2
1
April 2004
Handout 4925 Resource Economics
Finn R. Førsund
Hydropower
Electricity generators can use water, fossil fuels, bio-fuels, nuclear fuels, wind and geothermal
energy as primary energy sources to run the turbines producing electricity. Hydro power is
based on water driving the turbines. The primary energy is provided by gravity and the height
HYDROPOWER
the water falls down on to the turbine. We will assume the existence of a reservoir. The
potential for electricity generation of one unit of water (a cubic meter) is usually expressed by
the height from the dam level to the turbine level. The reservoir level will change when water
1. Introduction
is released and thus influence the electricity production. Electricity production is also
influenced by how processed water is transported away from the turbine allowing fresh water
Electricity
to enter the turbine. The turbine is constructed for an optimal flow of water. Lower or higher
Electricity is one of the key goods in a modern economy. The nature of electricity is such that
inflow of water will reduce the electricity per unit of water.
supply and demand must be in a continuous physical equilibrium. The system breaks down in
a relatively short time if demand exceeds supply and vice versa. The system equilibrium is
The key question in hydro power production is the time pattern of the use of the water in the
therefore demand-driven. The spatial configuration of supply and demand is important for
reservoir. Given enough storage capacity the water used today can alternatively be used
understanding the electricity system. The generators and consumers are connected by a
tomorrow. The analysis of hydro power is therefore essentially a dynamic one. This is in
network for transport of electricity. There is energy loss in the network. Physical laws govern
contrast with a fossil fuel (e.g. coal) generator. The question then is how to utilize the given
the flows through the networks and the energy losses. The electricity is characterised by
production capacity for each time period. Assuming that the market for the primary energy
voltage (220 - 240 V) and the Hertz number (50 ± 2 for alternate current) and measured as
source functions smoothly this is not a dynamic problem, but is a problem solved period for
effect (W), i.e. instantaneous energy, and energy (kWh), i.e. the amount of electricity during a
period.
time period (the integral of the effect over the time period in question). The capacity rating of
the turbines of the generators is in effect units. The network, or grid, has capacity limits in
effect units for a given spatial configuration of supply and demand nodes in the network.
The key variables we are going to use are reservoir Rt , inflow of water wt and electricity
production etH . Flow variables in small letters are understood to refer to the period, while
The time period used in a study of the electricity system is of crucial importance for how the
stock variables in capital letters refer to the end of the period, i.e. water inflow wt takes place
system is modelled. If the time resolution is one hour we can portray the demand by looking
during period t, while the content of the reservoir Rt refers to the water at the end of period t.
at the variation in energy use hour by hour during a day. The demand varies typically over
Release of water during period t is converted to electricity etH measured in kWh according to a
the day with the lowest energy consumption during the night and peaks at breakfast time and
fixed transformation coefficient, reflecting the vertical height from the centre of gravity of the
the start of the working day, and again a peak round dinner time. To see the need for effect
dam and to the turbines. Water reservoir and inflow can also be measured in kWh using the
capacity it is common to look at the demand for one year and sort the 8760 hours according to
same conversion. The reduced electricity conversion efficiency due to a reduced height (head)
the highest demand and then decreasing. The hours with the highest energy demand are
the water falls as the reservoir is used is disregarded. For the Norwegian system with high
called the peak load, and the hours with the lowest demand are called the base load. In
differences in elevations between dams and turbine stations of most of the dams, and having
between we have the shoulder.
few river stations, this is an acceptable simplification at our level of aggregation.
4
3
The transformation of water into electricity can be captured in the simplest way by the
production function
1
e ≤ rt
a
2. The basic hydro model
(1)
H
t
where rt is the release of water from the reservoir during time period t and a is the fabrication
coefficient for water. As mentioned above the coefficient may vary with the utilisation of the
reservoir, and also with the release of water due to the construction of the turbine giving
maximal productivity at a certain water flow. By assumption there are no other current costs.
This is a very realistic assumption for hydropower. We will in the following assume that the
production function (8) holds with equality and therefore we can drop this relation and
measure water in electricity units.
Social optimum
We will assume that there is unlimited transferability of water between the periods of the
given total amount of water available as in (3) above. The horizon is a seasonal cycle (one
year) from spring to spring. In Norway the snow smelting during a few spring weeks fills the
reservoirs with about 70 percent of the yearly total. Water is measures in energy units, kWh,
and no conversion from water to electricity is shown. The energy consumption in each period
is evaluated by utility functions, which can be thought of as valid either for a representative
consumer, or constitute a welfare function. There is no discounting (the horizon is too short
for discounting to be of significance). The utility functions are:
The dynamics of water management is based on the filling and emptying of the reservoir:
Rt ≤ Rt −1 + wt − etH , t ∈ T
(2)
Strict inequality means that there is overflow.
