J Austral Math Soc Ser B 35 pp1--22, 1993.

Braess's paradox and power-law nonlinearities in networks

Bruce Calvert and Grant Keady

Abstract

We study flows in physical networks with a potential function defined over the nodes and a flow defined over the arcs. The networks have the further property that the flow on an arc a, is a given increasing function of the difference in potential between its initial and terminal node. An example is the equilibrium flow in water-supply pipe networks where the potential is the head and the Hazen-Williams rule gives the flow as a numerical factor k_a times the head difference to a power s > 0 (and s @ 0.54) In the pipe-network problem with Hazen-Williams nonlinearities, having the same s > 0 on each arc, given the consumptions and supplies, the power usage is a decreasing function of the conductivity factors k_a. There is also a converse to this. Approximately stated, it is: if every relationship between flow and head difference is not a power law, with the same s on each arc, given at least 6 pipes, one can arrange (lengths of) them so that Braess's paradox occurs, i.e. one can increase the conductivity of an individual pipe yet require more power to maintain the same consumptions.

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Authors

Bruce Calvert

Mathematics Department, University of Auckland, New Zealand.

Grant Keady

Mathematics Department, University of Western Australia, Nedlands 6009.