1. Price-Based Unit Commitment

2. PBUC FORMULATION

3.  maximize the profit

4. PBUC FORMULATION  System Constraints  These constraints represent a GENCO’s special requirements.  For example, a GENCO may have minimum and maximum generation requirements in order to play the game in the energy market.  Because of reliability requirements, a GENCO may pose lower and upper limits on its spinning and no-spinning reserves.  These constraints can be relaxed otherwise.

5. PBUC  System Fuel Constraints (For a “FT” type of fuel)  System Emission Constraint

6. PBUC  Unit Constraints

7. PBUC  Unit Minimum ON/OFF Durations  Unit Ramping Constraints  Unit Fuel Constraints

8. PBUC SOLUTION  Lagrangian relaxation is used to solve PBUC.  The basic idea is to relax coupling constraints (i.e., coupling either units, time periods, or both) into the objective function by using Lagrangian multipliers.  The relaxed problem is then decomposed into subproblems for each unit.  The dynamic programming process is used to search the optimal commitment for each unit.  Lagrangian multipliers are then updated based on violations of coupling constraints

9. Solution without Emission or Fuel Constraints  Using Lagrangian multipliers to relax system constraints (i.e., energy and reserve), we write the Lagrangian function as

10. Single-Unit DP  The Lagrangian term for one unit at a single period is given as follows  The separable single-unit problem is formulated as

11. Optimality Condition  When the unit is ON, the derivatives of the Lagrangian function with respect to P, R, and N are

12. optimality condition  when the unit is ON, the optimality condition is

15. Optimality Condition when the Unit is OFF

16. Multipliers Update

19. Economic Dispatch  Once the unit commitment status is determined, an economic dispatch problem is formulated and solved to ensure the feasibility of the original unit commitment solution.  subject to energy, reserve, and unit generation limits  quadratic or linear programming can be applied to solve this problem

20. Economic Dispatch for Non-spinning Reserve

22. Economic Dispatch for Spinning Reserve

24. Economic Dispatch for Energy

26. Convergence Criterion  Suppose that the solution from unit commitment is SU and the solution from economic dispatch is SE  Substituting SU into the Lagrangian function, we would get the Lagrangian value, LU.  Substituting SE into the Lagrangian function we would get the Lagrangian value, denoted as LE  The relative duality gap (RDG)

27. Solution with Emission and Fuel Constraints

30. Optimality Condition

34. Multipliers Update for Emission and Fuel Constraints

36. Economic Dispatch

37. Energy Purchase

38. Derivation of Steps in Update of Multipliers

40. Optimality Condition

41. Bidding Strategy Based on PBUC

42. Bidding Strategy

53. Case Study of 5-Unit System

54. Case 1: Impact of the Energy Market Price

56. Case 2: Impact of Ramp Rates

58. Case 3: Impact of Fuel Price Variations

60. Case 5: Impact of Different LMPs