Harmonic Transfer-Function Model of Three-Phase Synchronous Reference Frame PLL under Unbalanced Grid Conditions

Abstract

The synchronous reference frame (SRF) phase-locked loop (PLL) is widely used in three-phase grid-connected converter systems. For the balanced grid condition, the PLL can be linearized in the dq frame, and the derived small-signal model is linear time-invariant (LTI), which can be readily analyzed with classical control theories. However, in the unbalanced grid, the presence of the negative-sequence voltage leads to frequency-coupling dynamics in the SRF-PLL, which cannot be characterized by the dq-frame LTI model. This paper thus proposes a harmonic transfer-function model of the SRF-PLL under three-phase unbalanced grid conditions based on the linear time-periodic theory. It is found that the negative-sequence voltage can result in considerable phase error, which yields three time-periodic coefficients in the dynamic model of SRF-PLL. Simulations and experimental results corroborate the effectiveness of the model. The frequency-coupling effects are verified in the time domain and the frequency domain.