Effects of the Transmission Network
on Electricity Markets
© 2011 D. Kirschen and the University of Washington
1
Introduction
• No longer assume that all generators and loads are
connected to the same bus
• Need to consider:
– Congestion, constraints on flows
– Losses
• Two forms of trading
– Bilateral or decentralized trading
– Pool or centralized trading
© 2011 D. Kirschen and the University of Washington
2
Bilateral or decentralized trading
• Transactions involves only buyer and seller
• Agree on price, quantity and other conditions
• System operator
– Does not get involved directly in trading
– Maintains balance and security of the system
• Buys or sells limited amounts of energy to keep load and
generation in balance
• Limits the amount of power that generators can inject at
some nodes if security cannot be ensured by other means
© 2011 D. Kirschen and the University of Washington
3
Example of bilateral trading
Bus A
Bus B
G1
L1
G2
L2
•
•
•
•
G3
G1 sold 300 MW to L1
G2 sold 200 MW to L2
Prices are a private matter
Quantities must be reported to system operator so it can check
security
© 2011 D. Kirschen and the University of Washington
4
Example of bilateral trading
Bus A
Bus B
G1
L1
G2
L2
•
•
•
•
G3
G1 sold 300 MW to L1
G2 sold 200 MW to L2
If capacity of corridor ≥ 500 MW  No problem
If capacity of corridor < 500 MW  some of these transactions
may have to be curtailed
© 2011 D. Kirschen and the University of Washington
5
But curtail which one?
• Could use administrative procedures
– These procedures consider:
• Firm vs. non-firm transactions
• Order in which they were registered
• Historical considerations
– Do not consider relative economic benefits
– Economically inefficient
– Better to let the participants themselves decide what makes
sense
• Participants should purchase right to use the network
when arranging a trade in energy
– Physical transmission rights
– Support actual transmission of power over a given link
© 2011 D. Kirschen and the University of Washington
6
Physical transmission rights
Bus A
Bus B
G1
L1
G2
L2
•
•
•
•
•
G3
G1 sold 300 MW to L1 at 30 $/MWh
G2 sold 200 MW to L2 at 32 $/MWh
G3 selling energy at 35 $/MWh
L2 should not pay more than 3 $/MWh for transmission rights
L1 should not pay more than 5 $/MWh for transmission rights
© 2011 D. Kirschen and the University of Washington
7
Problems with physical rights
• Parallel paths
• Market power
© 2011 D. Kirschen and the University of Washington
8
Parallel paths
xA
P
FA
1
2
P
FB
xB
F
A
=
xB
xA + xB
© 2011 D. Kirschen and the University of Washington
P
F
B
=
xA
xA + xB
P
9
Parallel paths
C
A
B
Branch
Z
1
Reactance
Capacity
[p.u.]
[MW]
1-2
0.2
126
1-3
0.2
250
2-3
0.1
130
2
3
Y
D
© 2011 D. Kirschen and the University of Washington
10
Parallel paths
C
A
B
I
Z
II
1
2
3
Y
D
400 MW transaction between B and Y
Need to buy transmission rights on all lines
© 2011 D. Kirschen and the University of Washington
11
Parallel paths
Branch
C
A
B
I
Z
II
1
2
Reactance
Capacity
[p.u.]
[MW]
1-2
0.2
126
1-3
0.2
250
2-3
0.1
130
3
Y
D
F
I
=
0.2
0.2 + 0.3
400 MW transaction between B and Y
´ 400 = 160 MW
F
II
=
0.3
0.2 + 0.3
´ 400 = 240 MW
Not possible because exceeds capacities of lines 1-2 and 2-3
© 2011 D. Kirschen and the University of Washington
12
Counter-flows
C
A
B
200 MW transaction
between D and Z
III
Z
1
2
IV
F
III
F
IV
=
3
Y
D
© 2011 D. Kirschen and the University of Washington
=
0.2
0.2 + 0.3
0.3
0.2 + 0.3
´ 200 = 80 MW
´ 200 = 120 MW
13
Resultant flows
Branch
C
A
B
Z
1
2
Reactance
Capacity
[p.u.]
