Natural Gas Flow Solvers Using Convex Relaxation

Abstract

The vast infrastructure development, gas flow (GF) 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 that require careful initialization and uniqueness has been an open question. In this context, this article considers the nonlinear steady-state version of the 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 not appear in cycles, the GF task is posed as n 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 probably exact in NGNs with nonoverlapping 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.