Transmission1 ECON 4925 Autumn 2007 Electricity Economics Lecture 8 Lecturer: Finn R. Førsund.

Transmission 1

ECON 4925 Autumn 2007 Electricity Economics Lecture 8

Lecturer:

Finn R. Førsund

Transmission 2

A radial transmission network Two nodes with one line the simplest case Converting variables to energy units (kWh) The energy balance

, 1,..,

consumption in kWh

production in kWh

lossin kWh

H Lt t t

t

Ht

Lt

x e e t T

x

e

e

Transmission 3

A radial transmission network, cont. Expressing loss on a line supplying a single

consumer node from a single hydropower generator node

Loss is increasing in consumption, and with a positive second-order derivative, according to Ohm’s law

A thermal capacity on the line:

2

2

( ) ( )( ), 0, 0, 1,..,

L LL L t t t tt t t

t t

e x e xe e x t T

x x

tx x

Transmission 4

The social planning problem for the two-node case for two time periods

2

1 0

1

max ( )

subject to

( )

, , 0, 1,2

, , given

tx

tt z

Ht t t t

t

L Ht t t

L Lt t t

t

H Lt t t

o

p z dz

R R w e

R R

x e e

e e x

x x

x e e t

R R x

Transmission 5

The Lagrangian function

2

1 0

2

11

2

1

2

1

2

1

( )

( )

( )

( ( ) )

( )

tx

tt z

Ht t t t t

t

t tt

L Ht t t t t

t

t tt

L p z dz

R R w e

R R

x e x e

x x

Transmission 6

1

1

( ) 0 ( 0 for 0)

0 ( 0 for 0)

0 ( 0 for 0)

0( 0 for )

0( 0 for )

0( 0 for ) , 1,2

Lt

t t t t t tt t

Ht t tH

t

t t t tt

Ht t t t t

t t

t t

eLp x x

x x

Le

e

LR

R

R R w e

R R

x x t

The Kuhn – Tucker conditions

Transmission 7

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

The difference between the social price and the water value

( ) 0, 1,2Lt

t t t t tt

ep x t

x

( ) 0, 1,2Lt

t t t t tt

ep x t

x

Transmission 8

Interpretations, cont.

Difference in social price between periods

Difference even if water values are equal (reservoir constraint not binding)

2 12 2 1 1 2 1 2 1

2 1

( ) ( ) ( )L Le e

p x p xx x

2 12 2 1 1 2 1

2 1

( ) ( ) ( ) ( )L Le e

p x p xx x

Transmission 9

A bathtub illustration without congestion

Total available water Ro + w1 + w2

p1

λ

DM D'A

λ

p2

A'

Loss 2

Period 1 Period 2

Loss 1

B C

11 1

1

( )Le

p xx

2 12 2 1 1

2 1

( ) ( ) ( )L Le e

p x p xx x

Transmission 10

A bathtub illustration with binding reservoir constraint, but without congestion

Total available water Ro + w1 + w2

p1

λ1

DD'A

λ2

p2

A'

Period 1 Period 2

B C

1

Transmission 11

Bathtub with congestion

p1

λ

DB D'A

λ

Total available water

p2

A'

Period 1 Period 2

2

C

x

Transmission 12

Three nodes and two periods

Electricity flow

Generating node 1 Generating node 2

Consumption node

Electricity flow

Transmission 13

The social planning problem2

1 0

, 1

2

1

2

max ( )

subject to

( )

, , , , 0

, , , given , free, 1,2 , 1,2

tx

tt z

Hjt j t jt jt

jt j

L Hjt jt jt

t jtj

L Ljt jt jt

jt j

H Ljt t jt jt jt

jt jo j j j

p z dz

R R w e

R R

x e e

x x

e e x

x x

R x x e e

w R R x R j t

Transmission 14

The Lagrangian function

2

12

1 0

2 2

, 11 1

2 2

1 1

2 2

1 1

( )

( )

( )

( ( ) )

jtjx

tt z

Hjt jt j t jt jt

t j

jt jt jt j

L Hjt jt jt jt jt

t j

L p z dz

R R w e

R R

x e x e

Transmission 15

The Kuhn – Tucker conditions

, 1

, 1

( ) 0 ( 0 for 0)

0 ( 0 for 0)

0 ( 0 for 0)

0( 0 for )

0( 0 for )

0( 0 for ) , 1,2 , 1,2

Ljt

t t jt jt jt jtjt jt

Hjt jt jtH

jt

jt j t jt jtjt

Hjt jt j t jt jt

jt jt j

jt jt j

eLp x x

x x

Le

e

LR

R

R R w e

R R

x x j t

Transmission 16

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

( ) , 1,2 , 1,2Ljt

t t j j jtjt

ep x j t

x

Transmission 17

Interpretations, cont. Difference between water values

The plant with highest sum of marginal loss and congestion will have the lowest water value

1 21 1 1 2 2 2

1 2

2 11 2 2 2 1 1

2 1

( )

( ), 1,2

Ljt

t t j j jtjt

L Lt t

t tt t

L Lt t

t tt t

ep x

x

e e

x x

e et

x x

Transmission 18

Interpretations, cont.

Differences between consumer prices

The highest consumer price will be in the period with the highest value of the sum of marginal loss and congestion term

2 12 2 1 1 2 1

2 1

( ) ( ) ( ), 1,2L Lj j

j j j jj j

e ep x p x j

x x

Transmission 19

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 20

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

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

1 2 , 1,2jo j j jR w w x j

Transmission 21

Production levels for the two periods Both plants will produce more in the high-

demand period and less in the low-demand period due to marginal loss increasing in energy

Disregarding congestion if plant 1 produces more in the high-demand period so must plant 2

Introducing congestion does not change this situation

1 21 1 2 2

1 2

(1 ) (1 ) , 1,2L Lt t

t tt t

e et

x x

Transmission 22

Implication of transmission for utilisation of the hydro plants Transmission causes higher price in the high-

demand 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 demand-period to the high-demand period, and opposite for the plant with relatively greater marginal loss

Transmission 23

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 24

Loop-flows (meshed network)

Electricity flow

Generating node 1 Generating node 2

Consumption node

Electricity flow

Electricity flow