  We formulate the optimal flow problem in a multi-area integrated electrical
and gas system as a mixed-integer optimization problem by approximating the
non-linear gas flows with piece-wise affine functions, thus resulting in a set
of mixed-integer linear constraints. For its solution, we propose a novel
algorithm that consists in one stage for solving a convexified problem and a
second stage for recovering a mixed-integer solution. The latter exploits the
gas flow model and requires solving a linear program. We provide an optimality
certificate for the computed solution and show the advantages of our algorithm
compared with respect to the state-of-the-art method via numerical simulations.
