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

TL;DR

This paper develops a robust numerical method for solving steady-state network flow equations for non-ideal gases, proving solution uniqueness and demonstrating practical advantages over ideal gas models.

Contribution

It introduces a non-dimensionalization approach and a Newton-Raphson algorithm for non-ideal gas network flows, ensuring convergence and solution validity.

Findings

The algorithm converges robustly on benchmark instances.

Non-dimensionalization improves Newton-Raphson convergence.

Non-ideal gas models are necessary for accurate pressure-flow solutions.

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 steady-state 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. 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.