[MW]
1-2
0.2
126
1-3
0.2
250
2-3
0.1
130
3
Y
I
D
F12 = F23 = F - F
F13 = F
II
-F
IV
III
= 160 - 80 = 80 MW
= 240 - 120 = 120 MW
The resultant flows are within the limits
© 2011 D. Kirschen and the University of Washington
14
Physical rights and parallel paths
• Counter-flows create additional physical transmission
rights
• Economic efficiency requires that these rights be
considered
• Decentralized trading:
– System operator only checks overall feasibility
– Participants trade physical rights bilaterally
– Theory:
• Enough participants  market discovers optimum
– Practice:
• Complexity and amount of information involved are such that it is
unlikely that this optimum can be found in time
© 2011 D. Kirschen and the University of Washington
15
Physical rights and market power
Bus A
•
•
•
•
•
•
Bus B
G1
L1
G2
L2
G3
G3 only generator at bus B
G3 purchases transmission rights from A to B
G3 does not use or resell these rights
Effectively reduces capacity from A to B
Allows G3 to increase price at B
“Use them or loose them” provision for transmission rights:
difficult to enforce in a timely manner
© 2011 D. Kirschen and the University of Washington
16
Centralized or Pool Trading
• Producers and consumers submit bids and offers
to a central market
• Independent system operator selects the winning
bids and offers in a way that:
– Optimally clears the market
– Respects security constraints imposed by the network
• No congestion and no losses: uniform price
• Congestion or losses: price depend on location
where generator or load is connected
© 2011 D. Kirschen and the University of Washington
17
Borduria-Syldavia Interconnection
Borduria
DB= 500MW
Syldavia
DS= 1500 MW
• Perfect competition within each country
• No congestion or losses within each country
– Single price for electrical energy for each country
– Price = marginal cost of production
© 2011 D. Kirschen and the University of Washington
18
Borduria-Syldavia Interconnection
Borduria
Syldavia
DB= 500MW
DS= 1500 MW
p B = MC B = 10 + 0.01PB [$ / MWh]
p S = MC S = 13 + 0.02 P S [$ / MWh]
$/MWh
$/MWh
43
15
10
13
500
MW
p B = MC B = 10 + 0.01´ 500 = 15 $ / MWh
© 2011 D. Kirschen and the University of Washington
1500
MW
p S = MC S = 13 + 0.02 ´ 1500 = 43 $ / MWh
19
Borduria-Syldavia Interconnection
Borduria
DB= 500MW
Syldavia
DS= 1500 MW
Economic effect of an interconnection?
© 2011 D. Kirschen and the University of Washington
20
Can Borduria supply all the demand?
Borduria
DB= 500MW
PB = 2000MW
PS = 0MW
Syldavia
DS= 1500 MW
MC B = 30$ / MWh
MC S = 13$ / MWh
• Generators in Syldavia can sell at a lower price than generators
in Borduria
• Situation is not tenable
• Not a market equilibrium
© 2011 D. Kirschen and the University of Washington
21
Market equilibrium
Borduria
Syldavia
DB= 500MW
DS= 1500 MW
p =pB =p S
PB + PS = D B + D S = 500 + 1500 = 2000MW
p B = MC B = 10 + 0.01PB [$ / MWh]
p S = MC S = 13 + 0.02 P S [$ / MWh]
p = p B = p S = 24.30$ / MWh
PB = 1433MW
PS = 567MW
© 2011 D. Kirschen and the University of Washington
22
Flow at the market equilibrium
Borduria
Syldavia
DB= 500MW
DS= 1500 MW
PB = 1433MW
PS = 567MW
FBS = PB - D B = DS - PS = 933MW
© 2011 D. Kirschen and the University of Washington
23
Graphical representation
p S = MC S
p B = MC B
Supply curve for Syldavia
Supply curve for Borduria
24.3 $/MWh
24.3 $/MWh
PB= 1433 MW
PS = 567 MW
FBS= 933 MW
D B = 500 MW
D S = 1500 MW
D B + D S = 2000 MW
© 2011 D. Kirschen and the University of Washington
24
Constrained transmission
• What if the interconnection can carry only 400
MW?
