Cheapo Electrons is an electricity retailer. Table 4.14 shows the load that it forecast
its consumers would use over a 6-h period. Cheapo Electrons purchased on the
forward market and the power exchange exactly enough energy to cover this
forecast. The table shows the average price that it paid for this energy for each
hour. As one might expect, the actual consumption of its customers did not exactly
match the load forecast and it had to purchase or sell the difference on the spot
market at the prices indicated. Assuming that Cheapo Electrons sells energy to its
customers at a flat rate of 24.00 $/MWh, calculate the profit or loss that it made
during this 6-h period. What would be the rate that it should have charged its
customers to break even?
- solutions
4.2
The input–output curve of a gas-fired generating unit is approximated by the
following function:
H P 120 9:3P 0:0025P2 MJ=h
This unit has a minimum stable generation of 200 MW and a maximum output
of 500 MW. The cost of gas is 1.20 $/MJ. Over a 6-h period, the output of this unit
is sold on a market for electrical energy at the prices shown in Table 4.15.
Assuming that this unit is optimally dispatched, is initially on-line, and cannot
be shut down, calculate its operational profit or loss for this period.
4.3
Repeat the calculation of Problem 4.2 assuming that the cost curve is replaced by a
three-segment piecewise linear approximation whose values correspond with
those given by the quadratic function for 200, 300, 400, and 500 MW.
4.4
Assume that the unit of Problem 4.2 has a startup cost of $500 and that it is initially
shutdown. Given the same prices as in Problem 4.2, when should this unit be
brought on-line and when should it be shutdown to maximize its operational
Table 4.14 Data for Problem 4.1.
Period
1
2
3
4
5
6
Load forecast (MWh)
120
230
310
240
135
110
Average cost ($/MWh)
22.5
24.5
29.3
25.2
23.1
21.9
Actual load (MWh)
110
225
330
250
125
105
Spot price ($/MWh)
21.6
25.1
32
25.9
22.5
21.5
Table 4.15 Data for Problem 4.2.
Period
1
2
3
4
5
6
Price ($/MWh)
12.5
10
13
13.5
15
11
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4 Participating in Markets for Electrical Energy
profit? Assume that dynamic constraints do not affect the optimal dispatch of this
generating unit.
4.5
Repeat Problem 4.4 taking into account that the minimum up-time of this unit is
4 h.
4.6
Borduria Generation owns three generating units that have the following cost
functions:
Unit A: 15 + 1.4 PA + 0.04 P2
A $/h
Unit B: 25 + 1.6 PB + 0.05 P2
B $/h
Unit C: 20 + 1.8 PC + 0.02 P2
C $/h.
How should these units be dispatched if Borduria Generation must supply a load
of 350 MW at minimum cost?
4.7
How would the dispatch of Problem 4.6 change if Borduria Generation had the
opportunity to buy some of this energy on the spot market at a price of 8.20 $/MWh?
4.8
If, in addition to supplying a 350 MW load, Borduria Generation had the
opportunity to sell energy on the electricity market at a price of 10.20 $/MWh,
what is the optimal amount of power that it should sell? What profit would it derive
from this sale?
4.9
Repeat Problem 4.8 if the outputs of the generating units are limited as follows:
Pmax = 100 MW
A
Pmax = 80 MW
B
Pmax = 250 MW.
C
4.10
Consider a market for electrical energy that is supplied by two generating
companies whose cost functions are as follows:
CA 36 ? PA $=h
CB 31 ? PB $=h
The inverse demand curve for this market is estimated to be:
π 120 � D $=MWh
Assuming a Cournot model of competition, use a table similar to the one used in
Example 4.10 and calculate the equilibrium point of this market (price, quantity,
production, and profit of each firm).
(Hint: Use a spreadsheet. A resolution of 5 MW is acceptable.)
4.11
Write and solve the optimality conditions for Problem 4.10.
4.12
Consider a pumped hydro plant with an energy storage capacity of 1000 MWh and
an efficiency of 75%. Assume that it takes 4 h to completely empty or fill the upper
reservoir of this plant if the plant operates at rated power and that the upper
138
Fundamentals of Power System Economics
Table 4.16 Data for Problem 4.12.
