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

TL;DR

This work tackles the problem of scalable, decentralized stability certification for grid-forming converters in converter-dominated grids, where traditional small-gain and small-phase criteria struggle due to non-sectorial low-frequency dynamics. It introduces a generalized loop-shaping transformation that blends power-polar and rectangular frames, with a frequency-dependent design and a weighting term $W_i(s)$, to enforce sectoriality and tighten phase bounds across frequencies. The approach is demonstrated on an infinite-bus system with virtual admittance compensation and on the IEEE 14-bus network, showing improved conservativeness and the ability to identify unstable converters; a representative unstable case validates the need for joint gain/phase consideration. The results offer a practical pathway to less conservative, decentralized stability certificates in grid-forming dominated grids and point to future work integrating network topology and mixed converter types for broader applicability.

Abstract

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.

Figures (10)

• Figure 1: Diagram of the feedback connection.
• Figure 2: Mixed small gain–phase condition for a GFM converter. If a system is not sectorial, its phase is represented by the full interval $[-2\pi,\, 2\pi]$. The small-gain condition is violated if the converter gain exceeds the network gain. The small-phase condition is violated if the converter phase overlap the network phase shifted by $\pm\pi$.
• Figure 3: Small-signal model of a generic converter with synchronization frame embedding.
• Figure 4: Proposed transformation as loop-shaping transformation
• Figure 5: Small phase condition for a GFM converter in Power-Polar frame

Theorems & Definitions (7)

• Theorem 2.1: Mixed Small Gain-Phase Theorem
• Theorem 2.2: Decentralized Mixed Small Gain-Phase Theorem
• Proposition 3.1
• proof
• Remark 1
• Remark 2
• Remark 3