Consum Consumers ers’’ gros grosss surpl surplus: us: $168 $168 000 000 Consumer Consumers’ s’ net surplus: surplus: $72 000
Kirschen and Goran Goran Strbac Fundamentals of Power System Economics Daniel Kirschen John Wiley Wiley & Sons, Sons, Ltd Ltd ISBN: 0-4700-470-84572-4 84572-4
Producers’ revenue: $96 000 Producers’ profit: $36 000 Global welfare: $108 000 2.4 a.
π = 900 $/widget q = 110
Consumers’ net surplus: $60 500 Producers’ profit: $46 750 Global welfare: $107 250 b.
π = 600 $/widget q = 80
Consumers’ net surplus: $80 000 Producers’ profit: $16 000 Global welfare: $96 000 c.
π = 1100 $/widget q = 90
Consumers’ net surplus: $40 500 Producers’ profit: $20 250 Tax revenue: $40 500 Global welfare: $101 250 2.5 q q
2.6
ε11 ε12 ε22 ε21
= = = =
=
π
200
−
π
ε
q
=
π
10 000/ π ε
0
200
−∞
∞
−1
50
150
−3
200
−1
100
100
−1
100
−1
150
50
−1/3
66.6
−1
200
0
0
50
−1
−0.120 0.048 −0.108 0.160
2.8 a.
Since the average production cost must be lower than the price, we get 65 ≤ y ≤ 155; y opt = 110
b.
The fixed cost is so high that there is no range of production at which the firm would make a profit.
267
APPENDIX: ANSWERS TO SELECTED PROBLEMS
Chapter 3 3.2
Pool Price ($/MWh)
a.
NSPCo
16
Produces 200 MWh Receives $3200 from the pool
Consumes 200 MWh Pays $3200 to the pool
18
Produces 200 MWh Receives $3600 from the pool Pays $400 to SAlCo
Consumes 200 MWh Pays $3600 to the pool Receives $400 from NSPCo
13
Produces 200 MWh Receives $2600 from the pool Receives $600 from SAlCo
Consumes 200 MWh Pays $2600 to the pool Pays $600 to NSPCo
b.
18
Produces 50MWh Receives $900 from Consumes 200 MWh Pays $3600 to the pool Pays $400 to SAlCo the pool Receives $400 from NSPCo
c.
13
Produces 200 MWh Receives $2600 from the pool Receives $600 from SAlCo
3.3
3.4 a.
SAlCo
Consumes 100 MWh Pays $1300 to the pool Pays $600 to NSPCo
Company
Profit ($)
Red Green Blue Yellow Magenta Purple
650 1280 1325 515 287.50 125
The supply curve is piecewise constant and is as follows in tabular form: Company
Amount (MWh)
Cumulative Amount (MW)
Price ($/MWh)
Blue
200
0–200
10.5
Red
100
200–300
12.5
Blue
200
300–500
13
Green
50
500–550
13.5
100
550–650
14
Green
50
650–700
14.5
Blue
100
700–800
15
Green
50
800–850
15.5
Red
50
850–900
18
Red
268
APPENDIX: ANSWERS TO SELECTED PROBLEMS
b. Forecast Load (MW)
Demand (MW)
Price ($/MWh)
Blue Blue Production Revenue (MWh) ($)
Red Red Production Revenue (MWh) ($)
Green Green Production Revenue (MWh) ($)
400
400
13.00
300
3900
100
1300
0
0
600
600
14.00
400
5600
150
2100
50
700
875
875
18.00
500
9000
225
4050
150
2700
Forecast Load (MW)
Demand (MW)
Price ($/MWh)
Blue Production (MWh)
Blue Revenue ($)
Red Production (MWh)
Red Revenue ($)
Green Production (MWh)
Green Revenue ($)
400
348
13.00
248
3224
100
1300
0
0
600
546
13.50
400
5400
100
1350
46
621
875
813
15.50
500
7750
200
3100
113
c.
1751.50
3.5 a. Item
Energy bought (MWh)
Industrial customer
Energy sold (MWh)
Price ($)
Expenses ($)
Revenue ($)
50
19.00
950.00
Other customers
1150
21.75
25 012.50
Future contract
200
21.00
4200.00
Put option
200
23.50
4700.00
Long-term contract
600
20.00
12 000.00
Future contract
100
22.00
2200.00
Call option
150
20.50
3075.00
Generation
300
21.25
6375.00
Spot market purchase
450
21.50
9675.00
150 MW Call option fee
1.00
150
200 MW Put option fee
1.00
200
300 MW Call option fee
1.00
300
Profit Balance
887.50 1600
1600
34 862.50
34 862.50
269
APPENDIX: ANSWERS TO SELECTED PROBLEMS
b.
If the spot price increases to $23.47, the cost of the 450 MWh purchase on the spot market would offset the profit. The 20.50 $/MWh call option and the 23.50 $/MWh put option would still be in the money. The 24.00 $/MWh call option would still be out of the money.
3.6 Since the market operator accepted 175 MW of bids, using the supply curve, we determine that the spot market price was 21.00 $/MWh. a. Item
Energy bought (MWh)
Energy sold (MWh)
Expenses ($)
Future T4
600
20.00
12 000.00
Nuclear unit
400
16.00
6400.00
Gas-fired unit
200
18.00
3600.00
Forward T1
Revenue ($)
50
21.00
1050.00
Long-term T3
350
20.00
7000.00
Forward T5
100
22.00
2200.00
Exercise Put option T6
250
23.50
5875.00
50
21.00
1050.00
Residential customers
300
25.50
7650.00
Commercial customers
200
25.00
5000.00
Spot sale T9
Balancing spot purchase
100
21.00
2100
Fee option T6
2.00
500.00
Fee option T7
2.00
400.00
Fee option T8
2.00
200.00
Profit Balance
b.
