## 4 Stability Analysis of GCI Controlled by Different Domains  
While the stability analysis can be performed using the frequency-coupled admittance matrix and the GNC, the analysis process is complicated. Under the condition that the same phase margin is met, the frequency-coupled admittance matrix can be simplified to a single input-single output (SISO) sequence impedance model, thus simplifying the subsequent stability analysis. The defined simplified SISO sequence impedance model is specified as follows:  
\[\left[\begin{array}{c}\mathbf{V}_{\mathrm{P}}\\ \mathbf{V}_{\mathrm{n}}\end{array}\right]=\left[\begin{array}{cc}\mathrm{Z}_ {\mathrm{P}}(s)&0\\ 0&\mathrm{Z}_{\mathrm{n}}(s)\end{array}\right]\left[\begin{array}{c}\mathrm{I }_{\mathrm{P}}\\ \mathrm{I}_{\mathrm{n}}\end{array}\right] \tag{41}\]  
In the formula, \(Z_{\mathrm{P}}\)(s) and \(Z_{\mathrm{n}}\)(s) denote the positive sequence and negative sequence impedances of the grid-connected inverters, respectively; \(\mathbf{V}_{\mathrm{n}}\) and \(\mathbf{I}_{\mathrm{n}}\) are the injected negative sequence disturbance voltage and the respective responses of the negative sequence current.  
By simplifying the established frequency-coupled admittance through the grid impedance \(Z_{\mathrm{g}}\), the positive and negative sequence impedances of the inverter are obtained as follows:  
\[\left\{\begin{array}{l}Z_{\mathrm{P}}(s)=\frac{1}{\gamma_{11}-\frac{\gamma_{ 12}\gamma_{2}\gamma_{2}\gamma_{2}\gamma_{2}\gamma_{2}\gamma_{2}\gamma_{2}(s-2 \omega)  
In Figure 4a, when the grid impedance is 1 mH and 5 mH, the Nyquist plots for the positive sequence impedance ratio \(ZR_{\text{p}}\)(s) and the negative sequence impedance ratio \(ZR_{\text{n}}\)(s) do not encompass the point \((-1,\text{j0})\). However, when the grid impedance increases to 16 mH, the Nyquist diagrams of the positive sequence impedance ratio \(ZR_{\text{p}}\)(s) encircle the point \((-1,\text{j0})\).  
In Figure 4b, when the grid impedance is 1 mH, 5 mH, and 16 mH, the Nyquist plots for \(ZR_{\text{p}}\)(s) and \(ZR_{\text{n}}\)(s) do not encompass the point \((-1,\text{j0})\). Therefore, the stability of GCIs using time-domain PR control is slightly better than that of inverters using \(dq\)-domain PI control under weak grid conditions, and their performances are very close.  
In Figure 5a, when the grid impedance is 1 mH, the Nyquist plots for \(ZR_{\text{p}}\)(s) and \(ZR_{\text{n}}\)(s) do not encompass the point \((-1,\text{j0})\). When the grid impedance is 45 mH and 5 mH, the Nyquist diagrams of \(ZR_{\text{p}}\)(s) and \(ZR_{\text{n}}\)(s) encompass the point \((-1,\text{j0})\).  
In Figure 5b, when the network impedance is 1 mH and 5 mH, the Nyquist diagrams for \(ZR_{\text{p}}\)(s) and \(ZR_{\text{n}}\)(s) do not encompass the point \((-1,\text{j0})\). However, when the grid impedance is 45 mH, the Nyquist diagrams for \(ZR_{\text{p}}\)(s) and \(ZR_{\text{n}}\)(s) encompass the point \((-1,\text{j0})\). By comparison with Figure 5, it is evident that considering power grid voltage feedforward control decreases the stability of GCIs controlled by PR control in the time domain and PI control in the \(dq\) domain when connected to a weak power grid.