Modelling inflows for water valuation
Dr. Geoffrey PritchardUniversity of Auckland / EPOC
Inflows – where it all starts
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).
CATCHMENTSCATCHMENTS
hydro generation
thermal generation
transmission consumption
reservoirs
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.
SDDP for hydro-thermal scheduling
Week 6 Week 7 Week 8
Week 6 Week 7 Week 8
min (present cost) + E[ future cost ]
s.t. (satisfy demand, etc.)
SDDP for hydro-thermal scheduling
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 won’t do.
Week 6 Week 7 Week 8
min (present cost) + E[ future cost ]
s.t. (satisfy demand, etc.) ps
s
SDDP for hydro-thermal scheduling
Inflow scenarios for a single week
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
(1/7/2014 – 7/7/2014, say)
• Each scenario has its own curve.
• Any number of scenarios, possibly with unequal probabilities.
- computationally intensive models
• Smooth seasonal variation.
- model interpretation
Scenarios by quantile regression
Serial dependence
Inflow scenarios for successive weeks should not just be sampled independently.
(Model simulated for 100 x 62 years, independent weekly inflows.)
Serial dependence affects the distribution of total inflow
over periods longer than 1 week.
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.
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.
(Model simulated for 100 x 62 years, independent weekly inflows with vif=2.2.)
With variance inflation, inflow distribution is wrong over 1 week –
but not bad over 4 months.
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 = log Xt-1 + + t (t random)
• General linear form (ideal for SDDP):
Xt = At + BtXt-1 (At, Bt random)
Serially dependent models
Models fitted to all data, shown for week beginning 2013-08-31
62 scenarios derived from regression residuals
16 scenarios fitted by quantile regression
(Model simulated for 100 x 62 years, dependent weekly inflows, general linear form.)
Serially dependent model can match inflow distribution over 1-week and 4-month timescales.
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 $1000 / MWh
Test problem: a dry winter.
• 35 weeks (2 April – 2 December)
• Initial storage 336 GWh
• Initial inflow 500 MW (~56% of average)
Solved with Doasa 2.0 (EPOC’s SDDP code).
Results – optimal strategy
Inflow modelLost load
(MW, probability)
Spill
(MW, probability)
Energy price
($/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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