Constrained optimization of local sources generation in smart grids by SDP approximation

We consider a smart grid scenario including many local sources of power and a connection to a main source provided by a power utility. The power provided by each source has an entangled cost which is related to the source characteristics (e.g., the type of consumed fuel, the availability of renewable energy, the cost of energy storage, etc.). We assume that the cost is proportional to the provided active power, and aim at optimizing the use of resources, with the objective of minimizing the total cost while meeting the requirements of the loads connected to the network. A quality constraint is also included, intended for limiting the amount of reactive power exchanged with the main source. The optimization is made non trivial by both the power losses on the transmission lines (which depend on the locations where currents are injected) and the constraints related to local sources capabilities (i.e., maximum provided active power). The problem turns out to be non-convex quadratic with non-convex quadratic constraints. In order to reduce the complexity of its solution, we consider a semi definite programming relaxation. Numerical results are provided comparing the proposed approach with both the optimal solution and commonly used unoptimized approaches.