U t (etH ) , U t '(etH ) ≥ 0 , U t ''(etH ) < 0 , t ∈ T
(4)
Note that the utility functions are concave. We will define the marginal utility U t ' measured
in monetary units as the marginal willingness to pay, pt, i.e. the demand function for
Some studies of hydropower at a high level of aggregation (references) disregards the storage
process and specifies directly the available water within a yearly weather cycle. The
assumptions are then no spill of water and no binding upper reservoir constraint. Since the
electricity:
U t '(etH ) = pt (etH ) ,
(5)
where pt will also be referred to as the “price” of electricity for short below.
time horizon for water management problems seldom is more than one year, it is common to
disregard discounting. The period concept may be as crude as two periods (summer and
The social optimisation problem can be formulated as follows:
winter season based on difference in inflow and/or release profile), and anything from month,
Max
weeks, days and hours. Although it is obvious that the world continues after one year, it is
s.t.
also usual not to specify any terminal value of the reservoir or to operate with a ”scrap” value
∑e
for the reservoir content of the last period. We can then simply add up the water releases to
obtain the water constraint for the chosen horizon of the periods:
∑e
t∈T
H
t
= Ro + ∑ wt = W
(3)
t∈T
where W is the total available inflows (including any water Ro in the reservoir at the
∑ U (e
t∈T
t∈T
t
H
t
)
(6)
H
t
≤W
The Lagrangian function is:
L = ∑ U t (etH ) −ν (∑ etH − W )
t∈T
(7)
t∈T
Necessary conditions are1:
beginning of the first period from the past).
1
The use of ” ⊥ ” is a short-hand notation for the conditions
(1999), p. 100).
∂L
∂L
≤ 0, etH
= 0 (Sydsæter, Strøm and Berck
∂etH
∂etH
5
6
∂L
= U t '(etH ) −ν ≤ 0 ⊥ etH ≥ 0 , t ∈ T
∂etH
periods by using a “bath tub” diagram showing total available energy for the two periods and
(8)
ν ≥ 0 (= 0 if ∑ etH < W )
the two marginal willingness to pay-functions measured along one vertical axis each. The
economic interpretation of the solution to the allocation problem is that energy should be
t∈T
From the Kuhn – Tucker conditions we know that if we have an interior solution for the
energy consumption for period t, etH > 0 , then the shadow price on the energy constraint must
be positive, ν > 0, and by complementary slackness the energy constraint must be binding. A
sufficient condition for a maximum is that the Lagrangian (7) is concave, which is satisfied
under our assumptions.
allocated on the periods in such a way that the shadow price of energy (i.e. the increase in the
objective function of a marginal increase in the given amount of total energy) is equal to the
marginal utility of energy in each period, thus the marginal utilities become equal. In the
illustration in Figure 1 if Period 1 is summer and Period 2 winter, the marginal utility should
be equal. Although the marginal utility of energy consumption may be higher in the winter
than in the summer for any level of consumption, the marginal utility in the winter should not
Result 1: The law of one price. The price of electricity is constant and equal for all periods if
marginal willingness to pay remains positive for all periods.
become greater than in the summer.
Constraints in hydropower modelling
Demonstration: According to (8) we have a binding constraint if the marginal willingness to
pay, the price, remains positive for all periods, and it is then equal to the common shadow
price on the energy constraint. By complementary slackness the shadow price is positive.
Result 1 is the Hotelling’s rule for our model. We do not discount, and by arbitrage the price
must be the same for all periods.
There are many constraints on how to operate a dam. A fundamental constraint is the maximal
amount of water that can be stored. This constraint will have a crucial importance for how the
dam can be operated. Environmental concerns may impose a lower limit on how much the
dam can be emptied. Empty dams create eyesores in the landscape, and can create bad smells
from rotting organic material along the exposed shores. Fish may have problems surviving or
spawning at too low water levels. The environmental lower constraint has a time index,
because the environmental problems may vary with season. In Norway where the dams are
The typical interior solution for both periods is illustrated in Figure 1 in the case of two
covered by ice in the winter season the lower level may be less then than in the summer.
The effect capacity of a power station may be constrained by the installed turbines or the
Period 1
diameter of the pipe from the reservoir to the turbines. Such a constraint has no period
Period 2
subscript. The effect concept will follow the period definition. For example, if the period
length is one hour the effect constraint is measured in kWh, by using the maximal kW for one
hour.
U1’ = ν
U2’ = ν
In aggregated analyses it is common not to specify the transmission system. But a constraint
on the transmission can be represented the same way as for effect capacity constraint, except
that a time index may be used on the constraint to indicate that transmission capacity within
Total available energy
Figure 1. Optimal allocation of energy on the two periods
some limits is an endogenous variable governed by physical laws of electrical flows of active
and reactive power in a multi-link grid system between input and output nodes.
7
8
The Lagrangian is:
Table 1. Constraints in the hydropower model
L=
Expression
∑ U (e
t∈T
t
H
t
)
−∑ λt ( Rt − Rt −1 − wt + etH )
Constraint type
(10)
t∈T
Max Reservoir
Rt ≤ R
Environmental concerns,
Rt ≥ Rt
−∑ γ t ( Rt − R )
t∈T
Min Reservoir
Max Effect capacity
etH ≤ e H
Max Transmission capacity
etH ≤ et H
Water flows, environment
et ≤ e ≤ et
H
Ramping up
0 < e −e
≤ rt
Ramping down
H
Necessary first order conditions are:
∂L
= U t '(etH ) − λt ≤ 0 ⊥ etH ≥ 0
∂etH
H
t
H
t
H
t −1
(11)
∂L
= −λt + λt +1 − γ t ≤ 0 ⊥ Rt ≥ 0
∂Rt
u
Assume positive production in all periods and use that U t '(etH ) = pt (etH ) , t ∈ T :
0 < etH−1 − etH ≤ rt d
pt (etH ) = λt ,
−λt + λt +1 − γ t ≤ 0 ⊥ Rt ≥ 0, t ∈ T
(12)
There may be environmental concerns about the size of the release from a reservoir. If the
The shadow price of the stored water is termed the water value2. According to Bellman’s
release occurs into a river system there may be concerns both about the lower and the higher
principle for solving dynamic programming problems with discrete time we start searching
amount of water that should be released due to impacts on the environment downstream.
for the optimal solution from the last period and then work our way towards the first period.
