www.siena.edu/booker water demand, risk, and optimal reservoir storage james f. booker with...

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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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