Relaxed convex model for optimal location and sizing of DGs in DC grids using sequential quadratic programming and random hyperplane approaches

Abstract This report addresses the problem of optimal location and sizing of constant power sources (distributed generators (DGs)) in direct current (DC) networks for improving network performance in terms of voltage profiles and energy efficiency. An exact mixed-integer nonlinear programming (MINLP) method is proposed to represent this problem, considering the minimization of total power losses as the objective function. Furthermore, the power balance per node, voltage regulation limits, DG capabilities, and maximum penetration of the DG are considered as constraints. To solve the MINLP model, a convex relaxation is proposed using a Taylor series expansion, in conjunction with the transformation of the binary variables into continuous variables. The solution of the relaxed convex model is constructed using a sequential quadratic programming approach to minimize the linearization error using the Taylor series method. The solution of the relaxed convex model is used as the input for a heuristic random hyperplane method that facilitates the recovery of binary variables that solve the original MINLP model. Two DC distribution feeders, one having 21 and the other having 69 nodes, were used as test systems. Simulation results were obtained using the MATLAB/quadprog package and contrasted with the large-scale nonlinear solvers available for General algebraic modeling system (GAMS) software metaheuristic optimization approaches to demonstrate the robustness and effectiveness of our proposed methodology.