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Braess's paradox and power-law nonlinearities in networks

Published online by Cambridge University Press:  17 February 2009

Bruce Calvert  and
Grant Keady
Bruce Calvert
Affiliation:
Mathematics Department, University of Auckland, New Zealand
Grant Keady
Affiliation:
Mathematics Department, University of Western Australia, Nedlands 6009
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Abstract

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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 ka 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 ka. 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.

Type
Research Article
Information
The ANZIAM Journal , Volume 35 , Issue 1 , July 1993 , pp. 1 - 22
Copyright
Copyright © Australian Mathematical Society 1993

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