Applications of Stochastic Programming in the Energy Industry

1. Applications of Stochastic Programming in the Energy Industry Chonawee Supatgiat Research Group Enron Corp. INFORMS Houston Chapter Meeting August 2, 2001

2. Outline • Stochastic Program • Colombia Hydro-thermal System • Model • Solution techniques • Nested Decomposition • Abridged Nested Decomposition • Example result • Fuel Inventory and Electric Generation • Model • Solution techniques • Bender Decomposition • Lagrangian Relaxation • Example result

3. x y x Stochastic Program • Mathematical program where some of the data incorporated into the objective or constraints is uncertain • Recourse program: some decisions or recourse actions can be taken after uncertainty is disclosed

4. Two-Stage Stochastic Linear Program Stage 2 Stage 1 x y x

5. Extensive Form y(x1) x1 x x2 y(x2) x3 y(x3)

6. Multi-Stage Stochastic Linear Program • QN+1(xN) = 0, for all xN

7. Application 1Colombia Long-Term Power Planning(joint work with John Birge, Northwestern University)

8. Power System Area 2 Area 1 • Colombia System: • 8 areas • 47 hydro units • 70 thermal units • 28 fuel types • hydro generation ~50% Area 3

11. Problem Central Dispatch Planning Problem • Colombia government need to determine capacity payment • Socially optimal dispatch planning of Colombia hydro-thermal generating units • Decision: in each period, water release (hydro) and generation (thermal), and export/import power flow between areas • Stochastic water inflow in each location

12. Model for Colombia Power • Objective: • minimize total generation costs + outage penalty + alert water level penalty • subject to the operating constraints: • meet load constraints • water balance constraints • thermal capacity constraints • hydro maximum/minimum flow constraints • export/import capacity constraints • minimum/maximum reservoir level

13. Stage 2 Stage 3 Stage 1 Inflow Scenario Tree: Serial Correlation

14. Multi-Stage Stochastic Linear Program • QN+1(xN) = 0, for all xN

15. Solution Technique: Nested Decomposition Basic Idea • Solve each subproblem separately • xt is passed forward to the Stage t+1 subproblem • Function Qt is passed backward to Stage t-1 subproblem • Updating x and functions Q until converge Q3(x2,) Q2 (x2,) Q3(x2,) x2 x1

16. xt Feasibility when Passing Forward • If xt* from Stage t subproblem makes a subproblem at Stage t+1 infeasible • Stage t+1 subproblem sends a message to Stage t that this xt* is a bad solution

17. Qt(xt-1) x Nested Decomposition • Forward Pass: • Starting at the root node and proceeding forward through the scenario tree, solve each node subproblem • Add feasibility cuts as infeasibilities arise • Backward Pass • Starting in top node of Stage t = N-1, use optimal dual values in descendant Stage t+1 nodes to construct new optimality cut. Repeat for all nodes in Stage t, resolve all Stage t nodes, then tt-1. • Repeat until converge

18. Nested Decomposition (ND) v.s. Dynamic Programming (DP) • DP: • starting from the last nodes and evaluating Q for all possible values of x • move backward when get complete information of Q • ND: • evaluating Q only for one value of x in each iteration • move forward to generate new value of x Q Qt(xt-1) x

19. Abridged Nested Decomposition • Incorporates sampling into the general framework of Nested Decomposition • Assumes • relatively complete recourse: a feasible solution exists for every feasible solution in the previous stage • serial independence: the stochastic data in each stage is independent of the realized values in the previous stages • Samples both the subproblems to solve and the solutions to continue from in the forward pass

20. Abridged Nested Decomposition Forward Pass 1. Solve root node subproblem 2. Sample Stage 2 subproblems and solve selected subset 3. Sample Stage 2 subproblem solutions and branch in Stage 3 only from selected subset (i.e., nodes 1 and 2) 4. For each selected Stage t-1 subproblem solution, sample Stage t subproblems and solve selected subset 5. Sample Stage t subproblem solutions and branch in Stage t+1 only from selected subset

21. Abridged Nested Decomposition Backward Pass Starting in first branching node of Stage t = N-1, solve all Stage t+1 descendant nodes and construct new optimality cut for all stage t subproblems. Repeat for all sampled nodes in Stage t, then repeat for t = t - 1

22. NDUM and CPLEX v. No. of Scenarios 19.4 hrs 2.8 hrs

23. Example Results (selected plants) MWh

24. Example: Hydro Generation

25. Example: Thermal Generation

26. Example: Dual Prices

27. Example: Max MW (selected plants)

28. Application 2Energy Marketer with Gas Storage and Generators(joint work with Samer Takriti and Lilian Wu, IBM Research)

29. Coordinating Fuel Inventory with Electricity Generation Energy marketer Gas-Turbine generation plants Natural gas market Gas storage buy fuel sell gas demand sell power Power market Gas customers

30. Properties of Gas Turbine Generators • Minimum up time • Minimum down time • Start up cost • Quadratic gas consumption gas consumption = a + b (generation) + c (generation)2 • Operating level • Sell power at market (bid) price

31. Gas Storage • Buy gas at market ask price, sell gas at market bid price • Storage holding cost • Inject/withdraw fees • Inject/withdraw limits • Storage capacity • Gas loss

32. Randomness and Decisions • Have multiple forecasts for natural gas demand, natural gas prices, electricity prices • Observe spot gas (bid/ask) prices, spot electricity (bid) price, and current gas demand • State: current gas storage level, status of the electric generators • Decide on the amount to buy/sell natural gas and the electricity generation

33. Scenario Tree p, g, d p, g, d p, g, d p, g, d p, g, d p, g, d p, g, d p, g, d p, g, d p, g, d p, g, d p, g, d p, g, d

34. Full Model • Large stochastic mixed integer program Maxtotal discounted expected future revenue from gas and power selling minus future operation costs from gas storage and generators and gas buying cost S.t. Minimum up/down time constraints (integer) Min/max power generation levels (conditional) Gas to power conversion equation (integer & possibly non-linear) Gas inventory balance constraints Gas storage capacity Gas injection/withdraw capacity

35. Bender Decomposition Maxtotal discounted expected future revenue from gas and power selling minus future operation costs from gas storage and generators and gas buying cost S.t. Minimum up/down time constraints (integer) Min/max power generation levels (conditional) Gas to power conversion equation (integer & non-linear) Gas inventory balance constraints Gas storage capacity Gas injection/withdraw capacity

36. Bender Decomposition • First Stage: Unit commitment problem: stochastic integer program • Second Stage: Inventory problem: stochastic linear program integer conditional Integer & non-linear Bender cuts To be solved by simple LP

37. Gen 1 Gen 2 Gen 3 Gen 1 Gen 2 Gen 3 Unit Commitment Problem • Lagrangian Relaxation of the Bender cuts Max L(l), l >= 0

38. Lagrangian Relaxation of Unit Commitment Problem • Max SiLi(l)+pbl l >= 0 • where Vector of avg. gas price of generator i in each node Individual generator problem to be solved by stochastic DP

39. Solution Technique Summary • Pure B&B • Use OSL B&B to solve the full problem in one shot • Bender + B&B • Decompose into two stages • Solve first stage by B&B and second stage by OSL LP • Bender + Lagrangian • Decompose into two stages • Relax the Bender cuts and decompose the first stage. Solve its sub-problems by DP • Solve the second stage by OSL LP

40. Numbers of Bender’s cuts needed are between 7-38