Transcript Lecture #7
ECON 4925 Autumn 2007
Electricity Economics
Lecture 7
Lecturer:
Finn R. Førsund
Trade and transmission
1
Trade between Hydro and Thermal
The cooperative social planning problem
T
max
xtH
H
[
p
t ( z ) dz
t 1 z 0
xtTh
ptTh ( z ) dz c(etTh )]
z 0
subject to
xtH etH eThXI,t eHXI,t
xtTh etTh eThXI,t eHXI,t
T
H
e
t W
t 1
etTh e Th
xtH , etH , eThXI,t , eHXI,t 0
T ,W , e Th given , t 1,.., T
Trade and transmission
2
The Lagrangian function
Inserting the energy balances
Export for one country is import for the other
T
XI
XI
etH eTh
,t eH ,t
t 1
z 0
L [
XI
XI
etTh eTh
,t eH ,t
ptH ( z ) dz
ptTh ( z ) dz c(etTh )]
z 0
T
t (etTh e Th )
t 1
T
( etH W )
t 1
Trade and transmission
3
The Kuhn – Tucker conditions
L
H
H
H
p
(
x
)
0
(
0
for
e
t
t
t 0)
H
et
L
H
H
Th
Th
XI
p
(
x
)
p
(
x
)
0
(
0
for
e
t
t
t
t
H ,t 0)
XI
eH ,t
L
Th
Th
Th
Th
p
(
x
)
c
'(
e
)
0
(
0
for
e
t
t
t
t
t 0)
Th
et
L
H
H
Th
Th
XI
p
(
x
)
p
(
x
)
0
(
0
for
e
t
t
t
t
Th ,t 0)
XI
eTh ,t
T
0 ( 0 for etH W )
t 1
t 0 ( 0 for etTh e Th )
Trade and transmission
4
Combining the bathtub diagram and the
thermal diagram for two periods
Period 2
Period 1
θ2
p2Th=p2H=
p1Th=p1H=
c'
c'
Export
Import
A' A
Expor
t
Thermal
M'
M B'
Hydro
Trade and transmission
B
Import
Thermal
5
Trade Hydro –Thermal with reservoir
constraint
Period 2
Period 1
θ2
γ1
p1Th=p1H=1
c'
c'
Import
Export A' A
Thermal
p2Th=p2H=2
Export
B
C D' D
Hydro
Trade and transmission
Import
Thermal
6
Transmission
The model of Lord Kelvin from 1881 (Smith,
1961)
A single production node connected with a single
consumption node
Consumption node
Generating node
Electricity flow
Assumptions
Voltage at consumption node given
No binding capacity limit on the line
Trade and transmission
7
The physical laws of transmission
Ohm’s law
PL I R
2
2L
R
A
Symbols
PL = loss in kW
I = current in amps
R = resistance on the line in
ohms
L = length of line
A = area of cross section
ρ = specific resistance of the
metal
Trade and transmission
8
The physical laws of transmission, cont.
Constancy of energy
Pi PL Po
Kirchhoff’s laws
Symbols
Pi = power produced
(kW)
PL = loss on the line
(kW)
Po = power received
(kW)
Current flow into a node must
be equal to current flow out
(energy cannot be lost)
Voltage drops around any
loop sum to zero (relevant for
loop flow networks)
Ohm’s and Kirchhoff’s
laws
Flows distribute within loops
proportional to impedance on
lines
Trade and transmission
9
The connection between voltage and
current
Definition for AC
Po Vo I cos
Po
I
Vo cos
Symbols
Po = power at consumption
node in kW
Vo = voltage at
consumption node
I = current in amps
cosφ = power factor of the
consumer’s load
φ = lag between voltage
and current variation in
an alternating-current
circuit
Trade and transmission
10
The transmission production function
Inserting in the power balance
2
Po 2L
Po Pi PL Pi I R Pi
Vo cos A
2
Introducing the weight of the cable
K = 2dLA, d= specific weight
Renaming Po and Pi , x and e, multiplying
each term above with K
2
4
L
d
2
F ( x, e, K ) K (e x) kx 0 , k
(Vo cos ) 2
Trade and transmission
11
Substitution between capital and power
input
Ex ante MRS (marginal rate of substitution)
dK
K
MRS
0
de e x
The explicit ex ante production function
K
ke 12
x f (e, K )
(1 4 ) 1
2k
K
Scale properties ex ante and ex post
Ex ante: constant returns to scale
Ex post (fixed capital): decreasing returns to scale
Trade and transmission
12