1. 1 Optimization Based Power Generation Scheduling Xiaohong Guan Tsinghua / Xian Jiaotong University

2. 2 In this talk Introduction to Power Generation Scheduling  Motivations and background  Difficulties  Current approaches Problem Formulation Solution Based on Lagrangian Relaxation Numerical Testing Results Issues of Homogenous Units and Resolution

3. 3 Power Generation Scheduling (Unit commitment)  Background and Motivations  Many generating units in a power system connected through transmission network to supply demand  Scheduling unit on/off and generation levels to meet the system demand, reserve and individual constraints  Minimizing the total generation cost  Potential for significant cost savings: over 10 millions of US dollars per year for a large generation company  Hot research topics for decades

4. 4  Difficulties  Complicated discrete and continuous and dynamic constraints of individual units  hybrid dynamics and constraints  System wide constraints coupling the operation of individual units  NP-hard mixed integer programming problem: extremely difficult to obtain the optimal schedule  Integrated with bidding problems in the market environment Power Generation Scheduling (Unit commitment)

5. 5  Current approaches  Priority list and other heuristics  little control on schedule quality  Enumeration such as branch and bound  Dynamic programming  Bender’s decomposition Problem of computational complexity  Lagrangian relaxation  our approach Power Generation Scheduling (Unit commitment)

6. 6 Lagrangian Relaxation Based Scheduling Algorithms Relax system wide constraints to form a two level optimization problems Solve low level individual subproblems with much less efforts Update Lagrange multipliers at the high level Modify dual solution into a feasible schedule Quantitative estimate of solution quality since the dual cost is the lower bound of the primal cost

7. 7 ’’  Dual Cost Primal Cost Duality Gap and Solution Quality

8. 8 In this talk Introduction to Power Generation Scheduling  Motivations and background  Difficulties  Current approaches Problem Formulation Solution Based on Lagrangian Relaxation Numerical Testing Results Issues of Homogenous Units and Resolution

9. 9 Problem Formulation Objective function Subject to C, with C = –System demand System wide constraints:

10. –Reserve requirements generation level reserve contribution generation range minimum generation generation capacity

11. Individual unit constraints (thermal units): –Discrete state transitions if discrete decision variable of unit i at time t, “1” for up, “-1” for down

12. – Operating regions of thermal units –Minimum up/down time if

13. Ramping constraint  continuous dynamics feasible region of t+1 minimum generation maximum generation t

14. – Operating regions of hydro units –Reservoir level limit or –Initial and terminal reservoir levels –Reservoir dynamics Individual unit constraints (hydro units):

15. –Hydraulic coupling

16. 16 In this talk Introduction to Power Generation Scheduling  Motivations and background  Difficulties  Current approaches Problem Formulation Solution Based on Lagrangian Relaxation Numerical Testing Results Issues of Homogenous Units and Resolution

17. 17 Lagrangian Relaxation Lagrangian function

18. Lagrangian relaxation framework Update Multipliers 18 Subproblems for thermal units Subproblems for hydro units Subproblems for other special units Obtain feasible schedule Multipliers Generation levels

19. 19 Solve Thermal Subproblems Objective function Individual constraints of thermal units –Discrete dynamic state transactions –Minimum down/up times –Discontinuous operating regions Major method: Dynamic Programming Min L i with

20. Optimal generation at a particular hour (ignoring ramping constraints) Stage-wise cost p ti (t) p* ti (t) total stage-wise cost generation cost c ti (p ti (t)) shadow cost - (t) p ti (t)-  (t) p ti (t) Obtain optimal stage-wise cost and generation by optimizing a single variable function !

