## 3 Stability Analysis under Weak Grid
### Frequency Response of LCL Filter under Grid Impedance Change  
In this subsection, the influence of the grid impedance variation on the resonant frequency is presented. Considering that the series configuration of \(L_{2}\) and \(L_{g}\) is combined as \(L_{2g}=L_{2}+L_{g}\), the transfer function from the inverter voltage to the grid-side current is obtained from Figure 2 as [5; 30]:  
\[G_{LCL}(s)=\frac{1}{s^{3}L_{1}L_{2g}\mathbb{C}_{f}+s(L_{1}+L_{2g})}. \tag{17}\]The LCL filter produces a resonance peak in frequency response which leads to an instability. The resonance frequency of the LCL filter is determined as:  
\[\omega_{R}=\sqrt{\frac{L_{1}+L_{2g}}{L_{1}L_{2g}C_{f}}}. \tag{18}\]  
Whereas the resonance frequency in Equation (18) is given as a fixed parameter in stiff grid, it is represented as a function of the grid inductance \(L_{g}\) under weak grid condition. In fact, as the grid inductance is increased, the resonance peak is shifted toward low frequency region. Thus, since the variation of \(L_{g}\) is often uncertain and unpredictable, the damping method designed at a fixed frequency is not effective under weak grid condition. To comprehensively analyze the performance of inverter system under the grid impedance variation, three LCL filter prototypes are designed to locate the resonance frequencies at three regions as given in Table 1, which are lower than the critical frequency (low region), around the critical frequency (critical region), and higher than critical frequency (high region). According to the criterion on selecting the resonance frequency of the LCL filter [3], the resonance frequencies in high region, critical region, and low region are selected as 2991 Hz, 2006 Hz, and 1158 Hz, respectively, with the critical frequency of 1667 Hz. Thus, it is obvious that the selected cases cover all possible LCL filter parameter designs. The grid impedance gradually increases in range of stiff grid (\(L_{g}\) = 0 mH) to very weak grid (\(L_{g}\) = 21 mH). Figure 4 shows the frequency responses of three LCL filter parameters without the grid impedance variation. Three different resonance frequencies are clearly shown with respect to the critical frequency. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Cases** & **LCL Filter Parameters** & \(f_{R}\) \\ \hline Case 1 & \(L_{1}\) = 1.7 mH, \(L_{2}\) = 1.0 mH, \(C_{f}\) = 4.5 μF & 2991 Hz \\ \hline Case 2 & \(L_{1}\) = 1.7 mH, \(L_{2}\) = 1.0 mH, \(C_{f}\) = 10 μF & 2006 Hz \\ \hline Case 3 & \(L_{1}\) = 1.7 mH, \(L_{2}\) = 1.0 mH, \(C_{f}\) = 30 μF & 1158 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: LCL (inductive-capacitive-inductive) filter parameters with grid impedance under weak grid. Figure 4: Frequency responses of LCL filter parameters without grid impedance for Cases 1, 2, and 3. Figure 5 shows the frequency responses when the grid inductance \(L_{g}\) is varied in the LCL filter parameter of Case 1. As the grid inductance is increased, the resonance frequency of the LCL filter is reduced and shifted toward the critical frequency. It is worthwhile to note that the LCL filter in Case 1 has the resonance frequency quite far from the critical frequency (high region). Even under severe weak grid condition, this resonance frequency is not shifted to the low region. On the contrary, the LCL filter in Case 2 represents that \(f_{R}\) possibly moves from the high region to the low region depending on the value of \(L_{g}\). Finally, the LCL filter parameter set in Case 3 is also employed to validate the controllers under the low region. Closed-Loop Stability of Direct Grid Current Control with Active Damping under Grid Impedance Change  
In this subsection, the impact of the grid impedance variation on the stability of direct grid current controller with capacitor current feedback active damping is discussed. The stability analysis of the closed-loop LCL-filtered inverter system is achieved by investigating the location of the closed-loop poles in the discrete-time domain.