Impacts on fishing and recreational activity and pressure from tourism may be relevant.
Our horizon ends at T, we must therefore have λT +1 = 0 . For period T we have two
Erosion of riverbanks and temperature change for agricultural activity nearby may also count.
possibilities as to the utilisation of the water in the reservoir, either it is emptied, RT = 0 , or
Then there is concern about navigation and flood control. All these effects may also be
some water is remaining, RT > 0 . Since the water has no value from T+1 on the last situation
present when releases change, so upper constraints may be introduced both on ramping up and
ramping down.
can only be optimal if the marginal utility of electricity becomes zero before the bottom of the
reservoir is reached. We will adopt the other alternative that the marginal utility of electricity
remains positive to the last drop. This means that we will have a situation of scarcity in the
last period, T. Scarcity gives economic value to the water in the last period:
3. Hydro with reservoir constraints
λT +1 = 0, RT = 0 ⇒ −λT − γ T < 0, γ T = 0 ⇒ λT > 0
(13)
The social optimum
The social planning problem is:
Max
∑ U (e
t∈T
t
H
t
Since we cannot have a situation of scarcity at the same time as the upper limit on the
reservoir is reached the shadow price on the upper constraint is set to zero. For all periods up
)
s.t.
Rt ≤ Rt −1 + wt − etH , Rt ≤ R , t ∈ T
(9)
2
But remember that in our simplified model water is measured in electricity units. We should really measure
water in m3 to use the expression. This can easily be done by introducing (1) into the model.
9
10
to T we will then have the water values positive if the upper constraint on the reservoir is not
The first equality follows from the Kuhn-Tucker condition in (11) when there is a positive
reached in any of the periods:
amount of water in the reservoir. The shadow price on water λs is zero if there is overflow.
This follows from the complementary slackness condition for the second term in (11). Then
Result 2. The law of one price: The price of electricity remains positive and constant for all
the shadow price n the reservoir constraint is equal to the water value induced by the scarcity
periods including the last period T if the reservoir is emptied only in the last period and the
period, i.e. λt . This is the benefit of increasing the reservoir constraint with one unit. If there
upper constraint on the reservoir is never reached.
is no spillage and the water is just maintained at the maximal level the water value λs will
Demonstration: Assume λT > 0 and γ u = 0, Ru −1 > 0, u = 1,.., T . We then have from the
typically be positive. In any case the water value λs is smaller than the water value λt for later
periods up to the scarcity period t. The first condition in (13) tells us that a zero water value
necessary conditions (11) that λu > 0∀u = 1,.., T .
can only go together with a zero value of the willingness to pay for electricity. This situation
will typically not be realistic in a world without uncertainty and willingness to pay functions
Scarcity during the period T
Assume that there is a scarcity in period t, and 1 < t < T , but no overflow situation. Using
condition (11) and result (13) we have:
The situation with two periods is illustrated in Figure 2. In the first period we have an inflow
−λt + λt +1 − γ t ≤ 0 ⊥ Rt ≥ 0
λt +1 = λT > 0, Rt = 0, γ t = 0, Ru > 0, γ u = 0, u = 1,.., t − 1
⇒ λu ≥ λT , u = 1,.., t
based on actually estimated demand functions.
(14)
equal to AC, and in the second period an inflow equal to CD. The capacity of the reservoir is
BC. The optimal allocation is to store the maximal amount BC in period 1 and consume what
Result 3: The law of different prices equalling the number of scarcity periods.
Assume the overflow or threat of overflow never occurs. Then each period between two
Period 2
Period 1
scarcity episodes will get its separate price. The period between the starting period and the
λ2
first period with scarcity will get the highest price, and then the price will typically fall for
γ1
each scarcity period experienced.
λ1
Demonstration: Remembering that marginal willingness to pay is defined as the price we
have directly from the first order condition in (11) that the prices are equal to the water values
assuming that a positive amount of electricity will be produced in every period.
Overflow or threat of overflow
Assume overflow or threat of overflow (reservoir completely filled) for a period s < t, where t
A
B
is the first scarcity period after s. We then have from the necessary conditions (11):
Rs > 0, λs +1 > 0, γ s > 0, s < t ⇒
−λs + λs +1 − γ s = 0 ⇒
λs = λs +1 − γ s ≥ 0 ⇒ λs < λs +1 = λt
(15)
Figure 2. Social optimum with reservoir constraint
C
D
11
12
cannot be stores, AB. In the second period the reservoir, containing BC from the first period
The variable etXI is net export and is positive if we have export and negative if we have
and an inflow of CD in the second period, is emptied. Using (15) above we have that
imports. We assume that in one period we can only have either exports or imports, or both are
λ1 = λ2 − γ 1 .
zero.
