Natural Gas Flow Equations: Uniqueness and an MI-SOCP Solver

TL;DR

This paper proves the uniqueness of steady-state gas flow solutions in natural gas networks and introduces an MI-SOCP solver that efficiently finds these solutions without requiring prior flow direction knowledge.

Contribution

It is the first to establish solution uniqueness over the entire feasible domain and develops a relaxation-based MI-SOCP solver that is exact under certain topologies.

Findings

The gas flow equations have a unique solution if feasible.

The MI-SOCP solver reliably finds solutions without initialization.

Relaxation remains exact even outside specific conditions.

Abstract

The critical role of gas fired-plants to compensate renewable generation has increased the operational variability in natural gas networks (GN). Towards developing more reliable and efficient computational tools for GN monitoring, control, and planning, this work considers the task of solving the nonlinear equations governing steady-state flows and pressures in GNs. It is first shown that if the gas flow equations are feasible, they enjoy a unique solution. To the best of our knowledge, this is the first result proving uniqueness of the steady-state gas flow solution over the entire feasible domain of gas injections. To find this solution, we put forth a mixed-integer second-order cone program (MI-SOCP)-based solver relying on a relaxation of the gas flow equations. This relaxation is provably exact under specific network topologies. Unlike existing alternatives, the devised solver does not need proper initialization or knowing the gas flow directions beforehand, and can handle gas networks with compressors. Numerical tests on tree and meshed networks with random gas injections indicate that the relaxation is exact even when the derived conditions are not met.