– PB = 500 MW + 400 MW = 900 MW
– PS = 1500 MW - 400 MW = 1100 MW
p B = MC B = 10 + 0.01´ 900 = 19 $ / MWh
p S = MC S = 13 + 0.02 ´ 1100 = 35 $ / MWh
• Price difference between the two locations
• Locational marginal pricing or nodal pricing
© 2011 D. Kirschen and the University of Washington
25
Graphical representation
p S = MC S
p B = MC B
35 $/MWh
16 $/MWh
PB= 900 MW
PS= 1100 MW
FBS= 400 MW
D B= 500 MW
D S = 1500 MW
D B + D S = 2000 MW
© 2011 D. Kirschen and the University of Washington
26
Summary
Separate markets
Single market
PB [MW]
500
1,433
Single market
with congestion
900
p B [$/MWh]
15
24.33
19
R B [$/h]
7,500
34,865
17,100
E B [$/h]
7,500
12,165
9,500
PS [MW]
1500
567
1100
43
24.33
35
R S [$/h]
64,500
13,795
38,500
E S [$/h]
64,500
36,495
52,500
0
933
400
R TOTAL = R B + R S
72,000
48,660
55,600
E TOTAL = E B + E S
72,000
48,660
62,000
p S [$/MWh]
F BS [MW]
© 2011 D. Kirschen and the University of Washington
27
Winners and Losers
• Winners:
– Bordurian generators
– Syldavian consumers
• Losers
– Bordurian consumers
– Syldavian generators
• Congestion in the interconnection reduces these
benefits
© 2011 D. Kirschen and the University of Washington
28
Congestion surplus
Consumer payments:
E TOTAL = p B × DB + p S × D S
Producers revenues:
RTOTAL = p B × PB + p S × PS = p B ×( D B + FBS ) + p S ×( D S - F BS )
Congestion or merchandising surplus:
E TOTAL - R TOTAL = p S × D S + p B × DB - p S × P S - p
B
× PB
= p S ×( D S - P S ) + p B × ( DB - P B )
= p S × F BS + p B ×( - F BS )
= ( p S - p B ) × F BS
© 2011 D. Kirschen and the University of Washington
29
Congestion surplus
© 2011 D. Kirschen and the University of Washington
30
Congestion surplus
• Collected by the market operator in pool trading
• Should not be kept by market operator in pool
trading because it gives a perverse incentive
• Should not be returned directly to network users
because that would blunt the economic incentive
provided by nodal pricing
© 2011 D. Kirschen and the University of Washington
31
Pool trading in a three-bus example
C
A
Branch
Reactance
Capacity
[p.u.]
[MW]
1-2
0.2
126
1-3
0.2
250
2-3
0.1
130
B
1
2
50 MW
60 MW
3
Generator
Capacity
Marginal Cost
[MW]
[$/MWh]
A
140
7.5
B
285
6
C
90
14
D
85
10
D
300 MW
© 2011 D. Kirschen and the University of Washington
32
Economic dispatch
A
C
125 MW
0 MW
B
285 MW
F12
1
F13
2
F 23
50 MW
60 MW
3
0 MW
D
300 MW
© 2011 D. Kirschen and the University of Washington
33
Superposition
60 MW
360 MW
1
2
3
300 MW
300 MW
1
2
3
300 MW
60 MW
60 MW
1
2
3
© 2011 D. Kirschen and the University of Washington
34
Flows with economic dispatch
A
C
125 MW
0 MW
B
285 MW
156 MW
1
2
204 MW
96 MW
50 MW
60 MW
3
0 MW
D
300 MW
© 2011 D. Kirschen and the University of Washington
35
Overload!
