref}=(G_{\rm vf}(s_{2})v_{\rm dc}-v_{\rm ref} )H_{\rm v}(s_{2})\\ i_{\rm qref}=0\end{array} \right. \tag{13}\]  
In the formula, \(H_{\rm v}(s_{2})=K_{\rm p}+K_{\rm i}/s\) is the PI controller used in the voltage outer loop. As shown in the control structure diagram of the GCI regulated by phase-domain PR in Figure 1b, the reference value for the grid-connected current along the \(dq\) axis is subjected to an inverse transformation, and the frequency domain representation for the a-phase grid-connected current is specified as follows:  
\[i_{\rm aref}=i_{\rm dref}[f]\otimes\cos\theta_{\rm PLL}[f]-i_{\rm qref}[f] \otimes\sin\theta_{\rm PLL}[f] \tag{14}\]  
In the formula, \(\bigotimes\) convolution operation is denoted. It can be deduced that:  
\[I_{aref}[f]=\left\{ \begin{array}{l}{\bf I}_{1},f=\pm f_{1}\\ \frac{1}{2}[v_{\rm dc}[f]G_{\rm vf}(s_{2})H_{\rm v}(s_{2})\\ +j_{1}(T_{\rm PLL}(s_{2})G_{\rm vf}(s){\bf V}_{\rm p}+T_{\rm PLL1}(s_{2})G_{ \rm vf}(s_{\rm p2}){\bf V}_{\rm p2})],\;f=\pm f_{p}\\ \frac{1}{2}[v_{dc}[f]G_{\rm vf}(s_{2})H_{\rm v}(s_{2})\\ -j_{1}(T_{\rm PLL}(s_{2})G_{\rm vf}(s){\bf V}_{\rm p}+T_{\rm PLL1}(s_{2})G_{ \rm vf}(s_{\rm p2}){\bf V}_{\rm p2})],\;f=\pm(f_{p}-2f_{1})\end{array} \right. \tag{15}\]  
Through the PR controller and the voltage feedforward link, the modulation signal's expression is delineated as follows:  
\[c_{a}[f]=\left\{ \begin{array}{l}H_{\rm PR}(s)(I_{\rm aref}[f]-G_{\rm if}(s){ \bf I}_{\rm p})+K_{\rm f}G_{\rm vf}(s){\bf V}_{\rm p},\;f=\pm f_{p}\\ H_{\rm PR}(s_{\rm p2})(I_{\rm aref}[f]-G_{\rm if}(s_{\rm p2}){\bf I}_{\rm p2})+K_ {\rm f}G_{\rm vf}(s_{\rm p2}){\bf V}_{\rm p2},\;f=\pm(f_{p}-2f_{1})\end{array} \right. \tag{16}\]  
Based on Figure 1, the subsequent relationships can be derived:  
\[sL_{\rm f}i_{a}=K_{\rm pwm}v_{\rm dc}G_{\rm del}(s)(c_{a}+K_{\rm f}G_{\rm vf}( s)v_{a})-v_{a} \tag{17}\]In the formula, control delay \(G_{\text{del}}(s)=\text{e}^{-1.518S}\). Combined with the formula, the following can be obtained:  
\[\left[ \begin{array}{c}\mathbf{I}_{\text{p}}\\ \mathbf{I}_{\text{p2}}\end{array} \right]=- \begin{bmatrix}\text{Y}_{11\_PR}& \text{Y}_{12\_PR}\\ \text{Y}_{21\_PR}&\text{Y}_{22\_PR}\end{bmatrix} \left[ \begin{array}{c} \mathbf{V}_{\text{p}}\\ \mathbf{V}_{\text{p2}}\end{array} \right] \tag{18}\]  
\[\left[ \begin{array}{c}H_{\text{p}}\\ H_{\text{p2}}\end{array} \right]=\left[ \begin{array}{c}v_{\text{dc}}G_{\text {vf}}(s_{2})H_{\text{v}}(s_{2})H_{\text{PR}}(s)+2(c_{1}+K_{\text{f}}G_{\text{ vf}}(s_{1})\mathbf{V}_{1})\\ v_{\text{dc}}G_{\text{vf}}(s_{2})H_{\text{v}}(s_{2})H_{\text{PR}}(s_{\text{p2}}) +2(c_{1}+K_{\text{f}}G_{\text{vf}}(s_{1})\mathbf{V}_{1})^{*}\end{array} \right] \tag{19}\]  
\[\left\{ \begin{array}{c}X_{\text{v1}}=v_{\text{dc}}G_{\text{vf}}(s)[0.5]I_{1} T_{\text{PLL}}(s_{2})H_{\text{PR}}(s)+K_{\text{f}}]\\ X_{\text{v2}}=v_{\text{dc}}G_{\text{vf}}(s)[0.5]I_{1}T_{\text{PLL}}(s_{2})H_{ \text{PR}}(s)]\\ X_{\text{v3}}=v_{\text{dc}}G_{\text{vf}}(s_{\text{p2}})[-0.5]I_{1}T_{\text{PLL}}(s_{2})H_{ \text{PR}}(s_{\text{p2}})]\\ X_{\text{v4}}=v_{\text{dc}}G_{\text{vf}}(s_{\text{p2}})[-0.5]I_{1}T_{\text{PLL}}(s_{2})H_{ \text{PR}}(s)+K_{\text{f}}]\end{array} \right. \tag{20}\]  
\[\left\{ \begin{array}{c}J_{\text{i}}=-H_{\text{PR}}(s)v_{\text{dc}}G_{\text{ if}}(s)\\ J_{\text{i2}}=-H_{\text{PR}}(s_{\text{p2}})v_{\text{dc}}G_{\text{if}}(s_{\text{p2}}) \end{array} \right. \tag{21}\]  
\[