Decentralized Small Gain and Phase Stability Conditions for Grid-Forming Converters: Limitations and Extensions

The increasing share of converter based resources in power systems calls for scalable methods to analyse stability without relying on exhaustive system wide simulations. Decentralized small gain and small-phase criteria have recently been proposed for this purpose, but their applicability to grid forming converters is severely limited by the sectoriality assumption, which is not typically satisfied at low frequencies. This work revisits and extends mixed gain phase conditions by introducing loop shaping transformations that reformulate converter and network models in alternative coordinate frames. The proposed approach resolves intrinsic non sectoriality at low frequencies and reduces conservativeness, thereby improving the applicability of decentralized stability certification. Analytical results are illustrated using an infinite bus system first and then extended to the IEEE 14 bus network, demonstrating the practicality and scalability of the method. These findings provide a pathway toward less conservative and more widely applicable decentralized stability certificates in power grids.