Overflow or threat of overflow between two periods with scarcity
The Lagrangian is:
Assume that overflow threatens in period s between two periods t1 and t2 with scarcity;
L = ∑ (U t ( xt ) + ptXI etXI )
1 < t1 < s < t2 . For the other periods up to t2 we have neither overflow nor scarcity. From
earlier results we know the situation for the water values up to t1, at s and from s+1 to t2. We
t∈T
−∑ν t ( xt − etH + etXI )
need to establish the situation from t1+1 to s -1:
−λ (∑ etH − W )
Ru > 0, γ u = 0 for u = t1 + 1,.., s − 1, Rs > 0, γ s > 0, − λs + λs +1 − γ s = 0, λs +1 = λt2
The necessary first order conditions are:
⇒ −λs −1 + λs − γ s −1 = 0 ⇒ λs −1 = λs = λt2 − γ s ≥ 0
(18)
t∈T
t∈T
(16)
pt ( xt ) −ν t ≤ 0 ⊥ xt ≥ 0
ν t − λ ≤ 0 ⊥ etH ≥ 0
(19)
ptXI −ν t = 0
Result 4: Cycling prices between periods with overflow and periods with scarcity.
Assume that the first period of scarcity comes before the first period of overflow or threat of
Using the reasonable assumption that xt > 0 ∀t ∈ T we get an adaptation of the foreign price
overflow, and after this episode we have interchanging scarcity and overflow periods in
regime.
between periods with neither scarcity nor overflow. The price will then cycle from higher
values in periods after an overflow (or threat of overflow) episode to the next scarcity period
Result 5: the law of adapting the foreign price regime. If electricity is used in all periods
to a lower price after a scarcity period and until the next overflow (or threat of overflow)
and there is no restriction on import/export capacity, then the foreign price regime will
period.
completely determine the home price.
Demonstration: The marginal willingness to pay, or the price, is equal to the water value
Demonstration: We have directly from the last condition in (19) that
according to the condition (12) assuming that electricity is produced in each period.
pt ( xt ) = ptXI ∀t ∈ T
(20)
But notice that we do not necessarily use hydropower in all periods. If the shadow price on
4. Trade
home consumption is less than the shadow price on water, no water shall be used for
hydropower production in that period, we just import. Without any constraint on the
The social optimum without constraints on transmission/trade
possibility to store water the model is too extreme because we will only export in one period,
The social planning problem is:
the period with the highest export price. The shadow price on water will be set equal to this
⎧
⎫
Max ⎨∑ ⎣⎡U t ( xt ) + ptXI etXI ⎤⎦ ⎬
⎩ t∈T
⎭
s.t.
x t = etH − etXI , ∑ etH ≤ W , t ∈ T
t∈T
maximum price:
(17)
λ = max t∈T { ptXI }
(21)
The total export will be:
etXI* = W − pt* (etH* )
(22)
14
13
where t* is the period corresponding to the period with the maximal export price above. In all
other periods we will only import to the going foreign price. Unlimited trade is therefore only
condition can be written:
of practical interest together with constraints on the possibility to store water. We will not
pt ( xt ) = λ = ν t = ptXI − α t + β t ∀t ∈ T
(26)
develop the case of monopoly for this case. It is obvious that with exogenously given foreign
prices the monopolist will also only export in one period, and he will reduce demand in all
The import/export capacity will typically be fully utilised, and we can divide the periods into
other periods by charging a price higher than the import price in accordance with marginal
export and import periods:
revenue equal to the shadow price on water, which in this case will be the same as above.
Export periods at full capacity
EXP = {t : λ − ptXI < 0} , t ∈ T (α t > 0, β t = 0)
The social optimum with constraints on transmission/trade
Import periods at full capacity:
The social planning problem is:
⎧
⎫
Max ⎨∑ ⎡⎣U t ( xt ) + ptXI etXI ⎤⎦ ⎬
⎩ t∈T
⎭
s.t.
x t = etH − etXI , ∑ etH ≤ W ,
(27)
IMP = {t : λ − ptXI > 0} , t ∈ T (α t = 0, βt > 0)
(28)
We note that the home price can never be lower than the import price in import periods. A
(23)
home price equal to the import price implies that imports will be less than the maximal
capacity. For export periods the home price can never be higher than the export price. A
t∈T
≤ e ≤ e , t ∈T
home price equal to the export price implies that exports will be less than the maximal
The corresponding Lagrangian is
capacity. Notice that the social solution is independent of the values of the export/import price
−e
XI
XI
t
XI
as long as the inequalities in (27) and (28) are fulfilled.
L = ∑ (U t ( xt ) + p e )
XI XI
t
t
t∈T
−∑ν t ( xt − etH + etXI )
The solution for the water value must obviously be positive in a realistic case (extreme
t∈T
−λ (∑ e − W )
t∈T
H
t
(24)
t∈T
abundance of water may result in a home price of zero for all periods and maximal exports in
all periods). The optimal water value is set such that the total available water, W, is just used
−∑ α t (e − e )
XI
t
XI
up on home consumption and exports. An illustration is provided in Figure 3.
