1
Behavioral Model Using Conjectural Variation in
Power Markets Considering the Effect of Contracts
Sangamesh G. Sakri, Student Member, IEEE, N. Sasi Kiran, and S. A. Khaparde, Senior Member, IEEE
Abstract— Introduction of competition in the generation sector
has resulted in an interest for modeling the behavior of competitors for all strategic decisions. The behavioral model using
Conjectural Variation (CV) will play a key role in the prediction
of behavior of all constituents of power market. The conjecture
of a firm is defined as its belief or expectation about how its
opponents will react to a change in its output. Since there is an
absence of any confidential information, the CV based model has
to be based on past data. In this paper, the following applications
are reported. (a) An optimal bidding strategy, based on CV, to
obtain the coefficients of affine supply function. For this, the
elasticity of demand is also considered. The method, as applied
to a practical system, is illustrated. (b) The effect of contracts on
a pool based spot market. In this analysis, we study the effect of
contracts on bids of generating companies, and the market power
exercised by them. Simulation results for different sample systems
are presented, which substantiate the analytic conclusions.
Index Terms— Conjectural variation, forward market, market
power, optimal bidding, supply function equilibrium.
I. I NTRODUCTION
ITH restructuring of electricity markets taking place
around the globe, many new problems have been posed
with regard to the operation and design of these markets. The
strategic decisions now necessarily involve economics, law and
behavioral modeling of participants in addition to forecasting.
The major source of errors in the decision making process
arise from inaccuracies in modeling. Further, the information
available for modeling is limited because of the introduction of
competition in electricity market. This available information is
also imperfect and the uncertainty inherent to it has to be taken
care of. Hence, a proper model for the behavior of market
participants is now a paramount requirement, as has been never
before, for solving different problems – optimal bidding is one
such.
Several approaches have been proposed for modeling the
behavior of system operator and market players under different
market conditions to solve the optimal bidding problem.
However, use of either game theory, dynamic programming or
Markov decision process suffers from the curse of dimensionality. An optimal bidding approach with modeling of imperfect
information has been proposed in [1]. In this, the bidding is
in terms of coefficients of affine function relating price and
quantity. The bids of other competitors are taken as a normal
distribution and a simulation method based on the Monte Carlo
W
Sangamesh G. Sakri is with PDA College of Engineering Gulbarga,
Karnataka, India, Email: [email protected].
N. Sasikiran obtained his Masters Degree from IIT Bombay, Mumbai, India,
Email: [email protected].
S. A. Khaparde is with IIT Bombay, Mumbai, Email: [email protected].
method has been used. However, it is not definitely known
whether the bids of competitors follow a normal distribution.
An actor-critic learning algorithm has been presented in [2]
wherein a Genco continuously learns from the bidding process.
However, this method requires estimates of cost curves of the
competing generators.
In some behavioral models, the whole system is taken
into consideration for analyzing market power and learning
capabilities of each of the individual market players. In [3],
a fuzzy methodology has been proposed for strategic bidding,
which takes into account the uncertainty in parameters like
load demand, generator bid and cost. The multi-level fuzzy
system extracts the market behaviors from historical data and
this is used in the estimation of the generator bids. In [4], the
behavior of the market participants has been modeled using
a set of bidding actions for each possible discrete state of
the state-space observed by the participant. Genetic Algorithm
(GA) has been used to show that the agents learn and coevolve from the bidding process. However, the state-space
of the trading agents becomes very large as the number of
generators in the system is increased, thereby, increasing the
computational complexity.
The market power exercised by the generators is as much
an issue of concern for the regulator as optimal bidding for the
generators. Market power is the ability of a generator to raise
the market price above the marginal price to earn more profit,
leading to production inefficiencies and providing inefficient
signals for new investment [5]. In [6], a comprehensive review
of market power related issues in emerging electricity markets
is given. Supply Function Equilibrium (SFE) [7] approach
is used for the analysis of equilibrium, market power and
electricity pricing estimation.
A duopoly market has been analyzed using the SFE based
pool spot market and a forward market modeled using CV in
[8], however, the CVs have been assumed and not estimated.
This idea has been extended to include multiple Gencos with
affine marginal costs in [9]. In [10], a linear asymmetric SFE
model, with transmission constraints, has been proposed to
develop bidding strategies considering forward contracts. A
similar approach has also been used in [11] to show that
holding of transmission rights has influence on the market
power. In [12], the method of supply functions has been used
and bidding coefficients obtained using Conjectural Supply
Function Equilibrium (CSFE) assuming inelastic demand.
Generally, the SFE solution contains multiple equilibria and
it is difficult to identify which of them represents the firm’s
strategic behavior. The existence and uniqueness of a solution
is difficult to establish except under very simple versions of
©2008 IEEE.
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2
SFE models. Further, presence of differential equations in the
SFE model increases the computational requirements [13].
