Stochastic Optimization in Electricity Systems

1. Stochastic Optimization in Electricity Systems Andy Philpott The University of Auckland www.esc.auckland.ac.nz/epoc

2. Electricity optimization Optimal power flow [Wood and Wollenberg, 1984,1996, Bonnans, 1997,1998] Economic dispatch [Wood and Wollenberg, 1984,1996] Unit commitment Lagrangian relaxation [Muckstadt & Koenig, 1977, Sheble & Fahd, 1994] Multi-stage SIP [Carpentier et al 1996, Takriti et al 1996, Caroe et al 1999, Romisch et al 1996-] Market models [Hobbs et al, 2001, Philpott & Schultz, 2006] Hydro-thermal scheduling Dynamic programming [Massé*, 1944, Turgeon, 1980, Read,1981] Multi-stage SP [Jacobs et al, 1995] SDDP [ Pereira & Pinto, 1991] Market models [Scott & Read, 1996, Bushnell, 2000] Capacity expansion of generation and transmission LP [Massé & Gibrat, 1957] SLP [Murphy et al, 1982] Multi-stage SP [Dantzig & Infanger,1993] Multi-stage SIP [Ahmed et al, 2006, Singh et al, 2006] Market models [Murphy & Smeers, 2005] * P. Massé, Applications des probabilités en chaîne à l’hydrologiestatistique et au jeu des réservoirsJournal de la Société de Statistique de Paris, 1944

3. Uncertainty in electricity systems System uncertainties • Long-term electricity demand (years) • Inflows to hydro-electric reservoirs (weeks/months) • Short-term electricity demand (days) • Intermittent (e.g. wind) supply (minutes/hours) • Plant and line outages (seconds/minutes) User uncertainties (various time scales) • Electricity prices • Behaviour of market participants • Government regulation

4. What to expect in this talk… • I will try to address three questions: • What stochastic programming models are being used by modellers in electricity companies? • How are they being used? • What will be the features of the next generation of models? • I will not talk about financial models in perfectly competitive markets (see previous tutorial speakers). • I will (probably) not talk about capacity expansion models. • Warning: this is not a “how-to-solve-it” tutorial.

5. Economic dispatch model

6. Uncertainty in economic dispatch • Plant and line outages (seconds/minutes) • Spinning reserve (N-1 security standard) • Uncertain demand/supply(e.g. wind) • Frequency keeping stations (small variations) • Re-dispatch (large variations) • Opportunity for stochastic programming (see Pritchard et al WIND model)

7. Unit commitment formulation

8. Stochastic unit commitment model

9. Lagrangian relaxation decouples by unit See sequence of papers by Romisch, Growe-Kuska, and others (1996 -)

10. Hydro-thermal scheduling

11. Hydro-thermal scheduling literature • Dynamic programming Massé (1944)* Turgeon (1980) Read (1981) • Multi-stage SP Jacobs et al (1995) • SDDP Pereira & Pinto (1991) • Market models Scott & Read (1996) Bushnell (2000) * P. Massé, Applications des probabilités en chaîne à l’hydrologiestatistique et au jeu des réservoirsJournal de la Société de Statistique de Paris, 1944

12. (Over-?) simplifying assumptions • Small number of reservoirs (<20) • System is centrally dispatched. • Relatively complete recourse. • Stage-wise independence of inflow process. • A convex dispatch problem in each stage.

13. p21 w1(2) w2(2) w1(1) w3(2) p11 p21 p12 w2(1) p13 w1(2) p21 w2(2) w3(1) w3(2)

14. Outer approximation

15. Outer approximation of Ct+1(y) Θ(t+1) θt+1 ≥ αt+1(k) + βt+1(k)Ty, k Reservoir storage,x(t+1)

16. Cut calculation

17. Sampling algorithm

18. w2(1) w2(2) w3(3) w1(2) w2(2) w1(1) w3(2) p11 p12 w2(1) p13 w3(1)

19. w1(1) p11 p12 w2(1) p13 w3(1)

20. w2(1) w2(2) w1(3) p21 w1(2) w2(2) w1(1) w3(2) p11 p21 w2(1) p13 w1(2) p21 w2(2) w3(1) w3(2)

21. w2(1) w2(2) w3(3) p21 w1(2) w2(2) w1(1) w3(2) p11 p21 w2(1) p13 w1(2) p21 w2(2) w3(1) w3(2)

23. HVDC line HAW MAN TPO Case study: New Zealand system

24. HAW MAN TPO A simplified network model demand N S demand

25. 2005-2006 policy simulated with historical inflow sequences

26. Computational results: NZ model • 10 reservoirs • 52 weekly stages • 30 inflow outcomes per stage • Model written in AMPL/CPLEX • Takes 100 iterations and 2 hours on a standard Windows PC to converge • Larger models have slow convergence

