Supply Function Equilibrium in Electricity Markets using
Smooth Capacity Constraints
Morten Hørmann ([email protected]) and
Mikkel T. Kromann ([email protected]).
COWI A/S, Parallelvej 2, DK-2880 Lyngby, Denmark
Abstract. In this working paper it is shown how the Supply Function Equilibrium
(SFE) concept can be extended to describe smooth capacity constraints using strict
convexly increasing marginal costs and supply functions. The SFE model framework
with convexly increasing marginal costs and supply functions is simulated numerically. The smooth capacity constraint concept can also be used for transmission
constraints, a completely novel feature for SFE models. Finally, as the simulation
model has a very appealing simulation time, it has also been possible to implement
the intertemporal decisions of price setting hydro producers.
Market power, electricity markets, Supply Function Equilibrium, SFE,
capacity constraints, transmission constraints, hydro power, intertemporal optimisation.
Keywords:
1. Introduction
Imperfect competition on the electricity market has been the subject of
intense investigation since the 1990's, where a wave of electricity market
liberalisations started. Two diering model traditions, Cournot-Nash
Equilibrium and Supply Function Equilibrium have been applied to
analyse the eects of potential imperfect competition in the liberalised
markets.
The supply function approach introduced by Grossman (1981) and
Klemperer and Meyer (1989) stipulates that in some instances rms
may not compete in quantities as in Cournot or in prices as in Bertrand.
Rather they submit bid-like continuous functions representing their
willingness to supply a given quantity at a given price. Especially when
facing uncertainty of demand, it may be preferable for the rms to state
a exible strategy, such that low and high demand are not met by the
same supply or the same price.
Unfortunately, the analytical framework of supply functions turns
out to be much more complex than that of Cournot and Bertrand.
The equilibrium solution to a market of
N
functions must then derived from a set of
rms competing in supply
N
rst order dierential
equations.
c 2005
Kluwer Academic Publishers. Printed in the Netherlands.
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M. Hørmann and M. Kromann
In order to reduce the complexity of the optimization problem, a
specic functional form of the supply functions is often assumed ex ante.
Green and Newberry (1992), Green (1996), Halseth (1998), Baldick,
Grant and Kahn (2004), Konkurrencestyrelsen (2004) and Eltra (2003)
all assume that rms compete in linear supply curves. The strategies of
the rms are thus reduced to deciding either the slope or the intersection
of the supply curve.
The merits of the assumtion of linear supply curves lie in the simplicity of the analytical framework. However, the assumption of linearity
is also associated with some shortcomings, especially in relation to the
modelling of markets for electricity. Because of the demand that supply
functions be continuously dierentiable, it is impossible for linear supply
curves to reect constraints on capacity. Furthermore, the composition
of generating companies from many dierent production technologies
implies an upward sloping and (not considering the discrete nature of
electricity generation MC curves) strictly convex marginal cost curve.
This is often assumed to imply that supply functions as well should
be upwards sloping and strictly convex. Finally, the linearity of supply
curves may understate the use of market power in situations of high
demand.
This paper departs from the assumption of linearity of supply functions and instead adopts so-called
smooth capacity constraints.
For the
supply functions and marginal costs of generation second order polynomial curves are applied. This will allow for the modelling of capacity constraints and upwards sloping stricly convex supply curves.
Other continous functional forms can also be applied. The paper also
briey describes how transmission capacity and intertemporal hydro
power generation constraints can be implemented using continuous constraints.
This paper does not address uncertainty of demand. As Green (1996)
states, the variation in demand for electricity over time is much more
signicant than any random variation. Since rms submit supply functions for one hour at a time, it is assumed that any random variation
within this time frame is limited.
In section 2, the model framework is described including derivations
of restricted polynomial marginal costs and supply functions and rst
order conditions of prot maximization using supply functions. Section
3 describes the implementation of the model framework into a working numerical simulation model.
1
In section 4, results of the numerical
simulation are presented and discussed, and in section 5 potential trans-
1
The model is called seems2c (Supply function Equilibrium in Electricity Markets using Smooth Capacity Constraints) and has been implemented using the
simulation software package Gams.
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SFE and Continuous Constraints
3
mission and hydro power extensions of the model are presented. Section
6 briey summarises the results of this paper.
