| The direct method was presented based on the classical model in the beginning. For convenience the power network is reduced, it just keeps the internal node. The model can be expressed as a group of ODE equations mathematically.Although the investigation about the classical model is comparatively mature, big error may exist because of the excessive briefoess.furthermore, the energy analysis of the whole topology cannot be taken into account owing to the reduction of the power network, for example, the characteristics of the loads and generators can not be added. In chapter 2, the history and its main branches of SPM are reviewed at first, then the SPM was analyzed and induced according to the relevant theories, ultimately the research field of SPM was also prospected.In chapter 3, using the Lagrange Mean Value Theorem and the LaSalle Invariance Principle the author presented a method to construct a closed hyper-ball that strictly resides in transient stability domain. This method is termed the first order upper bounding function method. The author derives the mathematical expression and computational steps of the approximated stability region based on the classical model of multi-machine power systems with uniform and non-uniform damping. Numerical examples are provided.For the purpose of improving the character and increase the precision of the first order upper bounding function method, the second order upper bounding function method is put forth in chapter 4 based on the Taylor Theorem and the LaSalle Invariance Principle, the results reveal that the method in this paper has improved the performance in conservative character in contrast to the first order upper bounding function method.The remarkable merit of these methods is its simplicity and reliability although it is conservative comparatively, and furthermore, the method avoids the difficulty of computing UEP that is essential in traditional transient energy function methods.
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