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Geometric Decentralized Stability Certificate for Power Systems Based on Projecting DW Shells

Linbin Huang, Liangxiao Luo, Ruohan Leng, Huanhai Xin, Dan Wang, Florian Dörfler

TL;DR

Problem: decentralized stability analysis for heterogeneous multi-converter networks is difficult due to high-dimensional interactions and conservativeness of sectorial-based small-phase conditions. Approach: introduce a geometric framework using $DW$ shells and their projections (including 2D $W(A)$ and 1D $x$-$z$ graphs) to interpret small-gain and small-phase theorems and to derive a decentralized stability certificate for block-diagonal $G$ and positive-definite $H$. Contributions: (i) DW shell separation and projection-based criteria, (ii) mixed geometric stability condition across frequency ranges, (iii) a concrete decentralized stability theorem, and (iv) validation on single- and multi-converter cases with time-domain simulations showing practical effectiveness and the impact of switching to grid-forming control. Significance: offers scalable, modular tools for stability assessment and controller design in large-scale converter-dominated power systems.

Abstract

The development of decentralized stability conditions has gained considerable attention due to the need to analyze multi-agent network systems, such as heterogeneous multi-converter power systems. A recent advance is the application of the small-phase theorem, which extends the passivity theory. However, it requires the transfer function matrix to be sectorial, which may not hold in some frequency range and will result in conservativeness. To address this issue, this paper proposes a geometric decentralized stability condition based on Davis-Wielandt (DW) shell and its projections. Our approach provides a geometric interpretation of the small-gain and small-phase theorems and enables decentralized stability analysis of power systems. It serves as a visualization method to understand the closed-loop interactions and assess the stability of large-scale network systems in a scalable and modular manner.

Geometric Decentralized Stability Certificate for Power Systems Based on Projecting DW Shells

TL;DR

Problem: decentralized stability analysis for heterogeneous multi-converter networks is difficult due to high-dimensional interactions and conservativeness of sectorial-based small-phase conditions. Approach: introduce a geometric framework using shells and their projections (including 2D and 1D - graphs) to interpret small-gain and small-phase theorems and to derive a decentralized stability certificate for block-diagonal and positive-definite . Contributions: (i) DW shell separation and projection-based criteria, (ii) mixed geometric stability condition across frequency ranges, (iii) a concrete decentralized stability theorem, and (iv) validation on single- and multi-converter cases with time-domain simulations showing practical effectiveness and the impact of switching to grid-forming control. Significance: offers scalable, modular tools for stability assessment and controller design in large-scale converter-dominated power systems.

Abstract

The development of decentralized stability conditions has gained considerable attention due to the need to analyze multi-agent network systems, such as heterogeneous multi-converter power systems. A recent advance is the application of the small-phase theorem, which extends the passivity theory. However, it requires the transfer function matrix to be sectorial, which may not hold in some frequency range and will result in conservativeness. To address this issue, this paper proposes a geometric decentralized stability condition based on Davis-Wielandt (DW) shell and its projections. Our approach provides a geometric interpretation of the small-gain and small-phase theorems and enables decentralized stability analysis of power systems. It serves as a visualization method to understand the closed-loop interactions and assess the stability of large-scale network systems in a scalable and modular manner.

Paper Structure

This paper contains 13 sections, 7 theorems, 19 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Modeling of Multi-Converter Systems
  4. Geometric Stability Conditions
  5. DW Shell Separation
  6. Geometric Separation Based on Projecting DW Shells
  7. Mixed Geometric Stability Condition
  8. Geometric Decentralized Stability Condition
  9. Applications of Geometric Decentralized Stability Condition in Power Systems
  10. Case Studies of a Single-Converter System
  11. Case Studies of a Three-Converter System
  12. Time-Domain Simulation Results
  13. Conclusions

Key Result

Lemma 4.1

Consider $G, H \in\mathcal{RH}_{\infty}^{m\times m}$. The system $G(s)\#H(s)$ is stable if, for each $\omega \in [0,\infty)$, that is, the DW shell of $G(j\omega)$ is separated from the DW shell of $H^{-1}(j\omega)$ scaled by $-\frac{1}{\tau}$.

Figures (14)

  • Figure 1: Illustration of the DW shell $DW(A)$, numerical range $W(A)$, and the phases $\underline\phi(A)$, $\overline{\phi}(A)$ of a sectorial matrix $A$. The (2D) numerical range is the projection of the (3D) DW shell onto the $x$-$y$ plane.
  • Figure 2: Illustration of a multi-converter system.
  • Figure 3: Closed-loop interaction of network and converters.
  • Figure 4: A standard closed-loop system $G(s)\#H(s)$.
  • Figure 5: Illustration of the DW shell trajectory of $-\tfrac{1}{\tau}H^{-1}$ with $\tau \in (0,1]$ when $H$ is 3-by-3 (i.e., the red area). The eigenvalues of a positive definite matrix $A \in \mathbb{R}^{n \times n}$ are organized as $\lambda_1(A) \le \lambda_2(A) \le \dots \le \lambda_n(A)$.
  • ...and 9 more figures

Theorems & Definitions (7)

  • Lemma 4.1: DW Shell Separation
  • Lemma 4.2: Numerical Range Separation
  • Lemma 4.3: $\bm x$-$\bm z$ Graph Separation
  • Lemma 4.4: Small Gain
  • Lemma 4.5: Small Phase chen2024phase
  • Theorem 4.6: Geometric Stability Condition
  • Theorem 4.7: Decentralized Stability Condition