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Modelling inflows for SDDP. Dr. Geoffrey Pritchard University of Auckland / EPOC. Inflows – where it all starts. CATCHMENTS. thermal generation. reservoirs. hydro generation. transmission. consumption. - PowerPoint PPT Presentation
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Modelling inflows for SDDP
Dr. Geoffrey PritchardUniversity of Auckland / EPOC
Inflows – where it all starts
In hydro-dominated power systems, all modelling and evaluation depends ultimately on stochastic models of natural inflow.
CATCHMENTSCATCHMENTS
hydro generation
thermal generation
transmission consumption
reservoirs
Why models?
• Raw historical inflow sequences get us only so far.
- they can’t 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.
What does a model need?
1. Seasonal dependence.
- Everything depends what time of year it is.
Waitaki catchment (above Benmore dam) 1948-2010
2. Serial dependence.
- Weather patterns persist, increasing probability of shortage/spill.
- Typical correlation length ~ several weeks (but varying seasonally).
What does a model need?
Iterated function systems
(numerical values are only to illustrate the form of the model).
Make this a Markov process by applying randomly-chosen linear transformations, as in:
ty
x
t
t week in inflow Island South
inflow Island North
Let
chance 50% , 7.0
4.0
9.01.0
5.08.0
chance 50% , 6.0
5.0
6.07.0
4.02.0
1
1
t
t
t
t
t
t
y
x
y
x
y
x
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.
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.
1 tttt XSRX
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.
Exact mean model inflows
1 tttt XSRX
• 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 :
1 tttt msrm 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.
(Model simulated for 100 x 62 years, dependent weekly inflows, general linear form.)
Inflow distribution over 4-month timescale.
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?
Week 6 Week 7 Week 8
Structure of SDDP
Week 6 Week 7 Week 8
min (present cost) + E[ future cost ]
s.t. (satisfy demand, etc.)
Structure of SDDP
- Stage subproblem is (essentially) a linear program with discrete scenarios.
Week 6 Week 7 Week 8
min (present cost) + E[ future cost ]
s.t. (satisfy demand, etc.) ps
s
Structure of SDDP
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.