A Recursive Conic Approximation for Solving the Optimal Power Flow Problem in Bipolar Direct Current Grids

This paper proposes a recursive conic approximation methodology to deal with the optimal power flow (OPF) problem in unbalanced bipolar DC networks. The OPF problem is formulated through a nonlinear programming (NLP) representation, where the objective function corresponds to the minimization of the expected grid power losses for a particular load scenario. The NLP formulation has a non-convex structure due to the hyperbolic equality constraints that define the current injection/absorption in the constant power terminals as a function of the powers and voltages. To obtain an approximate convex model that represents the OPF problem in bipolar asymmetric distribution networks, the conic relation associated with the product of two positive variables is applied to all nodes with constant power loads. In the case of nodes with dispersed generation, a direct replacement of the voltage variables for their expected operating point is used. An iterative solution procedure is implemented in order to minimize the error introduced by the voltage linearization in the dispersed generation sources. The 21-bus grid is employed for all numerical validations. To validate the effectiveness of the proposed conic model, the power flow problem is solved, considering that the neutral wire is floating and grounded, and obtaining the same numerical results as the traditional power flow methods (successive approximations, triangular-based, and Taylor-based approaches): expected power losses of 95.4237 and 91.2701 kW, respectively. To validate the effectiveness of the proposed convex model for solving the OPF problem, three combinatorial optimization methods are implemented: the sine-cosine algorithm (SCA), the black-hole optimizer (BHO), and the vortex search algorithm (VSA). Numerical results show that the proposed convex model finds the global optimal solution with a value of 22.985 kW, followed by the VSA with a value of 22.986 kW. At the same time, the BHO and SCA are stuck in locally optimal solutions (23.066 and 23.054 kW, respectively). All simulations were carried out in a MATLAB programming environment.