Maximum Throughput Problem in Dissipative Flow Networks with Application to Natural Gas Systems

TL;DR

This paper investigates the maximum throughput problem in dissipative flow networks, particularly applied to natural gas systems, establishing conditions for solution uniqueness, formulating convex optimization models, and validating results through simulations.

Contribution

It introduces a convex optimization framework for dissipative flow networks, analyzes maximum throughput with potential bounds, and applies the theory to natural gas networks with validation.

Findings

Solution uniqueness under monotonic flow functions

Convex and biconvex optimization formulations

Validation through natural gas network simulations

Abstract

We consider a dissipative flow network that obeys the standard linear nodal flow conservation, and where flows on edges are driven by potential difference between adjacent nodes. We show that in the case when the flow is a monotonically increasing function of the potential difference, solution of the network flow equations is unique and can be equivalently recast as the solution of a strictly convex optimization problem. We also analyze the maximum throughput problem on such networks seeking to maximize the amount of flow that can be delivered to the loads while satisfying bounds on the node potentials. When the dissipation function is differentiable we develop a representation of the maximum throughput problem in the form of a twice differentiable biconvex optimization problem exploiting the variational representation of the network flow equations. In the process we prove a special case of a certain monotonicity property of dissipative flow networks. When the dissipation function follows a power law with exponent greater than one, we suggest a mixed integer convex relaxation of the maximum throughput problem. Finally, we illustrate application of these general results to balanced, i.e. steady, natural gas networks also validating the theory results through simulations on a test case.