Dynamic Phasor Modeling and Stability Analysis of SRF-PLL-Based Grid-Tie Inverter Under Islanded Conditions

Synchronous reference frame (SRF) phase locked loop (PLL) is a widely employed scheme for grid-synchronization of a three-phase grid-tied inverter (GTI). The SRF-PLL model considered for analysis in literature typically assumes the grid to be always present at the point of common coupling (PCC) in the form of a sinusoidal voltage source. The behavior of the SRF-PLL-based GTI system along with the local loads in case of a grid outage is typically not considered for the analysis. However, this is critical for the study of an unintentional island that is subsequently formed after grid-disconnection. This is required for ceasing system operation for safety reasons. In this work, a dynamic-phasor (DP)-based complex domain model of the GTI system is proposed for studying the behavior of an unintentionally islanded GTI, which leads to analytical solutions of the system. Using the proposed DP model, it is shown that the PCC voltage behavior is governed predominantly by the dynamics of two subsystems, namely, the local load characteristics and the SRF-PLL. The equilibrium solutions of the model are shown to yield the steady-state operating frequency of the islanded system with good accuracy, for various GTI power-factor and load quality-factors. Small-signal analysis is performed to linearize the DP equations. This yields insights into the stability of the multiple possible frequency solutions of the islanded system, which is analytically established using eigenvalue analysis. Simulation and islanding experimental results conducted on a 4.5 kVA three-phase GTI system are presented that validate the proposed DP model of the GTI, accuracy of the analytical frequency, and stability analysis of the islanded system.