  We propose the application of occupation measure theory to the classical
problem of transient stability analysis for power systems. This enables the
computation of certified inner and outer approximations for the region of
attraction of a nominal operating point. In order to determine whether a
post-disturbance point requires corrective actions to ensure stability, one
would then simply need to check the sign of a polynomial evaluated at that
point. Thus, computationally expensive dynamical simulations are only required
for post-disturbance points in the region between the inner and outer
approximations. We focus on the nonlinear swing equations but voltage dynamics
could also be included. The proposed approach is formulated as a hierarchy of
semidefinite programs stemming from an infinite-dimensional linear program in a
measure space, with a natural dual sum-of-squares perspective. On the
theoretical side, this paper lays the groundwork for exploiting the oscillatory
structure of power systems by using Hermitian (instead of real) sums-of-squares
and connects the proposed approach to recent results from algebraic geometry.
