"Pum ping W ater to Com pete in Electricity Markets" C. - - PowerPoint PPT Presentation

"Pum ping W ater to Com pete in Electricity Markets"

• C. Cram pes and M. Moreaux

Energy Centre Workshop, February 2007

Energy Centre Workshop, February 2007 2

1) Introduction

Energy Centre Workshop, February 2007 3

Energy Centre Workshop, February 2007 4

Energy Centre Workshop, February 2007 5

Energy Centre Workshop, February 2007 6

2) Model setting

* Steady state with two elementary periods t = 1, 2 * Two plants: thermal plant with cost ( ) c q ,

' ''

0, c c > ≥ . hydro plant constrained by pumped water

1 2 1 2 H H

q q f f + ≤ + pumping technology: f α thermal kWh are necessary to pump the quantity of water f that will be transformed into f hydro kWh (where 1 α > ). * Final consumers:

( )

T H t t t

u q q +

strictly concave * We successively consider efficient mix first best monopoly

• pen-loop Cournot game

Energy Centre Workshop, February 2007 7

• 3. Efficient production schemes

When the management of the two plants is integrated, the operator chooses a mix that solves

1 1 2 2 ( , , , 1,2)

min ( ) ( )

H T t t t

T T q q f t

c q f c q f α α

=

+ + +

s.t.

1 2 1 2 H H

q q f f μ + ≤ + 1,2

H T t t t t

q q q t γ + ≥ =

≥

H t

q , ≥

T t

q , ≥

t

f 2 , 1 = t

t

ν

Energy Centre Workshop, February 2007 8

* Lemma 1 Assume that

t

q > . Then cost minimization implies:

T H t t t

q q q + = and

1 2 1 2 H H

q q f f + = + ;

H t t

f q × = . * Implication: electricity from the hydro-producer at any period t is actually coming from water pumped during period 1 t − . And since thermal marginal cost is increasing …

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* Lemma 2 ∀ t and ' t ( , ' 1,2 and ' t t t t = ≠ ),

' t t

q q ≥ > implies that

' H t

q = . * Implication: if

' t t

q q =

, then

' H H t t

q q = =

and consequently

'

0.

t t

f f = =

Actually, the hydro-system is a storage device or transfer device of the energy produced at one period to the next at a cost represented by the fraction of the energy lost during the transfer.

Energy Centre Workshop, February 2007 10

Hence, it must be used if and only if, without transfer, the marginal cost differential resulting from the production of quantity

t

q exclusively during period t is higher than

the marginal loss implied by the transfer. Lemma 3 Suppose

' t t

q q ≥ >

, ( , '

1,2 and ' t t t t = ≠

). Then

'

'( ) '( )

t t

c q c q α ≤

is a necessary and sufficient condition for

'

0.

t t

f f = =

Energy Centre Workshop, February 2007 11

period 2 is peak

2 1

q q >

"hydro zone"

"no-hydro zone"

( )

2 1 m

q q

45°

2

q

1

q

Let

( )

2 1 m

q q

be the solution to

2 1

'( ) '( ) c q c q α =

Example: if

'( ) c q c cq = +

( ) ( )

2 1 1

1

m

c q q q c α α = − +

Energy Centre Workshop, February 2007 12

"no-hydro zone"

α = ∞

2

q

1

q

( ) ( )

2 1 1

1

m

c q q q c α α = − +

1 α =

"hydro zone"

2

q

1

q

"no-hydro zone"

Energy Centre Workshop, February 2007 13

• 4. Integrated management

* First best:

( , , , 1,2)

1,2

max ( ) ( )

H T q q f t t t t

T H T t t t t t t

u q q c q f α

=

=

+ − +

∑

s.t.

2 1 2 1

f f q q

H H

+ ≤ + , ≥

H t

q , ≥

T t

q , ≥

t

f 2 , 1 = t As 2 = t is the peak period,

1 2

= =

H

q f in any case. Proposition 1: First best dispatch is: either all-thermal:

2 1

= =

H

q f and

u ti T t

q q = defined by

' '

( ) ( )

u u t ti ti

u q c q =

, 2 , 1 = t if ) ( ) (

1 ' 2 ' 2 u i u i

q c q u α < .

• r a mixed solution:

) ( ) (

2 1 ' 1 ' 1 H T T

q q c q u α + = and ) ( ) ( ) ( '

2 ' 2 2 ' 2 2 1 T H T H T

q c q q u q q c = + = +α α

• therwise.

