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

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