−∑ β t ( − etXI − e XI )
t∈T
p
The first order conditions are:
pt ( xt ) −ν t ≤ 0 ⊥ xt ≥ 0
ν t − λ ≤ 0 ⊥ etH ≥ 0
(25)
p −ν t − α t + β t = 0
XI
t
Result 6: The law of one home price. If there is a limited import/export possibility then the
price will be constant for all periods and different from the import/ export price.
A’
A
α
B
b
C
C’
x
Demonstration: Assume that xt , etH > 0 ∀t ∈ T . Then from the first condition in (25) we have
that pt ( xt ) = ν t and from the second constraint we have that ν t = λ . By substitution the last
B’
β
Figure 3. Social optimum and export/import constraints
16
15
Two possible locations of price of imports/exports at A and C compared with water value at
Min∑ ci (eitT )
i∈N
B:
(i)
(30)
s.t.
Higher import/export price at A than the water value at B:
Optimal use of water: Do maximal exports AA’ and use water BB’ for home
∑ eitT ≥ etT , eitT ≤ eiT ,
i∈N
production realising point B’ on the demand curve. Home price is equal to the
(ii)
water value.
For each total generation level we get a set of plants producing positive output, and a set
Lower import/export price at C than water value at B:
being idle according to the marginal cost levels. If the range of variation in the marginal costs
Optimal use of water: Import the maximal electricity corresponding to Bb, use own
for each plant is sufficiently small all but one plant will be utilised to full capacity, and there
water in amount bB’ realising point B’ on the demand curve. Home price is equal
will be a marginal unit partially utilised. We can perform a merit order ranking of the active
to the water value.
units according to average costs at full capacity utilisation. Finally, the sequence of individual
cost curves can be simplified or approximated by a smooth function:
Sum of total use of water must obey the water constraint. For two periods with identical
ct = c(etT ), c ' > 0, c '' > 0, etT ≤ ∑ eiT = e T
at C the other period for the illustration to be correct.
(31)
i∈N
demand functions we must have 2BB’ = W with export price at A one period and import price
Social solution of mixed hydro and thermal capacity
The basic hydro model (6) without constraints on reservoirs, but only on total availability of
water, is adopted. The optimisation problem faced by a system planner is:
5. Thermal plants
We introduce plant specific variable cost functions for the generation of electricity based on
thermal energy sources. Each plant has an upper capacity for generation, eitT , that can only be
changed by investments. The cost functions are not dated for simplicity, but the cost function
may change between periods due to different fuel prices. Fuels may be more expensive in the
high demand season:
⎧
⎫
Max ⎨∑ [(U t ( xt ) − c(etT )]⎬ s.t.
⎩ t∈T
⎭
xt = etH + etT , ∑ etH ≤ W , etT ≤ e T
t∈T
The Lagrangian function is:
L = ∑ (U t ( xt ) − c(etT ))
t∈T
−∑ν t ( xt − etH − etT )
t∈T
(29)
−∑ θt (etT − e T )
The plant may be designed to have the smallest marginal cost at close to full capacity
−λ (∑ etH − W )
cit = ci (eitT ), ci ' > 0, eitT ≤ eiT , i ∈ N
utilisation. We disregard costs of ramping up or down plants, and especially going from a cold
to a spinning state. But having a phase of declining marginal costs may capture a start-up
effect.
(33)
t∈T
t∈T
The necessary conditions are:
pt ( xt ) −ν t ≤ 0 ⊥ xt ≥ 0
ν t − λ ≤ 0 ⊥ etH ≥ 0
The set of individual thermal plants can be aggregated to a thermal sector by the following
(32)
(34)
−c '(etT ) + ν t − θt ≤ 0 ⊥ etT ≥ 0
least cost procedure satisfying a total generating requirement of etT for each period:
Assuming that electricity must be produced in all periods we have:
pt ( xt ) = ν t
(35)
18
17
We must then in any period either activate hydro or thermal, or both. Thermal will not be used
p
for periods where:
c '(0) < ν t = λ
(36)
Hydro will not be used in periods where:
A
c '(e ) = ν t < λ
(37)
T
t
B
For periods where both hydro and thermal is used we have:
pt ( xt ) = λ = c '(etT ) + θ t
c’
b
B’
(38)
C
We have the basic result of the law of one price if hydro is used in all periods.
x
e
Result 6: The law of one price with mixed hydro and thermal capacity. In a situation with
T
ēT
eH
no reservoir constraints and assuming that hydro will be used in every period the price will be
Figure 4. Hydro and thermal. Social optimum
constant for all periods.
Demonstration: The result follows directly from (38).
average availability of water being bB’ the one period solution shown in the figure will be
Result 7: Thermal capacity as constant base load. If the maximal thermal capacity is not
utilised in any period we will have a constant load of thermal for all periods.