Energy Centre Workshop, February 2007 14

2 u i

q

1 u i

q

' c α

€/kWh

quantities

' c

' 2

u

' 1

u

) ( ) (

1 ' 2 ' 2 u i u i

q c q u α <

NO HYDRO

Energy Centre Workshop, February 2007 15

2 2 T H

q q +

2 T

q

2 u i

q

1 T

q

1 1 T

q f α +

' 1

u

1 u i

q

' c α

€/kWh

quantities

' c

' 2

u

' ' 2 2 1

( ) ( )

u u i i

u q c q α >

THERMAL + HYDRO

welfare loss welfare gain

Energy Centre Workshop, February 2007 16

* Private monopoly: Standard result as regards outputs: under-provision of energy Concerning the energy mix, all cases are possible

4 3 "hydro zone"

"no-hydro zone"

2

2

q

1

q

1

Energy Centre Workshop, February 2007 17

• 5. Cournot competition

control sets: T decides on

1 2

and

T T

q q H decides on

1 2

and

H H

q q Who decides on

1 2

and f f ? alternative institutional arrangements concerning

1 2

and f f : T decides on its total output H is an eligible consumer: it decides on inflows

• 1

2

and f f are strategic variables chosen by the social planner: TBC

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5.1. The thermal firm controls its total output: Let

def T T t t t

y q f α = + The thermal firm solves

( )

( ) ( )

, 1 2

1,2

max

T T y y

H T T T t t t t t t t

p q y f y c y α

=

+ − −

∑

The hydro firm solves

( )( )

( , 1,2)

1,2

max

H q f t t t

H T H t t t t t t t

p q y f q f α α

=

=

+ − −

∑

s.t.

2 1 2 1

f f q q

H H

+ ≤ + μ ≥

H t

q , ≥

t

f 2 , 1 = t ν

Energy Centre Workshop, February 2007 19

• hydro firm’s FOCs:

:

H t

q

'

( )

H H t t t t t

p q f p α ν μ + − + − = 2 , 1 = t :

t

f

'

( )

H f t t t t t

p q f p α α α μ ν − − − + + = 2 , 1 = t Having both

H t

q > and

t

f > would require

( )

' '

( ) ( )

H H t t t t t t t t

p q f p p q f p α μ α α + − = = + − that is μ = or 1 α = Therefore, in this institutional setting,

H t

q > and

t

f > cannot be simultaneously true. Analysis of the Cournot equilibrium: TBC

Energy Centre Workshop, February 2007 20

5.2. The hydro producer is an eligible consumer: The thermal firm solves

( )

( )( ) ( )

, 1 2

1,2

max

T T q q

H T T T t t t t t t t t

p q q q f c q f α α

=

+ + − +

∑

The hydro firm solves

( )( )

( , 1,2)

1,2

max

H q f t t t

H T H t t t t t t

p q q q f α

=

=

+ −

∑

s.t.

2 1 2 1

f f q q

H H

+ ≤ + μ ≥

H t

q , ≥

t

f 2 , 1 = t ν

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Two types of equilibrium when H is active most likely:

1 1 2 2

0, 0,

H H

q f q f = = > = but we cannot exclude

1 1 2 2

0, 0, 0,

H H

q f q f > > > =

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• hydro firm’s FOCs:

:

H t

q

'

( )

H H t t t t t

p q f p α ν μ + − + − = 2 , 1 = t :

t

f

f t t

p α μ ν − + + = 2 , 1 = t

• Note the difference with the case where T controls

def T T t t t

y q f α = + :

t

f

'

( )

H f t t t t t

p q f p α α α μ ν − − − + + =

Energy Centre Workshop, February 2007 23

1 1 2 2

0, 0, 0,

H H

q f q f > > > =

• Having both

H t

q > and

t

f > requires

'

( )

H t t t t t

p q f p p α μ α + − = =

• r

( )

'

( ) 1

H t t t t

q f p p α α − = − , which means

H t t

q f α <

• Not possible at the peak period but it can be the solution at the off-peak period:

1 H

q > when

1

f > requires

( )

' 1 1 1 1

p f p p α μ α + − > =

• r

( )

( )

( )

' 2 1 1 1 1

1

H T T

q p q p q α α − > −

Energy Centre Workshop, February 2007 24

• actually

1 H

q > helps to maintain the cost of water

1

p α low;

• Most likely when

2 H

q is expected to be large α is close to 1

' 1

p is large

Energy Centre Workshop, February 2007 25

1 1 2 2

0, 0,

H H

q f q f = = > = illustration: choice of the thermal unit given

H

q2 ) ( '

2

q c total demand

H

q2 α

1

q ) ( '

1

q c

2

q

H

q2 residual demand

• ff peak period

peak period

Energy Centre Workshop, February 2007 26

Conclusions and extensions

• key role of hydro production units thanks to flexibility
• pump storage, a way to store electricity: costly but efficient under some

circumstances

• inefficient use of pump storage under competition
• extensions
• hydro capacity constraints
• restrictions to thermal flexibility (ramping rates, warming-up)
• more competition with several firms, each controlling both technologies
• random natural inflows
• …