Data-Driven Computational Methods for the Domain of Attraction and Zubov's Equation
This work addresses the challenge of stability analysis in power systems and other high-dimensional dynamical systems, offering incremental improvements through new computational methods.
The paper tackles the problem of characterizing the domain of attraction for systems of ordinary differential equations by deriving an integral form solution to Zubov's equation and developing two data-driven computational methods, including a deep learning approach applied to a New England 10-generator power system model with proven approximation error bounds as a cubic polynomial of the number of generators.
This paper deals with a special type of Lyapunov functions, namely the solution of Zubov's equation. Such a function can be used to characterize the domain of attraction for systems of ordinary differential equations. We derive and prove an integral form solution to Zubov's equation. For numerical computation, we develop two data-driven methods. One is based on the integration of an augmented system of differential equations; and the other one is based on deep learning. The former is effective for systems with a relatively low state space dimension and the latter is developed for high dimensional problems. The deep learning method is applied to a New England 10-generator power system model. We prove that a neural network approximation exists for the Lyapunov function of power systems such that the approximation error is a cubic polynomial of the number of generators. The error convergence rate as a function of n, the number of neurons, is proved.