For two periods we may use the bath-tub diagram to illustrate the allocation of the two types
of power on the two periods. In Figure 5 the length of the bath-tub (bd) is extended at each
Demonstration: Rearranging (38) with the shadow price θt = 0 yields:
c '(etT ) = λ = pt ( xt ) ⇒ etT = c '−1 (λ ) = const.
repeated each period.
end with the thermal capacity. Using the result (39) we have that the thermal extension of the
(39)
bath-tub is equal at each end; with (ab) in period 1 and (de) in period 2. We have that (ab) =
(de). The equilibrium allocation is at point c, resulting in an allocation of (ab) thermal and
An illustration for one period is shown in Figure 4. The marginal cost curve, c’, for thermal
capacity starts at C and ends at the full capacity value, e . Assuming b’ to be the available
T
water the optimal solution is the price at level B equal to the shadow price of water, and a
thermal contribution of Bb = eT and a hydro contribution of bB’= xH.
If we assume that the figure is representing just one of many periods it is meaningful to
introduce two alternative water values by the dotted lines at C and A. For water vales from
levels A to a the full capacity of thermal units will be utilised. For water values higher than at
the level a only thermal capacity will be used. For water values lower than at level C no
thermal capacity will be used. In a multi-period setting with identical demand functions and
(bc) hydro in period 1, and (cd) hydro and (de) thermal in period 2.
19
20
In the case of a constraint on the reservoirs the profit maximisation problem of a producer
Period 1
Period
1
(dropping the producer subscript) is:
Period 2
Period 2
Max
∑pe
t∈T
p1(.)
H
t t
(40)
s.t.
p2(.)
Rt ≤ Rt −1 + wt − etH , Rt ≤ R , t ∈ T
c’
c’
The Lagrangian for the problem is:
L=
∑pe
t∈T
H
t t
−∑ λt ( Rt − Rt −1 − wt + etH )
(41)
t∈T
−∑ γ t ( Rt − R )
t∈T
For notational ease we have used the same symbols for shadow prices as in the social
a
b
c
d
e
Hydro energy
Thermal extension
Figure 5. Energy bath tub with thermal-extended walls of the hydro bath-tub
planning case with a single producer. The shadow prices are now plant specific. The
necessary conditions are:
∂L
= pt − λt ≤ 0 ⊥ etH ≥ 0
∂etH
∂L
= −λt + λt +1 − γ t ≤ 0 ⊥ Rt ≥ 0
∂Rt
(42)
Let us assume that there is a positive market price in every period. The producer will not
supply any electricity if the water value is higher than the market price. For the periods he
will supply a positive amount the market price has to be equal to his water value. In general
the producer will strive to sell all his energy at the period with the highest price, but he is
prevented to do this by the upper constraint on his reservoir. When overflow threatens his
6. Market organisations
water value will be adjusted downwards for that period. He is willing to sell to a lower price
than a higher price in a later period. But to the right price he may sell in an earlier period and
prevent an overflow situation happening. For the last period the obvious strategy for a
Free competition
In Section 2 we treated the hydro supply side as an aggregated single unit. We now assume
now that we are studying one among several suppliers selling electricity at a spot market for
every period. There is no uncertainty, so the period prices pi are known. Given the capacity of
each producer and the size of his reservoir he will in the situation of no (active) constraint on
his reservoir obviously choose to deliver all his electricity in the period with the highest price
in order to maximise profits. Therefore, in order to have positive total supply in all periods the
prices must be equal over periods in a market equilibrium. The allocation over periods is then
completely demand driven.
producer is to sell all available electricity in the reservoir:
−λT − γ T ≤ 0 ⇒ λT = pT ,
(43)
assuming there is no overflow in the last period. The shadow price will then be equal for all
the plants. If the price in period T-1 is higher than the price in period T then all the producers
will only sell the yearly inflows in period T. This can only be an equilibrium outcome if the
sum of the last period inflows is equal to the total demand at the price of period T. The water
value in period T-1 is then:
−λT −1 + λT − γ T −1 ≤ 0 ⇒ λT −1 ≥ pT ,
(44)
21
22
assuming that there is no overflow in period T-1. We assumed above that the market price in
period T-1 was greater than the market price in period T. This can only be possible if the
reservoir is emptied in period T-1 and the plant specific water values becomes equal and equal
to the market price in period T-1. But again we have the coordination problem: the demand to
∑ p (e
Max
t∈T
t
H
t
) ⋅ etH
(46)
s.t.
∑ etH ≤ W
t∈T
The Lagrangian is:
the price in period T-1 must correspond to the sum of all reservoirs.
∑ p (e
Let us use the situation in Figure 2 to discuss how a competitive market may work. For the
last period the water values are determined according to (43). In the first period overflow
threatens implying that the market price must be lower in period 1 than in period 2. Plants
with sufficient reservoir capacity will then not offer any electricity in period 1 but sell all in
period 2. Their water values for period 1 are greater than the price in period 1. But there must
be at least one plant that is forced to sell in period 1 due to threat of overflow. For such a plant
t∈T
t
H
t
) ⋅ etH −ν (∑ etH − W )
(47)
t∈T
The necessary first order conditions are:
∂L
= pt '(etH )etH + pt (etH ) − ν ≤ 0 ⊥ etH ≥ 0, t ∈ T
∂etH
(48)
Assuming that the monopolist will produce electricity in all periods the conditions may be
written:
(
(
pt (etH )(1 + ηt ) = pt ' (etH' )(1 + ηt ' ) , t , t ' ∈ T
(45)
(49)
(
In the expression for the marginal revenue we have introduced the demand flexibility, ηt ,
But since the plant is selling in period 1 we have from the first condition in (42) that the price
which is negative (the inverse of the demand elasticity). The condition is that the marginal
must be equal to the water value implying that all the water values for plants threatened with
revenue should be equal for all the periods and equal to the shadow price on stored water. The
overflow and delivering in period 1 will have the same water value. The plants selling in the
absolute value of the demand flexibilities must be less than (or equal to) one.