21. Typical start up cost functions Start up cost S i (x ti (t)) x ti (t) exponential Saturate linear Minimum down time Cold start time

22. Optimal states across time obtained with efficiently with only a few states and transitions States t t+1 Up min time Up one hour Up two hour Down one hour Down two hour Down min down time Down cold start time generation costs c ti (p ti (t)) Start up costs S ti (x ti (t))

23. 23 Dealing with ramping constraints Difficulties of ramping constraints  Ramping couples generation levels across time  Continuous dynamics  The optimal generation of an “Up State” can no longer be a single point  Dynamic programming on discrete states can no longer be applied straightforwardly

24. Approach to resolve the issue Heuristics Discretizing generation levels  greatly increasing the number of states and computational efforts Relaxation of ramping constraints  three level optimization structure Constructive dynamic programming for continuous optimal generation level and regular dynamic programming for optimal discrete states best algorithm so far Double dynamic programming method for solving subproblems with ramping constraints  best algorithm so far

25. Ideas of constructive dynamic programming  Optimal generation level can only be at the corner points of the cost function or the active points of the ramping constraints  The optimal generation levels of the previous or next stage w.r.t the above points can be mapped across time systematically  The possible optimal generation levels are constructed backwardly without discretization

26. tt+1 mapping of optimal generation levels The number of states would increase but not significantly The method is efficient

27.  Redefine the discrete state as an “up” or “running” cycle  Apply constructive dynamic programming to obtain optimal general levels and cost for all running cycles  Apply dynamic programming to obtain the optimal cycle

28. U i : the number of hours before the unit committed (up) for the ith time, Di: as the number of hours before the unit decommitted (down) for the ith time.

29. 29 Solve Hydro Subproblems (including Pumped Storage) Objective function Individual constraints of hydro units –Water balance –Reservoir levels –Discontinuous operating regions –Discrete operating constraints such as minimum down times Major difficulties –Hydraulic coupling integrated with discontinuous operating regions and discrete dynamic constraints min L j, with

30. Method 1: Network flow optimization  Ignore discontinuous operating regions and discrete operating constraints  Apply minimum cost flow optimization to schedule generation levels with water balance and reservoir level constraints  Meet other constraint by heuristics  Two reservoir example: w1(t)w1(t) w2(t)w2(t) v1(t)v1(t) v2(t)v2(t) t t+1 v1(·)v1(·) v2(·)v2(·) -w 1 (t) w1(t)w1(t) -w 2 (t)

31. Method 2: Relaxation of reservoir level constraints  Substitute continuous hydro dynamics and relax limits on reservoir levels  Solve subproblems w.r.t. individual hydro units using dynamic programming similar to thermal subproblems  Apply minimum cost flow optimization with fixed discrete states to schedule generation levels to meet water balance and reservoir level constraints

32. Method 3: General mixed integer programming  Solve the hydro subproblems as a mixed integer programming problem using solver such as CPLEX  Efficiency closely related to the problem formulation

33. 33 Solve High Level Dual Problem Subgradient with  ( (t),  (t)) =

34. 34 Updating Multipliers The multipliers are updated using an efficient subgradient algorithm –Adaptive step sizing –Good initial multipliers using priority list scheduling

35. 35 Over generation  Price down Time Total generation Under generation  price up System demand Updating Multipliers

36. 36 Obtaining Feasible Schedules Goal: to satisfy once relaxed system demand and reserve requirement constraints Heuristics should be applied If possible, satisfy these constraints by adjusting generation levels only  economic dispatch –For piece-wise linear cost function, sorting all power blocks of all scheduled “up” units and piling these blocks up till the the system wide constraints satisfied

37. Adjust discrete operating (commitment) states. Calculate the “opportunity cost” of state change based on the state transition and cost-to-go information in the dual solution for all units Adjust the commitment state of a unit with the smallest cost increase Repeat if sufficient and necessary feasibility conditions not satisfied

38. 38 In this talk Introduction to Power Generation Scheduling  Motivations and background  Difficulties  Current approaches Problem Formulation Solution Based on Lagrangian Relaxation Numerical Testing Results Issues of Homogenous Units and Resolution