A
C
125 MW
0 MW
FMAX = 126 MW
B
285 MW
156 MW
1
2
204 MW
96 MW
50 MW
60 MW
3
0 MW
D
300 MW
© 2011 D. Kirschen and the University of Washington
36
Correcting the economic dispatch
Additional generation at bus 2
0.6 MW
1 MW
1 MW
1
2
0.4 MW
3
© 2011 D. Kirschen and the University of Washington
37
Superposition
1
360 MW
2
60 MW
156 MW
96 MW
204 MW
3
300 MW
50 MW
50 MW
30 MW
1
2
20MW
1
3
310 MW
2
10 MW
126 MW
116 MW
184 MW
3
300 MW
© 2011 D. Kirschen and the University of Washington
38
Correcting the economic dispatch
Additional generation at bus 3
1 MW
0.4 MW
1
2
0.6 MW
3
1 MW
© 2011 D. Kirschen and the University of Washington
39
Superposition
1
360 MW
2
60 MW
156 MW
96 MW
204 MW
3
300 MW
75 MW
30 MW
1
45 MW
2
1
2
3
75 MW
285 MW
60 MW
126 MW
66 MW
159 MW
3
225 MW
© 2011 D. Kirschen and the University of Washington
40
Cost of the dispatches
•
•
•
•
Economic dispatch:
Redispatch generator 2:
Redispatch generator 3:
Cost of security:
© 2011 D. Kirschen and the University of Washington
2,647.50 $/h
2,972.50 $/h
2,835.00 $/h
187.50 $/h
41
Security constrained dispatch
A
C
50 MW
0 MW
B
285 MW
126 MW
1
2
159 MW
66 MW
50 MW
60 MW
3
75 MW
D
300 MW
© 2011 D. Kirschen and the University of Washington
42
Nodal prices
• Cost of supplying an additional MW of load at a
particular node without violating the security
constraints
• Start from the security constrained dispatch
© 2011 D. Kirschen and the University of Washington
43
Nodal prices
A
B
C
0 MW
50 MW
285 MW
126
MW
1
2
159 MW
66 MW
50 MW
60 MW
3
75 MW
D
300 MW
© 2011 D. Kirschen and the University of Washington
• Node 1:
• A is cheapest
generator
• p 1 = MC A = 7.50 $ / MWh
44
Nodal prices
A
B
C
0 MW
50 MW
285 MW
126
MW
1
2
159 MW
60 MW
66 MW
50 MW
3
75 MW
D
300 MW
© 2011 D. Kirschen and the University of Washington
• Node 3
• A is cheaper than D
• Increasing A would overload
line 1-2
• D is cheaper than C
• Increase D by 1 MW
p 3 = MC D = 10 $ / MWh
45
Nodal prices
A
B
C
0 MW
50 MW
285 MW
126
MW
1
2
159 MW
66 MW
50 MW
60 MW
3
75 MW
D
300 MW
© 2011 D. Kirschen and the University of Washington
• Node 2
• C is very expensive
• Increasing A or D would
overload line 1-2
• ?
46
Nodal price at node 2
0.6 MW
1 MW
1 MW
1
0.4 MW
2
3
1 MW
0.2 MW
1
0.8 MW
2
3
1
MW
© 2011 D. Kirschen and the University of Washington
47
Nodal price at node 2
• Increase generation at node 3 AND decrease
generation at node 1
DF12 =0
DP2 =
DP1
1
1 MW
2
3
© 2011 D. Kirschen and the University of Washington
DP3
48
Nodal price using superposition
0.6 MW
1 MW
1 MW
1
0.4 MW
DP1 + DP 3 = DP2 = 1 MW
0.6 DP1 + 0.2 DP3 = DF12 = 0 MW
2
DP1 = -0.5 MW
3
1 MW
DP3 = 1.5 MW
0.2 MW
1
2
0.8 MW
3
1 MW
© 2011 D. Kirschen and the University of Washington
p 2 = 1.5× MC D - 0.5 × MC A = 11.25 $ / MWh
49
Observations
• Generators A and D are marginal generators
because they supply the next MW of load at the
bus where they are located
• Generators B and C are not marginal
• Unconstrained system: 1 marginal generator
• m constraints: m+1 marginal generators
• Prices at nodes where there is no marginal
generator are set by a linear combination of the
prices at the other nodes
© 2011 D. Kirschen and the University of Washington
50
Summary for three-bus system
Bus 1
Consumption [MW]
Bus 2
Bus 3
System
50
60
300
410
Production [MW]
335
0
75
410
Nodal marginal price [$/MWh]
7.50
11.25
10.00
-
375.00
675.00
3,000.00
4,050.00
2,512.50
0.00
750.00
3,262.50
Consumer payments [$/h]
Producer revenues [$/h]
Merchandising surplus [$/h]
787.50
(= congestion surplus)
© 2011 D. Kirschen and the University of Washington
51
Counter-intuitive flows
A
B
50 MW
C
π1=7.50 $/MWh
0 MW
285 MW
π2=11.25 $/MWh
126 MW
1
2
159 MW
66 MW
50 MW
60 MW
π3=10.00 $/MWh
3
75 MW
Power flows from
high price to low
price!
