We study the steady-state expansion problem for potential-based flow networks. Constructing a cost-minimal network that admits a flow satisfying the underlying physical laws is a central problem in the design of gas, hydrogen, water, and electricity infrastructures. The physical behavior of such networks is governed by nonlinear relations between arc flows and the potential differences at their endpoints.
We derive a novel class of valid inequalities for network expansion problems that include compressor arcs, which may increase the potential in their prescribed direction and are essential for transporting flow over long distances. For fixed terminal sets we further show that the corresponding separation problem can be solved in polynomial time by minimum s-t cut computations in an auxiliary graph.