Modelling%20inflows%20for%20SDDP

1
Modelling inflows for SDDP
Dr. Geoffrey Pritchard University of Auckland /
EPOC
2
Inflows where it all starts
CATCHMENTS
thermal generation
reservoirs
hydro generation
transmission
consumption
In hydro-dominated power systems, all modelling
and evaluation depends ultimately on stochastic
models of natural inflow.
3
Why models?

• Raw historical inflow sequences get us only so
  far.
• - they cant deal with situations that have
  never happened before.

• Autumn 2014
• - Mar 1620 MW
• - Apr 2280 MW
• - May 4010 MW
• Past years (if any) with this exact sequence are
  not a reliable forecast for June 2014.

4
What does a model need?
1. Seasonal dependence. - Everything depends
what time of year it is.
Waitaki catchment (above Benmore dam) 1948-2010
5
What does a model need?
2. Serial dependence. - Weather patterns
persist, increasing probability of
shortage/spill. - Typical correlation length
several weeks (but varying seasonally).
6
Iterated function systems
Let
Make this a Markov process by applying
randomly-chosen linear transformations, as in
(numerical values are only to illustrate the form
of the model).
7
(No Transcript)
8
(No Transcript)
9
(No Transcript)
10
(No Transcript)
11
(No Transcript)
12
(No Transcript)
13
(No Transcript)
14
(No Transcript)
15
(No Transcript)
16
(No Transcript)
17
IFS inflow models

• Differences from IFS applications in computer
  graphics
• Seasonal dependence
• - the image varies periodically, a
  repeating loop.
• Serial dependence
• - the order in which points are generated
  matters.

18
Single-catchment version
Model for inflow Xt in week t
- where (Rt, St) is chosen at random from a
small collection of (seasonally-varying)
scenarios. The possible (Rt, St) pairs can be
devised by quantile regression - each
scenario corresponds to a different inflow
quantile.
19
Scenario functions for the Waitaki
High-flow scenarios differ in intercept (current
rainfall). Low-flow scenarios differ mainly in
slope. Extreme scenarios have their own
dependence structure.
20
Exact mean model inflows

• We can specify the exact mean of the IFS inflow
  model.
• Inflow Xt in week t

Take averages to obtain mean inflow mt in
week t

• where (rt, st) are the averages of (Rt, St)
  across scenarios.
• Usually we know what we want mt (and mt-1) to
  be the resulting constraint on (rt, st) can be
  incorporated into the model fitting process,
  guaranteeing an unbiased model.
• Similarly variances.
• Control variates in simulation.

21
Inflow distribution over 4-month timescale.
(Model simulated for 100 x 62 years, dependent
weekly inflows, general linear form.)
22
Hydro-thermal scheduling by SDDP

• 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.
• What does the answer look like?

23
Structure of SDDP
Week 7
Week 8
Week 6
24
Structure of SDDP
Week 7
Week 8
Week 6
min (present cost) E future cost s.t.
(satisfy demand, etc.)
25
Structure of SDDP
Week 7
Week 8
Week 6
min (present cost) E future cost s.t.
(satisfy demand, etc.)
S ps
s

• Stage subproblem is (essentially) a linear
  program with discrete scenarios.

26
Why IFS for SDDP inflows?

• The SDDP stage subproblem is (essentially) a
  linear program with discrete scenarios.
• Most stochastic inflow models must be
  modified/approximated to make them fit this form,
  but ...
• the IFS inflow model already has the final
  form required to be usable in SDDP.