630 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 44, NO. 7, JULY 1997 A Novel Method to Compute the Closest Unstable Equilibrium Point for Transient Stability Region Estimate in Power Systems Chih-Wen Liu and James S. Thorp, Fellow, IEEE Abstract—It is well known that the threat of instability governs most aspects of modern power system operation. This being the case, any improvements made in determining the imminence of an unstable condition, or controlling it has great benefits to modern interconnected power networks. In this paper, we first review the well-known closest unstable equilibrium point (closest UEP) method for transient stability region estimate. One key step involved in the closest UEP method is to search for the closest UEP. This paper presents a new search algorithm for the closest UEP so that one can avoid the fractal nature of the Newton–Raphson method which is usually used in the closest UEP method. Finally, we applied the algorithm to three-machine system and obtained satisfactory simulation results. Index Terms—Power systems, transient stability, UEP. I. INTRODUCTION I T IS WELL KNOWN that the threat of instability governs most aspects of modern power system operation. This being the case, any improvements made in determining the imminence of an unstable condition, or controlling it has great benefits to modern interconnected networks. Transient stability of a power system is concerned with system’s capability to withstand severe disturbance, like short circuit, lighting, and loss of generators. To determine transient stability of a power system, the conventional, and still the standard, method is to solve the system equations to obtain a time solution of the system variables and parameters, for a given scenario of events. The alternate method, which is used in this paper, is to determine stability “directly.” A good survey paper on this approach is by Ribben–Pavella [1]. It appears that in order to make energy function methods practically attractive it is necessary to define “suitable” stability region es- timates, i.e., estimates combining accuracy and computational efficiency. By the mid-1970’s research efforts concentrated on defining such “suitable” stability region estimates which would be more practical than trying to find the “theoretical” stability region . This resulted in the development of Manuscript received June 29, 1995; revised September 23, 1996. This paper was recommended by Associate Editor M. Ilic. C.-W. Liu is with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan (e-mail: cwliu@cc.ee.ntu.edu.tw). J. S. Thorp is with the School of Electrical Engineering, Cornell University, Ithaca, NY 14850 USA (e-mail: thorp@gonzo.ee.cornell.edu). Publisher Item Identifier S 1057-7122(97)03484-3. the well-known closest unstable equilibrium point (closest UEP) method [1]. In [2], the stability region estimated by the closest UEP is shown to be optimal in the sense that it is the largest region within which can be characterized by the corresponding energy function. In this paper, we first review the closest UEP method. Next a novel numerical algorithm is derived and justified theo- retically. Finally, we applied the algorithm to three-machine system and obtained satisfactory simulation results. II. MODEL,ENERGY FUNCTIONS AND THE CLOSEST UEP METHOD In this section, we review power system dynamic models, corresponding energy functions, and the closest UEP method. We assumed that the postfault dynamic model is represented by the classical generator model with constant impedance load. We assume that the transfer admittance is purely imaginary, i.e., the system is lossless, and the transmission lines are purely reactive. For the derivation of the model, refer to [3], and for its usage in a variety of analytical studies, particularly the transient stability problem, refer to [1]–[9]. The dynamics of the th generator of a total of generators can be described by the following swing equations: (2.1) (2.2) where denotes the number of generators. The parameters are as follows: inertia constant; damping constants; constant mechanical power input; angle of internal complex voltage of th machine; rotor angle velocity of the th machine with respect to the reference frequency of the power system 60 rad/s); transmission susceptances. Define the function, , as the following: (2.3) 1057–7122/97$10.00  1997 IEEE Authorized licensed use limited to: National Taiwan University. Downloaded on March 9, 2009 at 04:01 from IEEE Xplore. Restrictions apply.