2. Model Specication
This section describes three models of an electricity market: A Perfect
Competition model, which serves as a baseline for comparison of the
two other forms of competition assumed, namely Cournot competition
and Supply Function Equilibrium (SFE).
In this paper we show how convexly increasing Supply Functions
and marginal costs with capacity constraints can be modelled within
the Supply Function Equilibrium framework. To our knowledge, this
has not been demonstrated before, even though exactly this type of
cost and bid structure is generally agreed to be very characteristic of
electricity markets.
We start by presenting our assumptions on demand and cost structure. Then we proceed to a perfect competition and Cournot competition setting before we present a single period version of our SFE
model.
2.1.
Demand and Cost Structure
In the interest of providing an easily interpretable introduction to the
dynamics of the model, the basic scenario describes a market where
demand is linear.
D (P ) = Qmax − sP ⇔ P (Q) =
Qmax − Q
s
(1)
Alternatively, the demand is represented by a constant elasticity
demand function, for elastic demand with
e < 0, and also with inelastic
demand.
e
D (P ) = f P e ⇔ P (Q) = Q
f
D (P ) = Qmax for P ∈ <+
for constantly elastic demand
for inelastic demand
Convexly increasing marginal costs with a capacity constraint is often the better empirical approximation to electricity producers' cost
structure. We apply a polynomial of second order to represent the
inverse function of the marginal cost function, c.f. equation 2.
Such a polynomial is described by three parameters which can be
chosen freely by the modeller. However, these three parameters can be
meaningfully calibrated by three important characteristica of any electricity producer: the marginal cost of the cheapest production facility
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4
M. Hørmann and M. Kromann
Figure 1. Marginal costs structure
M Cl , the marginal cost of the most expensive production facility M Ch ,
and the total generating capacity available to the producer, k . This cost
structure is illustrated grapically in gure 2.1.
The following polynomial specication of the cost structure relates
quantities to marginal costs:
Q (M C) = −aM C 2 + bM C − c
We can now determine the parameters
where
a, b, c ≥ 0
(2)
a, b, c using the characteristics
of the marginal costs mentioned above. The inverse of the marginal
costs curve
determine
b
Q (M C)
M Ch which can be used
Q0 (M C) = −2aM C + b:
has its maximum at
by utilising its derivative
Q0 (M Ch ) = 0 ⇔ b = 2aM Ch
Similarily,
a
(3)
can be determined through above derivation of
the capacity constraint
k
by utilising that
to
b
and
Q (M Ch ) = k :
−aM Ch2 + 2aM Ch2 − c = k ⇔ a =
k+c
M Ch2
(4)
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5
SFE and Continuous Constraints
c
Finally,
can be determined from the marginal cost of the cheapest
production unit where
Q (M Cl ) = 0,
c = −aM Cl2 + 2aM Ch M Cl =
c = (k + c) d =
MC
k+c
M Cl (2M Ch − M Cl )
M Ch2
kd
1−d where
Rearranging the relation between
for
giving
d=
MC
2M Ch M Cl −M Cl2
M Ch2
and
Q,
(5)
a closed expression
can be obtatined, although this expression is meaningful only
for the intervals
Q ∈ [0; k]
and
M C ∈ [M Cl ; M Ch ]:
−aM C 2 + bM C − c = Q
−4a −aM C 2 + bM C
2
2
MC
over
= −4a (c + Q) ⇔
b + 4a M C − 4abM C = b2 − 4a (c + Q) ⇔
(b − 2aM C)2 = b2 − 4a (c + Q) ⇔
p
b − b2 − 4a (c + Q)
M C (Q) =
2a
Integrating
2
Q
gives the total cost function
T C (Q),
which
can be shown to be
T C (Q) =
=
=
RQ
0 M C (Q) dQ
3
2 1
b
2 − 4a (c + Q) 2 −1
Q
−
b
2a
3 2a
4a
3
b
1
2
2
2a Q + 12a2 b − 4a (c + Q)
(6)
With this description of demand and marginal costs, we now turn to
the producer behaviour. The prot maximisation problem is
max πi = P (Q) Si − T Ci (Si )
where
Q=
PN
i=1 Si . This problem can be stated as the following rst
order condition
P (Q) +
∂P ∂Q
Si − M Ci ≤ 0
∂Q ∂Si
and noting that under linear demand,
sections focus the attention on
∂Q/∂Si
∂P/∂Q = −1/s
(7)
the following
which describes the aggregate
reaction of the other rms in the market. It will be shown that Perfect,
Cournot and Supply Function competition are very dierent in their
specication of the reaction of the competitors.
