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SOStab: a Matlab Toolbox for Transient Stability Analysis

Stéphane Drobot, Matteo Tacchi, Carmen Cardozo, Colin N. Jones

TL;DR

The paper introduces SOStab, a MATLAB toolbox that automates Sum-of-Squares-based computation of finite-horizon Regions of Attraction (RoA) for nonlinear polynomial dynamics, with a focus on transient stability in power systems. By formulating RoA problems within the Moment-SOS hierarchy and targeting ellipsoidal sets, SOStab provides inner and outer RoA certificates via degree-$d$ polynomials and outputs turnkey stability classifiers $v_d$ and $w_d$, along with visualizations. The authors demonstrate the method on a phase-locked loop and a single-machine infinite-bus model, highlighting how increasing the degree improves accuracy but incurs higher computational costs, and discuss scalability challenges. The work offers a practical, automated tool that lowers the barrier to SOS-based RoA analysis, enabling rapid experimentation and reproducibility, while outlining limitations and future directions toward scaling to more complex, higher-dimensional systems using structure-exploiting techniques.

Abstract

This paper presents a new Matlab toolbox, aimed at facilitating the use of polynomial optimization for stability analysis of nonlinear systems. In the past decade several decisive contributions made it possible to recast this type of problems as convex optimization ones that are tractable in modest dimensions. However, available software requires their user to be fluent in Sum-of-Squares programming, preventing them from being more widely explored by practitioners. To address this issue, SOStab entirely automates the writing and solving of optimization problems, and directly outputs relevant data for the user, while requiring minimal input. In particular, no specific knowledge of optimization is needed for implementation. The toolbox allows a user to obtain outer and inner approximates of the \ac{roa} of the operating point of different grid connected devices such as synchronous machines and power converters.

SOStab: a Matlab Toolbox for Transient Stability Analysis

TL;DR

The paper introduces SOStab, a MATLAB toolbox that automates Sum-of-Squares-based computation of finite-horizon Regions of Attraction (RoA) for nonlinear polynomial dynamics, with a focus on transient stability in power systems. By formulating RoA problems within the Moment-SOS hierarchy and targeting ellipsoidal sets, SOStab provides inner and outer RoA certificates via degree- polynomials and outputs turnkey stability classifiers and , along with visualizations. The authors demonstrate the method on a phase-locked loop and a single-machine infinite-bus model, highlighting how increasing the degree improves accuracy but incurs higher computational costs, and discuss scalability challenges. The work offers a practical, automated tool that lowers the barrier to SOS-based RoA analysis, enabling rapid experimentation and reproducibility, while outlining limitations and future directions toward scaling to more complex, higher-dimensional systems using structure-exploiting techniques.

Abstract

This paper presents a new Matlab toolbox, aimed at facilitating the use of polynomial optimization for stability analysis of nonlinear systems. In the past decade several decisive contributions made it possible to recast this type of problems as convex optimization ones that are tractable in modest dimensions. However, available software requires their user to be fluent in Sum-of-Squares programming, preventing them from being more widely explored by practitioners. To address this issue, SOStab entirely automates the writing and solving of optimization problems, and directly outputs relevant data for the user, while requiring minimal input. In particular, no specific knowledge of optimization is needed for implementation. The toolbox allows a user to obtain outer and inner approximates of the \ac{roa} of the operating point of different grid connected devices such as synchronous machines and power converters.
Paper Structure (20 sections, 1 theorem, 27 equations, 8 figures, 3 tables)

This paper contains 20 sections, 1 theorem, 27 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Methodological Framework
  3. Models for case study
  4. Phase Locked Loop 2$^{\text{nd}}$ order model
  5. Single machine - infinite bus with regulations
  6. Computing RoA with SOStab
  7. Installation
  8. Overview
  9. Initialization and dynamics specification
  10. RoA approximation
  11. Case study
  12. RoA of the Phase Locked Loop system
  13. RoA of the single machine - infinite bus system
  14. A curse of dimensionality
  15. Discussions and conclusions
  16. ...and 5 more sections

Key Result

Proposition 1

Let $(v,w)$ be feasible for eq:LP. Then,

Figures (8)

  • Figure 1: Block scheme of a PLL system.
  • Figure 2: A synchronous machine connected to an infinite bus.
  • Figure 3: Flowchart of the toolbox workings
  • Figure 4: Inner and outer RoA approximations for the PLL system.
  • Figure 5: Plot of $w_d^{in}$ for the PLL system
  • ...and 3 more figures

Theorems & Definitions (6)

  • Definition 1: roa henrionOuterROAkordaInnerROA
  • Remark 1
  • Definition 2
  • Remark 2
  • Proposition 1: henrionOuterROA
  • Definition 3