04-Economic Dispatch 2. EE570 Energy Utilization & Conservation Professor Henry Louie

Transcription

1 04-Economic Dispatch EE570 Energy Utilization & Conservation Professor Henry Louie 1

2 Topics Example 1 Example Dr. Henry Louie

3 Consider two generators with the following cost curves and constraints: C 1 (P 1 ) = P P 1 C (P ) = P P 0 P P 350 The total demand is 700MW. Find the optimal dispatch of P 1 and P

4 We will solve this problem using MATLAB in an iterative process where we start by guessing the incremental cost, l and revise it while enforcing generator constraints

5 Begin by defining system variables Because this is an iterative method, We do not expect to arrive at the exact solution. However, we can set an error bound. We will make Sure that the solution is such that the total demand is met within 0.1 MW (699.9 to 700.1)

6 Start with an initial guess of l. A reasonable guess is near the b coefficient of one of the cost functions. C 1 (P 1 ) = P P 1 C (P ) = P P

7 Find an equation that relates a value of l to generation values dc IC l P dp1 dc IC l P P P dp rearranging l l Recall that:

8 Define variables that are used in administering the algorithm Placing bounds on what l can be Will speed up our algorithm. l will never Be negative, and we are reasonably sure It will not exceed 100. However, the upper bound can be set to a much greater value if unsure.

9 We will use a while loop, since we do not know when the solution will be reached

10 Now update l based on whether or not the previous guess of l resulted in too much or too little generation relative to demand Note: this is within the while loop

11 Update P 1 to correspond to the new l and enforce generation limits, if needed

12 Update P to correspond to the new l and enforce generation limits, if needed

13 Display the updated generation values and end the while loop pause lets you see the solution for this iteration. Press any key to advance to the next iteration. It is not needed to include the pause

14 We converge on a solution at the 14 th iteration: Generator 1 outputs 400 MW (its limit) Generator outputs MW Note: P1 + P = MW, which is within 0.1MW of the demand

15 The incremental costs are: Generator 1: $4.6/hr Generator : $44./hr Incremental costs are not equal because generator 1 is at its limit

16 Visualizing the results

17 Example Consider the system without generator limits IC 0.007P 4.1 $/MWh IC 0.007P 4.1 $/MWh P P 50 MW P P L D1 D 1 G1 G G 300 MW 50 MW Find the optimal generation for each generator and the power loss in the transmission line 1 Note: the incremental costs are the same 17 Dr. Henry Louie

18 Example First find P P L G now find the penalty factor L L 1 18 Dr. Henry Louie

19 Example First find P P L G 0.00 P P 0.1 G G now find the penalty factor L 1 L P G L L i i,...,m PL 1 P Gi At the solution, L 1 xic 1 = L xic = l 19 Dr. Henry Louie

20 Example We then have: IC 0.007P 4.1 $/MWh IC 0.007P 4.1 $/MWh L 1 L 1 1 G1 G P According to the optimal dispatch rule: P 4.1 l G P G G 0.007P 4.1 G l 0 Dr. Henry Louie

21 Example Start with a guess for l, and update if needed (depending on the corresponding generation values) P 4.1 G P G l 0.007P 4.1 it is easier to rewrite the equations as: G1 l P P G1 G l l l 1 Dr. Henry Louie

22 Example Start with a guess of l = 5.0 P P P G1 G L Dr. Henry Louie

23 Example Start with a guess of l = PG PG L P Power to the load is = 10 MW (the load is 350 MW) We need more generation to meet the load, so increase l to Dr. Henry Louie

24 Example l = 6.0, what are the corresponding power and loss? P P P G1 G L 4 Dr. Henry Louie

25 Example l = 6.0, what are the corresponding power and loss? PG PG L P Power to the load is = MW (the load is 350 MW) Now there is too much generation; we need to decrease l 5 Dr. Henry Louie

26 Example After some more iterations, we find that l = P G1 = MW P G = MW P L = 4.9 MW IC 1 = $5.74/MWh IC = $4.94/MWh L = 1.16 This is close enough to the exact answer Even though the generators have the same IC, due to losses, it is cheaper to have generator 1 supply more of the load 1 Note: P G1 + P G = P D + P L 300 MW 50 MW 6 Dr. Henry Louie

27 Economics Summary The economic dispatch (ED) problem comes in several flavors depending on loss considerations and generator limits The goal is to minimize the fuel costs needed to serve the load We use a solution approach based on the incremental costs of generators The solution approaches presented have used approximate system models (voltage, reactive power, line constraints, etc have been ignored) 7 Dr. Henry Louie