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We consider general, steady, balanced flows of a commodity over a networkwhere an instance of the network flow is characterized by edge flows and nodalpotentials. Edge flows in and out of a node are assumed to be conserved, thusrepresenting standard network flow relations. The remaining freedom in the flowdistribution over the network is constrained by potentials so that thedifference of potentials at the head and the tail of an edge is expressed as anonlinear function of the edge flow. We consider networks with nodes dividedinto three categories: sources that inject flows into the network for a certaincost, terminals which buy the flow at a fixed price and "internal" customerseach withdrawing an uncertain amount of flow, which has a priority and thus itis not priced. Our aim is to operate the network such that the profit, i.e.amount of flow sold to terminals minus cost of injection, is maximized, whilemaintaining the potentials within prescribed bounds. We also require that theoperating point is robust with respect to the uncertainty of customers'withdrawals. In this setting we prove that potentials are monotonic functionsof the withdrawals. This observation enables us to replace in the maximumprofit optimization infinitely many nodal constraints, each representing aparticular value of withdrawal uncertainty, by only two constraintsrepresenting the cases where all nodes with uncertainty consume their minimumand maximum amounts respectively. We illustrate this general result on exampleof the natural gas transmission network. In this enabling example gaswithdrawals by consumers are assumed uncertain, the potentials are gaspressures squared, the potential drop functions are bilinear in the flow andits intensity with an added tunable factor representing compression.

Monotonicity of Dissipative Flow Networks Renders Robust Maximum Profit Problem Tractable: General Analysis and Application to Natural Gas Flows