In this paper we produce a Γ-convergence result for a class of energies F k ε,a modeled on the Ambrosio-Tortorelli functional. For the choice k = 1 we show that F 1 ε,a Γ-converges to a branched transportation energy whose cost per unit length is a function f n−1 a depending on a parameter a > 0 and on the codimension n − 1. The limit cost f a (m) is bounded from below by 1 + m so that the limit functional controls the mass and the length of the limit object. In the limit a ↓ 0 we recover the Steiner energy. We then generalize the approach to any dimension and codimension. The limit objects are now k-currents with prescribed boundary, the limit functional controls both their masses and sizes. In the limit a ↓ 0, we recover the Plateau energy defined on k-currents, k < n. The energies F k ε,a then can be used for the numerical treatment of the k-Plateau problem.