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In a smart microgrid currents injected by distributed energy resources (DERs) and by the point of common coupling can be adapted to minimize the energy cost. Design and quality constraints usually make the problem grow fast with the number of nodes in the network. In this paper we provide a solution to the optimization problem having a significantly reduced complexity with respect to existing techniques. The efficiency of the proposed solution stems by modeling the smart microgrid as a linear network where loads are approximated as impedances. This simplification allows avoiding explicit use of power flow equations, and having a number of equation proportional to the number of DERs rather than to the total number of nodes (loads and DERs). The optimal power flow problem is then solved by a semidefinite programming (SDP) relaxation, which provides the initial point for the search of a feasible solution by a …