D
300 MW
© 2011 D. Kirschen and the University of Washington
52
Counter-intuitive prices
• Prices at nodes without a marginal generator can
be higher or lower than prices at the other nodes
• Nodal prices can even be negative!
• Predicting nodal prices requires calculations
• Strategically placed generators can control prices
• Network congestion helps generators exert market
power
© 2011 D. Kirschen and the University of Washington
53
Effect of losses on prices
1
2
G
D
L
variable
æ Sö
2
=I R»
èV ø
2
R=
P2 +Q2
V
G( D) = D + L = D + K × D
2
×R »
R
V
2
×P = K× P
2
2
2
DG = G( D + DD) - G( D) = DD + 2 DD× D×K = (1+ 2 D× K ) DD
DC = c (1 + 2D× K ) DD
DC
DD
= c (1 + 2D× K )
© 2011 D. Kirschen and the University of Washington
p1 = c
p 2 = p 1 (1 + 2 D×K )
54
Losses between Borduria & Syldavia
PS = D S - FBS
PB = D B + FBS + K × FBS
2
Minimization of the total cost
37500
With losses
No losses
37000
Generation Cost [$/h]
36500
36000
35500
35000
34500
80
10
60
10
40
10
20
10
0
00
10
98
0
96
0
94
0
92
0
90
0
88
0
86
0
84
0
82
0
80
0
78
0
76
0
74
0
72
70
0
34000
Power Transfer [MW]
© 2011 D. Kirschen and the University of Washington
55
Mathematical Formulation
of Nodal Pricing
© 2011 D. Kirschen and the University of Washington
56
Introduction
• Independent System Operator needs systematic
method to calculate prices
• Constrained optimization problem
– Maximization of global welfare
• Assume perfect competition
© 2011 D. Kirschen and the University of Washington
57
One-bus network
D
Total demand of the consumers
P
Total production of the generators
B(D)
C(P)
Consumers’ benefit function
Producers’ cost function
B(D) - C(P) Global welfare
Maximize B(D) - C(P)
Subject to:
P-D= 0
© 2011 D. Kirschen and the University of Washington
58
One-bus network
Lagrangian:
(D, P, p ) = B(D) - C(P) + p (P - D)
Optimality conditions:
¶
dB
º
-p = 0
¶D dD
¶
dC
º+p = 0
¶P
dP
¶
º P-D=0
¶p
© 2011 D. Kirschen and the University of Washington
dB dC
=
=p
dD dP
Consumption and production increase
up to the point where marginal value =
marginal cost = price
59
Network of infinite capacity with losses
I k = Pk - Dk : net injection at bus k
Network creates economic welfare by allowing trades between
nodes with positive injections and nodes with negative injections
Wk (I k )
Ik < 0
Benefits of consumers at node k
Ik > 0
- Cost of producers at node k
n
W = åWk (I k )
: Global welfare
k=1
© 2011 D. Kirschen and the University of Washington
60
Network of infinite capacity with losses
Welfare maximization:
é n
ù
max (W ) = max êåWk (I k )ú
Ik
Ik
ë k=1
û
Alternative formulations:
ìn
ü
ìn
ü
min ( -W ) = min íå[ -Wk (I k )]ý = min íå Ck (I k )ý
Ik
Ik
î k=1
þ I k î k=1
þ
Assumes that:
• Demands are insensitive to prices
• Loads are constant
• Hence consumers’ benefits are constant
Equivalent to Optimal Power Flow problem
© 2011 D. Kirschen and the University of Washington
61
Network of infinite capacity with losses
ì n
ü
min íå Ck (I k )ý
Ik
î k=1
þ
Constraints:
• No constraints on network flows because infinite capacity
• Total generation = total load + losses
or
• Net injection = total losses in the branches of the network
n
åI
k
= L(I1 , I 2 , .. I n-1 )
k=1
(Bus n is the slack bus)
© 2011 D. Kirschen and the University of Washington
62
Network of infinite capacity with losses
é
ù
= å Ck (I k ) + p ê L ( I1 , I 2 , .. , I n-1 ) - å I k ú
