Optimized Design of the PLL with Reconstructed Singular Return Ratio Matrix for Grid-Connected Inverter

Phase-locked loop (PLL) is essential for the grid-connected inverter. Taking the effect of PLL into consideration, this paper derives the mathematical model and the equivalent circuit of the grid-connected inverter system. According to impedance-based stability criterion, the return ratio matrix, which is the product of the inverter output admittance and grid impedance, should satisfy the generalized Nyquist criterion to ensure the system stability. However, the return ratio matrix is non-singular and non-diagonal, with two complex none-zero eigenvalues. It is hard to guide PLL-parameters design. Considering that PLL does not track d-axis of PCC voltage, the admittance introduced by the PLL has one column of zero. By equivalent transformation, the return ratio matrix can be reconstructed as the product of the admittance introduced by PLL and others. In this way, the reconstructed return ratio matrix is singular, which has only one none-zero eigenvalue. Then, the relationship between the non-zero eigenvalue and the stability margin is derived for optimizing design of the PLL. Finally, a 10 kVA three-phase LCL-type grid-connected inverter is fabricated and tested in the lab, and the experimental results are provided to verify the validity of the PLL parameter design method.