Price ($)
4625.00 1300
1300
29 825.00
29 825.00
Borduria Energy’s deficit for that period would increase from 100 MW to 500 MW. The spot price would increase from 21.00 to 28.00 $/MW. The cost of spot purchases would increase from $2000 to $14 000 but the cost of operating the nuclear power plant would drop to zero. Borduria Energy would therefore incur a loss of $975.00
Chapter 4 4.1 Cheapo Electrons makes a $1738.50 loss. The breakeven rate is 25.52 $/MWh. 4.2 The unit makes an operational profit of $690.07.
270
APPENDIX: ANSWERS TO SELECTED PROBLEMS
4.3 4.4
The unit makes an operational profit of $688.00 The unit should be brought on-line at the beginning of Period 3 and shutdown at the end of Period 5. Its operational profit would then be $976.43. 4.5 The unit should be brought on-line at the beginning of Period 3 and shutdown at the end of Period 6. Its operational profit would then be $680.43. 4.6 P A = 95.3 MW; P B = 74.2 MW; P C = 180.5 MW. Total hourly cost = 1927.15 $/h 4.7 P A = 85 MW; P B = 66 MW; P C = 160 MW; Market purchase: 39 MW Total hourly cost = 1911.20 $/h 4.8 P A = 110 MW; P B = 86 MW; P C = 210 MW; Market sale: 56 MW Profit from the sale: $33.03 4.9 P A = 100 MW; P B = 80 MW; P C = 210 MW; Market sale: 40 MW Profit from the sale: $27.23 4.10 P A = 25 MW; P B = 30 MW; D = 55 MW; π = 65 $/MWh; A = $725; B = $1, 020 4.11 P A = 26.33 MW; P B = 31.33 MW; D = 55 MW; π = 57.66 $/MWh; A = $694; B = $982 4.12 Profit: $5235; Efficiency that reduces profit to zero: 66.33%
πA = 80 $/MWh; πB = 35 $/MWh; P A = 2000 MW; P B = 1000 MW; F AB = 0
b.
πA = πB = 53 $/MWh; P A = 1100 MW; P B = 1900 MW; F AB = −900 MW
c.
πA = πB = 65 $/MWh; P A = 1500 MW; P B = 1500 MW; F AB = −500 MW
271
APPENDIX: ANSWERS TO SELECTED PROBLEMS
d.
πA = πB = 57 $/MWh; P A = 900 MW; P B = 2100 MW; F AB = −1100 MW
e.
πA = 62 $/MWh; πB = 47 $/MWh; P A = 1400 MW; P B = 1600 MW; F AB = −600 MW
6.3
Case:
a
b
c
d
e
EA ($)
160 000
106 000
130 000
114 000
124 000
EB ($)
35 000
53 000
65 000
57 000
47 000
RA ($)
160 000
58 300
97 500
51 300
86 800
RB ($)
35 000
100 700
97 500
62 700
75 200
The generator at B and the demand at A benefit from the line because it increases the price at B and lowers the price at A. 6.4 $9000. The congestion surplus is equal to zero when the flow is equal to zero and when it is equal to the unconstrained value of −900 MW. 6.5
P A = 0 MW; P B = 0 MW; P C = 120 MW; P D = 400 MW π1 = π2 = π3 = 10 $/MWh
6.6
F 21 = 120 MW; F 31 = 280 MW; F 32 = 200 MW
Line 1 –3 is overloaded by 30 MW. 6.7
Method 1: P A = 0 MW; P B = 48 MW; P C = 72 MW; P D = 400 MW F 21 = 102 MW; F 31 = 250 MW; F 32 = 182 MW
Increase in cost: $240 Method 2: P A = 80 MW; P B = 0 MW; P C = 40 MW; P D = 400 MW F 21 = 150 MW; F 31 = 250 MW; F 32 = 150 MW
Increase in cost: $160 Method 2 is preferable because it is cheaper. 6.8
6.16 62.5 MW of flowgate rights on branch 3-1. 6.17 37.5 MW of flowgate rights on branch 2-1 and 62.5 MW of flowgate rights on branch 3-1.
Chapter 7 7.1 7.2 7.3 7.4 7.5
12.14%; 32.28 $/MWh 11.17% 14.13%; 12.33%; Yes, because the IRR is 12.49%. The investment is higher if technology A is adopted, but the Incremental Internal Rate of Return on the additional investment is 14.13%, which is higher than the Minimum Acceptable Rate of Return. 7.6 The plant should continue operating because it continues to generate a positive cash flow of $32 524 128 per year. Borduria Power would not have built the plant because it would not have achieved its MARR. 7.7 If the plant has 20 years of expected life left, Borduria Power should repair it because the Internal Rate of Return on the investment required for the repair is 12.17%, which is above the MARR used by the company. If the plant has only 15 years left, the IRR is only 10.51% and the plant should be closed down. 7.8 Minimum price: 78.80 $/MWh. Average production cost: 37.70 $/MWh.
Chapter 8 8.3 8.4
8.00 $/MWh Short run marginal value of transmission 50
40
30
20
10
0 0
100
200
300
400
500
600
700
−10
Transmission capacity (MW)
800
900
1000
274 8.5 8.6 8.7 8.8 8.9
APPENDIX: ANSWERS TO SELECTED PROBLEMS
πT = 45 − 0.05 · F BA
12.00 $/MWh 660 MW 750 MW; $78 750 000. The two amounts are identical. 500 MW; $90 000 000 versus $52 500 000 1000 MW; $35 000 000 versus $105 000 000