A Reconstructed Singular Return Ratio Matrix for Optimizing Design of the PLL in Grid-Connected Inverters

Phase-locked loop (PLL) is essential for the grid-connected inverter to ensure grid synchronization. Since the PLL introduces a negative-resistive admittance to be in parallel with the original inverter output admittance, it poses an instability risk for the grid-connected inverter under weak grid. The system stability can be evaluated by the return ratio matrix, which is the ratio of the inverter output admittance to the grid admittance. Due to the nonsingular and nondiagonal properties of this matrix, its eigenvalues are irrational functions, leading to inconvenience of stability analysis and PLL parameters design. To address this issue, this article reconstructs the return ratio matrix to make it singular. Thus, the eigenvalues become simpler, including a zero eigenvalue and a nonzero eigenvalue. Based on the nonzero eigenvalue, the relation between the gain margin and the PLL parameters is established, and an optimizing design method of the PLL parameters is proposed to achieve system stability and satisfactory PLL dynamic response. Finally, experimental results from a 10-kVA prototype of three-phase grid-connected inverter are provided to verify the validity of the proposed design method.