Abstract –– In this dissertation a new advancement in the development of time-varying multi-area representations of large-scale complex systems is presented. The new approach combines on-line trajectory clustering techniques and Linear OMIB theory and can be used to prove the asymptotically stable condition. Inspired by the idea of Time varying OMIB response a linear technique for transient stability assessment of stressed power systems involving complex inter-area oscillations are developed. Keywords –– Equal-area criterion, OMIB equivalent, Transient stability assessment, Linear Stability Theory, Phase Plane Analysis. I. INTRODUCTION Over the last few years, an energy loss studies have been developed and applied to large-scale power system models for transient stability assessment and control with varied success [1]. Much of the early work on energy loss identification for transient stability analysis was based on the use of Inter-Area Mode between relative rotor speed Time Varying OMIB deviations of generators loosing stability and the subsequent transformation to a simple system involving one or two machines [2] , [3] and [4]. The stability analysis of large power system models can require considerable computational effort, and can result in inaccurate estimates of the proximity of the system to an unstable condition. In this paper a method for determining phase plane of Linear OMIB system is suggested [1]. In this case the linear system approximation is considered. This process provide the stability information about the OMIB model [5]. Using this method analytical estimates of the critical conditions are then derived that overcome many of the existing approaches. Various properties of the model such as the multi-swing instability conditions, and the nature and character of system oscillations in the aftermath of a critical contingency are established. Physical and mathematical conditions for the existence and stability of the limiting operating conditions are also given and simplifications of the model are analyzed The OMIB equivalent is based on the observation of the separation of the multimachine power system originates from the irrevocable separation into a critical cluster and the non critical cluster [6]. In this work a Time-varying one-machine equivalents (TV-OMIB) identify a linear-order dynamic model of the system that preserves the essential dynamics of the system. The basic theory is summarized here for completeness (details of this type of models are given in [5]). The development of time-varying based OMIB equivalent was motivated by several factors:  Reduced-order dynamic descriptions that are inherently capable of capturing linear structure are of great practical importance in an increasing variety of power system applications. The most successful algorithms for dynamic reduction of large-scale power system models have been OMIB-based approaches  Most existing approaches are based on rather heuristic considerations often resulting from generalizations of the equal-area criterion. This limits their applicability to the study of first and second swing stability. The generalization of these approaches to multi-swing stability analysis is difficult and may result in large errors  Recently, on line dynamic security tools have been developed to examine the nature of system behavior after critical contingencies. Time-varying OMIB system has been successfully applied to determine the mode of instability The need for improved modeling and analysis procedures in the study of transient stability analysis has been recently pointed out by several investigators [7] and [8]. Various approximations have been utilized in the modeling of weakly nonlinear systems. The method is particularly attractive for the study of linear effects arising from the series expansion of the original nonlinear power system representation and is amenable to computer implementation. More recent approaches for transient stability characterization have focused on the use of exact differential analysis. Linear system analysis has been very successfully applied to particular classes of system, and problems contraction analysis was recently proposed to as a new time- frequency analysis tool for non-stationary signal [1] and [9]. The proposed technique is applied to analyze a full- scale stressed power system that exhibits complex inter-area mode separation following major contingencies Analysis of Power System Stability using Phase Plane Analysis of Linear OMIB Equivalents C. Juarez 1 and I. Martinez 1 1 Professional Academy of UAEMex Tianguistenco, Mexico Phone (01 717) 132 46 00 E-mail: toledocarlos@hotmail.com