27. Computational results:Brazilian system • 283 hydro plants • AR-6 streamflow model • about two thousand state variables • 271 thermal plants • 219 stages • 80 sequences in the forward simulation • 30 scenarios (“openings”) for each state in the backward recursion • 7 iterations • 11 hours CPU (Pentium IV-HT 2.8 GHz 1 Gbyte RAM ) Source: Reproduced with permission of Luiz Barossa, PSR

28. Electricity pool markets • Chile (1970s) • England and Wales (1990) (NETA 2001) • Nordpool (1996) • New Zealand (1996) • Australia (1997) • Colombia, Brazil, … • Pennsylvania-New Jersey-Maryland (PJM) • New York (1999) • New England (1999) • Ontario (May 1, 2002) • Texas (ERCOT, full LMP by 2009)

29. price T1(q) quantity demand Uniform price auction (single node) price T2(q) p quantity price combined offer stack p quantity

30. p Tm(q) q Nodal dispatch-pricing formulation [pi]

31. Residual demand curve for a generator S(p) = total supply curve from other generators D(p) = demand function c(q) = cost of generating q R(q,p) = profit = qp – c(q) p Residual demand curve = D(p) – S(p) Optimal dispatch point to maximize profit q

32. A distribution of residual demand curves p e D(p) – S(p) + e (Residual demand shifted by random demand shock e ) Optimal dispatch point to maximize profit q

33. One supply curve optimizes for all demand realizations The offer curve is a “wait-and-see” solution. It is independent of the probability distribution of e

34. This doesn’t always work There is no nondecreasing offer curve passing through both points. Optimization in this case requires a risk measure. We will use the expectation of profit with respect to the probability distribution of e.

35. Monotonicity Theorem[Anderson & P, 2002] If (S-D)-1 is a log concave function of q and c(q) is convex then a single monotonic supply curve exists that maximizes profit for all realizations of e. p q

36. Define: y(q,p) = Pr [D(p)+ e – S(p) < q] = F(q + S(p) – D(p)) = Pr [an offer of (q,p) is not fully dispatched] = Pr [residual demand curve passes below (q,p)] price p q quantity The market distribution function[Anderson & P, 2002] • S(p) = supply curve from • other generators • D(p) = demand function • = random demand shock F = cdf of random shock

37. p(t) q(t) Expected profit from curve (q(t),p(t)) price quantity

38. Finding empirical y • Use small dispatch model • Aggregated demand • DC-load flow dispatch • Piecewise linear losses • Solved in ampl/cplex • Draw a sample from demand • Draw a sample from other generators offers • Solve dispatch model with different offers q • Increment the locations where dispatch occur by 1

39. Estimation of y using simulation Sampled residual demand curve Dispatch count on segment increases by 1

40. The real world • Transmission congestion gives different prices at different nodes. • Generators own plant at different nodes. • Generators in New Zealand are vertically integrated with electricity retailers, with demand at a different node. • Generators have contracts with purchasers at different nodes. • Maintenance and outages affect generation and transmission capacity.

41. Contracts A contract for differences (or hedge contract) for a quantity Q at an agreed strike price f is an agreement for one party (the contract holder) to pay the other (the contract writer) the amount Q(f-p) where p is the electricity price at an agreed node. A generator having written a contract for Q seeks to maximize E[R(q,p)] = E[qp - c(q) + Q(f-p)]

42. Generator’s real objective Owner of HLY station might want to maximize gross revenue at HLY + TOK • $35/MWh fuel cost at HLY • cost of purchases to cover retail base of • 25% at OTA • 5% at ISL • 5% at HWB accounting for hedge contracts at $50/MWh of • 250MW at OTA • 150MW at HAY • 50 MW at HWB (Numbers are illustrative only!)

43. Implementation in the real world • BOOMER code [Pritchard, 2006] • Single period/single station simulation/optimization model. • Construct discrete y on a rectangular grid. • For every grid segment record all the relevant dispatch information (e.g. nodal prices at contract nodes) • Use dynamic programming to construct a step function maximizing expected profit. • A longest path problem through acyclic directed graph, where increment on each edge is the overall profit function times the probability of being dispatched on this segment

44. Longest path gives maximum expected profit

45. without retail and contracts with retail and 450MW of contracts

46. with retail customers moved to be more remote

47. What is wrong with this model? • Single period • Competitors response not modelled • Extreme solutions: no “comfort factor” • Can be used as a benchmark for traders

48. Challenges for SP • Electricity systems have been a happy hunting ground for stochastic optimization. • What are the SP success stories in electricity? • Tractability is only part of the story – model veracity is more important. • In markets the dual problem is as important as the primal (e.g. WIND model). • Are the assumptions of the models valid e.g. perfect competition? • Are the answers simple enough to verify (e.g. by out-of-sample simulation)? • Models are used differently from their intended application.

49. The End