Recent work on DOASA
Andy Philpott
Electric Power Optimization Centre (EPOC)
University of Auckland
(www.epoc.org.nz)
joint work with
Anes Dallagi, Emmanuel Gallet, Ziming Guan, Vitor de Matos
ONS SDDP Workshop, August 17, 2011
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DOASA
Dynamic Outer Approximation Sampling Algorithm
• EPOC version of SDDP with some differences
• Version 1.0 (P. and Guan, 2008)
–
–
–
–
Written in AMPL/Cplex
Very flexible
Used in NZ dairy production/inventory problems
Takes 8 hours for 200 cuts on NZEM problem
• Version 2.0 (P. and de Matos, 2010)
– Written in C++/Cplex with NZEM focus
– Adaptive dynamic risk aversion
– Takes 8 hours for 5000 cuts on NZEM problem
ONS SDDP Workshop, August 17, 2011
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Notation for DOASA
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Hydro-thermal scheduling
SDDP (PSR) versus DOASA
SDDP (NZ model)
DOASA
Fixed sample of N openings
in each stage. Solves all.
Fixed sample of N openings in
each stage. Solves all.
Fixed sample of forward pass
scenarios (50 or 200)
Resamples forward pass
scenarios (1 at a time)
High fidelity physical model
Low fidelity physical model
Loose convergence criterion
Stricter convergence criterion
Risk models (None for NZ)
Risk model (Markov chain)
ONS SDDP Workshop, August 17, 2011
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DOASA
Overview of this talk
This talk should be about optimization…
• A Markov Chain inflow model
• Risk modelling example in DOASA
• River chain optimization
My next talk(?) is about benchmarking
electricity markets using SDDP.
ONS SDDP Workshop, August 17, 2011
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Part 1
Markov chains and risk aversion
(joint work with Vitor de Matos, UFSC)
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Electricity sector by energy supply in 2009
4%1%
11%
7%
57%
20%
ONS SDDP Workshop, August 17, 2011
HYDRO
GAS
COAL
GEOTHERMAL
WIND
OTHER
http://www.med.govt.nz/
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New Zealand electricity mix
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Experiments in NZ system
9 reservoir model
WKO
MAN
HAW
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Inflow modelling
Benmore inflows over 1981-1985
Source: [Harte and Thomson, 2007]
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Markov-chain model
• DOASA model assumes stagewise
independence
• SDDP models use PAR(p) models.
• NZ reservoir inflows display regime jumps.
• Can model this using “Hidden Markov
models” ( [Baum et al, 1966])
ONS SDDP Workshop, August 17, 2011
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Hidden Markov model with 2 climate states
DRY
WET
p11
w1
p26
w2
w3
w4
w5
w6
INFLOWS
ONS SDDP Workshop, August 17, 2011
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Hidden Markov model with AR1
(Buckle, Haugh, Thomson, 2004)
Yt is log of inflows
St a Markov Chain with 4 states
Zt is an AR1 process
ONS SDDP Workshop, August 17, 2011
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Hidden Markov model with AR1
Benmore inflows in-sample test
Source: [Harte and Thomson, 2007]
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Markov Model with 2 climate states
Aim: test if we can optimize with Markov states
DRY
WET
p11
w1
p26
w2
w3
WET INFLOWS
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w4
w5
w6
DRY INFLOWS
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Transition matrix P
P=
ONS SDDP Workshop, August 17, 2011
q
1-p
1-q
p
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Markov-chain DOASA
This gives a scenario tree
ONS SDDP Workshop, August 17, 2011
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Markov-chain model for experiments
• Climate state for each island in New Zealand
(W or D)
• State space is (WW, DW, WD, DD).
• Assume state is known.
• Sampled inflows are drawn from historical record
corresponding to climate state e.g. WW.
• Record a set of cutting planes for each state.
• Report experiments with a 4-state model:
– (WW, DW, WD, DD).
