ECON 4925 Autumn 2007
Electricity Economics
Lecture 8
Lecturer:
Finn R. Førsund
Transmission
1
A radial transmission network



Two nodes with one line the simplest case
Converting variables to energy units (kWh)
The energy balance
xt  e  e , t  1,.., T
H
t
L
t
xt  consumption in kWh
etH  production in kWh
etL  lossin kWh
Transmission
2
A radial transmission network, cont.

Expressing loss on a line supplying a single
consumer node from a single hydropower
generator node
L
2 L

e
(
x
)

et ( xt )
L
L
t
t
et  et ( xt ),
 0,
 0, t  1,.., T
2
xt
xt


Loss is increasing in consumption, and with
a positive second-order derivative, according
to Ohm’s law
A thermal capacity on the line: xt  x
Transmission
3
The social planning problem for the twonode case for two time periods
2
max
xt

t 1 z  0
pt ( z )dz
subject to
Rt  Rt 1  wt  etH
Rt  R
xt  etL  etH
etL  etL ( xt )
xt  x
xt , etH , etL  0, t  1, 2
Ro , R , x given
Transmission
4
The Lagrangian function
2
L
xt

t 1 z  0
pt ( z )dz
2
 t ( Rt  Rt 1  wt  etH )
t 1
2
  t ( Rt  R )
t 1
2
 t ( xt  etL ( xt )  etH )
t 1
2
 t ( xt  x )
t 1
Transmission
5
The Kuhn – Tucker conditions
etL
L
 pt ( xt )   t   t
 t  0 ( 0 for xt  0)
xt
xt
L
H






0
(

0
for
e
t
t
t  0)
H
et
L
 t  t 1   t  0 ( 0 for Rt  0)
Rt
t  0( 0 for Rt  Rt 1  wt  etH )
 t  0( 0 for Rt  R )
t  0( 0 for xt  x ) , t  1, 2
Transmission
6
Interpretation of the first-order conditions



Assuming positive production in both periods
The shadow prices on the energy balance will
then be positive and equal to the water
values
L
et
pt ( xt )  t  t
 t  0, t  1, 2
xt
The difference between the social price and
the water value
etL
pt ( xt )  t  t
 t  0, t  1, 2
xt
Transmission
7
Interpretations, cont.

Difference in social price between periods
e2L
e1L
p2 ( x2 )  p1 ( x1 )  2
 1
 ( 2  1 )
x2
x1

Difference even if water values are equal
(reservoir constraint not binding)
e2L e1L
p2 ( x2 )  p1 ( x1 )   (

)  ( 2  1 )
x2 x1
Transmission
8
A bathtub illustration without congestion
Period 2
Period 1
p2
p1
p2 ( x2 )  p1 ( x1 )   (
λ
p1 ( x1 )  
Loss 1
A
A'
B
M
C
e2L e1L

)
x2 x1
e1L
x1
λ
Loss 2
D'
D
Total available water
Ro + w 1 + w 2
Transmission
9
A bathtub illustration with binding reservoir
constraint, but without congestion
Period 2
Period 1
p2
p1
1
λ1
A
A'
λ2
C
B
D'
D
Total available water
Ro + w1 + w2
Transmission
10
Bathtub with congestion
Period 2
Period 1
2
p2
p1
λ
λ
x
A
A'
C
B
D'
D
Total available water
Transmission
11
Three nodes and two periods
Generating node 1
Generating node 2
Electricity flow
Electricity flow
Consumption node
Transmission
12
The social planning problem
2
max
xt

t 1 z  0
pt ( z )dz
subject to
R jt  R j ,t 1  w jt  e Hjt
R jt  R j
x jt  e Ljt  e Hjt
2
xt   x jt
j 1
e Ljt  e Ljt ( x jt )
x jt  x j
R jt , xt , x jt , e Hjt , e Ljt  0
w jt , R jo , R j , x j given , R j 2 free, j  1, 2 , t  1, 2
Transmission
13
The Lagrangian function
 j1 x jt
2
2
L
t 1
2

pt ( z )dz
z 0
2
  jt ( R jt  R j ,t 1  w jt  e Hjt )
t 1 j 1
2
2
  jt ( R jt  R j )
t 1 j 1
2
2
 jt ( x jt  e Ljt ( x jt )  e Hjt )
t 1 j 1
Transmission
14
The Kuhn – Tucker conditions
e Ljt
L
 pt ( xt )   jt   jt
  jt  0 ( 0 for x jt  0)
x jt
x jt
L
H






0
(

0
for
e
jt
jt
jt  0)
H
e jt
L
  jt   j ,t 1   jt  0 ( 0 for R jt  0)
R jt
 jt  0( 0 for R jt  R j ,t 1  w jt  e Hjt )
 jt  0( 0 for R jt  R j )
 jt  0( 0 for x jt  x j ) , j  1, 2 , t  1, 2
Transmission
15
Interpretations of the first-order conditions

Assumptions:


Positive production in the first period at both
plants (both empty the reservoirs in the second
period)
No threat of overflow in the first period


Water values for a plant the same for both periods
Difference between consumer price and
water values
e Ljt
pt ( xt )   j   j
  jt , j  1, 2 , t  1, 2
x jt
Transmission
16
Interpretations, cont.

Difference between water values
pt ( xt )   j   j
e
L
jt
x jt
  jt 
e1Lt
e2Lt
1  1
 1t  2  2
 2t 
x1t
x2t
e2Lt
e1Lt
1  2  2
  2t  (1
 1t ), t  1, 2
x2t
x1t

The plant with highest sum of marginal loss
and congestion will have the lowest water
value
Transmission
17
Interpretations, cont.

Differences between consumer prices
p2 ( x2 )  p1 ( x1 )   j

e Lj 2
x j 2
  j 2  ( j
e Lj1
x j1
  j1 ), j  1, 2
The highest consumer price will be in the
period with the highest value of the sum of
marginal loss and congestion term
Transmission
18
Nodal prices




Prices and water values are specific to each
node
Consumer price greater than water values for
each time period
Water values differ due to loss and
congestion between plants for each time
period
Marginal loss evaluated at water values plus
congestion is equal for each time period
Transmission
19
Congestion in the two-period case

Assumptions:




A line is at most congested in the high-demand
period only
There is no lock-in of water due to congestion
R jo  w j1  w j 2  x j , j  1,2
Period 1 is the low-demand period and period 2
the high-demand period
Immediate implication: at least one plant
must produce more in the high-demand
period than the low-demand -period
Transmission
20
Production levels for the two periods

Both plants will produce more in the highdemand period and less in the low-demand
period due to marginal loss increasing in
energy
e1Lt
e2Lt
1 (1 
)  1t  2 (1 
)  2t , t  1, 2
x1t
x2t


Disregarding congestion if plant 1 produces
more in the high-demand period so must
plant 2
Introducing congestion does not change this
situation
Transmission
21
Implication of transmission for utilisation
of the hydro plants


Transmission causes higher price in the highdemand period 2 and leads to a relatively
greater use of water in period 1 than
compared to no transmission
The plant with relatively less marginal loss
will shift production from the low demandperiod to the high-demand period, and
opposite for the plant with relatively greater
marginal loss
Transmission
22
Implications, cont.


The plant with relatively higher marginal loss
plus congestion in one period will also have a
relatively higher loss plus congestion in the
other period (plant water value constant)
The plant with the lowest water value is used
relatively more in the low-demand period, and
the plant with the highest water value
relatively more in the high-demand period
Transmission
23
Loop-flows (meshed network)
Generating node 1
Generating node 2
Electricity flow
Electricity flow
Electricity flow
Consumption node
Transmission
24