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6
M. Hørmann and M. Kromann
2.2.
Perfect Competition
In perfect competition, all producers assume that their choice of price
or quantity does not aect the market price because the size of each
producer is insignicant compared to the total size of the market. This
translates into
∂Q/∂Si = 0.
Thus, the rst order condition for prot
maximisation is
P = M Ci
(8)
This is also known as so-called marginal cost pricing, i.e. the producer
increases his supply if the price is higher than his marginal cost, and
decrease production if the price is lower than marginal costs.
2.3.
Cournot-Nash Equilibrium
In Cournot-Nash Equilibrium the with market power are assumed to
use quantities as their strategic variable. This means that they commit
to supplying a certain quantity before the market opens and the market price is realised. The quantity commitment is made simultaneously
among the rms, and therefore each rm rightly believe that their choice
of quantity does not aect the other rms' choice of quantities. Thus,
their belief is that a change in their own quantity will be fully reected
in the total market supply, i.e.
∂Q/∂Si = 1.
Thus, the Cournot com-
petitors rst order condition (with the linear demand described above)
is
1
P (Q) − Si − M Ci ≤ 0
s
2.4.
(9)
Supply Function Equilibrium
The electricity spot markets known today does, however, not resemble
the quantity commitments of Cournot-Nash Equilibrium. Rather, the
producers sell their electricity in an auction like setting, where they
must submit a so-called bid schedule, specifying the quantities they are
willing to produce given dierent realisations of the market price.
The Supply Function Equilibrium resembles exactly that. The
N
rms commit to a so-called Supply Function, which species the quantity supplied by the rm at the realised market price. And not only is
the rm committed to its own Supply Function, it also knows that its
competitors are also committted to their Supply Functions, and thus
change their supply in reaction to changing market price. Therefore,
when the rm changes its supply (according to its Supply Function
as a response to a market price change), it must also expect that the
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SFE and Continuous Constraints
7
competing rms will change their supply in response to the changing
market price.
For this reason
0 < ∂Q/∂Si < 1 in SFE, depending on the particular
form of the Supply Function. Thus the SFE lies in between Perfect and
Cournot competition.
The following structure of the rms' Supply Functions will accommodate the implied capacity constraints of the marginal cost functions:
Si (P ) = −αi P 2 + βi P − γi
for
αi , βi , γi ≥ 0
(10)
Obviously, this structure resembles the structure of the marginal
costs. Contrary to the marginal costs formulation, we wish to maintain
a degree of freedom for each rm to choose the shape of this supply
functiton that maximises its prots. Thus, we will bind only two of the
three parameters to physical characteristics of the producers generating
equipment.
In the previous SFE litterature, it has been observed that the exercise of market power especially takes place in situations with very
high capacity utilisation, whereas the market seems rather competitive
when the capacity utilisation is low. It therefore makes sense to bind
the supply functions to the marginal cost of the cheapest technology
and to the total capacity constraint.
At the capacity constraint the slope of the supply function is vertical.
P̄ , accordβi
. Inserting
= 0 ⇔ P̄i = 2α
i
bind α to β and γ through the
The price chosen by the rm running at maximum capacity,
ing to its supply function is thus
Si0 P̄
this in the supply function allows us to
capacity constraint:
Si
β
2α
= ki ⇔
−βi2
β2
+ i − γi = ki ⇔ βi2 = 4αi (ki + γi )
4α
2αi
(11)
The second binding of the supply function concerns the lowest price
at which the rm is willing to supply a positive quantity. As mentioned
above, the literature indicates that competition is most intense when
capacity utilisation is low. We thus assume that the rm will start to
supply the market when the market price is equal to the cost of its
cheapest production unit,
M Cl :
2
Si (M Cl,i ) = 0 ⇔ γi = −αi M Cl,i
+ βi M Cl,i
(12)
The following set of equations thus dene the restricted second order
polynomial supply function:
Si (P ) = −αi P 2 + βi P − γi
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8
M. Hørmann and M. Kromann
βi2
4 (ki + γi )
2
γi = −αi M Cl,i
+ βi M Cl,i
αi =
Green (1996) derives rst order conditions of prot maximzation for
N
asymmetric rms competing in supply functions. The resulting set
of equations does not lend itself to an analytical solution. However,
as we intend to solve the problem numerically, it is sucient to state
the rst order conditions along with equilibrium conditions. Under the
assumption of competition in supply functions, the strategic variable of
the rms is a combination of price and quantity. The prots of rm
i
may be written as:
πi (p) = pSi (p) + Ci (Si (p))
The supply of rm
P
j6=i Sj (p).
