  In branched transportation problems mass has to be transported from a given
initial distribution to a given final distribution, where the cost of the
transport is proportional to the transport distance, but subadditive in the
transported mass. As a consequence, mass transport is cheaper the more mass is
transported together, which leads to the emergence of hierarchically branching
transport networks. We here consider transport costs that are piecewise affine
in the transported mass with N affine segments, in which case the resulting
network can be interpreted as a street network composed of N different types of
streets. In two spatial dimensions we propose a phase field approximation of
this street network using N phase fields and a function approximating the mass
flux through the network. We prove the corresponding $\Gamma$-convergence and
show some numerical simulation results.
