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

We formulate a steady-state network flow problem for nonideal 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 the uniqueness of a generalized solution for a broad class of nonideal pressure-density relations that satisfy a monotonicity property. Furthermore, we develop a Newton-Raphson algorithm for numerical solution of the steady-state problem, which is made possible by a systematic nondimensionalization 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 nondimensionalization 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 nonphysical generalized solution found by a nonlinear solver implies nonexistence of a physical solution, i.e., infeasibility of the problem. Finally, we compare pressure and flow solutions based on ideal and nonideal 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.

Automated summary

In this paper, we formulate and solve a steady-state network flow problem for nonideal gas. We present a theorem on the uniqueness of the solution for a wide range of nonideal pressure-density relations and develop a Newton-Raphson algorithm for numerical solution. The algorithm is robust and the nondimensionalization of the equations is found to be crucial for its convergence. We also compare pressure and flow solutions based on ideal and nonideal equations of state, demonstrating the importance of applying the latter in practice.