Numerical Solution of the Steady-State Network Flow Equations for a Non-Ideal Gas

Abstract

We formulate a steady-state network flow problem for non-ideal gas that relates injection rates and nodal pressures in the network to flows in pipes. For this problem, we present and prove a theorem on uniqueness of generalized solution for a broad class of non-ideal pressure-density relations that satisfy a monotonicity property. Further, we develop a Newton-Raphson algorithm for numerical solution of the steady-state problem, which is made possible by a systematic non-dimensionalization of the equations. The developed algorithm has been extensively tested on benchmark instances and shown to converge robustly to a generalized solution. Previous results indicate that the steadystate network flow equations for an ideal gas are difficult to solve by the Newton-Raphson method because of its extreme sensitivity to the initial guess [1]-[4]. In contrast, we find that non-dimensionalization of the steady-state problem is key to robust convergence of the Newton-Raphson method. We identify criteria based on the uniqueness of solutions under which the existence of a non-physical generalized solution found by a non-linear solver implies non-existence of a physical solution, i.e., infeasibility of the problem. Finally, we compare pressure and flow solutions based on ideal and non-ideal equations of state to demonstrate the need to apply the latter in practice. The solver developed in this article is open-source and is made available for both the academic and research communities as well as the industry.

Optimization of Hydrogen Blending in Natural Gas Networks for Carbon Emissions Reduction

“…Using the ideal gas approximation significantly simplifies our exposition of the economic optimization presented here. The assumptions used to obtain equations (1a)-(1b) can in principle be relaxed to extend the results to the regime of non-ideal gases, which better approximates the conditions of gas transportation pipelines [15].…”

“…We non-dimensionalize the governing equations prior to solving problem ( 14) in order to avoid numerical issues [15]. In addition, we re-scale (1a) because the wave speed V in blended gas is orders of magnitude larger than other variables in the equation.…”