Modelling inflows for water valuation

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Modelling inflows for water valuation
Dr. Geoffrey Pritchard University of Auckland /
EPOC
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Inflows where it all starts
CATCHMENTS
thermal generation
reservoirs
hydro generation
transmission
consumption
We want inflow scenarios for use with
generation/power system models - in a form
useful for optimization. Historical inflow
sequences work for back-testing of a known
strategy - but not for optimization (will be
clairvoyant).
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Hydro-thermal scheduling

• The problem Operate a combination of hydro and
  thermal power stations
• - meeting demand, etc.
• - at least cost (fuel, shortage).
• Assume a mechanism (wholesale market, or single
  system operator) capable of solving this problem.

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SDDP for hydro-thermal scheduling
Week 7
Week 8
Week 6
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SDDP for hydro-thermal scheduling
Week 7
Week 8
Week 6
min (present cost) E future cost s.t.
(satisfy demand, etc.)
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SDDP for hydro-thermal scheduling
Week 7
Week 8
Week 6
min (present cost) E future cost s.t.
(satisfy demand, etc.)
S ps
s
The critical step requires estimating expected
cost (the expected future cost for earlier
stages) so uncertainty (from inflows) must be
modelled with discrete scenarios. - lognormal,
Pearson III, or other continuous distributions
wont do.
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Inflow scenarios for a single week
(1/7/2014 7/7/2014, say)
The historical record gives one scenario per
historical year - may be too many scenarios, or
too few - historical extreme events can recur,
but only in the identical week of the year
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Scenarios by quantile regression

• Each scenario has its own curve.
• Any number of scenarios, possibly with unequal
  probabilities.
• - computationally intensive models
• Smooth seasonal variation.
• - model interpretation

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Serial dependence
Inflow scenarios for successive weeks should not
just be sampled independently.
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Serial dependence affects the distribution of
total inflow over periods longer than 1 week.
(Model simulated for 100 x 62 years, independent
weekly inflows.)
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Variance inflation

• Keep the assumed independence of weekly inflows,
  but modify their distribution to increase its
  variance.
• Wrong in two ways, but hopefully the errors
  cancel.
• Used e.g. in SPECTRA.

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Variance inflation

• Keep the assumed independence of weekly inflows,
  but modify their distribution to increase its
  variance.
• Wrong in two ways, but hopefully the errors
  cancel.
• Used e.g. in SPECTRA.

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With variance inflation, inflow distribution is
wrong over 1 week but not bad over 4 months.
(Model simulated for 100 x 62 years, independent
weekly inflows with vif2.2.)
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Explicit serial dependence

• Inflow is a random linear (or concave) function
  of inflow (or a state variable) from previous
  stage(s)
• Xt Ft(Xt-1) (Ft
  random, i.e. scenario-dependent)
• Commonest type is log-autoregressive
• log Xt b log Xt-1 a et
  (et random)
• General linear form (ideal for SDDP)
• Xt At BtXt-1
  (At, Bt random)

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Serially dependent models
62 scenarios derived from regression residuals
16 scenarios fitted by quantile regression
Models fitted to all data, shown for week
beginning 2013-08-31
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Serially dependent model can match inflow
distribution over 1-week and 4-month timescales.
(Model simulated for 100 x 62 years, dependent
weekly inflows, general linear form.)
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A test problem

• Challenging fictional system based on Waitaki
  catchment inflows.
• Storage capacity 1000 GWh (cf. real Waitaki
  lakes 2800 GWh)
• Generation capacity 1749 MW hydro, 900 MW
  thermal
• Demand 1550 MW, constant
• Thermal fuel 50 / MWh, VOLL 1000 / MWh
• Test problem a dry winter.
• 35 weeks (2 April 2 December)
• Initial storage 336 GWh
• Initial inflow 500 MW (56 of average)
• Solved with Doasa 2.0 (EPOCs SDDP code).

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Results optimal strategy
Inflow model Lost load (MW, probability) Spill (MW, probability) Energy price (/MWh)
dependent (general linear) 9.37, 28 2.90, 6 251
independent (vif) 8.71, 23 3.21, 12 220
independent (uncorrected) 1.59, 9 0.14, 1 112
(Quantities are expected averages over full time
horizon probabilities are for any shortage/spill
within time horizon.)
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