Period
1
2
3
4
5
6
Price ($/MWh)
40.92
39.39
39.18
40.65
45.42
56.34
Period
7
8
9
10
11
12
Price ($/MWh)
58.05
60.15
63.39
59.85
54.54
49.50
reservoir is initially empty. Suppose that the operator of this plant has decided to
go through a full cycle during the 12-h period using a very simple strategy: pump
water to the upper reservoir during the 4 h with the lowest energy prices and
release it during the 4 h with the highest energy prices. Table 4.16 shows the prices
during this 12-h period.
Calculate the profit or loss that this plant would make during this cycle of
operation. Determine the value of the plant efficiency that would make the profit or
loss equal to zero.
(Hint: Use a spreadsheet.)
4.13
Dragon Power Ltd is considering the construction of a new power plant in addition
to the 50 MW power plant it already operates in the Syldavian electricity market.
Possible capacities for the new plant are 50, 100, and 150 MW. The marginal cost
of production of the existing plant and the new plant is 25 $/MWh. Dragon
Power’s analysis of the Syldavian market shows that the following:
? Its competitors currently operate 200 MW of generation capacity.
? The incremental cost of operation of the generation capacity of its competitors is
30 $/MWh.
? The demand can be represented by the demand curve π = 450 � D, where D is
the total demand and π is the market price.
a Using a Cournot model, determine the capacity of the new plant that would
maximize the total hourly operating profit of Dragon Power. What is the total
operating profit of Dragon Power with this plant in service?
b What would be the effect of the construction of a new plant of the optimal
capacity on the hourly profit of Dragon Power’s competitors?
c Assuming that this plant is built and that the demand is 300 MW, estimate how
much imperfect competition in the Syldavian market costs consumers at each hour.
d Another team of analysts estimates that the demand curve of the Syldavian
market is π = 440 � 1.2 D. How does this revised estimate affect the profitability
of the optimal new plant? (Assume that the plant must make an hourly profit of
6000 $/h to cover its fixed costs.) What can you conclude about the robustness
of the Cournot model?
5.1
Consider the power system shown in Figure P5.1. Assuming that the only limita
tions imposed by the network are imposed by the thermal capacity of the
transmission lines and that reactive power flows are negligible, check that the
sets of transactions shown in Table P5.1 are simultaneously feasible.
Table P5.1 Sets of simultaneous transactions for Problem 5.1.
Seller
Buyer
Amount
Set 1
B
X
200
A
Z
400
C
Y
300
Set 2
B
Z
600
A
X
300
A
Y
200
A
Z
200
Set 3
C
X
1000
X
Y
400
B
C
300
A
C
200
A
Z
100
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Fundamentals of Power System Economics
Figure P5.1 Three-bus power system for Problem 5.1.
5.2
Consider the two-bus power system shown in Figure P5.2. The marginal cost of
production of the generators connected to buses A and B are given, respectively, by
the following expressions:
MCA 20 0:03PA $=MWh
MCB 15 0:02PB $=MWh
Assume that the demand is constant and insensitive to price and that energy is
sold at its marginal cost of production and that there are no limits on the output of
the generators. Calculate the price of electricity at each bus, the production of each
generator and the flow on the line for the following cases:
a The line between buses A and B is disconnected.
b The line between buses A and B is in service and has an unlimited capacity.
c The line between buses A and B is in service and has an unlimited capacity, but
the maximum output of generator B is 1500 MW.
d The line between buses A and B is in service and has an unlimited capacity, but
the maximum output of generator A is 900 MW. The output of generator B is
unlimited.
e The line between buses A and B is in service but its capacity is limited to
600 MW. The output of the generators is unlimited.
Figure P5.2 Two-bus power system for Problems 5.2–5.4, 5.10, and 5.11.
197
5 Transmission Networks and Electricity Markets
5.3
Calculate the generator revenues and the consumer payments for all the cases
considered in Problem 5.2. Who benefits from the line connecting these two
buses?
