Resumen
This research addresses the efficient integration and sizing of flexible alternating current transmission systems (FACTS) in electrical distribution networks via a convex optimization approach. The exact mixed-integer nonlinear programming (MINLP) model associated with FACTS siting and sizing aims for the minimization of the expected annual operating costs of the network (i.e., energy losses and FACTS purchasing costs). The constraints of this problem include power equilibrium equalities, voltage regulation bounds, and device capacities, among others. Due to the power equilibrium constraints per node and period, the MINLP model is a non-convex optimization problem. To transform the exact MINLP model into a mixed-integer convex one, the approximation of the product between two variables in the complex domain is relaxed through its hyperbolic equivalent, which generates a set of convex cones. The main advantage of the proposed mixed-integer convex model is that it ensures the global optimum of the problem, even when considering objective multiplexes. Numerical simulations in the IEEE 33-, 69-, and 85-bus grids demonstrate the effectiveness and robustness of FACTS integration via the proposed convex approach in comparison with the exact solution of the MINLP model in the GAMS software as well as with combinatorial optimization algorithms (i.e., the black widow optimizer and the vortex search algorithm). All simulations were carried out in MATLAB with Yalmip optimization and the Gurobi and Mosek solvers. The simulation results show that, for a fixed operation of the FACTS devices (i.e., a VAR compensator) during the day, the annual operating costs are reduced by 12.63%, 13.97%, and 26.53% for the IEEE 33-, 69-, and 85-bus test systems, respectively, while for the operation variable, the reductions are by 14.24%, 15.79%, and 30.31%, respectively.