ONS SDDP Workshop, August 17, 2011
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Markov-chain SDDP
(c.f. Mo et al 2001)
P is a transition matrix for S climate states, each with inflows wti
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Ruszczynzki/Shapiro risk measure construction
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Coherent risk measure construction
Two-stage version
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Coherent risk measure construction
Multi-stage version (single Markov state)
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State-dependent risk aversion
We can choose lambda according to Markov state
lt+1(i) =
0.25, i=1,
0.75, i=2.
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State-dependent risk aversion
“4 Lambdas” model in experiments
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Experiments
Nine reservoir model (+ four Markov states)
Reservoir inflow samples drawn from 1970-2005 inflow data
Each case solved with 4000 cuts
Simulated with 4000 Markov Chain scenarios for 2006 inflows
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Experiments
Average storage trajectories
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Experiments
Fuel and shortage cost in 200 most expensive scenarios
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Experiments
Fuel and shortage cost in 200 least expensive scenarios
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Experiments
Number of minzone violations
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Experiments
Expected cost compared with least expensive policy
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Part 2
Mid-term scheduling of river chains
(joint work with Anes Dallagi and Emmanuel Gallet at EDF)
ONS SDDP Workshop, August 17, 2011
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Mid-term scheduling of river chains
What is the problem?
• EDF mid-term model gives system
marginal price scenarios from
decomposition model.
• Given price scenarios and uncertain
inflows how should we schedule each
river chain over 12 months?
• Test SDDP against a reservoir
aggregation heuristic
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Case study 1
A parallel system of three reservoirs
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Case study 2
A cascade system of four reservoirs
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Case studies
Initial assumptions
• weekly stages t=1,2,…,52
• no head effects
• linear turbine curves
• reservoir bounds are 0 and capacity
• full plant availability
• known price sequence, 21 per stage
• stagewise independent inflows
• 41 inflow outcomes per stage
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Mid-term scheduling of river chains
Revenue maximization model
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DOASA stage problem SP(x,w(t))
Outer approximation using cutting planes
V(x,w(t)) =
Θt+1
Reservoir storage, x(t+1)
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Heuristic uses reduction to single reservoirs
Convert water values into one-dimensional cuts
xi0
xi1
xi2
xi3
i0+i0 xi1
i1
i0
ONS SDDP Workshop, August 17, 2011
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Results for parallel system
Upper bound from DOASA with 100 iterations
460
455
450
445
440
435
430
0
10
20
30
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40
50
60
70
80
90
100
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Results for parallel system
Difference in value DOASA - Heuristic policy
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
-0.300
-0.200
-0.100
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0
0.000
0.100
0.200
0.300
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Results cascade system
Upper bound from DOASA with 100 iterations
745
740
735
730
725
720
715
1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96
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Results: cascade system
Difference in value DOASA - Heuristic policy
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-1
0
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1
2
3
4
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Case studies
New assumptions
• weekly stages t=1,2,…,52
• include head effects
• nonlinear production functions
• reservoir bounds are 0 and capacity
• full plant availability
• known price sequence, 21 per stage
• stagewise independent inflows
• 41 inflow outcomes per stage
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Modelling head effects
Piecewise linear production functions vary with volume
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Modelling head effects
A major problem for DOASA?
• For cutting plane method we need the future cost
to be a convex function of reservoir volume.
• So the marginal value of more water is
decreasing with volume.
• With head effect water is more efficiently used
the more we have, so marginal value of water
might increase, losing convexity.
• We assume that in the worst case, head effects
make the marginal value of water constant at
high reservoir levels.
• If this is not true then we have essentially
convexified C at high values of x.
ONS SDDP Workshop, August 17, 2011
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Modelling head effects
Convexification
• Assume that the slopes of the production
functions increase linearly with reservoir
volume, so
Denergy = .Dvolume.flow
• In the stage problem, the marginal value of
increasing reservoir volume at the start of
the week is from the future cost savings (as
before) plus the marginal extra revenue we
get in the current stage from more efficient
generation.
• So we add a term p(t)..E[h(w)] to the
marginal water value at volume x.
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Modelling head effects: cascade system
Difference in value: DOASA - Heuristic policy
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Modelling head effects: casade system
Top reservoir volume - Heuristic policy
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Modelling head effects: casade system
Top reservoir volume - DOASA policy
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FIM
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