Recently, CV based methods have been proposed to estimate
the strategic behavior in electricity markets. In [14], the
concept of CV was applied to electricity markets for the first
time to optimize the dispatch of generators. The mathematical
formulation for CV as applicable to electricity markets has
also been given and the same has been used in our work. A
CV based learning method has been proposed in [15], [16] in
which each firm learns and dynamically modify its conjectures
according to available information and then makes its optimal
generation decision based on an updated CV. A theory and
methodology of estimating the CVs of Gencos and the analysis
of dynamic oligopoly behavior underlying market power has
been developed in [17].
CV based methods have, so far, been used to obtain optimal
dispatch. The main contribution of our work has been to obtain
generator bids as parameters of linear supply function using
CV and examine the CV based market model in the presence
of contracts. The effect of contracts on the bids has been
shown. The method of mitigating the market power exercised
by the generators by use of contracts is analyzed using CV. A
study of how a generator improves its optimal dispatch with
forward contracts and the factors that could affect the strategic
contracting behavior is also carried out.
The following is the organization of the paper. In section
II, a multi-player market model based on the concept of CV
is presented. This model also incorporates the contracts that
a generating company enters into. In section III, an optimal
bidding problem is formulated. Its application to a market is
illustrated and the results obtained are discussed. Section IV
deals with the effect of contracts on the market with respect to
bids made by generators and market power exercised by them.
The results of simulations carried out on sample systems are
presented. Finally, we draw our conclusions in section V.
II. M ODEL FOR M ULTI - PLAYER M ARKET WITH
C ONTRACTS
In this section, a mathematical model based on Conjectural
Variation based Bidding Strategy (CVBS) is proposed for
generators to improve their strategic behavior. The relation
for CV considering contracts is derived. Conjecture of a firm
is its belief or expectation of how its opponents will react to
a change of its output. In this work, forward contracts are
considered.
The model can be considered as a two-stage game of a
forward contract market and a bid based pool spot market.
Uniform pricing market model that is used in many power
markets (e.g., England and Wales, PJM and NEMMCO) has
been considered. Here, the spot market is modeled with CV.
In the first stage, trading takes place in the forward market,
then production occurs in the second stage and agents meet in
the spot market. The optimal decisions are derived considering
the positions initiated in the forward market.
Consider an n firm spot electricity market with the following
inverse demand function,
p = r − sQe
(1)
Here, p is the price of the electricity, s is the slope of the the
inverse demand curve, r is its intercept and Q e is the total
generation in the system.
Generator i, producing q i is offering a contract quantity x i
at a price f . This generator has a production cost of C i (q). It
will, therefore, have a profit function,
πi = pqi + (f − p)xi − Ci (qi )
(2)
= p(qi − xi ) + f xi − Ci (qi )
i = 1, 2, · · · , n
Using the following expression for the cost function,
1 2
ci q + ai qi
(3)
2 i
Using (1) and (3) in 2), we have the following expression for
the profit,
Ci (qi ) =
1
πi = (r − sQe )(qi − xi )+ f xi − ci qi2 − ai qi
2
i = 1, 2, · · · , n
(4)
Assuming that each generation firm is rationally aiming
at maximizing its profit, then the corresponding optimization
problem for firm i can be defined as,
max πi
s.t.
Qe =
n
qi
i=1
qmin ≤ qi ≤ qmax
i
For maximizing the profit of a generator, dπ
dqi = 0
⎞
⎛
n
dq
dπi
j⎠
− ci qi − ai
= (r − sQe ) − s(qi − xi ) ⎝1 +
dqi
dqi
j=1,j=i
(5)
By definition, conjectural variation γ i of a firm i is the
response of its competitors to change in its own production
[14],
n
dqj
dq−i
γi =
=
(6)
dqi
dqi
j=1,j=i
Where q−i is the quantity of a pseudo-generator representing
all other generators considered together.
r−s
qi =
⇒ qi =
n
j=1,j=i
qj − ai + xi s(1 +
n
j=1,j=i
γij )
s(2 + γi ) + ci
r − sq−i − ai + xi s(1 + γi )
s(2 + γi ) + ci
i = 1, 2, · · · , n
(7)
Therefore, the relation for CV using (7) is,
γi =
p − M Ci
−1
s(qi − xi )
In the absence of contracts,
p − M Ci
−1
γi =
s qi
(8)
(9)
For each generator i, p, q i and xi can be obtained from
historical data and the marginal cost (M C i ) is known to itself.
Moreover, the parameters for a linear demand function are
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3
known from historical records. Thus, a generator can estimate
its CV.
It can be noted from (7) that an increase in the i th generator’s forward sales directly affects its expected equilibrium
output level, since q i is directly dependent on x i .
III. O PTIMAL B IDDING USING CV
Optimal bidding is one of the tasks that a generator
has to carry out in restructured electricity markets. This is
challenging because of the limited information available to
develop the bids. Reasonable assumptions have to be made
about the unknown information or it has to be obtained from
past data. However, since CV can be estimated from the
past data, a CV based optimal bidding method will be more
appropriate. (Hence, optimal bidding based on CV can serve
as a better method since CV can be reliably estimated from
past data.) Some of the Independent System Operators (ISOs),
e.g., NEMMCO, around the world are making the past bid data
publicly available which can be used for estimating the future
bids. In this work, we have used the past dispatch data of the
generators for the purpose of validation.
