ons sddp workshop, august 17, 2011 slide 1 of 50 andy philpott electric power optimization centre...
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ONS SDDP Workshop, August 17, 2011 Slide 1 of 50
Andy PhilpottElectric Power Optimization Centre (EPOC)
University of Auckland(www.epoc.org.nz)
joint work with
Anes Dallagi, Emmanuel Gallet, Ziming Guan, Vitor de Matos
Recent work on DOASA
ONS SDDP Workshop, August 17, 2011 Slide 2 of 50
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
DOASA
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Notation for DOASA
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SDDP (PSR) versus DOASA
Hydro-thermal scheduling
SDDP (NZ model) DOASA
Fixed sample of N openingsin 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)
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Overview of this talk
This talk should be about optimization…
• A Markov Chain inflow model• Risk modelling example in DOASA• River chain optimization
DOASA
My next talk(?) is about benchmarking electricity markets using SDDP.
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Part 1
Markov chains and risk aversion
(joint work with Vitor de Matos, UFSC)
ONS SDDP Workshop, August 17, 2011 Slide 7 of 50http://www.med.govt.nz/
57%
20%
7%
11%4%1%
HYDRO
GAS
COAL
GEOTHERMAL
WIND
OTHER
Electricity sector by energy supply in 2009
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New Zealand electricity mix
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9 reservoir model
MAN
HAW
WKO
Experiments in NZ system
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Benmore inflows over 1981-1985
Inflow modelling
Source: [Harte and Thomson, 2007]
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• 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])
Markov-chain model
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Hidden Markov model with 2 climate states
1 2 3 4 5 6
p11 p26
DRYWET
INFLOWS
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Hidden Markov model with AR1 (Buckle, Haugh, Thomson, 2004)
Yt is log of inflowsSt a Markov Chain with 4 statesZt is an AR1 process
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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
1 2 3 4 5 6
p11 p26
DRYWET
WET INFLOWS DRY INFLOWS
Aim: test if we can optimize with Markov states
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Transition matrix P
q 1-q 1-p p
P =
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Markov-chain DOASA This gives a scenario tree
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• 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).
Markov-chain model for experiments
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Markov-chain SDDP
P is a transition matrix for S climate states, each with inflows ti
(c.f. Mo et al 2001)
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Ruszczynzki/Shapiro risk measure construction
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Coherent risk measure constructionTwo-stage version
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Multi-stage version (single Markov state)Coherent risk measure construction
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State-dependent risk aversionWe can choose lambda according to Markov state
t+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
Reservoir inflow samples drawn from 1970-2005 inflow dataEach case solved with 4000 cutsSimulated with 4000 Markov Chain scenarios for 2006 inflows
Nine reservoir model (+ four Markov states)
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Average storage trajectoriesExperiments
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ExperimentsFuel and shortage cost in 200 most expensive scenarios
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ExperimentsFuel and shortage cost in 200 least expensive scenarios
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ExperimentsNumber of minzone violations
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ExperimentsExpected 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)
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What is the problem?
Mid-term scheduling of river chains
• 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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A parallel system of three reservoirs
Case study 1
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A cascade system of four reservoirs
Case study 2
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• 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
Case studiesInitial assumptions
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Revenue maximization modelMid-term scheduling of river chains
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DOASA stage problem SP(x,(t))Outer approximation using cutting planes
Θt+1
Reservoir storage, x(t+1)
V(x,(t)) =
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xi0xi1 xi2
i0+i0 xi1
xi3
i0
i1
Heuristic uses reduction to single reservoirsConvert water values into one-dimensional cuts
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Upper bound from DOASA with 100 iterations Results for parallel system
430
435
440
445
450
455
460
0 10 20 30 40 50 60 70 80 90 100
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Difference in value DOASA
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.300 -0.200 -0.100 0.000 0.100 0.200 0.300
Difference in value DOASA - Heuristic policyResults for parallel system
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Upper bound from DOASA with 100 iterations Results cascade system
715
720
725
730
735
740
745
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
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-1 0 1 2 3 4
Difference in value DOASA - Heuristic policy
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• 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
Case studiesNew assumptions
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Modelling head effectsPiecewise linear production functions vary with volume
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Modelling head effectsA 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.
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Modelling head effectsConvexification
• Assume that the slopes of the production functions increase linearly with reservoir volume, so
energy = volume.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()] to the marginal water value at volume x.
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Modelling head effects: cascade systemDifference in value: DOASA - Heuristic policy
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Modelling head effects: casade systemTop reservoir volume - Heuristic policy
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Modelling head effects: casade systemTop reservoir volume - DOASA policy
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FIM