  In electrical distribution grids, the constantly increasing number of power
generation devices based on renewables demands a transition from a centralized
to a distributed generation paradigm. In fact, power injection from Distributed
Energy Resources (DERs) can be selectively controlled to achieve other
objectives beyond supporting loads, such as the minimization of the power
losses along the distribution lines and the subsequent increase of the grid
hosting capacity. However, these technical achievements are only possible if
alongside electrical optimization schemes, a suitable market model is set up to
promote cooperation from the end users. In contrast with the existing
literature, where energy trading and electrical optimization of the grid are
often treated separately or the trading strategy is tailored to a specific
electrical optimization objective, in this work we consider their joint
optimization. Specifically, we present a multi-objective optimization problem
accounting for energy trading, where: 1) DERs try to maximize their profit,
resulting from selling their surplus energy, 2) the loads try to minimize their
expense, and 3) the main power supplier aims at maximizing the electrical grid
efficiency through a suitable discount policy. This optimization problem is
proved to be non convex, and an equivalent convex formulation is derived.
Centralized solutions are discussed first, and are subsequently distributed.
Numerical results to demonstrate the effectiveness of the so obtained optimal
policies are then presented.