ë
û
k=1
k=1
n
æ ¶L ö
¶
dCk
º
+p ç
-1÷ = 0
¶I k dI k
è ¶I k ø
¶
dCn
º
-p = 0
¶I n dI n
n
æ
dCk dCn æ
¶L ö
¶L ö
=
1= p ç 1ç
÷
dI k
dI n è ¶I k ø
è ¶I k ÷ø
k = 1, .. n - 1
n
¶
º L ( I1 , I 2 , .. , I n-1 ) - å I k = 0
¶p
k=1
© 2011 D. Kirschen and the University of Washington
63
Network of infinite capacity with losses
æ
dCk dCn æ
¶L ö
¶L ö
=
1= p ç 1ç
÷
dI k
dI n è ¶I k ø
è ¶I k ÷ø
k = 1, .. n -1
Nodal price at bus k is related to the nodal price at the slack bus
¶L
>0
¶I k
dCk dCn
<
dI k
dI n
If the injection at a bus increases the losses, the price at that node
will be less than the price at the slack bus
• Penalizes the generators at that bus
• Encourages consumers at that bus
© 2011 D. Kirschen and the University of Washington
64
Network of finite capacity
Limits on line flows:
Fl ( I1, I 2 , .. I n-1 ) £ Fl
l = 1, .. m
max
é
ù
= å Ck (I k ) + p ê L ( I1 , I 2 , .. , I n-1 ) - å I k ú
k=1
ë
k=1
û
n
n
m
+ å ml éë Fl
l=1
© 2011 D. Kirschen and the University of Washington
max
- Fl ( I1 , I 2 , .. , I n-1 ) ùû
65
Network of finite capacity
n
é
ù m
= å Ck (I k ) + p ê L ( I1 , I 2 , .. , I n-1 ) - å I k ú + å ml éë Fl max - Fl ( I1 , I 2 , .. , I n-1 ) ùû
ë
û l=1
k=1
k=1
n
æ ¶L ö m ¶Fl
¶
dCk
º
+p ç
- 1÷ - å ml
=0
¶I k dI k
è ¶I k ø l=1 ¶I k
¶
dCn
º
-p = 0
¶I n dI n
k = 1, .. n - 1
n
¶
º L ( I1 , I 2 , .. , I n-1 ) - å I k = 0
¶p
k=1
¶
º Fl max - Fl ( I1 , I 2 , .. I n-1 ) ³ 0
¶ ml
l = 1, .. m
ml × éë Fl max - Fl ( I1, I 2 , .. I n-1 ) ùû = 0 ; ml ³ 0
© 2011 D. Kirschen and the University of Washington
l = 1, .. m
66
Assume that only line i is congested
mi ³ 0; m j = 0 for j ¹ i
æ
dCk
¶L ö
¶Fi
= p ç 1+ mi
÷
dI k
¶I k
è ¶I k ø
k = 1, .. n -1
dCn
=p
dI n
n
åI
k=1
k
Price at all buses (except
slack bus) is affected by
the congestion on one line.
= L ( I1 , I 2 , .. , I n-1 )
Fi ( I1, I 2 , .. I n-1 ) = Fi
© 2011 D. Kirschen and the University of Washington
max
; mi > 0
67
Network of finite capacity: DC model
n
DC power flow:
(
j=1
(
Fij = yij q i - q j
Line flows:
)
i = 1, .. n
)
i, j = 1, .. n
I i = å yij q i - q j
Line flow constraints:
(
)
yij q i - q j £ F
max
ij
i, j = 1, .. n
Lagrangian function
é n
ù n n
= å Ci (I i ) + å p i ê å yij (q i - q j ) - I i ú + å å mij éë Fijmax - yij (q i - q j )ùû
i=1
i=1
ë j=1
û i=1 j=1
n
n
© 2011 D. Kirschen and the University of Washington
68
Network of finite capacity: DC model
é n
ù n n
max
é
= å Ci (I i ) + å p i ê å yij (q i - q j ) - I i ú + å å mij ë Fij - yij (q i - q j )ùû
i=1
i=1
ë j=1
û i=1 j=1
¶
dCi
º
- p i = 0 i = 1, .. n
¶I i dI i
n
¶
º å yij p i - p j + mij - m ji = 0 i = 1, .. n -1 (Slack at bus n)
¶q i j=1
n
¶
º å yij q i - q j - I i = 0 i = 1, .. n
¶p i j=1
¶
º Fijmax - yij (q i - q j ) ³ 0 i, j = 1, .. n
¶ mij
n
n
(
)
(
)
mij × éë Fijmax - yij (q i - q j ) ùû = 0 ; mij ³ 0
© 2011 D. Kirschen and the University of Washington
i, j = 1, .. n
69
Implementation
• m binding constraints  m+1 marginal generators
– Price at these buses determined using
– m+1 known prices
dCi
- pi = 0
dIi
• n-m-1 unknown prices
• m unknown Lagrange multipliers mij
• Use the second optimality condition to determine
these prices and shadow prices:
n
¶
º å yij p i - p j + mij - m ji = 0
¶q i j=1
(
© 2011 D. Kirschen and the University of Washington
)
i = 1, .. n - 1 (Slack at bus n)
70
Implementation
n
å y (p
ij
i
)
- p j + mij - m ji = 0
