Hydropower Economics: An Overview

Finn R FørsundProfessor EmeritusUniversity of Oslo

*Slides prepared for

ECON4925 - Resource EconomicsOctober 5 2017

Hydropower economics 1

• CurriculumHydropower Economics:

An Overview.

Encyclopedia of energy,

Natural Resource and

Environmental Economics,

pp. 200-208

Hydropower economics 2

10 top hydroelectricity produers

Hydropower economics 3

Renewable power 10 largest producers

Rank Country Year Total

renewable Hydropower

Wind

power

Biomass

Solar

power

Geother

mal

1 China[1]

2014 1,300.0 1,066.1 160.0[2] 42.0 43.2 -

2 United States[3] 2015 549.5 251.2 190.9 64.2 38.6 16.8

3 Brazil

2012 451.5 411.2 5.0 35.3 - -

4 Canada 2012 397.3 376.7 11.3 9.0 0.5 -

5 India[4]

2014 199.1 131.6 37.1 25.5 4.9 -

6 Germany[5] 2014 168.4 25.4 57.4 49.4 36.1 0.1

7 Russia

2012 167.9 164.4 - 3.0 - 0.5

8 Japan

2014 148.6 81.0 5.0 35.5 24.5 2.6

9 Norway[6][5]

2014 139.0 136.6 2.2 0.2 - -

10 Italy

2014 122.6 58.0 15.2 21.2 22.3 5.9

Hydropower economics 4

Weekly inflow and production of hydropower in Norway 2003

Weeks

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Inflow

Production

GW

h

Source: OED: Fakta 2005

Hydropower economics 5

The basic model for management of hydropower with reservoir

• Hydroelectricity production function

– a: water per kWh

• Water accumulation equation

• Reservoir constraint

tR R

1H

t te ra

1 1

1 , 1,..,

H

t t t t t t t

H

t t t t

R R w r R w ae

R R w e t T

Hydropower economics 6

The social planning problem

• The objective function

• Maximising consumer plus producer surplus

• Area under the demand curve from 0 to optimal value of

1 0

( )

HteT

t

t z

p z dz

Hydropower economics 7

H

te

The optimisation problem The Lagrangian

• Shadow prices

• The envelope theorem

1 0

1

max ( )

subject to

, 0 , 1,..,

, , , given, free

HteT

t

t z

H

t t t t

t

H

t t

t o T

p z dz

R R w e

R R

R e t T

T w R R R

1 0

1

1

1

( )

( )

( )

HteT

t

t z

TH

t t t t t

t

T

t t

t

L p z dz

R R w e

R R

Hydropower economics 8

• The necessary first-order conditions

• Dynamics of shadow price determination

– Water values are the same for t and t+1 if the reservoir constraint is not binding

1

1

( ) 0 ( 0 for 0)

0 ( 0 for 0)

0( 0 for )

0( 0 for ) , 1,..,

H H

t t t tH

t

t t t t

t

H

t t t t t

t t

Lp e e

e

LR

R

R R w e

R R t T

Hydropower economics 9

• The interior solution for the price of electricity

• The price will only change if the water value changes, and this will only happen if the reservoir constraint is binding (empty or full)

Hydropower economics 10

Bathtub diagram for two periodsInterior solution for reservoir levels

Hydropower economics 11

p1

M D C B A R

1

He

1

He

1 1( )Hp e 2 2( )Hp e

Period 2 Period 1

p2=λ2

p2

2

He

2

He

p1=λ1

Total available water

1 2oR w w

Upper reservoir constraint binding in period 1

Hydropower economics 12

Bellman’s backward induction

• Using up all the available water in the terminal period T

• Inserting in the demand function gives the optimal price in period T

– We have to solve for

• Range of transfer

Hydropower economics 13

1

H

T T Te R w

1( ) ( )H

T T T T T Tp p e p R w

1 [0, ]TR R

1TR

• Stepping one period back from T

• Assuming interior solution for

• Solving for

• Solution for if we have the solution for

Hydropower economics 14

1 2 1 1

H

T T T Te R R w

1TR

1 1 1

1

1 1 1

( ) ( )

( ( ))

H

T T T T T

H

T T T T T

p e p R w

e p p R w

2TR

2 1 1 1

1

1 1 1 1( ( ))

H

T T T T

T T T T T T

R R w e

R w p p R w

• Repeating going backward to period 1 assuming interior solutions for the reservoir levels

• We know , can then solve for and then all prices, reservoir levels and output levels

