Gain and Phase: Decentralized Stability Conditions for Power Electronics-Dominated Power Systems

TL;DR

This work tackles small-signal stability in power systems dominated by power electronics by introducing decentralized stability conditions that couple converter-level dynamics with network dynamics via a mixed gain–phase framework. It models each converter with a 2×2 admittance $Y_{C,i}(s)$ in a global $dq$ frame and represents the network through a Kron-reduced transfer function $Y_{grid}(s)$, enabling a modular, scalable analysis that accommodates GFL, GFM, and SG components. The core contributions are the decentralized gain and phase conditions, a corollary for identical $R/X$ networks using gSCR, and an extension to SG–GFM–GFL hybrids through equivalent subsystems. Simulation results on 9- and 68-bus systems validate the theory, showing how GFM and network-strength enhancements improve stability and how PLL design choices influence the stability margins. Overall, the method offers a less conservative, interpretable, and scalable alternative to eigenvalue or Nyquist-based analyses for large-scale converter-dominated power systems, with practical design guidelines for stability assurance.

Abstract

This paper proposes decentralized stability conditions for multi-converter systems based on the combination of the small gain theorem and the small phase theorem. Instead of directly computing the closed-loop dynamics, e.g., eigenvalues of the state-space matrix, or using the generalized Nyquist stability criterion, the proposed stability conditions are more scalable and computationally lighter, which aim at evaluating the closed-loop system stability by comparing the individual converter dynamics with the network dynamics in a decentralized and open-loop manner. Moreover, our approach can handle heterogeneous converters' dynamics and is suitable to analyze large-scale multi-converter power systems that contain grid-following (GFL), grid-forming (GFM) converters, and synchronous generators. Compared with other decentralized stability conditions, e.g., passivity-based stability conditions, the proposed conditions are significantly less conservative and can be generally satisfied in practice across the whole frequency range.

Figures (21)

• Figure 1: Illustration of a multi-converter power system.
• Figure 2: Basic implementation of a grid-following (GFL) power converter. A more advanced PLL implementation (including voltage normalization and rated frequency feed-forward) will be considered in Example \ref{['ex:68_bus']}.
• Figure 3: Closed-loop representation of a multi- device system by combining \ref{['eq:admittance_devices']} and \ref{['eq:Y_grid']}.
• Figure 4: Phases of a sectorial matrix $A$ are contained between $[\underline\phi(A),\overline\phi(A)]$, and the gray area $W(A)$ is the numerical range of $A$.
• Figure 5: A standard closed-loop system $G(s)\#H(s)$.

Theorems & Definitions (15)

• Example 1: Transmission network with an identical R/X ratio
• Lemma 3.1
• Proposition 4.1: Decentralized stability conditions
• proof
• Corollary 4.2: Power grid strength
• proof
• Example 2: A single-converter system
• Example 3: A three-converter system
• Lemma 5.1: Equivalent subsystems
• proof