# Natural Gas Flow Solvers using Convex Relaxation

## TL;DR

This paper introduces a convex relaxation-based solver for steady-state natural gas flow problems, proving its exactness under certain conditions and demonstrating its scalability and robustness through numerical tests.

## Contribution

It develops a novel convex relaxation and a mixed-integer quadratic programming solver for natural gas flow equations, ensuring solution uniqueness and broad applicability.

## Key findings

- The solution to the gas flow problem is unique under arbitrary network configurations.
- The convex relaxation is exact in networks with non-overlapping cycles and a single reference node.
- The MI-QCQP solver scales well and remains accurate even when sufficient conditions are not met.

## Abstract

The vast infrastructure development, gas flow dynamics, and complex interdependence of gas with electric power networks call for advanced computational tools. Solving the equations relating gas injections to pressures and pipeline flows lies at the heart of natural gas network (NGN) operation, yet existing solvers require careful initialization and uniqueness has been an open question. In this context, this work considers the nonlinear steady-state version of the gas flow (GF) problem. It first establishes that the solution to the GF problem is unique under arbitrary NGN topologies, compressor types, and sets of specifications. For GF setups where pressure is specified on a single (reference) node and compressors do no appear in cycles, the GF task is posed as an convex minimization. To handle more general setups, a GF solver relying on a mixed-integer quadratically-constrained quadratic program (MI-QCQP) is also devised. This solver can be used for any GF setup at any NGN. It introduces binary variables to capture flow directions; relaxes the pressure drop equations to quadratic inequality constraints; and uses a carefully selected objective to promote the exactness of this relaxation. The relaxation is provably exact in NGNs with non-overlapping cycles and a single fixed-pressure node. The solver handles efficiently the involved bilinear terms through McCormick linearization. Numerical tests validate our claims, demonstrate that the MI-QCQP solver scales well, and that the relaxation is exact even when the sufficient conditions are violated, such as in NGNs with overlapping cycles and multiple fixed-pressure nodes.