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Andy Philpott Electric Power Optimization Centre (EPOC) University of Auckland ( epoc.nz )

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. EPOC version of SDDP with some differences Version 1.0 (P. and Guan, 2008)

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Andy Philpott Electric Power Optimization Centre (EPOC) University of Auckland ( epoc.nz )

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  1. 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

  2. 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 Dynamic Outer Approximation Sampling Algorithm

  3. Notation for DOASA

  4. Hydro-thermal scheduling SDDP (PSR) versus DOASA

  5. This talk should be about optimization… A Markov Chain inflow model Risk modelling example in DOASA River chain optimization DOASA Overview of this talk My next talk(?) is about benchmarking electricity markets using SDDP.

  6. Part 1 Markov chains and risk aversion (joint work with Vitor de Matos, UFSC)

  7. Electricity sector by energy supply in 2009 http://www.med.govt.nz/

  8. New Zealand electricity mix

  9. Experiments in NZ system HAW MAN WKO 9 reservoir model

  10. Inflow modelling Benmore inflows over 1981-1985 Source: [Harte and Thomson, 2007]

  11. 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

  12. Hidden Markov model with 2 climate states DRY WET p11 p26 w1 w2 w3 w4 w5 w6 INFLOWS

  13. 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

  14. Hidden Markov model with AR1 Benmore inflows in-sample test Source: [Harte and Thomson, 2007]

  15. Markov Model with 2 climate states Aim: test if we can optimize with Markov states DRY WET p11 p26 w1 w2 w3 w4 w5 w6 WET INFLOWS DRY INFLOWS

  16. Transition matrix P q 1-q 1-p p P =

  17. Markov-chain DOASA This gives a scenario tree

  18. 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

  19. Markov-chain SDDP (c.f. Mo et al 2001) P is a transition matrix for S climate states, each with inflows wti

  20. Ruszczynzki/Shapiro risk measure construction

  21. Coherent risk measure construction Two-stage version

  22. Coherent risk measure construction Multi-stage version (single Markov state)

  23. State-dependent risk aversion We can choose lambda according to Markov state lt+1(i) = 0.25, i=1, 0.75, i=2.

  24. State-dependent risk aversion “4 Lambdas” model in experiments

  25. Experiments Reservoir inflow samples drawn from 1970-2005 inflow data Each case solved with 4000 cuts Simulated with 4000 Markov Chain scenarios for 2006 inflows Nine reservoir model (+ four Markov states)

  26. Experiments Average storage trajectories

  27. Experiments Fuel and shortage cost in 200 most expensive scenarios

  28. Experiments Fuel and shortage cost in 200 least expensive scenarios

  29. Experiments Number of minzone violations

  30. Experiments Expected cost compared with least expensive policy

  31. Part 2 Mid-term scheduling of river chains (joint work with Anes Dallagi and Emmanuel Gallet at EDF)

  32. 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

  33. Case study 1 A parallel system of three reservoirs

  34. Case study 2 A cascade system of four reservoirs

  35. 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

  36. Mid-term scheduling of river chains Revenue maximization model

  37. DOASA stage problem SP(x,w(t)) Θt+1 Reservoir storage,x(t+1) Outer approximation using cutting planes V(x,w(t)) =

  38. Heuristic uses reduction to single reservoirs Convert water values into one-dimensional cuts

  39. Results for parallel system Upper bound from DOASA with 100 iterations

  40. Results for parallel system Difference in value DOASA - Heuristic policy Difference in value DOASA

  41. Results cascade system Upper bound from DOASA with 100 iterations

  42. Results: cascade system Difference in value DOASA - Heuristic policy

  43. 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

  44. Modelling head effects Piecewise linear production functions vary with volume

  45. 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.

  46. Assume that the slopes of the production functions increase linearly with reservoir volume, so Denergy = b.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).b.E[h(w)] to the marginal water value at volume x. Modelling head effects Convexification

  47. Modelling head effects: cascade system Difference in value: DOASA - Heuristic policy

  48. Modelling head effects: casade system Top reservoir volume - Heuristic policy

  49. Modelling head effects: casade system Top reservoir volume - DOASA policy

  50. FIM

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