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