Small-signal stability analysis of mixed power system integrating grid forming and following inverters-based resources

This paper presents a comprehensive small-signal stability analysis of multi-bus power systems with high penetration of inverter-based resources (IBRs). A phasor-domain model is employed to capture the key steady-state dynamics of IBRs, particularly the behavior of outer control loops such as active and reactive power controllers. While this modeling approach simplifies the representation by neglecting the fast dynamics of inner voltage and current control loops, it effectively focuses on the slower outer-loop dynamics. This allows for a significant reduction in computational burden when working with large-scale systems, while maintaining adequate accuracy compared to electromagnetic transient (EMT) simulations. The framework is applied to both single- and multi-bus systems, and theoretical stability analysis is carried out using eigenvalue methods, supported by mathematical justifications. Validation is performed through case studies on three benchmark systems: the single-machine infinite-bus (SMIB) system, the Western System Coordinating Council (WSCC) IEEE 9-bus system, and the more complex IEEE 39-bus system. The results reveal the influence of IBR integration on small-signal stability and show that approximately 30%–40% grid-forming inverters (GFMs) are necessary to ensure stable and efficient operation in 100% IBR scenarios.