Economic Dispatch for Power System Analysis

1. ECE 476POWER SYSTEM ANALYSIS Lecture 16 Economic Dispatch Professor Tom Overbye Department of Electrical andComputer Engineering

2. Announcements • Homework 7 is 6.46, 6.49, 6.52, 11.19, 11.21, 11.27; due date is Thursday October 30 • Potential spring courses: ECE 431 and ECE 398RES (Renewable Electric Energy Systems) • Be reading Chapter 11, concentrating on sections 11.4 and 11.5 • Design Project is assigned today (see website for details). Due date is Nov 20.

3. In the News: Wind Blade Failure Last Wednesday a 140 foot, 6.5 ton blade fell off from a Suzlon Energy wind turbine.The wind turbine is located in a small wind farm near Peoria. Suzlon Energy is oneof the world’s largest wind turbine manufacturers; its shares fell 39% following theaccident. No one was hurtand wind turbines failuresare extremely rare events. Photo source: Peoria Journal Star

4. Thermal Plants Can Fail As Well: CWLP Dallman Explosion, Fall 2007

5. CWLP Dallman Explosion, Fall 2007

6. Economic Dispatch: Formulation • The goal of economic dispatch is to determine the generation dispatch that minimizes the instantaneous operating cost, subject to the constraint that total generation = total load + losses Initially we'll ignore generator limits and the losses

7. Unconstrained Minimization • This is a minimization problem with a single inequality constraint • For an unconstrained minimization a necessary (but not sufficient) condition for a minimum is the gradient of the function must be zero, • The gradient generalizes the first derivative for multi-variable problems:

8. Minimization with Equality Constraint • When the minimization is constrained with an equality constraint we can solve the problem using the method of Lagrange Multipliers • Key idea is to modify a constrained minimization problem to be an unconstrained problem

9. Economic Dispatch Lagrangian

10. Minimization with Equality Constraint • When the minimization is constrained with an equality constraint we can solve the problem using the method of Lagrange Multipliers • Key idea is to modify a constrained minimization problem to be an unconstrained problem

11. Economic Dispatch Lagrangian

12. Economic Dispatch Example

13. Economic Dispatch Example, cont’d

14. Lambda-Iteration Solution Method • The direct solution only works well if the incremental cost curves are linear and no generators are at their limits • A more general method is known as the lambda-iteration • the method requires that there be a unique mapping between a value of lambda and each generator’s MW output • the method then starts with values of lambda below and above the optimal value, and then iteratively brackets the optimal value

15. Lambda-Iteration Algorithm

16. Lambda-Iteration: Graphical View In the graph shown below for each value of lambda there is a unique PGi for each generator. This relationship is the PGi() function.

17. Lambda-Iteration Example

18. Lambda-Iteration Example, cont’d

19. Lambda-Iteration Example, cont’d

20. Lambda-Iteration Example, cont’d

21. Lambda-Iteration Solution Method • The direct solution only works well if the incremental cost curves are linear and no generators are at their limits • A more general method is known as the lambda-iteration • the method requires that there be a unique mapping between a value of lambda and each generator’s MW output • the method then starts with values of lambda below and above the optimal value, and then iteratively brackets the optimal value

22. Generator MW Limits • Generators have limits on the minimum and maximum amount of power they can produce • Often times the minimum limit is not zero. This represents a limit on the generator’s operation with the desired fuel type • Because of varying system economics usually many generators in a system are operated at their maximum MW limits.

23. Lambda-Iteration with Gen Limits

24. Lambda-Iteration Gen Limit Example

25. Lambda-Iteration Limit Example,cont’d

26. Lambda-Iteration Limit Example,cont’d

27. Lambda-Iteration Limit Example,cont’d

28. Thirty Bus ED Example Case is economically dispatched without considering the incremental impact of the system losses

29. Back of Envelope Values • Often times incremental costs can be approximated by a constant value: • $/MWhr = fuelcost * heatrate + variable O&M • Typical heatrate for a coal plant is 10, modern combustion turbine is 10, combined cycle plant is 7 to 8, older combustion turbine 15. • Fuel costs ($/MBtu) are quite variable, with current values around 2 for coal, 7 for natural gas, 0.5 for nuclear, probably 10 for fuel oil. • Hydro costs tend to be quite low, but are fuel (water) constrained

30. Aside: Levelized Cost of Generation Keep in mind these numbers involve LOTs of assumptionsthat can drastically affect the value, and that many technology costs are site dependent. Source: California Energy Commission: http://energyalmanac.ca.gov/electricity/levelized_costs.html

31. Inclusion of Transmission Losses • The losses on the transmission system are a function of the generation dispatch. In general, using generators closer to the load results in lower losses • This impact on losses should be included when doing the economic dispatch • Losses can be included by slightly rewriting the Lagrangian:

32. Impact of Transmission Losses

33. Impact of Transmission Losses The penalty factor at the slack bus is always unity!

34. Impact of Transmission Losses

35. Calculation of Penalty Factors

36. Two Bus Penalty Factor Example

37. Thirty Bus ED Example Because of the penalty factors the generator incremental costs are no longer identical.

38. Area Supply Curve The area supply curve shows the cost to produce the next MW of electricity, assuming area is economically dispatched Supply curve for thirty bus system

39. Economic Dispatch - Summary • Economic dispatch determines the best way to minimize the current generator operating costs • The lambda-iteration method is a good approach for solving the economic dispatch problem • generator limits are easily handled • penalty factors are used to consider the impact of losses • Economic dispatch is not concerned with determining which units to turn on/off (this is the unit commitment problem) • Economic dispatch ignores the transmission system limitations