Summing up 1

ECON 4930 Autumn 2007 Electricity Economics Lecture 12

Lecturer:

Finn R. Førsund

Summing up 2

The key theme: drivers of price changes Point of departure: water as a natural

resource in limited supply All prices equal, Hotelling’s rule

The price of electricity varies over both day and season

We must come up with explanations NB! price variation in the market may not be

optimal

Summing up 3

Constraints as price drivers

Reservoir constraint Threat of overflow Running empty

Production constraint Pipes, turbines, generator

Environmental constraints Ramping up and down Minimum production (max covered above)

Summing up 4

Threat of overflow and empty reservoir

p2(e2H)

e2H

p1(e1H)

DC

Period 2Period 1

λ1

λ2

BA e1H

γ1

p2p1

Total available waterRo + w1 + w2

Summing up 5

In between empty and full

M

λu+1=pT

pu pu+1

Period u Period u+1

B C DA

Water to period u+2

Summing up 6

Production constraint: constraint in period 2: peak demandp1

p1=λ1

DCA

Period 2Period 1

ρ2

p2

λ2

p2=λ2+ρ2

B B’ B’’

Summing up 7

Production constraint: constraint in period 1:threat of overflow

Spill

p1

p1=ρ1

DCA

Period 2Period 1

ρ1

p2

p2=λ2=γ1

BB'

γ1

λ1=0

Summing up 8

Unregulated generation

Run-of-the-river Full reservoirs turn a power station into a run-of-

the-river station Windmills

Range of fluctuation large Geothermal

May take time to regulate steam pressure, etc. Downward price jumps when hydro plants

can do pure accumulation and low demand

Summing up 9

Must take: Run-of-the-river and wind power

Total controllable inflow

p1

p2

B D

River flow

e1R

p2(x2)

e2H

p1(x1)

C

Period 2Period 1

λ1 = λ2

e1H

River flow

e2RA

Summing up 10

Hveding’s conjecture Assume independent hydropower plants with

one limited reservoir each, and perfect manoeuvrability of reservoirs, but plant-specific inflows

The plants can be regarded as a single aggregate plant and the reservoirs can be regarded as a single aggregate reservoir when finding the social optimal solution for operating the hydropower system. If overflow, then all reservoirs overflow at the

same time, if empty then all reservoirs are emptied at the same time

Summing up 11

Price fluctuation when hydro interacts with thermal Thermal sector is used according to merit

order ranking of marginal cost General rule: price equals water value equals

marginal cost of thermal Typical result: price variations less than in a

pure hydro system

Summing up 12

Bathtub diagram with thermal and hydro with reservoir constraint

p1

λ1

Period 2Period 1

Thermal extensions

p1

c'c'

A' A B C D D'Hydro energy

γ1

λ2

p2

p2

Summing up 13

The impact of trade on prices Unlimited trade

External exogenous prices determine the regional prices

Limited trade due to transmission Region may have different prices if trade is limited If import-constrained higher regional price If export-constrained lower regional price

Endogenous trade prices Equal prices without constraints on

interconnectors Different prices with constraints on

interconnectors

Summing up 14

Impact of trade on prices

p1

Prod.1 Cons.2Import Export

Total available water

p2

α2

β1

γ1pAU

Period 1 Period 2

x1A B C Dx2

p1=1

p1XI

p2XI

p2=2

Summing up 15

The impact of transmission on prices Transmission: connecting all generator nodes

and consumption nodes by lines Fundamental physics: loss of energy on lines

Implication: nodal prices as the optimal price structure; higher consumer price the higher the loss

Limited capacity of lines and congestion Thermal capacity (Ohm’s law) More general resistance; impedance when AC Loop flows and other electric mysteries with AC

Summing up 16

Bathtub with loss and congestion

p1

λ

DB D'A

λ

Total available water

p2

A'

Period 1 Period 2

2

C

x

Summing up 17

Uncertainty as price driver

Assuming all information for the present period to be known, price for the next period will be expected price The construction and use of expected water value

table, role of constraints When time progresses there will be a

continuous update of expected prices Realised price will typically differ from

expected price, implying fluctuating price

Summing up 18

A general case for period t

A' A B Mt|t-1 Mt C

ptPeriod t

pt

1{ }t t tE p R

1 1{ }t t tE p R

1 1{ }t t tp E p

Period t-1

Summing up 19

Market power as price driver

Monopoly No reservoir constraint, no spill

Shifting of water from relatively inelastic periods to more elastic periods

Price will then go down in periods where more is produced and up in periods with less production

Reservoir constraint, no spill Shifting if constraint not binding Social price solution if reservoir constraint is binding

Summing up 20

Bathtub illustration of two periods

p2M

Period 1 Period 2

p2S = λSp1

S= λS

p1M

λM

Summing up 21

The competitive solution

Second welfare theorem: any efficient allocation can be sustained by a competitive equilibrium

Problems: Electric externalities due to transmissions with

loop flows, reactive power, etc. Hydraulic externalities for plants along the same

river system Uncertainty and expectation formation

Summing up 22

The optimisation problem of producer j at period 1 with certainty for given prices

1

, 1

max

subject to

, 0 ,

, , , , , given, 1,.., ,

free

TH

t jtt

Hjt j t jt jt

jt j

H Hjt j

Hjt jt

Ht jt jo j j

jT

p e

R R w e

R R

e e

R e

T p w R R e t T

R

Summing up 23

The Lagrangian

1

, 11

1

1

( )

( )

( )

TH

t jtt

TH

jt jt j t jt jtt

T

jt jt jt

TH H

jt jt jt

L p e

R R w e

R R

e e

Summing up 24

The Kuhn – Tucker conditions

, 1

, 1

0 ( 0 for 0)

0 ( 0 for 0)

0( 0 for )

0( 0 for )

0( 0 for )

Ht jt jt jtH

jt

jt j t jt jtjt

Hjt jt j t jt jt

jt jt j

H Hjt jt j

Lp e

e

LR

R

R R w e

R R

e e

Summing up 25

Investments

Deregulation of electricity sector Unbundling generation and transmission

Investment in generation and investment in transmission made by independent organisations, but investments need coordination both over time and over space

Use of shadow prices on capacities as marginal investment signals