It is well-known in the power systems literature that the behavior of the transmission power system (under certain simplifying assumptions) can be used to study the post-fault dynamics of a power system and provide principled estimates on dynamic stability margins. In this paper, we study a special feature of the energy function that has previously received little attention: convexity. We prove that the energy function for structure preserving models of power systems is convex under certain reasonable conditions on phases and voltages. Beyond stability analysis, these convexity results have a number of applications, noticeably, building a provably convergent PF solver, which we discuss in detail in this paper. We also outline potential applications to reformulating Optimum Power Flow (OPF), Model Predictive Control (MPC) and identifying the most probable failure (instanton) as convex optimization problems.