j=1
i = 1, .. n -1
• K: set of buses where the price is known
• U: set of buses where the price is unknown
n
Yiip i - å yijp j + å yij ( mij - m ji ) = å yijp j
jÎU
j=1
jÎK
n
-å yijp j + å yij ( mij - m ji ) = -Yiip i + å yijp j
jÎU
j=1
© 2011 D. Kirschen and the University of Washington
i ÎU; i ¹ slack bus
i ÎK; i ¹ slack bus
jÎK
71
Example
C
A
B
1
2
3
D
Marginal generators at buses 1 & 3
Price at bus 2 is unknown
m12 is also unknown
© 2011 D. Kirschen and the University of Washington
dC A
p1 =
= 7.5$/MWh
dPA
dC D
p3 =
= 10.0 $/MWh
dPD
K = {1, 3}
U = {2}
72
Example
Choose bus 3 as the slack bus
n
Yiip i - å yijp j + å yij ( mij - m ji ) = å yijp j
jÎU
j=1
i ÎU; i ¹ slack bus
jÎK
K = {1, 3}
U = {2}
i = 2 : Y22p 2 - y12 m12 = y21p 1 + y23p 3
n
-å yijp j + å yij ( mij - m ji ) = -Yiip i + å yijp j
jÎU
i = 1:
j=1
i ÎK; i ¹ slack bus
jÎK
- y12p 2 + y12 m12 = -Y11p 1 + y13p 3
© 2011 D. Kirschen and the University of Washington
73
Pool trading in a three-bus example
C
A
Branch
Reactance
Capacity
[p.u.]
[MW]
1-2
0.2
126
1-3
0.2
250
2-3
0.1
130
B
1
2
3
D
æ -10 5
5
Y = ç 5 -15 10
ç
10 -15
è 5
© 2011 D. Kirschen and the University of Washington
ö
÷
÷
ø
74
Example
Y22p 2 - y12 m12 = y21p 1 + y23p 3
-y12p 2 + y12 m12 = -Y11p 1 + y13p 3
æ -10 5
5
Y = ç 5 -15 10
ç
10 -15
è 5
ö
÷
÷
ø
ì5p 2 - 5 m12 = 25
í
î-15p 2 + 5 m12 = -137.5
ìp 2 = 11.25 $/MWh
í
î m12 = 6.25 $/MWh
© 2011 D. Kirschen and the University of Washington
75
Financial Transmission Rights
© 2011 D. Kirschen and the University of Washington
76
Managing transmission risks
•
•
•
•
Congestion and losses affect nodal prices
Additional source of uncertainty and risk
Market participants seek ways of avoiding risks
Need financial instruments to deal with nodal
price risk
© 2011 D. Kirschen and the University of Washington
77
Contracts for difference
• Centralized market
– Producers must sell at their nodal price
– Consumers must buy at their nodal price
• Producers and consumers are allowed to enter
into bilateral financial contracts
– Contracts for difference
© 2011 D. Kirschen and the University of Washington
78
Example of contract for difference
Borduria
400 MW
Borduria
Power
Syldavia
Syldavia
Steel
400 MW
• Contract between Borduria Power and Syldavia Steel
– Quantity: 400 MW
– Strike price: 30 $/MWh
• Other participants also trade across the
interconnection
© 2011 D. Kirschen and the University of Washington
79
No congestion  market price is uniform
Borduria
400 MW
Borduria
Power
πB = 24.30 $/MWh
•
•
•
•
•
•
Syldavia
Syldavia
Steel
400 MW
πS = 24.30 $/MWh
Borduria Power sells 400 at 24.30  gets $9,720
Syldavia Steel buys 400 at 24.30 pays $9,720
Syldavia Steel pays 400 (30 - 24.30) = $2,280 to Borduria Power
Syldavia Steel’s net cost is $12,000
Borduria Power’s net revenue is $12,000
They have effectively traded 400 MW at 30 $/MWh
© 2011 D. Kirschen and the University of Washington
80
Congestion  Locational price differences
Borduria
400 MW
Borduria
Power
πB = 19 $/MWh
•
•
•
•
•
•
Syldavia
Syldavia
Steel
400 MW
πS = 35 $/MWh
Borduria Power sells 400 at 19.00  gets $7,600
Syldavia Steel buys 400 at 35.00  pays $14,000
Borduria Power expects 400 (30 -19) = $4,400 from Syldavia Steel
Syldavia Steel expects 400 (35 -30) = $2,000 from Borduria Power
Shortfall of $6,400
Basic contracts for difference break down with nodal pricing!
