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The objective of this work is development of techniques which provide on line decision on the stability of a large interconnected power system faced by assumed or actual disturbances. Since the system is very large and nonlinear and the time scale only a few seconds at best the only hope for results which are mathematically honest, computable on line, and of sufficient accuracy lies in a set of carefully coordinated approximations founded on precise mathematical theory. Such an approach is discussed in this paper. It divides the problem into four segments in order to narrow down those parts of the system which are actively involved and then represent the critical elements by using approximate transformations and graded precision for remote elements.