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Advances in the theory of positive polynomials, semidefinite programming, and sum of squares (SOS) decomposition have provided a very promising approach to the analysis and design of systems with polynomial vector fields. Recently, an algebraic reformulation technique has been proposed for the analysis of nonpolynomial vector fields, by recasting them into rational vector fields. Sum of squares decomposition techniques can then be applied to analyze the stability of the recasted system and to infer the properties of the original, nonpolynomial systems. I will demonstrate the application of these techniques to reformulate and improve the transient stability analysis of power systems and the estimation of the region of attraction of the stable operating point. First, I will discuss the limitations inherent in traditional approaches based on the energy function method and present an alternative which alleviates these limitations based on the algorithmic construction of Lyapunov functions using the SOS methodology. Second, I will introduce an algorithm that optimizes the estimate of the region of attraction of the stable operating point and I will compare its performance to the estimate provided by the energy function method. Third, for power systems with transfer conductances, for which energy functions cannot be mathematically defined, the proposed SOS algorithm will for the first time compute Lyapunov functions, estimate the region of attraction, and provided mathematical guarantees for local asymptotic stability. Finally, I will discuss how this approach can be generalized to design state feedback controllers and to estimate the disturbance rejection qualities of the controllers. Host: Aric Hagberg, CNLS 