High-Fidelity Modeling and Stability Analysis of Microgrids by Considering Time Delay

Microgrids (MGs) offer substantial environmental, economic, and technological benefits by supplying electrical energy to the grid or local consumers via power electronic inverter-interfaced Distributed Energy Resources (DERs). However, the design, control, and stability analysis of inverter-interfaced MGs present significant challenges, as numerous system parameters influence the overall stability of these MGs. While extensive research has been conducted on MG stability, predominantly through eigenvalue-based state-space models, further refinement is necessary for more accurate stability assessments. This paper provides an accurate and detailed stability analysis of MGs, focusing specifically on parallel-connected grid-forming inverters (GFIs) operating in island mode. The novelty of this paper lies in three key contributions: (1) accurately considering a time delay in regard to the dq-axis synchronous reference frame, (2) the enhanced accuracy of the small-signal model for the purpose of the voltage control loop, and (3) the enhanced accuracy of the small-signal model for the purpose of the current control loop. In the literature, digital control-induced time delays are typically incorporated directly into the dq-axis, while the effect of the phase shift is then neglected, leading to inaccurate stability assessment results. Accordingly, the motivation of this paper is to consider the time delay, which naturally exists in regard to the abc-axis, and accurately represent it in regard to the dq-axis by modeling the phase shift effect for precise stability analysis. These contributions enable a precise small-signal model to be developed and eigenvalue-based stability analysis to be carried out by not only incorporating an accurate representation of the time delay, but also considering the voltage control loop and the current control loop in regard to the dq-axis synchronous reference frame. To achieve this aim, a full-order state-space and small-signal model of GFIs is developed, considering all the subsystem dynamics. The theoretical analysis conducted within the MATLAB m-file code environment (other programming languages, such as C or Python, could also be employed) and the real-time simulation results obtained using PLECS show excellent agreement, verifying the accuracy of the proposed method and highlighting its superior precision compared to conventional stability analysis. The real-time simulation results show that the proposed small-signal model has less than 5% deviation in regard to both active and reactive power droop coefficient limits, while the conventional model deviates by 22% and 530% in regard to active and reactive power droop, respectively. Consequently, this study determines the critical boundaries of the control parameters that ensure MG stability, providing a more accurate foundation for stability analysis and control design.