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This article presents an optimal power flow (OPF) formulation based on the branch flow model for bipolar DC grids with asymmetric loading. The proposal considers the power injection formulation of the OPF to obtain the branch flow model equations that allow solving the power flow problem in bipolar DC networks. Furthermore, a nonlinear programming (NLP) optimization model is deduced and transformed into a second-order conic programming one, which involves convex optimization and guarantees a global optimum for the relaxed model, the uniqueness of this solution, and fast convergence. Finally, an approximation based on balanced operation is proposed to simplify the OPF. Several case studies in two test systems composed of 21 and 85 nodes with a bipolar structure demonstrate that the proposal is accurate, fast, and close to the exact solution of the NLP model when compared with competitive literature reports. All optimization models were implemented in the Python language. • Bipolar DC networks can operate with balanced and unbalanced conditions as a function of the demands connected between poles. • A convex approximation can be applied to bipolar DC networks via second order cone programming. • A simplification based on expected balanced operation ca be reached to obtain a relaxed convex model with low estimation errors. • Dispersed generators and battery energy storage systems can be added to bipolar DC networks using a optimal power flow formulation. [ABSTRACT FROM AUTHOR]