we have:
−λ1 + λ2 − γ 1 ≤ 0 ⇒ λ1 = pT − γ 1
first period will then sell the filled reservoirs in the second period. We still have the
coordination problem of balancing total supply and total demand at the market prices.
An illustration in the case of two periods is provided in Figure 6. The broken lines are the
marginal revenue curves. We see that in our case (the same demand curves as in Figure 1) the
We know that in principle a competitive equilibrium is compatible with the market prices
marginal revenue curves intersect for a positive value, i.e. it will not be optimal for the
being equal to the water values in the solution (11) to the social problem (10). But it is not so
easy to see how the producers are guided to make the correct allocation decision that will lead
Period 1
Period 2
to the shadow prices of the solution to the social problem being adopted as market prices in
the case of the reservoirs and inflows being different.
p2M
However, we will not pursue this here, but turn to the case of all hydro producers being part
p1S
of a monopoly, and further simplifying by assuming a single production unit.
p1M
p2S
Monopoly without binding reservoir constraints
We assume that the monopolist faces the demand functions pt = pt (etH ) , t ∈ T . The
optimisation problem of the monopolist in the basic case of Section 2 is:
Figure 6. The basic monopoly case
23
24
monopolist to spill any water. This value is the shadow value on water. But this result is
Let us first assume that the monopolist will not find it profitable to spill any water. In order
depending on the form of the demand functions. If we have spillage as an optimal solution,
for the stored water to become scarce in any period it is necessary that the consumers demand
then the shadow water value is zero. We see that the water value in general is smaller than the
the total amount of stored water to the price the monopolist is charging. Therefore, if the
shadow value for water in the social optimal case in Figure 1. Going up to the demand curves
water becomes scarce in the same period as in social optimum, the monopolist cannot charge
gives us the monopoly prices for the two periods. An important general result is that in the
a higher price than in the social optimum. It is the shadow value of water that must adjust
case of monopoly the market prices become different for the periods. For the period with the
downwards for this to be possible.
most inelastic demand the price becomes larger than the social optimal price, and for the most
elastic period the price becomes smaller. Thus we have a general shifting in the utilisation of
Can the monopolist use more water in elastic periods to achieve more periods with scarcity?
water from periods with relative inelastic demand to periods with relative elastic demand.
Can the monopolist choose to go empty in the period with the highest price that will empty
the reservoir? But is it in this period that the social solution also goes empty? Maximising
consumer surplus should lead to using as much water as possible when demand is high.
Monopoly and reservoir constraints:
The profit maximisation problem is:
Max
∑ p (e
H
t
t
t∈T
If the monopolist reallocates water to elastic periods, this is no point since price is determined
) etH
(50)
s.t.
Rt ≤ Rt −1 + wt − etH , Rt ≤ R , t ∈ T
∑ p (e
t∈T
t
H
t
the attempt to increase price in all periods.
)etH
−∑ λt ( Rt − Rt −1 − wt + etH )
becomes zero before the reservoir is filled. Notice that the monopolist does not have to empty
the reservoir to create value, just demand a positive price, it is only overflow that will break
The Lagrangian is:
L=
by water value when reservoir is emptied. Spilling water is profitable if marginal revenue
(51)
t∈T
−∑ γ t ( Rt − R )
Period 2
Period 1
t∈T
p2M
The necessary first order conditions are:
∂L
= pt '(etH )etH + pt (etH ) − λt ≤ 0 ⊥ etH ≥ 0
∂etH
∂L
= −λt + λt +1 − γ t ≤ 0 ⊥ Rt ≥ 0
∂Rt
(52)
p1M
(
Assuming electricity is always supplied and introducing the demand flexibilityηt = pt ' etH / pt :
(
pt '(etH )etH + pt (etH ) − λt = pt (etH )(1 + ηt ) − λt = 0
−λt + λt +1 − γ t ≤ ⊥ Rt ≥ 0 ∀t ∈ T
(53)
The marginal willingness to pay (the price) is substituted with the marginal revenue. The
A
B
C
discussion of the use of water is parallel to the social optimum case. But will a monopolist
choose the same time profile for the same inflows, demand functions, etc.?
Figure 7. Monopoly with reservoir constraint
D
26
25
In the illustration above the reservoir constraint is not binding, and we have no spillage. We
get the same type of solution as in Figure 2. But we note that the monopoly price in the period
with the relatively most elastic demand becomes lower than the social optimal price with a
binding reservoir constraint, and the monopoly price in the period with relatively inelastic
pM
p
demand becomes higher than in the social optimal case. This is the general shifting of water
c’
from periods with relative inelastic demand to periods with relatively elastic demand in the
case of market power.