© 2011 D. Kirschen and the University of Washington
81
Financial Transmission Rights (FTR)
• Observations:
– Shortfall in contracts for difference is equal to congestion
surplus
– Congestion surplus is collected by the system operator
• Concept:
– System operator sells financial transmission rights to users
– FTR contract for F MW between Borduria and Syldavia entitles
the owner to receive: F ×( p S - p B )
– Holders of FTRs are indifferent about where they trade energy
– System operator collects exactly enough money in congestion
surplus to cover the payments to holders of FTRs
© 2011 D. Kirschen and the University of Washington
82
Example of Financial Transmission Rights
Borduria
400 MW
Borduria
Power
Syldavia
Syldavia
Steel
400 MW
• Contract between Borduria Power and Syldavia Steel
– Quantity: 400 MW
– For delivery in Syldavia
– Strike price: 30 $/MWh
• To cover itself against location price risk, Borduria Power
purchases 400 MW of financial transmission rights from the
System Operator
© 2011 D. Kirschen and the University of Washington
83
Example of Financial Transmission Rights
Borduria
400 MW
Borduria
Power
πB = 19 $/MWh
•
•
•
•
•
•
•
Syldavia
400 MW
Syldavia
Steel
400 MW
πS = 35 $/MWh
Borduria Power sells 400 at 19.00  gets $7,600
Syldavia Steel buys 400 at 35.00  pays $14,000
The system operator collects 400 (35 -19) = $ 6,400 in congestion surplus
Borduria Power collects 400 (35 -19) = $6,400 from the system operator
Borduria Power pays Syldavia Steel 400 (35 -30) = $2,000
Syldavia Steel net cost is $12,000
The books balance!
Borduria power net revenue is $12,000
© 2011 D. Kirschen and the University of Washington
84
Financial transmission rights (FTR)
• FTRs provide a perfect hedge against variations in nodal
prices
• Auction transmission rights for the maximum
transmission capacity of the network
– The system operator cannot sell more transmission rights than
the amount of power that it can deliver
– If it does, it will lose money!
• Proceeds of the auction help cover the investment costs
of the transmission network
• Users of FTRs must estimate the value of the rights they
buy at auction
© 2011 D. Kirschen and the University of Washington
85
Financial transmission rights
• FTRs are defined from point-to-point
• No need for a direct branch connecting directly the points
between which the FTRs are defined
• FTRs automatically factor in the effect of Kirchoff’s
voltage law
• Problem:
– There are many possible point-to-point transmission rights
– Difficult to assess the value of all possible rights
– Difficult to set up a market for point-to-point transmission rights
© 2011 D. Kirschen and the University of Washington
86
Flowgate rights
• Observation:
– Typically, only a small number of branches are congested
• Concept:
– Buy transmission rights only on those lines that are congested
– Theoretically equivalent to point-to-point rights
• Advantage:
– Fewer rights need to be traded
– More liquid market
• Difficulty:
– Identify the branches that are likely to be congested
© 2011 D. Kirschen and the University of Washington
87