39. Numerical results  Based on Northeast Utilities system with 70 thermal units, 7 hydro units and 1 large pumped storage unit Data setFeb W2, 1991Feb W3, 1991 CPU time (s)408 # of iterations5040 Best feasible cost ($) 4,617,4647,883,044 Max dual cost ($)4,595,0557,831,425 Duality gap0.47%0.66 %

40.  Consistent convergence and near optimal schedules obtained  Only a few seconds on P-IV computer  Significant cost saving in comparison with the schedules by NU engineers  Production use for many years

41. 41 In this talk Introduction to Power Generation Scheduling  Motivations and background  Difficulties  Current approaches Problem Formulation Solution Based on Lagrangian Relaxation Numerical Testing Results Issues of Homogenous Units and Resolution

42. 42 Inherent Issues of Lagrangian Relaxation Based Scheduling Algorithms Homogeneous subproblem solutions to the subproblems of identical units  May deviate far away from the optimal schedule  Difficult to obtain feasible solution since not much information on solution structure  Long been recognized and considered as a major obstacle for applying Lagrangian approach Existing approaches to solving these issue  parameter perturbation (heuristics)

43. 43 Homogenous solution: 2-Unit Example Subject to Optimal solutions: with

44. 44 Dual solution with Lagrangian Relaxation Subject to Optimal dual solution: if

45. 45 Dual solution with Lagrangian Relaxation Dual solution patterns: or Primal optimal solution: or

46. 46 Dual solution with Lagrangian Relaxation Solutions oscillation around as the multiplier being updated Subproblem solutions far away from primal optimum Primal optimal solution never obtained

47. 47 Key Idea of the New Algorithm: Differentiate homogenous subproblems Add quadratic or piece-wise linear penalty terms to Larangian Solve individual subproblem successively with each high level iteration to keep decomposability Update Lagrange multipliers using surrogate subgradient at the high level

48. 48 New Algorithm: Successive Subproblem Solving Method (SSS) Initialization Let w = 0, solve standard LR Update multipliers Step 0 Step 1 Solve only one subproblems Step 2 Check Convergence Step 3

49. 49 Features of the New Algorithm Surrogate subgradient = proper search direction Dual cost still the lower bound of the primal cost Larger penalty weight  smaller constraint violation in the dual problem Rigorous convergence proof

50. 50 Numerical Testing for SSS Algorithm Testing results of the simple problem with two identical units Testing results of generation scheduling problem of 10-units with two groups of identical units Excellent results observed

51. 51 Numerical Testing Testing results of the simple problem Iteration l 1000100.0000 2000.4750100.9500 3000.9545101.9090 4001.4386102.8772 5001.9272103.8545 6202.4206104.0000

52. SLRSSS HourUnit 3~8Unit 3 Unit 4Unit 5 Unit 6 100 00 0 200 00 0 300 00 0 400 00 0 500 00 0 600 00 25 700 00 107.2 825.00 025 114.05 9120.90 25120.9 10127.750 11134.60 121.6134.6 12162.00 25144.0 162.0 13127.750 14120.90 25120.9 15107.20 025.0 107.2 1652.40 0 1738.70 025.0 38.7 1866.10 0 19107.20 025.0 107.2 20162.00 0 21120.90 0 22100.350 0 2325.00 0 2400 025.0 0 Testing results of the 10-units generation scheduling

53. 53 Degree of Constraint Violation Iterations

54. 54 Conclusions Lagrangian relaxation is an efficient and effective approach for solving generation scheduling problems, and other similar resource scheduling problems. Double dynamic programming is an efficient and effective method to solve subproblems with hybrid dynamic constraints. The new SSS algorithm is effective to resolve the issues of homogenous subproblems and quantitative measure of the solution quality can still be obtained. Numerical testing results demonstrated the efficiency and effectiveness of the above method. The methods have been applied to the scheduling problems of manufacturing, optical switching networks, etc.

55. 55 On-going and Future Work Forecasting, simulation, bidding strategy, game theoretic analysis and mechanism design for electric power markets Modeling and analysis of networked systems: power system collapse Computer network security Sensor networks