The emergence of distributed energy resources can lead to congestions in distribution grids. DC distribution grids are becoming more relevant as more sources and loads connected to the low voltage grid use dc. Bipolar dc distribution grids with asymmetric loading can experience partial congestion resulting in a nodal price difference between the two polarities. In order to take into account this price difference, this paper presents an exact optimal power flow (OPF) model, formulated in terms of voltage and current. In the case of bipolar dc distribution grids, the common ground plane assumption cannot be used as current can flow in the neutral conductors as well. Moreover, loads and sources can be connected between any two nodes in the network. The proposed OPF problem formulation includes bilinear cost functions, therefore the locational marginal prices (LMP) are solved in a second step.