i is equal
to the residual demand
Si (p) = D(p) −
Inserting this, the prot of rm i can be rewritten.
πi (p) = p D(p) −
j6=i Sj (p) + Ci D(p) −
P
P
j6=i Sj (p)
The prot is dierentiated with respect to price.


X
X ∂Sj (p)
∂πi

= D(p) −
Sj (p) + p D0 (p) −
∂p
∂p
j6=i
j6=i

−Ci0 D(p) −

X
Sj (p) D0 (p) −
j6=i
X ∂Sj (p)
j6=i
∂p


Setting equal to zero and rearranging yields the rst order condition
of prot maximization of rm i.

Si (p) = p − Ci0 (Si (p)) −D0 (p) +
X ∂Sj (p)
j6=i
∂p


3. A Numerical Simulation Model
The model framework described above has been simulated in GAMS.
This section describes how the framework is translated into a set of
equations that will yield equilibrium solutions in a mixed complimentarity solver. In order to analyse the eects of uctuating demand on
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9
SFE and Continuous Constraints
equilibrium and to accomodate future expansions of the model in regards to hydro power and inter temporal prot maximization, an index
for time has been incorporated in the model. Otherwise, the system
of equations described here should be recognizeable from the previous
section.
3.1.
Common Equations
The demand, marginal cost and market equilibrium equations are common to all three models.
Demand:
Dt (Pt ) =

 Qmax,t + sPt
f Pe
 t
Qmax,t
, Linear
, Constant Elasticity
, Inelastic
Marginal Demand:

 s
∂Dt (Pt )
eft P e−1
=

∂Pt
0
, Linear
, Constant Elasticity
, Inelastic
Marginal Cost:
2
qi,t (M Ci,t ) = −ai M Ci,t
+ bi M Ci,t − ci
If a is set to zero, then marginal costs become linear. In that case,
1
b and the intersection or
c
the marginal cost of the cheapest unit produced will be .
b
the slope of the marginal cost curve will be
Market Equilibrium:
Qt = Dt (Pt )
3.2.
Perfect Competition
The only equation which is specic to perfect competition is the rst
order condition of prot maximization.
F.O.C.
M Ci,t ≥ Pt ⊥ qi,t ≥ 0
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10
M. Hørmann and M. Kromann
3.3.
Cournot Competition
The only equation which is specic to Cournot competition is the rst
order condition of prot maximization.
F.O.C.
M Ci,t ≥ Pt +
3.4.
qi,t
⊥ qi,t ≥ 0
M Dt
Suplly Function Competition
The supply function problem is more complicated. First, the core of
the problem consists of two sets of complimentarities. Like the perfect
and Cournot competition cases, the decisisons of the rms are dictated
by their rst order conditions of prot maximization. Since the supply
function itself is not the strategic variable, it is necessary to further specify the relationship between the strategic variable
βi,t
and the supply
function.
Si,t (Pt ) ≥ (Pt + M Ci,t ) −M Dt +
−αi,t Pt2 + βi,t Pt − γi,t ≥ Si,t (Pt )
P
j6=i
∂Sj,t (Pt )
∂Pt
⊥ Si,t (Pt ) ≥ 0
⊥
βi,t ≥ 0
Specied along with these two sets of complimentary equations are
two supporting equations.
Bind Gamma
Bind Alpha
γi,t = −αi,t M Cl2 + βi,t M Cl
αi,t = β 2 /4(ki + γi,t )
4. Model Properties
In this section the various properties of the model are illustrated using
an simulations of articial market situations not based on actual data.
