A graph theoretic view on small signal stability of inverter based power grids

Jul 16, 2026

Dynamic grid stability is traditionally secured by synchronous generators, but modern power systems rely increasingly on inverter-based resources with grid-forming control for system-wide synchronization. The work addresses limitations of existing centralized and decentralized small-signal stability certificates that mostly provide only sufficient conditions. It constructs a necessary and sufficient small-signal stability criterion for lossless inverter-based grids of arbitrary topology. Asymptotic stability is shown equivalent to positive definiteness of a single matrix combining topology, operating point, and effective droop gains, leading to decentralized graph-theoretic certificates and quantified conservatism.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages10

Reading Time25 min

Introduction

Background and motivation

Scope and organization

Literature Review

Existing stability certificates

A graph theoretic view on small signal stability of inverter-based power grids Iva Bai, Jakob Niehues, Philipp C. Böttcher, Cai Dieball, Leonardo Rydin Gorjão, Andrea Benigni, Senior Member, IEEE, Frank Hellmann, Dirk Witthaut arXiv :2607 .08260v1 [ ee ss . SY] 9 Jul 2026 Abstract—Dynamic grid stability is traditionally ensured with synchronous generators. Modern grids rely substantially more on inverter-based resources, which require grid-forming control to guarantee adequate system-wide synchronization and stability. Small-signal stability has granted various centralized and decentralized stability certificates—but these have primarily been limited to sufficient criteria only. In this work, we construct a necessary and sufficient small-signal stability criterion for lossless inverter-based power grids with arbitrary topology. We show that asymptotic stability is equivalent to the positive definiteness of a single matrix that combines network topology, operating point, and effective droop gains. We derive graphtheoretic stability criteria based on an augmented cone graph and show that the contribution of graph cycles is typically small, as illustrated for three IEEE test cases. The resulting framework yields decentralized stability criteria, quantifies the conservatism introduced by decentralization, and may support the development of future grid codes. I. INTRODUCTION The ongoing energy transition poses difficult questions for the stable operation and design of power grids [1] . One central challenge is to ensure system-wide frequency synchronization and dynamic stability [2] . Traditionally, this was the task of synchronous generators (SGs) . As they are progressively replaced by inverter-based resources (IBRs), grid-forming control becomes essential to maintain system stability. Accordingly, grid codes are currently being revised to define the required grid-forming capabilities and to ensure that large populations of IBRs can collectively replace the stabilizing role of synchronous generators. While SGs are connected to the closely monitored and actively managed transmission grid, IBRs are predominantly connected at the distribution level. Operators have only limited observability and controllability of low- and medium-voltage distribution grids, and IBRs’ grid-forming controls will need to operate autonomously. These challenges have led to a renewed interest in the small-signal stability of power grids, I. Bai, P. C. Böttcher, A. Benigni and D. Witthaut are with the Forschungszentrum Jülich, Institute of Climate and Energy Systems: Energy Systems Engineering (ICE-1), D-52428 Jülich, Germany. I. Bai is with the Institute of Physics Belgrade, University of Belgrade, Serbia. J. Niehues and F. Hellmann are with the Potsdam Institute for Climate Impact Research (PIK), D-14412 Potsdam, Germany. J. Niehues is with Technische Universität Berlin, D-10623 Berlin, Germany. C. Dieball and D. Witthaut are with the Institute for Theoretical Physics, University of Cologne, D-50937 Köln, Germany. L. Rydin Gorjão is with the Department of Environmental Sciences, Open University of the Netherlands, Heerlen, The Netherlands & Institute of Physics, Norwegian University of Life Sciences, Ås, Norway. A. Benigni is with RWTH Aachen, Germany. and especially in finding local stability criteria suitable for implementation in grid codes. The field of small-signal stability of multi-machine power grids has a long tradition, going back to the seminal work of Bergen and Hill [3] . Since then, a plethora of results from power engineering [4], control theory [5], [6], and theoretical physics [1] have provided a broad range of analytical tools for studying the small-signal stability of power systems. In this paper, we derive a necessary and sufficient criterion for the small-signal stability of synchronization in lossless inverter-based power grids. For the first time, we provide necessary criteria without assuming a restrictive d

FAQs

What problem does the paper solve in inverter-based grid stability?

It derives a necessary and sufficient small-signal stability criterion for lossless inverter-based power grids, addressing the gap where most existing results provide only sufficient conditions.

How is stability characterized in the proposed framework?

Asymptotic stability is equivalent to the positive definiteness of a single matrix that integrates network topology, the operating point, and effective droop gains.

What is the role of the graph-theoretic perspective?

The criterion admits a graph-theoretic interpretation (augmented cone graph), enabling decentralized local stability certificates and allowing the conservatism of decentralization to be quantified via neglected stabilizing interactions.

Jul 16, 2026

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