5.4
Calculate the congestion surplus for case (e) of Problem 5.2. Check your answer
using the results of Problem 5.3. For what values of the flow on the line between
buses A and B is the congestion surplus equal to zero?
5.5
Consider the three-bus power system shown in Figure P5.5. Table P5.5 shows the
data about the generators connected to this system. Calculate the unconstrained
economic dispatch and the nodal prices for the loading conditions shown in
Figure P5.5.
Figure P5.5 Three-bus power system for Problems 5.5–5.9 and 5.12–5.17.
Table P5.5 Characteristics of the generators for Problem 5.5.
Generator
Capacity (MW)
Marginal cost ($/MWh)
A
B
C
D
150
200
150
400
12
15
10
8
5.6
Table P5.6 gives the branch data for the three-bus power system of Problem 5.5.
Using the superposition principle, calculate the flows that would result if the
generating units were dispatched as calculated in Problem 5.5. Identify all the
violations of transmission constraints.
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Fundamentals of Power System Economics
Table P5.6 Characteristics of the branches for Problem 5.6.
Branch
Reactance (p.u.)
Capacity (MW)
1–2
0.2
250
1–3
0.3
250
2–3
0.3
250
5.7
Determine two ways of removing the constraint violations that you identified
in Problem 5.6 by redispatching generating units. Which redispatch is
preferable?
5.8
Calculate the nodal prices for the three-bus power system of Problems 5.5 and 5.6
when the generating units have been optimally redispatched to relieve the
constraint violations identified in Problem 5.6 and corrected in Problem 5.7.
Calculate the merchandising surplus and show that it is equal to the sum of
the surpluses of each line.
5.9
Consider the three-bus power system described in Problems 5.5 and 5.6. Suppose
that the capacity of branch 1–2 is reduced to 140 MW, while the capacity of the
other lines remains unchanged. Calculate the optimal dispatch and the nodal
prices for these conditions.
(Hint: The optimal solution involves a redispatch of generating units at all three
buses.)
5.10
Consider the two-bus power system of Problem 5.2. Given that K
R
V 2 0:0001 MW�1 for the line connecting buses A and B and that there is no
limit on the capacity of this line, calculate the value of the flow that minimizes the total
variable cost of production. Assuming that a competitive electricity market operates
at both buses, calculate the nodal marginal prices and the merchandising surplus.
(Hint: Use a spreadsheet.)
5.11
Repeat Problem 5.10 for several values of K ranging from 0 to 0.0005. Plot the
optimal flow and the losses in the line, as well as the marginal cost of electrical
energy at both buses. Discuss your results.
5.12
Using the linearized mathematical formulation (DC power flow approximation),
calculate the nodal prices and the marginal cost of the inequality constraint for the
optimal redispatch that you obtained in Problem 5.7. Check that your results are
identical to those that you obtained in Problem 5.8. Use bus 3 as the slack bus.
5.13
Show that the choice of slack bus does not influence the nodal prices for the DC
power flow approximation by repeating Problem 5.12 using bus 1 and then bus 2 as
the slack bus.
5 Transmission Networks and Electricity Markets
199
5.14
Using the linearized mathematical formulation (DC power flow approximation),
calculate the marginal costs of the inequality constraints for the conditions of
Problem 5.9.
5.15
Consider the three-bus system shown in Figure P5.5. Suppose that generator D
and a consumer located at bus 1 have entered into a contract for difference for the
delivery of 100 MW at a strike price of 11.00 $/MWh with reference to the nodal
price at bus 1, Show that purchasing 100 MW of point-to-point financial rights
between buses 3 and 1 provides a perfect hedge to generator D for the conditions of
Problem 5.8.
5.16
What flowgate rights should generator D purchase to achieve the same perfect
hedge as in Problem 5.15?
5.17
Repeat Problems 5.15 and 5.16 for the conditions of Problem 5.9.
5.18
Determine whether trading is centralized or decentralized in your region or
country or in another area for which you have access to sufficient information.
Determine also the type(s) of transmission rights that are used to hedge against the
risks associated with network congestion.
