Trade and transmission 1

ECON 4925 Autumn 2007 Electricity Economics Lecture 7

Lecturer:Finn R. Førsund

Trade and transmission 2

Trade between Hydro and Thermal The cooperative social planning problem

1 0 0

, ,

, ,

1

, ,

max [ ( ) ( ) ( )]

subject to

, , , 0

, , given , 1,..,

H Tht tx xT

H Th Tht t t

t z z

H H XI XIt t Th t H t

Th Th XI XIt t Th t H t

THt

t

Th Tht

H H XI XIt t Th t H t

Th

p z dz p z dz c e

x e e e

x e e e

e W

e e

x e e e

T W e t T

Trade and transmission 3

The Lagrangian function

Inserting the energy balances Export for one country is import for the other

, , , ,

1 0 0

1

1

[ ( ) ( ) ( )]

( )

( )

H XI XI Th XI XIt Th t H t t Th t H te e e e e eT

H Th Tht t t

t z z

TTh Th

t tt

THt

t

L p z dz p z dz c e

e e

e W

Trade and transmission 4

The Kuhn – Tucker conditions

,,

,,

1

( ) 0 ( 0 for 0)

( ) ( ) 0 ( 0 for 0)

( ) '( ) 0 ( 0 for 0)

( ) ( ) 0 ( 0 for 0)

0 ( 0 for )

0 ( 0 fo

H H Ht t tH

t

H H Th Th XIt t t t H tXI

H t

Th Th Th Tht t t t tTh

t

H H Th Th XIt t t t Th tXI

Th t

THt

t

t

L p x eeL p x p x ee

L p x c e eeL p x p x ee

e W

r )Th Th

te e

Trade and transmission 5

Combining the bathtub diagram and the thermal diagram for two periods

θ2

Period 1 Period 2

c' c'

Import Export

p1Th=p1

H=

p2Th=p

2

H=

ImportExport

Hydro ThermalThermal

A' A M' M B' B

Trade and transmission 6

p1Th=

p1H=

1

Trade Hydro –Thermal with reservoir constraint

Period 1 Period 2

c' c'

Import

θ2

p2Th=

p2H=

2

A B C D ImportExport

Hydro ThermalThermal

Export

γ1

A' D'

Trade and transmission 7

Transmission The model of Lord Kelvin from 1881 (Smith,

1961) A single production node connected with a single

consumption node

Assumptions Voltage at consumption node given No binding capacity limit on the line

Generating node Consumption node

Electricity flow

Trade and transmission 8

The physical laws of transmission Ohm’s law 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

2LP I R

2LRA

Trade and transmission 9

The physical laws of transmission, cont.

Constancy of energy

Symbols Pi = power produced

(kW) PL = loss on the line

(kW) Po = power received

(kW)

Kirchhoff’s laws 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

i L oP P P

Trade and transmission 10

The connection between voltage and current Definition for AC Symbols

Po = power at consumption node in kW

Vo = voltage at consumption node

I = current in ampscosφ = power factor of the

consumer’s load φ = lag between voltage

and current variation in an alternating-current circuit

cos

cos

o o

o

o

P V IP

IV

Trade and transmission 11

The transmission production function Inserting in the power balance

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

2 2coso

o i L i io

P LP P P P I R PV A

22

2

4( , , ) ( ) 0 ,( cos )o

L dF x e K K e x kx kV

Trade and transmission 12

Substitution between capital and power input Ex ante MRS (marginal rate of substitution)

The explicit ex ante production function

Scale properties ex ante and ex post Ex ante: constant returns to scale Ex post (fixed capital): decreasing returns to scale

0dK KMRSde e x

12( , ) (1 4 ) 1

2K kex f e Kk K