ECON 4925 Autumn 2007
Electricity Economics
Lecture 2
Lecturer:
Finn R. Førsund
Reservoir constraint
1
Non-linear programming

Problem formulation (Sydsæter et al., 1999)
max f ( x1 ,..., xn ) subject to g j ( x1 ,..., xn )  b j , j  1,.., m
xi  0, i  1,.., n

The Lagrangian function
m
L( x )  f ( x )    j ( g j ( x )  b j )
j 1

Kuhn – Tucker necessary conditions
g j ( x)
L( x) f ( x) m

 j
 0( 0 for xi  0)
xi
xi
xi
j 1
 j  0( 0 for g j ( x)  b j ), j  1,.., m
Reservoir constraint
2
Applying Kuhn - Tucker

The Lagrangian
T
L
etH

t 1 z  0
pt ( z ) dz
T
 t ( Rt  Rt 1  wt  etH )
t 1
T
  t ( Rt  R )
t 1
Reservoir constraint
3
Applying Kuhn – Tucker, cont.

The necessary first-order conditions
L
H
H

p
(
e
)



0
(

0
for
e
t
t
t
t  0)
H
et
L
 t  t 1   t  0 ( 0 for Rt  0)
Rt
t  0 ( 0 for Rt  Rt 1  wt  etH )
 t  0 ( 0 for Rt  R ) ,
t  1,.., T
Reservoir constraint
4
Interpreting the Lagrangian
parameters

Shadow prices on the constraints

A shadow price tells us the change in the value of
the objective function (value function) when the
corresponding constraint is changed marginally

t : shadow price on stored water, water

value, the benefit of one unit of water more
t : shadow price on the reservoir constraint,
the benefit of one unit increase in capacity
Reservoir constraint
5
Interpreting the Kuhn – Tucker
conditions

Inequality in the first condition



Social price lower than the shadow price on
stored water
Nothing will be produced
Inequality in the second condition


Water value in the next period lower than the sum
of water value in the present period and the
shadow price on the reservoir constraint
May have overflow or just reaching reservoir
capacity
Reservoir constraint
6
Qualitative interpretations

Reasonable assumption


Positive production in each period →
Social price equal to the water value
pt (etH )  t


Assuming reservoir capacity is not reached→
Water value in the present period will be equal to
the water value in the next period
 t  0 ( Rt  R )  t  t 1
Reservoir constraint
7
Qualitative conclusion

For consecutive periods with the reservoir
constraint not binding the water values are all
the same
t  t 1   , t  T t0

The social price is the same for all periods
with non-binding reservoir constraints and
equal to the common water value
pt (e )  pt 1 (e )  , t T t 0
H
t
H
t 1
Reservoir constraint
8
Hourly price variation four days in 2005
NOK / MWh
250
230
Wed. 20.07
Mon. 24.01
210
Sun. 17.07
190
Sun. 23.01
170
150
1
3
5
7
9
11 13 15 17 19 21 23
Hours
Reservoir constraint
9
Price-duration curve Norway 2005
800
700
600
NOK
/MW
500
h
400
300
200
100
0
1
1000
2000
3000
4000
5000
6000
7000
8000
8760
Hours
Reservoir constraint
10
The bathtub diagram

Only two periods
Available water in the first period
Ro  w1

Available water in the second period

R1  w2

Adding together the two water-storage
equations
e  e  Ro  w1  w2
H
1
H
2
Reservoir constraint
11
The bathtub diagram for two periods
Period 2
Period 1
p2
p1
p1(e1H)
p2(e2H)
λ2
λ1
e1H
A
e1H
e2H
B
M
C
e2H
D
Total available water
Ro + w1 + w2
Reservoir constraint
12
Backwards induction, Bellman

Start with the terminal period
p2 (e2H )  2 (e2H  0),
2   2  0 ( 0 if R2  0)

Assumption: no satiation of demand
p2 ( w2  Max R1 )  p2 ( w2  R )  0

Implication
R2  0,  2  0, p2 (e2H )  2  0

No threat of overflow in period 1
R1  R   1  0, p1 (e1H )  1  2  0
Reservoir constraint
13