A convex mathematical model based on second-order cone programming (SOCP) for the optimal operation in direct current microgrids (DCMGs) with high-level penetration of renewable energies and battery energy storage systems (BESSs) is developed in this paper. The SOCP formulation allows converting the non-convex model of economic dispatch into a convex approach that guarantees the global optimum and has an easy implementation in specialized software, i.e., CVX. This conversion is accomplished by performing a mathematical relaxation to ensure the global optimum in DCMG. The SOCP model includes changeable energy purchase prices in the DCMG operation, which makes it in a suitable formulation to be implemented in real-time operation. An energy short-term forecasting model based on a receding horizon control (RHC) plus an artificial neural network (ANN) is used to forecast primary sources of renewable energy for periods of 0.5h. The proposed mathematical approach is compared to the non-convex model and semidefinite programming (SDP) in three simulation scenarios to validate its accuracy and efficiency.