EE 458, Economic Systems for Electric Power Planning,



EE 458, Economic Systems for Electric Power Planning,

Exam 1, Spring 2008, Dr. McCalley

Closed book, Closed Notes, No Calculator

1. (24 pts) Consider the generator characteristic given below.

[pic]

a. (6 pts) Compute the maximum efficiency of this generator.

Solution: Maximum efficiency occurs at minimum heat rate, which appears from the above plot to be at about 57MW, where H=10.45 MBTU/MWhr. To

get efficiency, we compute 3.41/H=3.41/10.45=32.63%.

b. (6 pts) Compute the fuel input when the power is 50 MW.

Solution: R=H*P=(10.5MBTU/MWhr)(50MW)=525 MBTU/hr

c. (6 pts) Compute the cost rate when the power is 50 MW and fuel costs $2.00/MBTU.

Solution: C=R*K=525MBTU/hr($2.00/MBTU)=$1050/hr.

d. (6 pts) Approximate the incremental cost rate between 50 MW and 80 MW.

Solution: To get this, we need to compute the cost rate when the power is 50 MW and when the power is 80 MW. We already made this computation for the case of 50 MW in part c. Therefore, for 80 MW, we have

R=H*P=(10.65MBTU/MWhr)(80MW)=852MBTU/hr

C=R*K=(852MBTU/hr)($2/MBTU)=1704$/hr

Then we ΔC/ΔP=(1704-1050)/(80-50)=21.8$/MWhr

2. (36 pts) Generator cost rate functions, in $/hr, for a three unit system are given as

[pic]

Limits on the generation levels are [pic][pic][pic] The three generators must supply a total demand of 975 MW.

a) (6 pts) Express the objective function for the cost minimization problem.

b) (6 pts) Express the LaGrangian function assuming no constraints are binding.

c) (6 pts) Identify the first order conditions assuming no constraints are binding.

d) (6 pts) Form the linear matrix equation necessary to solve the unconstrained optimization problem.

e) (6 pts) The solution to the unconstrained optimization problem is [pic] [pic] [pic]Assume P1,MAX=500 MW. Compute lambda.

f) (6 pts) Now assume that P1,MAX=450 MW. Form the linear matrix equation necessary to solve the next iteration of getting the solution to this problem.

Solution:

a) f(P1, P2, P3)=C1(P1)+C2(P2)+C3(P3)

b) F(P1, P2, P3,()=f(P1, P2, P3)-(( P1+P2+P3-975)

c) The KKT conditions assuming no binding constraints are:

[pic]

(d)

[pic]

(e)

We can use any of the [pic]to obtain (

[pic]

(f) Now since [pic] exceeds its limit, we need to bring in the corresponding constraint with its Lagrange Multiplier.

[pic]

And when we apply first-order conditions, we will get

[pic]

3. (18 pts) Let C(P)=P2 be the cost (cost-rate) function for a producer.

a. Find the inverse supply (i.e., marginal cost) function.

Solution:

[pic]

b. The price is $50. On a plot of price p vs. supply P, identify the supply necessary for the producer to maximize profits, and identify the amount of producer surplus.

Solution:

[pic]

c. The price is dropped to $30. On a plot of price p vs. supply P, identify the change in producer surplus and whether that change is an increase or decrease.

Solution:

[pic]

4. (21 pts) True-False:

a. Up until the 1970’s, a competitive electric power marketplace was not seriously considered because Averch-Johnson effects were thought to be outweighed by economies of scale benefits achievable by monopolistic firms.

b. One reason why smaller power plants became more economically attractive was that the highly efficient combine cycle plants could not be built too large due to the coupling between the combustion turbines and the heat recovery steam generators.

c. Dr. Fred Schweppe developed the theory underlying locational marginal pricing in electric networks.

d. Today, over 90% of electric energy is purchased through the day-ahead or balancing markets in regions of the US that operate such markets.

e. FERC’s 2003 Standard Market Design became formal law via the 2005 Energy Policy Act passed in the US congress.

f. Solution to an equality-constrained optimization problem occurs when the magnitude of the objective function gradient equals the magnitude of the equality constraint gradient.

g. Inclusion of exact cost-curves for combined cycle plants violates objective function convexity requirements for guaranteeing solution to an optimization problem is a global optimum.

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Name (1 pt): ______________________________________________________________

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