We consider the problem of optimal reactive power compensation for theminimization of power distribution losses in a smart microgrid. We firstpropose an approximate model for the power distribution network, which allowsus to cast the problem into the class of convex quadratic, linearlyconstrained, optimization problems. We then consider the specific problem ofcommanding the microgenerators connected to the microgrid, in order to achievethe optimal injection of reactive power. For this task, we design a randomized,gossip-like optimization algorithm. We show how a distributed approach ispossible, where microgenerators need to have only a partial knowledge of theproblem parameters and of the state, and can perform only local measurements.For the proposed algorithm, we provide conditions for convergence together withan analytic characterization of the convergence speed. The analysis shows that,in radial networks, the best performance can be achieved when we commandcooperation among units that are neighbors in the electric topology. Numericalsimulations are included to validate the proposed model and to confirm theanalytic results about the performance of the proposed algorithm.
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