Encyclopedia of Systems and Control DOI 10.1007/978-1-4471-5102-9_266-1 © Springer-Verlag London 2014 Lyapunov Methods in Power System Stability Hsiao-Dong Chiang  School of Electrical and Computer Engineering, Cornell University, Ithaca, NY, USA Introduction Energy functions, an extension of the Lyapunov functions, have been practically used in electric power systems for several applications. A comprehensive energy function theory for general nonlinear autonomous dynamical systems along with its applications to electric power systems will be summarized in this article. We consider a general nonlinear autonomous dynamical system described by the following equation: .t/ D f.x.t// (1) We say a function V W R n ! R is an energy function for the system (1) if the following three conditions are satisfied (Chiang et al. 1987): (E1): The derivative of the energy function V.x/ along any system trajectory x.t/ is nonpositive, i.e., P V .x.t//  0. (E2): If x.t/ is a nontrivial trajectory (i.e., x.t/ is not an equilibrium point), then along the nontrivial trajectory x.t/ the set ft 2 R W P V .x.t// D 0g has measure zero in R. (E3): That a trajectory x.t/ has a bounded value of V.x.t// for t 2 R C implies that the trajectory x.t/ is also bounded. Condition (E1) indicates that the value of an energy function is nonincreasing along its trajectory, but does not imply that the energy function is strictly decreasing along any trajectory. Conditions (E1) and (E2) imply that the energy function is strictly decreasing along any system trajectory. Property (E3) states that the energy function is a proper map along any system trajectory but need not be a proper map for the entire state space. Obviously, an energy function may not be a Lyapunov function. As an illustration of the energy function, we consider the following classical transient stability model and derive an energy function for the model. Consider a power system consisting of n generators. Let the loads be modeled as constant impedances. Under the assumption that the transfer conductance of the reduced network after eliminating all load buses is zero, the dynamics of the i th generator can be represented by the equations P ı i D ! i M i P ! i D P i  D i ! i  X j D1 V i V j B ij sin.ı i  ı j / (2)  E-mail: chiang@ece.cornell.edu Page 1 of 6