www.siena.edu/booker water demand, risk, and optimal reservoir storage james f. booker with...
TRANSCRIPT
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Water Demand, Risk, and Optimal Reservoir Storage
James F. Booker with contributions by John O’Neil
Siena College
Annual Conference of the University Council on Water Resources, Portland, Oregon, July 20-22, 2004
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or,
How dammed should the river be?
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Previous approaches
• Meet predetermined (inelastic) demand,
and find probability (and costs?) of failing
to “meet” the demand.
• Burness and Quirk, 1978: “The Theory of
the Dam: An Application to the Colorado
River” - uses elastic demand.
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Outline
• The basic scenario
• “Theory of the Dam”
• Fundamental intertemporal condition
• Optimal reservoir size
• Application: Colorado River
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My starting point
Think about getting the most out of a
predetermined resource --
go beyond meeting a predetermined
(inelastic) “demand” with a certain
reliability.
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“Optimal size”
maximize diversions
from a stochastic flow using
storage
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The Physical Problem
• Single stochastic inflow
• Reservoir storage upstream from use
• Loss (e.g. evaporation) is a function of
storage
• Single use below reservoir
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The Objective
Maximize the beneficial use of water over time, where marginal benefits of use in each time are defined by a demand function:
p(x) = x 1/ , where is the price elasticity of demand.
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Now do some math ...
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Solutions look like ...
(1,0.7); 5%; 1.0inflow N elasticity t inflow x Z MU ratio
66 1.05 1.30 0.1 0.77 1.05367 1.91 1.24 0.7 0.81 1.05368 1.25 1.18 0.8 0.85 1.05369 0.37 1.12 0.0 0.90 1.05370 1.96 1.30 0.7 0.77 0.8671 1.57 1.23 1.0 0.81 1.05372 1.56 1.17 1.3 0.85 1.053
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or ...
(1,0.7); 5%; 0.5inflow N elasticity t s x Z MU ratio
66 1.05 1.25 0.2 0.64 1.05367 1.91 1.22 0.9 0.67 1.05368 1.25 1.19 0.9 0.71 1.05369 0.37 1.16 0.1 0.75 1.05370 1.96 1.13 0.9 0.78 1.05371 1.57 1.10 1.3 0.83 1.05372 1.56 1.07 1.8 0.87 1.053
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or ...
t s x Z MU ratio66 1.05 1.09 1.4 10.58 1.05367 1.91 1.07 2.2 11.14 1.05368 1.25 1.06 2.3 11.73 1.05369 0.37 1.05 1.5 12.34 1.05370 1.96 1.04 2.3 12.99 1.05371 1.57 1.03 2.7 13.68 1.05372 1.56 1.02 3.1 14.40 1.053
(1,0.7); 5%; 0.2inflow N elasticity
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From here
• Generalize: An approach for an
arbitrary basin
• Application: The Colorado River Basin
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General solutions (numerical)evaporation loss = 0.05 * storage
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The Colorado
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442 year Lee Ferry Tree-Ring reconstruction; evaporation=3%
mean use
13.4013.30
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Conclusions
• Optimal reservoir storage is a function of the price elasticity
of demand, evaporation losses, and variance of inflow.
• Existing capacities may be greater than optimal given
evaporation losses.
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Future work
• Add more realistic decisionmaking: Monte Carlo
approach to future flows.
• Add more realistic inflow distributions, including
autocorrelation.
• Define more precisely “maximum” reservoir
size.
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