B
b
B’
Monopoly with hydro and thermal plants
Let us assume that a monopolist has full control over both hydro and thermal capacity. The
demand functions are pt ( xt ) . The optimisation problem is:
⎧
⎫
Max ⎨∑ [( pt ( xt ) xt − c(etT )]⎬ s.t.
⎩ t∈T
⎭
xt = etH + etT , ∑ etH ≤ W , etT ≤ e T
x
(54)
e
T
e
H
t∈T
Figure 8. Monopoly. Hydro and thermal capacity
The Lagrangian is:
L = ∑ ( pt ( xt ) xt − c(etT ))
For two periods we may again use the bath-tub diagram to illustrate the allocation of the two
t∈T
−∑ν t ( xt − etH − etT )
types of power on the two periods. In Figure 9 the length of the bath-tub (bd) is extended at
t∈T
(55)
−∑ θt (etT − e T )
of the bath-tub is equal at each end; with (ab) in period 1 and (de) in period 2. We have that
t∈T
−λ (∑ etH − W )
(ab) = (de). The equilibrium allocation is at point c, resulting in an allocation of (ab) thermal
t∈T
and (bc) hydro in period 1, and (cd) hydro and (de) thermal in period 2.
The necessary conditions are:
pt '( xt ) xt + pt ( xt ) −ν t ≤ 0 ⊥ xt ≥ 0
ν t − λ ≤ 0 ⊥ etH ≥ 0
(56)
−c '(e ) + ν t − θt ≤ 0 ⊥ e ≥ 0
T
t
each end with the thermal capacity. Using the result (39) we have that the thermal extension
T
t
Concentrating on periods where both hydro and thermal are used the general result is that
marginal revenue substitutes for the marginal willingness to pay in the social optimal solution:
(
(57)
pt ( xt )(1 + ηt ) = ν t = λ = c '(etT ) + θ t
The monopoly solution is illustrated in Figure 8.
27
28
References
Period 2
Period
2
Period 1
Period
1
Borenstein, S., Bushnell, J. B. and Wolak, F. A. (2002): “Measuring market
inefficiencies in California’s restructured wholesale electricity market,” American Economic
Review 92(5), 1376-1405.
p2 M
p
2
M
Borenstein, S., Bushnell, J. B. and Knittel, C. R. (1999): “Market power in electricity
markets: beyond concentration measures,” Energy Journal 20(4), 65-88.
Borenstein, S., Bushnell, J. B., and Stoft, S. (2000): “The competitive effects of
transmission capacity in a deregulated electricity market,” Rand Journal of Economics 31(2),
294-325.
MM
p1
p
1
c’
c’
a b
c
d
Hydro energy
Thermal extension
Figure 9. Two periods and monopoly, hydro and thermal
e
Bushnell, J. B. (1999): “Transmission rights and market power,” Electricity Journal
12(8), 77-85.
Bushnell, J. (2003): A mixed complementarity model of hydrothermal electricity
competition in the Western United States, Operations Research 2003, 51(1) JanuaryFebruary.
Crampes, C. and Moreaux, M. (2001): “Water resource and power generation,”
International Journal of Industrial Organization 19, 975-997.
Fleten, S.-E. (2000); Portfolio management emphasizing electricity market
applications. A stochastic programming approach, Dr. ing. Thesis January 2000, NTNU
Doktor Ingeniøravhandling 2000:16.
Fleten, S.-E. and Lie, T. T. (2000): “A stochastic Cournot model of the Scandinavian
electricity market,” Proceedings, Annual European Energy Conference, Bergen.
Fleten, S.-E., Wallace, S. W. and Ziemba, W. (2002): “Hedging electricity portfolios
via stochastic programming,” in: C. Greengard and A. Ruszczynski, eds.,
Decision-making under uncertainty: energy and power, Springer-Verlag, New York, 71-93.
Førsund, F. R. (1994): “Driftsoptimalisering i vannkraftsystemet”, SNF - rapport
29/94, SNF - Oslo.
Hogan, W. W. (1997): “A market power model with strategic interaction in electricity
networks,” Energy Journal 18(4), 107-141.
Johnsen, T. A. (2001): “Hydropower generation and storage, transmission constraints and
market power,” Utilities Policy 10, 63-73.
Joskow, P. L. and Kahn, E. (2002): “A quantitative analysis of pricing behavior in
California’s wholesale electricity market during summer 2000,” Energy
Journal 23(4), 1-35.
Report from the Nordic competition authorities (2003): “A powerful competition policy,”
Report No. 1/2003
29
Scott, T. J. and Read, E. G. (1996): Modelling hydro reservoir operation in a
deregulated electricity market,” International Transactions in Operational
Research 3(3/4) , 243-253.
Singh, B., T. Eldegard og J. Skaar (1999): ”Storskala kraftutveksling – hovedrapport”,
Rapport 66/99 SNF, Bergen.
Sweeney, J. L. (2002): “The California electricity crisis,” Hoover Institution Press
Publication No. 513.
Wallace, S. W. and Fleten, S.-E. (2002): “Stochastic programming models in energy,”
in: A. Ruszczynski and A. Shapiro, eds., Handbooks in Operations Research
and Management Science 10, North-Holland.