## 3 Sequence Admittance Modeling of Grid-Connected Inverter  
The frequency coupling effect significantly influences the admittance characteristic frequency of the GCI. Therefore, considering factors such as frequency coupling, control delay, and voltage feedforward link, the sequence admittance model of the GCI under different domain controls is derived. The filters of voltage and current sampling are described as follows:  
\[\left\{ \begin{array}{l}G_{\text{vf}}(s)=\frac{1}{(1+s/\omega_{\text{v}})}\\ G_{\text{if}}(s)=\frac{1}{(1+s/\omega_{\text{i}})}\end{array} \right. \tag{3}\]  
In the formula, \(\omega_{\text{v}}\) and \(\omega_{\text{i}}\) represent the angular frequencies of the voltage and current sampling filters, respectively. For simplified expressions, \(\text{s}=\pm\text{j2}\pi\text{f}_{\text{p}}\), \(\text{s}_{1}=\pm\text{j2}\pi\text{f}_{1}\), \(\text{s}_{\text{p1}}=\pm\text{j2}\pi\text{f}_{\text{p}}\), and \(\text{s}_{2}=\pm\text{j2}\pi\text{f}_{\text{p}}\text{f}_{1}\)) are defined. Figure 1: GCI system topology and control block diagram. (**a**) GCI system topology; (**b**) GCI system controlled by PI in the dq domain; (**c**) GCI system with PR control in the time domain. From the system main circuit topology diagram shown in Figure 1a, there is a relationship between the inverter bridge arm voltage, output voltage, and output current:  
\[L_{f}\frac{d}{dt} \begin{bmatrix}i_{a}\\ i_{b}\\ i_{c}\end{bmatrix} = \begin{bmatrix}e_{a}\\ e_{b}\\ e_{c}\end{bmatrix} - \begin{bmatrix}v_{a}\\ v_{b}\\ v_{c}\end{bmatrix}  \tag{4}\]  
According to the frequency coupling effect, when a positive sequence voltage disturbance at frequency \(f_{\rm p}\) is introduced at the grid connection point, it induces a positive sequence response current at the same frequency \(f_{\rm p}\) and a negative sequence response current at frequency \(f_{\rm p}\)-\(2f_{1}\). As the negative sequence response current moves through the grid impedance, it induces a matching negative sequence voltage disturbance at the same frequency. Consequently, the time-domain expressions for the voltage and current of phase A at the grid connection point are described as follows:  
\[v_{a}(t)= V_{1}\cos(\omega_{1}t)+V_{\rm p}\cos(\omega_{p}t+\varphi_{\rm vp}) \tag{5}\] \[+V_{\rm p2}\cos(\omega_{\rm p2}t+\varphi_{\rm vp2})\]  
\[i_{a}(t)= I_{1}\cos(\omega_{1}t+\varphi_{\rm i1})+I_{p}\cos(\omega_{\rm p}t+ \varphi_{\rm ip}) \tag{6}\] \[+I_{\rm p2}\cos(\omega_{\rm p2}t+\varphi_{\rm ip2})\]  
In the formula, the voltage amplitudes \(V_{1}\), \(V_{\rm p}\), and \(V_{\rm p2}\) represent the fundamental, positive sequence disturbance, and negative sequence coupling frequencies, respectively. The angular frequencies \(\omega_{1}\), \(\omega_{\rm p}\), and \(\omega_{\rm p2}\) similarly represent the fundamental, positive sequence disturbance, and negative sequence coupling frequencies. Additionally, the initial phase angles for the disturbance voltages \(\varphi_{\rm vp}\) and \(\varphi_{\rm vp2}\) are specified at the positive and negative sequence disturbance frequencies. The currents \(I_{1}\), \(I_{p}\), and \(I_{\rm p2}\) for these frequencies are matched with their respective initial phase angles \(\varphi_{\rm i1}\), \(\varphi_{\rm ip}\), and \(\varphi_{\rm ip2}\), highlighting the coherent response of the system across these varied conditions. According to Formulas (5) and (6), the frequency domain expression can be expressed as follows:  
\[V_{a}=\left\{ \begin{array}{ll}\mathbf{V}_{1},\ f=\pm f_{1}\\ \mathbf{V}_{\rm p},\ f=\pm f_{p}\\ \mathbf{V}_{\rm p2},\ f=\pm(f_{p}-2f_{1})\end{array} ,i_{a}=\left\{ \begin{array} []{ll}\mathbf{I}_{1},\ f=\pm f_{1}\\ \mathbf{I}_{\rm p},\ f=\pm f_{p}\\ \mathbf{I}_{\rm p2},\ f=\pm(f_{p}-2f_{1})\end{array} \right.