5.19
Determine how the cost of losses is allocated in your region or country or in
another area for which you have access to sufficient information.
5.20
Consider the small power system shown in Figure P5.20.
Assume that:
? Generating units A and B have the following constant marginal production
costs:
MCA 20 $=MWh
MCB 40 $=MWh
? The no-load and start-up costs are neglected.
? All three transmission lines have the same reactance and a negligible resistance.
? The DC power flow assumption is valid.
? The capacity of each generator is 500 MW.
? The capacity of the lines depends on the weather conditions. Under cold
weather conditions, each line is capable of carrying 400 MW. Under hot weather
conditions, this capacity is reduced to 240 MW.
? This system is operated under the N � 0 security criterion, i.e. we do not have to
consider line or generator outages.
a Calculate the optimal power flow under cold weather conditions.
b Calculate the optimal power flow under hot weather conditions.
c Calculate the hourly cost of security under cold weather conditions.
d Calculate the hourly cost of security under hot weather conditions.
200
Fundamentals of Power System Economics
Figure P5.20 Three-bus power system for Problems 5.20–5.22.
5.21
Repeat Problem 5.20 but assume that the reactance of the line between buses 1 and
2 is two times the reactance of the other two lines.
5.22
Consider the small power system shown in Figure P5.20 and assume that:
? The load at bus 3 is now 300 MW.
? Generating units A and B have the following constant marginal production
costs:
MCA 10 $=MWh
MCB 20 $=MWh
? All three transmission lines have the same impedance.
a Calculate the unconstrained optimal dispatch for these conditions.
b Calculate the hourly cost of this unconstrained dispatch.
c Calculate the power that would flow in each line if this dispatch was implemented.
d What is the marginal cost of energy at each node under these conditions?
e How should this unconstrained dispatch be modified if the flow in line 1–3 is
limited to 150 MW for operational reliability reasons?
f Calculate the hourly cost of this constrained dispatch and the hourly cost of
security.
g What is the marginal cost of energy at each node when the constraint on the
flow on line 1–3 is taken into consideration?
h Identify an economic paradox6 in this system.
i Assume that the Independent System Operator sells only point-to-point FTRs
from bus 1 to bus 3. What is the maximum amount of transmission rights that it
could sell without losing money?
5.23
Consider the three-bus power system shown in Figure P5.23a. Two identical
circuits of equal impedance and equal MW capacity connect each pair of buses.
The reactance and MW capacity of each circuit are given in Table P5.23.
6 Paradox (Noun): A statement, proposition, or situation that seems to be absurd or contradictory, but in
fact is or may be true.
201
5 Transmission Networks and Electricity Markets
Table P5.23 Characteristics of the circuits in Problem 5.23.
From bus
To bus
Circuit reactance (p.u.)
Circuit capacity (MW)
1
1
2
2
3
3
0.2
0.4
0.2
120
180
250
Figure P5.23b shows the marginal cost curves of the two generators. Startup and
no-load costs are assumed negligible. A DC (linear) transmission model is deemed
acceptable.
a Calculate the economic dispatch ignoring network constraints.
b Check that this economic dispatch does not violate any transmission constraint
when all the circuits are in service.
c Assuming that all generators bid at their marginal cost, what is the locational
marginal price at each node under these conditions?
The system operator must operate this system in a way that guarantees N � 1
security, i.e. there should be no line overload in the event of the outage of any
transmission circuit. (We do not consider generation contingencies.)
d For each branch, determine the single circuit contingencies that would result in
a flow constraint violation if this economic dispatch were implemented. (Take
into account the fact that the transmission capacity and reactance of a branch
change if one of its circuit is disconnected.)
Identify the contingency that would cause the worst violation of a flow
constraint and the circuit or branch that would be overloaded.
e Determine the least cost generation dispatch that would avoid a line overload
for the critical contingency identified in part (d). Assume that no postcontin
gency redispatch is allowed.
f What is the locational marginal price at each node under these conditions?
Figure P5.23a Three-bus power system for Problem 5.23.
202
Fundamentals of Power System Economics
Figure P5.23b Marginal cost curves of the generators of Problem 5.23.