  In this paper, we analyze minimal representations of
$(p,\boldsymbol{\alpha})$-power cones as simpler cones. We derive some new
results on the complexity of the representations, and we provide a procedure to
construct a minimal representation by means of second order cones in case
$\boldsymbol{\alpha}$ and $p$ are rational. The construction is based on the
identification of the cones with a graph, the mediated graph. Then, we develop
a mixed integer linear optimization formulation to obtain the optimal mediated
graph, and then, the minimal representation. We present the results of a series
of computational experiments in order to analyze the computational performance
of the approach, both to obtain the representation and its incorporation into a
practical conic optimization model that arises in facility location.