Instead, the artical markets are constructed to illustrate the model
properties in a clear way. We provide simulations of a 24 hour period
where the demand uctuates,
2
and where a small number (below 5) of
companies compete against each other.
2
A sinus function has been used to emulate uctuations in a linear inverse
demand curve. The demand is above the base level in period 1 to 12 and below
in period 13 to 24. The available total generation capacity is assumed constant over
the simulated period. The MC curves each rms generation capacities are dened
by their capacity, and their minimum and maximum MC.
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SFE and Continuous Constraints
4.1.
11
Linear or polynomial cost and supply functions
The novel feauture of the presented model is the ability to describe
the market using convexly increasing supply and MC curves instead
of linearly increasing ones. This has implications for the prices and
markups in the simulations as well as other variables.
For the interpretation of the results we rst note that the two linear
MC simulations have higher prices than the simulations with polynomial MC curves. This is not at all surprising as the polynomial curves
allways lie below the linear counterparts (they have the same MC at
zero and maximum capacity utilisation).
Figure 4.1 shows the simulated markups when using the four possible
combination of linear and polynomial MC and Supply Functions (SF).
It can be seen that the markups are relatively higher in periods with
high demand compared to periods with low demand. Thus our model
conforms to previous SFE ndings on this point.
It can also be seen that the polynomial SF show distinctly higher
markups, whereas linear SF gives much lower markups. The reason is
that convex Supply Functions have steeper slopes (reducing competitive
pressure) at lower price levels (tending not to curb consumer demand
as much as is the case with linear SF's).
Linear MC shows lower markups than linear ones in high demand
situations, whereas the opposite is the case in low demand situations.
This might partially be attributed to their higher costs associated with
linear MC curves, leaving less space for higher markups. The dierences
are, however, not very large.
When demand is low, the simulations show that there is much less
dierence between linear and polynomial marginal cost and supply functions. But contrary to the high demand situations, it is now linear MC
curves that gives the highest markups. This is because the linear SF's
are relatively steeper than their polynomial counterparts when demand
is low. The choice of cost function seems to be of less importance.
We can thus conclude that the assumptions made on the functional
form of the supply curves are rather important to the markups. The interpretation of a steeper supply curve is that the rms are less sensitive
to price changes. Using polynomial instead of linear supply curves thus
implicitly assumes a more intense competition with low demand and
less intense with high demand. The functional form of the MC curves
are of less importance, and no very general impression is found in the
presented simulations.
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12
M. Hørmann and M. Kromann
Figure 2. Markups with combinations of linear and polynomial MC and Supply
Functions
4.2.
Equilibrium Concept
In the model three equilibrium concepts are implemented: Perfectly
Competitive, Cournot, and Supply Function Equilibrium. It has also
been made possible to specify a representitive competitive fringe supplier in the simulations, i.e. a supplier which always increases its supplies
whenever the price is above his marginal costs. In gure 4.2 below,
the markups from simulations with dierent equilibrium concepts are
shown.
As can be seen from the gure, the Cournot equilibrium with three
rms and an additional fringe supplier tends to yield markups that
are orders of magnitude higher than SFE markups. Thus, modellers
choosing the Cournot Equilibrium concept for their modelling assume
the least competivite equilibrium concept.
4.3.
Consumer Demand Modelling
The consumer demand has been implemented using two dierent demand functions: linear demand and constant elasticity demand. Con-
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SFE and Continuous Constraints
13
Figure 3. Markups under low demand with combinations of linear and polynomial
MC and Supply Functions
3
trary to the Cournot Equilibrium, SFE allow for inelastic demand.
The simulations undertaken show that inelastic demand can lead to
extremely high markups, c.f. gure 4.3.
As can be seen from the gure, inelastic demand tends to give especially high markups when demand is very high compared to the available
capacity. In period 6 the capacity utilisation is at 90 per cent, compared
to around 25 per cent in period 18.
It should be noted that the assumed elasticity of the linear demand is
valid only for the base point of the linear demand. The realised demand
given the market price may have dierent elasticity.
5. Perspectives
The presented model is non-linear and can be implemented both as
a Mixed Complementarity Problem (MCP) and as a Constrained Nonlinear System (CNS). For both implementations, the calculation time is
3
With the linear demand function D(P ) = Qmax − sP inelastic demand is
represented by s = 0, which invalidates the Cournot prot maximsing rst order
condition, c.f. equation 9.
