Summing up1 ECON 4930 Autumn 2007 Electricity Economics Lecture 12 Lecturer: Finn R. Førsund.
Term paper 20071 ECON 4930 Term paper Finn R. Førsund.
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Transcript of Term paper 20071 ECON 4930 Term paper Finn R. Førsund.
Term paper 2007 1
ECON 4930 Term paper
Finn R. Førsund
Term paper 2007 2
1a. Define the situation of overflow of the reservoir The water accumulation equation
Strict inequality means that the amount of water at the end of period t is less than the sum of what was received from period t-1 plus inflow during period t subtracted the release during period t → overflow
Overflow implies that , the reservoir capacity
1t t t tR R w r
tR R
Term paper 2007 3
1b. What is the unit of measurement of the fabrication coefficient a? Explain the calibration of the coefficient Unit for a:
Calibration: The height of the fall from the reservoir to the
turbine, called head. Gravity gives the energy of water
The efficiency losses due to friction in pipes, turbine not perfect, totalling 10-15% loss
3
( )tHt
r ma
e kWh
Term paper 2007 4
1c. Converting variables measured in water to variables measured in kWh Inserting the production function into the
water accumulation equation:
1 1
1
Ht t t t t t t
Ht t tt
R R w r R w ae
R R we
a a a
Term paper 2007 5
2. The social planning problem
1 0
1
max ( )
subject to
, 0
, , , , given, free, 1,..,
HteT
tt z
Ht t t t
t
H Ht
Ht t
Ht o T
p z dz
R R w e
R R
e e
R e
T w R R e R t T
Term paper 2007 6
2a. Discuss the objective function for the planning problem The objective function is the area under the inverse demand curve (NB! Choke price finite) Demand function can be linked to utility function The model is partial because there are no links to
other activities, goods, etc. in the economy A typical general objective function is the consumer
plus producer surplus. Because variable production costs are zero we are left with the area under the demand curve.
Discounting is neglected due to short total time period
Term paper 2007 7
2b. Why is the planning problem formulated as a dynamic problem? Having a reservoir means that water used
today can alternatively be used tomorrow, water has an opportunity cost
Term paper 2007 8
2c. Discuss reasons for a constraint on production to be realistic Production measured in kWh can have an
upper limit for a period due to technical reasons The flow of water through the pipe hitting the
turbines is constrained by the diameter The conversion to electricity is constrained by
installed turbine capacity The production of electricity may be constrained
by the size of the generator
Term paper 2007 9
2d. Kuhn – Tucker conditions
The Lagrangian function
1 0
11
1
1
( )
( )
( )
( )
HteT
tt z
TH
t t t t tt
T
t tt
TH H
t tt
L p z dz
R R w e
R R
e e
Term paper 2007 10
2d., cont.
The Kuhn – Tucker conditions
1
1
( ) 0 ( 0 for 0)
0 ( 0 for 0)
0( 0 for )
0( 0 for )
0( 0 for ) , 1,..,
H Ht t t t tH
t
t t t tt
Ht t t t t
t t
H Ht t
Lp e e
e
LR
R
R R w e
R R
e e t T
Term paper 2007 11
2d., cont. Interpretation of shadow prices:
Change in the optimised objective function of a marginal change in the constraint, found by partial differentiation of the optimised Lagrangian
Shadow price on the water accumulation constraint Change in the objective function of a marginal change in
the constraint (i.e., change in Rt-1,wt)
1 0
1 1
( ( ) )*
HteT
tt z
tt t
p z dzL
R R
Term paper 2007 12
2d., cont.
Shadow price on the reservoir capacity constraint
Shadow price on the production constraint
1 0
( ( ) )*
HteT
tt z
t
p z dzL
R R
1 0
( ( ) )*
HteT
tt z
tH H
p z dzL
e e
Term paper 2007 13
2e. Circumstances that may lead to a binding constraint for production. Concept of locking in of water and manoeuvrability of the reservoir Constraining production
Satisfying consumption in a high-demand period Producing in order to prevent overflow
Locking in of water Impossible to prevent overflow physically
Manoeuvrability The rate of maximal production relative to
reservoir size
Term paper 2007 14
2f. Kuhn – Tucker conditions for period T
Realistic assumptions
No satiation of demand: price positive Binding production constraint in period T
Not realistic unless T is a high-demand period, prevention of overflow is not realistic in the last period
( ) 0 ( 0 for 0)
0 ( 0 for 0)
H HT T T T TH
T
T T TT
Lp e e
e
LR
R
0, 0, 0 0HT T T Te p
Term paper 2007 15
2g. A bathtub diagram illustration
pT-1
λT-1
M DCBA
Period TPeriod T-1
λT
Total available water
pT
Term paper 2007 16
2h. Events that may lead to social price and shadow price changes Threat of overflow t
Emptying the reservoir t
Binding production constraint t
1
1 1 1 1
0 ( 0), 0( 0)
, ,t t t t t
t t t t t t t t t
R
p p p p
1
1 1
0 ( 0), 0( )
,t t t t t t
t t t t
R R R
p p
( ) 0 ( 0), 0( )
( ) ( )
H Ht t t t t t
H Ht t t t t t t
p e e
p e p e
Term paper 2007 17
2i. Shadow prices on stored water for period u+2, u+1 and u Production constraint binding for period u+1,
but not for period u, and u+2 to T Reservoir in between full and empty from T-1
to u
2 1
2 1 1 1 1
,
,u u u T
u T u u u u T
u u T
p p p p
p
Term paper 2007 18
2i. Illustration: two-period bathtub diagram for periods u and u+1pu
pu=λu=pT
DCA
Period u+1Period u
ρu+1
Pu+1
pT=λu+1
Pu+1=λu+1+ρu+1
BB’’