Operating-point-dependent stability of inverter-dominated microgrids via analytical combination of GFL and GFM impedances

This paper presents an impedance-based framework for assessing the operating-point-dependent stability of microgrids composed exclusively of inverter-based resources (IBRs), including both grid-following (GFL) and grid-forming (GFM) converters. While dq-frame impedance models have been established for individual GFL and GFM converters, their analytical combination to evaluate closed-loop stability at the system level has not been sufficiently addressed. In the proposed framework, each IBR is represented as a modular subsystem using an analytically derived dq-frame impedance model, and the impedance model of a virtual-synchronous-generator-based GFM converter is newly constructed within a unified block-diagram formulation. The analytical impedances of the GFL and GFM subsystems are then combined to assess overall stability through the generalized Nyquist criterion, without constructing a full-system state-space model. The analytical models are validated against measurement-based identification and electromagnetic transient simulations, showing close agreement over a wide frequency range. Because the impedances are obtained as explicit analytical expressions, stability can be evaluated over the entire frequency range and across a large number of operating points at low computational cost, which is impractical using measurement- or simulation-based frequency sweeps alone. Exploiting this property, a stability map is constructed over the power-flow plane to reveal how the stability margin varies with the operating point, providing system operators with a practical basis for stability-oriented operational planning and monitoring. Case studies involving GFL and GFM IBRs in weak microgrids demonstrate the effectiveness of the proposed approach.