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Region of Attraction for Power Systems using Gaussian Process and Converse Lyapunov Function -- Part I: Theoretical Framework and Off-line Study

Chao Zhai, Hung D. Nguyen

TL;DR

A novel framework to construct the region of attraction (ROA) of a power system centered around a stable equilibrium by using stable state trajectories of system dynamics is introduced and can significantly enlarge the estimated ROA compared to that of the analytic Lyapunov counterpart.

Abstract

This paper introduces a novel framework to construct the region of attraction (ROA) of a power system centered around a stable equilibrium by using stable state trajectories of system dynamics. Most existing works on estimating ROA rely on analytical Lyapunov functions, which are subject to two limitations: the analytic Lyapunov functions may not be always readily available, and the resulting ROA may be overly conservative. This work overcomes these two limitations by leveraging the converse Lyapunov theorem in control theory to eliminate the need of an analytic Lyapunov function and learning the unknown Lyapunov function with the Gaussian Process (GP) approach. In addition, a Gaussian Process Upper Confidence Bound (GP-UCB) based sampling algorithm is designed to reconcile the trade-off between the exploitation for enlarging the ROA and the exploration for reducing the uncertainty of sampling region. Within the constructed ROA, it is guaranteed in probability that the system state will converge to the stable equilibrium with a confidence level. Numerical simulations are also conducted to validate the assessment approach for the ROA of the single machine infinite bus system and the New England $39$-bus system. Numerical results demonstrate that our approach can significantly enlarge the estimated ROA compared to that of the analytic Lyapunov counterpart.

Region of Attraction for Power Systems using Gaussian Process and Converse Lyapunov Function -- Part I: Theoretical Framework and Off-line Study

TL;DR

A novel framework to construct the region of attraction (ROA) of a power system centered around a stable equilibrium by using stable state trajectories of system dynamics is introduced and can significantly enlarge the estimated ROA compared to that of the analytic Lyapunov counterpart.

Abstract

This paper introduces a novel framework to construct the region of attraction (ROA) of a power system centered around a stable equilibrium by using stable state trajectories of system dynamics. Most existing works on estimating ROA rely on analytical Lyapunov functions, which are subject to two limitations: the analytic Lyapunov functions may not be always readily available, and the resulting ROA may be overly conservative. This work overcomes these two limitations by leveraging the converse Lyapunov theorem in control theory to eliminate the need of an analytic Lyapunov function and learning the unknown Lyapunov function with the Gaussian Process (GP) approach. In addition, a Gaussian Process Upper Confidence Bound (GP-UCB) based sampling algorithm is designed to reconcile the trade-off between the exploitation for enlarging the ROA and the exploration for reducing the uncertainty of sampling region. Within the constructed ROA, it is guaranteed in probability that the system state will converge to the stable equilibrium with a confidence level. Numerical simulations are also conducted to validate the assessment approach for the ROA of the single machine infinite bus system and the New England -bus system. Numerical results demonstrate that our approach can significantly enlarge the estimated ROA compared to that of the analytic Lyapunov counterpart.

Paper Structure

This paper contains 17 sections, 5 theorems, 42 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. The ROA of a General Dynamical System
  3. GP for Learning Unknown Dynamics
  4. Gaussian process and RKHS norm
  5. GP-UCB based algorithm
  6. Main Results
  7. Numerical Simulations
  8. Power system model
  9. Single machine infinite bus system
  10. IEEE 39 bus system
  11. Discussions on computational cost
  12. Conclusions and Future Work
  13. The class $\Gamma$ function
  14. An upper bound of discretizing error
  15. Information gain
  16. ...and 2 more sections

Key Result

Lemma 1

Without loss of generality, let $\mathbf{x=0}$ be an asymptotically stable equilibrium point for the nonlinear system $\dot{\mathbf{x}}\mathbf{=f(x)}$, where $\mathbf{f}: X\rightarrow{R^n}$ is locally Lipschitz, and $S$ is the region of attraction, then there is a continuous positive definite functi and with $\alpha(z)$ is a class $\Gamma$ function (see Appendix app:def), and the level set $\Ome

Figures (7)

  • Figure 1: Illustration on the level set of Lyapunov function. The red ellipse describes the level set $\{\mathbf{x}\in R^2~|~V(\mathbf{x})\leq C_1\}$, while the blue one denotes the level set $\{\mathbf{x}\in R^2~|~V(\mathbf{x})\leq C_2\}$ with $C_1<C_2$ and $\mathbf{x}=(x_1,x_2)$. The black dot represents a stable equilibrium point of dynamical system. Each state in the level set can converge to the equilibrium point on condition of $\dot{V}(\mathbf{x})<0$.
  • Figure 2: Examples of basic kernel functions: (a) Squared Exponential Kernel $k(\mathbf{x},\mathbf{x}')=e^{-\|\mathbf{x}-\mathbf{x}'\|^2/2l^2}$ with a length scale parameter $l$ and (b) Linear Kernel $k(\mathbf{x},\mathbf{x}')=\mathbf{x}^T\mathbf{x}'$ with $\mathbf{x}'=1$. Each kernel allows to approximate the unknown function with a certain "complexity".
  • Figure 3: Illustration on three different sampling rules. The blue line denotes mean values of the unknown function $V(\mathbf{x})$, and the gray shade describes the $95\%$ confidence interval. The three red circles $A$, $B$ and $C$ represent three different sampling rules, respectively. $A$ prefers the point with the maximum posterior mean value in order to achieve the large $V(\mathbf{x})$, and $B$ selects the point with the maximum posterior variance in order to reduce the uncertainty. $C$ aims to allow for both the exploitation for large $V(\mathbf{x})$ and the exploration for eliminating the uncertainty.
  • Figure 4: Sampling points and state trajectories of SMIB system. Red dots represents the unstable sampling points that fail to converge to the origin, while green dots refer to stable sampling points that converge to the origin. Blue lines indicate their state trajectories, and the arrows point in the direction of state trajectories. The red dashed ellipse denotes the certified ROA according to (\ref{['roa_est']}).
  • Figure 5: Confidence evaluation for the ROA of SMIB system with $100$ sampling points. The green dots denote the stable sampling points, and red ellipse region refers to the certified ROA with an existing Lyapunov function. The yellow region indicates the state that converges to the origin with the probability at least $95\%$.
  • ...and 2 more figures

Theorems & Definitions (19)

  • Lemma 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • proof
  • Remark 4
  • Theorem 2
  • proof
  • Remark 5
  • ...and 9 more