ONS SDDP Workshop, August 17, 2011 Slide 1 of 50 Andy Philpott Electric Power Optimization Centre (EPOC) University of Auckland ( 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

ONS SDDP Workshop, August 17, 2011 Slide 3 of 50 Notation for DOASA

ONS SDDP Workshop, August 17, 2011 Slide 4 of 50 SDDP (PSR) versus DOASA Hydro-thermal scheduling 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 modelLow fidelity physical model Loose convergence criterionStricter convergence criterion Risk models ( None for NZ )Risk model ( Markov chain )

ONS SDDP Workshop, August 17, 2011 Slide 5 of 50 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.

ONS SDDP Workshop, August 17, 2011 Slide 6 of 50 Part 1 Markov chains and risk aversion (joint work with Vitor de Matos, UFSC)

ONS SDDP Workshop, August 17, 2011 Slide 7 of 50 Electricity sector by energy supply in 2009

ONS SDDP Workshop, August 17, 2011 Slide 8 of 50 New Zealand electricity mix

ONS SDDP Workshop, August 17, 2011 Slide 9 of 50 9 reservoir model MANHAW WKO Experiments in NZ system

ONS SDDP Workshop, August 17, 2011 Slide 10 of 50 Benmore inflows over Inflow modelling Source: [Harte and Thomson, 2007]

ONS SDDP Workshop, August 17, 2011 Slide 11 of 50 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

ONS SDDP Workshop, August 17, 2011 Slide 12 of 50 Hidden Markov model with 2 climate states 11 22 33 44 55 66 p 11 p 26 DRY WET INFLOWS

ONS SDDP Workshop, August 17, 2011 Slide 13 of 50 Hidden Markov model with AR1 (Buckle, Haugh, Thomson, 2004) Y t is log of inflows S t a Markov Chain with 4 states Z t is an AR1 process

ONS SDDP Workshop, August 17, 2011 Slide 14 of 50 Hidden Markov model with AR1 Benmore inflows in-sample test Source: [Harte and Thomson, 2007]

ONS SDDP Workshop, August 17, 2011 Slide 15 of 50 Markov Model with 2 climate states 11 22 33 44 55 66 p 11 p 26 DRY WET WET INFLOWSDRY INFLOWS Aim: test if we can optimize with Markov states

ONS SDDP Workshop, August 17, 2011 Slide 16 of 50 Transition matrix P q 1-q 1-p p P =

ONS SDDP Workshop, August 17, 2011 Slide 17 of 50 Markov-chain DOASA This gives a scenario tree

ONS SDDP Workshop, August 17, 2011 Slide 18 of 50 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

ONS SDDP Workshop, August 17, 2011 Slide 19 of 50 Markov-chain SDDP P is a transition matrix for S climate states, each with inflows  ti (c.f. Mo et al 2001)

ONS SDDP Workshop, August 17, 2011 Slide 20 of 50 Ruszczynzki/Shapiro risk measure construction

ONS SDDP Workshop, August 17, 2011 Slide 21 of 50 Coherent risk measure construction Two-stage version

ONS SDDP Workshop, August 17, 2011 Slide 22 of 50 Multi-stage version (single Markov state) Coherent risk measure construction

ONS SDDP Workshop, August 17, 2011 Slide 23 of 50 State-dependent risk aversion We can choose lambda according to Markov state t+1 (i) = 0.25, i=1, 0.75, i=2.

ONS SDDP Workshop, August 17, 2011 Slide 24 of 50 State-dependent risk aversion “4 Lambdas” model in experiments

ONS SDDP Workshop, August 17, 2011 Slide 25 of 50 Experiments Reservoir inflow samples drawn from inflow data Each case solved with 4000 cuts Simulated with 4000 Markov Chain scenarios for 2006 inflows Nine reservoir model (+ four Markov states)

ONS SDDP Workshop, August 17, 2011 Slide 26 of 50 Average storage trajectories Experiments

ONS SDDP Workshop, August 17, 2011 Slide 27 of 50 Experiments Fuel and shortage cost in 200 most expensive scenarios

ONS SDDP Workshop, August 17, 2011 Slide 28 of 50 Experiments Fuel and shortage cost in 200 least expensive scenarios

ONS SDDP Workshop, August 17, 2011 Slide 29 of 50 Experiments Number of minzone violations

ONS SDDP Workshop, August 17, 2011 Slide 30 of 50 Experiments Expected cost compared with least expensive policy

ONS SDDP Workshop, August 17, 2011 Slide 31 of 50 Part 2 Mid-term scheduling of river chains (joint work with Anes Dallagi and Emmanuel Gallet at EDF)

ONS SDDP Workshop, August 17, 2011 Slide 32 of 50 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

ONS SDDP Workshop, August 17, 2011 Slide 33 of 50 A parallel system of three reservoirs Case study 1

ONS SDDP Workshop, August 17, 2011 Slide 34 of 50 A cascade system of four reservoirs Case study 2

ONS SDDP Workshop, August 17, 2011 Slide 35 of 50 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 studies Initial assumptions

ONS SDDP Workshop, August 17, 2011 Slide 36 of 50 Revenue maximization model Mid-term scheduling of river chains

ONS SDDP Workshop, August 17, 2011 Slide 37 of 50 DOASA stage problem SP(x,  (t)) Outer approximation using cutting planes Θ t+1 Reservoir storage, x(t+1) V(x,  (t)) =

ONS SDDP Workshop, August 17, 2011 Slide 38 of 50 Heuristic uses reduction to single reservoirs Convert water values into one-dimensional cuts

ONS SDDP Workshop, August 17, 2011 Slide 39 of 50 Upper bound from DOASA with 100 iterations Results for parallel system

ONS SDDP Workshop, August 17, 2011 Slide 40 of 50 Difference in value DOASADifference in value DOASA - Heuristic policy Results for parallel system

ONS SDDP Workshop, August 17, 2011 Slide 41 of 50 Upper bound from DOASA with 100 iterations Results cascade system

ONS SDDP Workshop, August 17, 2011 Slide 42 of 50 Results: cascade system Difference in value DOASA - Heuristic policy

ONS SDDP Workshop, August 17, 2011 Slide 43 of 50 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 studies New assumptions

ONS SDDP Workshop, August 17, 2011 Slide 44 of 50 Modelling head effects Piecewise linear production functions vary with volume

ONS SDDP Workshop, August 17, 2011 Slide 45 of 50 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 Slide 46 of 50 Modelling head effects Convexification 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.

ONS SDDP Workshop, August 17, 2011 Slide 47 of 50 Modelling head effects: cascade system Difference in value: DOASA - Heuristic policy

ONS SDDP Workshop, August 17, 2011 Slide 48 of 50 Modelling head effects: casade system Top reservoir volume - Heuristic policy

ONS SDDP Workshop, August 17, 2011 Slide 49 of 50 Modelling head effects: casade system Top reservoir volume - DOASA policy

ONS SDDP Workshop, August 17, 2011 Slide 50 of 50 FIM