Hydropower economics 15

1 11

0 1 1

1 1

( ( ))T T

T i i T T T

i i

R R w p p R w

0R 1TR

Binding reservoir constraints• Empty reservoir in period t+1 implies = 0

• Assuming interior solution for the reservoir from T to t+2 implies same price all periods, and price in period t+1 typically higher than this price

Hydropower economics 16

• Threat of overflow in period s, giving a full reservoir to period s+1, s+1<t, interior solutions backward from t to s+1, use period t+1 as the terminal period T

• Lower price in period s, if interior solutions from s-1 to t=1 then a lower price regime

Hydropower economics 17

1

1 1

1 1

1

1 1

1 1

( ( ))

( ( ))

t t

s t i i t t t

i s i s

t t

t i i t t t

i s i s

R R w p p R w

R R w p p R w

• Finding the lower price in period 1,…,s using

– R0 known, only Rs-1 unknown

Hydropower economics 18

1 11

0 1 1

1 1

( ( ))s s

s i i s s s

i i

R R w p p R w

Limit on production capacity period 2

, 1,..,H H

te e t T

Hydropower economics 19

p1

p1=λ1

D C

He

A 1

He

1 1( )Hp e 2 2( )Hp e

Period 2 Period 1

ρ2

p2

2

He

λ2

p2=λ2+ρ2

B B' B''

Multiple producers• Energy balance

• Necessary first-order conditions

Hydropower economics 20

1

, 1,.., , 1,..,N

H

t jt

j

x e j N t T

1

, 1

, 1

( ) 0( 0 for 0)

0( 0 for 0)

0( 0 for )

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

NH H

t jt jt jtHjjt

jt j t jt jt

jt

H

jt jt j t jt jt

jt jt j

Lp e e

e

LR

R

R R w e

R R t T j N

Hydro together with other technologies

• Thermal technology

• Intermittent technologies

• Coefficients

– Capacity factor; fraction of use of full installation of power capacity

Hydropower economics 21

( ),Th Th Th

t tc e e e

, [0,1], wind,solar,run-of-riverI I

it it i ite a e a i

ita

I

ie

• The optimisation problem of the social planner

Hydropower economics 22

1 0

1

max [ ( ) ( )]

subject to

, , , 0, 1,..,

, , , , , ( 1,..., ) given,

free

txTTh

t t

t z

H Th I

t t t t

H

t t t t

t

Th Th

t

H Th

t t t t

Th I

o t t

T

p z dz c e

x e e e

R R w e

R R

e e

x e R e t T

T R R e e w t T

R

• The Lagrangian

Hydropower economics 23

1 0

1

1

1

1

[ ( ) ( )]

( )

( )

( )

H Th It t te e eT

Th

t t

t z

TTh Th

t t

t

TH

t t t t t

t

T

t t

t

L p z dz c e

e e

R R w e

R R

• The necessary first-order conditions

Hydropower economics 24

1

1

( ) 0 ( 0 for 0)

( ) ( ) 0 ( 0 for 0)

0 ( 0 for 0)

0 ( 0 for )

0 ( 0 for )

0 ( 0 for ), 1,..,

H Th I H

t t t t t tH

t

H Th I Th Th

t t t t t t tTh

t

t t t t

t

H

t t t t t

t t

Th Th

t t

Lp e e e e

e

Lp e e e c e e

e

LR

R

R R w e

R R

e e t T

• Determination of price, interior solutions

• Period Tj : prices are the same

• Use of thermal capacity constant for T

• Hydro swing producer

• No use of hydro

Hydropower economics 25

( ) ( ) , , 1,...,H Th I Th

t t t t t t j jp e e e c e p t T j J

1 1 1( ) ( )

hydroswing demandchange intermittent change

, 1,...,

H H I I

t t t t t t

j

e e x x e e

t T j J

( )Th I

t t t tp e e

Bathtub of mix of technologiesIntermittent only in period 1

Hydropower economics 26

Intermittent only in period 2

Hydropower economics 27

Additional hydropower issues• Uncertainty

– Inflows and demand stochastic variables– Maximising the expected consumer plus producer

surplus

• Market power– The monopoly case: prices can be raised by

reducing output– But wasting water will be stopped by a regulator– The monopolist set flexibility-corrected prices

equal. More water is used when demand is elastic and less water when demand is inelastic

Hydropower economics 28

• Transmission and many production-and consumer nodes– Nodal pricing: optimal prices vary between nodes due

to loss in the network (Ohm’s law) and congestion on lines

– Problematic to implement in practice: In Norway 5 price regions

• Trade between countries– Export – import will make prices more equal

– Congestion of cables an issue: congestion rent

• Norway as the battery of Europe– Instantly available power capacity of hydropower

Hydropower economics 29