Data-Driven Computational Methods for the Domain of Attraction and Zubov's Equation

• 成果类型: Article
• 署名作者: Kang, Wei; Sun, Kai; Xu, Liang
• 署名单位: United States Department of Defense; United States Navy; Naval Postgraduate School; University of California System; University of California Santa Cruz; University of Tennessee System; University of Tennessee Knoxville; United States Department of Defense; United States Navy; United States Naval Research Laboratory
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2023.3326769
• 发表日期: 2024
• 页码: 1600-1611
• 关键词: Lyapunov methods; mathematical models; Power system stability; Deep learning; Integral equations; sun; computational modeling; domain of attractions (DOAs); Lyapunov functions; power systems

摘要

This article deals with a special type of Lyapunov functions, namely the solution of Zubov's equation.Such a function can be used to characterize the exact boundary of the domain of attraction for systems of ordinary differential equations. In Theorem 2, 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. A feedforward neural network is trained to approximate the corresponding Zubov's equation solution. The network characterizes the system's domain of attraction. 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 is O(n(-1/2)),where n is the number of neurons.