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Methods have been devised, in the study of second-order non-linear differential equations, for identifying the nature of the phase-plane trajectories, without having to find the solutions to the equations. These methods are applied to determine the transient stability limits for a power system with two machines and are shown to be equivalent to establishing the energy integrals of the system. Formulae are derived for the critical switching time taking resistance into account. The methods are then generalized for multi-machine systems both with and without an infinite busbar, and the energy integrals are given. The application of these integrals, in conjunction with step-by-step integration, to find the critical switching time for a fault is described. An example of a three-machine system solved by the conventional and by the new methods is given.
P.D. Aylett (Wed,) studied this question.