Solving the natural gas flow problem using semidefinite program relaxation

Decreasing gas prices and the pressing need for fast-responding electric power generators are currently transforming natural gas networks. The intermittent operation of gas-fired power plants to balance wind energy generation introduces spatiotemporal fluctuations of increasing volumes in gas demand. At the heart of modeling, monitoring, and control of gas networks is a set of nonlinear equations relating nodal gas injections and pressures to flows over pipelines. Given gas demands at all points of the network, the gas flow task aims at finding the rest of the physical quantities involved. For a tree-like network, the problem enjoys a closed-form solution; yet solving the equations for practical meshed networks is non-trivial. This problem is posed here as a feasibility problem involving quadratic equalities and inequalities, and is further relaxed to a convex semidefinite program (SDP) minimization. Drawing parallels to the power flow problem, the relaxation is shown to be exact if the cost function is judiciously designed using a set of frequently occurring network states. Numerical tests on a Belgian network corroborate the superiority of the novel method in recovering the actual gas system state over a Newton-Raphson solver.