The global dynamical behavior of a classical power system consisting of n generators is studied in this paper. Existence and uniqueness of an invariant curve in 2n-dimensional space under suitable conditions are proved. The invariant curve is globally attracting so that the system behaves exactly as a one-dimensional system. Furthermore, a rotation number is defined in the power system and then, it is proved that each generator has one rotation number, but n rotation numbers for the n generators are all equal. Moreover, the rotation number is used to determine the dynamical behavior of the system, in the sense that if it is a rational number, an attractor of the system is composed of subharmonics while if an irrational number, the attractor is composed of horizontal curves. As a consequence the system has no chaotic motion under these conditions. Finally, numerical simulations are used to verify the theoretical analysis. (~) 2004 Elsevier Ltd. All rights reserved. Keywords--Classical power system, Invariant curve, t-map, Rotation number.
1. Introduction
Considerable progress has been made in recent years in the study of power systems [1][2][3][4]. Most studies concentrate mainly on the stability region (or region of attraction) of stable equilibrium points [5-7], the various stabilities including power system stability in general and online transient stability and voltage stability in particular, by using classical methods, such as Lyapunov function method [8][9][10][11] and Lyapunov function approach which is based mainly on LaSalle's expansion of Lyapunov theory [12][13][14][15], and a variety of phenomena, for instance, bifurcation, periodic solutions, quasiperiodic solutions, and chaotic behaviors [16][17][18].
However, very little effort was made to study global behaviors of power systems, such as existence of global attractors and invariant manifolds. Yet, this study is significant since, if a