Energy function for power systems with transmission losses: extension of the invariance principle

In many engineering and physical problems, it is very hard to find a Lyapunov function satisfying the classical version of the LaSalle´s invariance principle. This obstacle has been a great problem in the application of energetic methods to the stability analysis of power systems with more realistic models. In this work, an extension of the invariance principle, which does not require the Lyapunov function to be negative semi-definite, is used to support theoretically the proposal of a new energy function for power systems with transmission losses