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14
M. Hørmann and M. Kromann
Figure 4. Markups with diering equilibrium concepts
very satisfying, as it is counted in milliseconds for the shown simulation
examples, as well as for 168 hour simulations.
Larger problems with more rms and time periods are solved in seconds or minutes. The reason for such small calculation time is that the
continous SFE model explicitly describes the reaction of the dierent
competitors.
The small calculation time has made possible yet unnished experiments with the model formulation. These show that it apparently is
possible to include two very important aspects of electricity markets in a
continous polynomial Supply Function model: multiple hydro producers
excercising market power and transmission constraints.
5.1.
Market Power of Multiple Hydro Producers
On the longer term the price setting hydro power producer has an
intertemporal problem of deciding when the production should be high
and low according to the prevailing electricity price in each period. This
operationalises to the following limitation for each hydro producer:
X
StH ≤ H̄
t
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SFE and Continuous Constraints
15
Figure 5. Markups with various demand modelling
where
H̄
is the maximum production capacity of the hydro producer
over the modelled period.
4
This limitation gives rise to an endogenous
shadow price on the water (the water value),
λ.
Alternatively, and perhaps most relevant to the modelling of relatively short time spans, the water value could be specied as exogenous
(meaning that the use of water in the modelled period is quite small
compared to the available amount of water). In this latter case, the
hydro producer's problem is equivalent to that of the other producers'.
The model eorts on the endogenous water value is still experimental. By now it is clear that the endogenous water value should not
be used to x the interception of the polynomial supply curve. As the
polynomial supply curve cannot be downwards sloping and it apparently
is optimal to use all the water, the optimal hydro supply curves becomes
very at, leading to problems of linear dependency with the marginal
cost curve (dened as
M C = λ).
Two formulations of this problem is in the working. The rst is to
assume that the long term value of water is zero and start the polynomial supply curve in
(0, 0).
Another solution is keep the endogenous
4
Minimum production and other hydro specic production requirements are not
yet considered.
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16
M. Hørmann and M. Kromann
water value as the intercept but allow a linear supply function to be
both upwards and downwards sloping. The latter solution still needs to
be evaluated against key assumptions of the SFE framework.
5.2.
Smooth Transmission Capacity Constraints
A requirement of a SFE model is that the MC and Supply Functions
and their rst order derivaties are continous. For this reason it is not
immediately possible to include hard transmission constraints with
associated shadow prices.
This problem can, however, be circumvented by adding a continous
transmission cost function to the MC curves of the rms. This cost
function must be very low when transmission capacity is available,
and prohibitively high when transmission capacity is fully utilised. A
suggestion for such a function is
XM C =
where
x¯r,s
1
x¯r,s − Xr,s
Xr,s is the actual
r and region s. Results for this implementa-
is the transmission capacity constraint and
transmission between region
tion is yet too unnished to be presented here, but the calculation time
are in the same scale as the simulations presented hitherto, and preliminary results of transmission cost simulations seem promising. Still
work remain to adjust the capacity constraint of the supply function to
also consider the transmission constraint.
6. Conclusions
This working paper has demonstrated how smooth capacity constraints
can be implemented in a Supply Function Equilibrium to emulate generation and transmission capacity constraints relevant for modelling of
the electricity market.
Furthermore, the presented model has been implemented as a numerical simulation problem. The simulation model shows to be very
computationally eective. This has allowed extending the investigations
into the intertemporal problem of hydro electricity producers' market
power, albeit these investigations are only at a very experimental, yet
promising level.
The simulation model was used for comparing various properties
of the Cournot and Supply Function Equilibrium concepts, as well
as investigating details about the modelling of consumer demand. Of
special importance, it was found that polynomial Supply Functions
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SFE and Continuous Constraints
17
tend to increase markups compared to linear Supply Functions. This
is especially prevalent when the capacity utilisation is high. It was also
demonstrated that Cournot equilibrium tends to give much higher costs
than the Supply Function Equilibrium.
Acknowledgements
This working paper and the construction and implementation of the
model has been partly funded by the Danish Energy Research Programme. The authors wishes to acknowledge valuable comments from
...
References
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Address for Oprints: <Address for oprints, printed at the end of the article>
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