## 3 Stability Analysis under Weak Grid
### Closed-Loop Stability of Integral-Resonant State Feedback LQR Control under Grid Impedance Change  
In this section, the stability of the integral-resonant state feedback current controller is presented under distorted grid and the grid impedance variation for three-phase LCL-filtered inverter. The locations of the closed-loop poles in the discrete-time domain which are obtained from Equations (13) and (15) are used for the stability analysis. To attenuate the current harmonic distortion caused by distorted grid, the resonant controls in the orders of 6 th and 12 th in the SRF are augmented in the state model as in Equation (12). Figure 9 shows the locations of the closed-loop poles by this control when \(L_{g}\) is gradually increased from stiff grid to weak grid. While most of the closed-loop system poles remain in the stable region in spite of large variation of \(L_{g}\), the poles of the 6 th resonant controller move toward the stability boundary of the unit circle as \(L_{g}\) is increased. However, as compared with Figure 6 which shows unstable poles at 10 mH of \(L_{g}\), Figure 9 shows that the system is only unstable when \(L_{g}\) is increased beyond 14 mH. Figure 6: Location of the closed-loop poles for direct grid current controller with \(L_{1}=1.7\) mH, \(L_{2}=1.0\) mH, and \(C_{f}=4.5\)\(\mu\)F under distorted grid and \(L_{g}\) variation. Figures 10 and 11 show the locations of the closed-loop poles with different \(C_{f}\) values of 10 \(\upmu\)F and 30 \(\upmu\)F, respectively, when \(L_{g}\) is varied from stiff grid to weak grid. These filter capacitors produce the resonance frequencies around the critical frequency, and lower than the critical frequency. Similarly, as the grid impedance increases, the poles of the 6 th resonant controller move outside the unit circle. However, in the integral-resonant state feedback control, the stability limit is much extended since the instability occurs with larger value of \(L_{g}\) than the direct grid current controller. As a result, the integral-resonant state feedback control can be regarded as more robust to uncertainty in the grid impedance change. In addition, it is confirmed that the current control is more likely to be unstable for small variation of \(L_{g}\) as the resonance frequency of the LCL filter gets smaller. Figure 8: Location of the closed-loop poles for direct grid current controller with \(L_{1}=1.7\) mH, \(L_{2}=1.0\) mH, and \(C_{f}=30\)\(\upmu\)F under distorted grid and \(L_{g}\) variation. Figure 7: Location of the closed-loop poles for direct grid current controller with \(L_{1}=1.7\) mH, \(L_{2}=1.0\) mH, and \(C_{f}=10\)\(\upmu\)F under distorted grid and \(L_{g}\) variation. Figure 10: Location of the closed-loop poles for augmented integral-resonant state feedback LQR controller with \(L_{1}=1.7\) mH, \(L_{2}=1.0\) mH, and \(C_{f}=10\) μF under distorted grid and \(L_{g}\) variation. Figure 9: Location of the closed-loop poles for augmented integral-resonant state feedback LQR controller with \(L_{1}=1.7\) mH, \(L_{2}=1.0\) mH, and \(C_{f}=4.5\) μF under distorted grid and \(L_{g}\) variation. To investigate the stability under different grid frequency, Figure 12 represents the location of the closed-loop poles for augmented integral-resonant state feedback LQR controller with \(L_{1}=1.7\) mH, \(L_{2}=1.0\) mH, and \(C_{f}=4.5\) uF without the grid impedance under three different grid frequencies of 60, 55, and 50 Hz. As shown in this figure, the closed-loop poles are almost overlapped in each frequency value, and are maintained in the stable region regardless of the frequency change. Figure 11: Location of the closed-loop poles for augmented integral-resonant state feedback LQR controller with \(L_{1}=1.7\) mH, \(L_{2}=1.0\) mH, and \(C_{f}=30\) μF under distorted grid and \(L_{g}\) variation. Figure 12: Location of the closed-loop poles for integral-resonant state feedback controller with \(L_{1}=1.7\) mH, \(L_{2}=1.0\) mH, and \(C_{f}=4.5\) μF under different grid frequencies of 60, 55, and 50 Hz.