Consider a system with n generators. The generators are
assumed to bid the coefficients of affine function relating
price and quantity – α (constant coefficient) and β (linear
coefficient), in the market. For any generator i with a dispatch
qi and Market Clearing Price (MCP) p, the following equation
holds,
i = 1, 2, · · · , n
(10)
αi + βi qi = p
Since the objective of any generator is to maximize its profit,
therefore, we have,
p qi − Ci (qi )
max
Considering all other competitors as a single pseudocompetitor, the dispatch of generator i can be obtained from
the CV estimation as,
q−i,est = q−i,mean + γi (qi − qi,mean )
where q−i,mean and qi,mean are the means of the dispatches
of the pseudo-generator and the generator under consideration
which are obtained from the past bidding data. γ i is the CV of
the ith generator which is calculated using (9). Finally, q −i,est
is the estimate of the dispatch of the pseudo-generator. The
mean values are considered for a time horizon representing
different system conditions and they cover consecutive bidding
periods over a time period. Thus, behavior of various market
participants as well as different system condition are reflected
in the CVs.
Using (15) and (16) in (14), we get,
Qo = q−i,mean + γi (qi − qi,mean ) + qi + K p
p=
[Qo − (q−i,mean − qi,mean γi − αi /βi (1 + γi ))]
(18)
(1/βi (1 + γi ) + K)
The other constraint to be considered is the maximum and
minimum generation limits. Thus, the formulation requires no
data about the cost function of other generators. It is dependent
only on the estimation of CV for each generator which can be
obtained from the past data. Thus, we obtain p and β which
will be submitted to the ISO as the bid along with α.
TABLE I
E STIMATED M ARKET S HARE OF F IRMS : S AMPLE S YSTEM 1 [18]
(11)
No.
1
2
3
4
5
6
7
8
9
10
11
The constraints are,
n
qi = Qe = q−i + qi
(17)
Rearranging the above equation and expressing in terms of
MCP, the demand-supply constraint becomes,
Here, Ci is the cost function of generator i. Using (10) in (11),
we have,
p − αi
p − αi
− Ci
(12)
max
p
βi
βi
qi,min ≤ qi ≤ qi,max
(16)
(13)
Generator ID
BW
ER
GSTONE
HWPS
LD
LOYY
LY
MP
STAN
TARONG
YWPS
Market share
8.40 %
6.36 %
5.36 %
6.44 %
6.81 %
4.13 %
8.15 %
7.76 %
4.96 %
5.63 %
5.93 %
(14)
i=1
where qi,min and qi,max are the minimum and maximum
generation limits and Q e represents the total generated power.
Here, we consider de-centralized power market model.
Therefore, Q e is the effective load as well, after considering
the demand-price elasticity and it is given by,
Qe = Qo − K p
(15)
where Qo is the maximum demand, K = 1/s is a nonnegative
constant and s is the slope of the inverse demand function. In
(12), α is fixed and the optimization is carried out to solve for
p and β with constraints (13) and (14).
A. Algorithm
The following is the algorithm for determining the bids for
a particular generator for a particular period.
• Firstly, each generator computes, from the past data, made
available by the ISO, the mean values of the power
dispatched by all the other generating companies in the
system and by itself along with the mean MCP.
• The marginal cost of the mean power dispatched by the
generator is then computed using its own cost curves.
• Next, the demand-price elasticity is determined from the
past data.
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4
TABLE II
S IMULATION R ESULTS OF S AMPLE S YSTEM 1 W ITHOUT C ONTRACTS
No.
1
2
3
4
5
6
7
8
9
10
11
•
•
•
Generator ID
BW
ER
GSTONE
HWPS
LD
LOYY
LY
MP
STAN
TARONG
YWPS
γi
2.085
0.985
2.680
8.237
3.604
19.584
4.596
2.515
4.915
4.625
12.773
p
Qe
q−i
qi
βi
profit
$/MWh
MWh
MWh
MWh
$/MWh/MW
$
30.573
30.636
30.576
30.649
30.621
30.534
30.642
30.590
30.639
30.654
30.596
4301.504
4297.988
4301.318
4297.305
4298.809
4303.680
4297.653
4300.556
4297.853
4297.000
4300.239
3764.600
3867.030
33952.185
3900.371
3868.745
4050.843
3791.202
3806.346
3983.710
3941.956
3937.161
536.904
430.957
349.134
396.934
430.064
252.837
506.451
494.209
314.143
355.045
363.078
0.062
0.037
0.071
0.224
0.102
0.448
0.132
0.073
0.132
0.129
0.320
16864.457
7243.515
8522.181
28792.328
16587.844
24431.185
28293.637
16434.048
11483.551
14071.629
34868.217
Using the above calculated parameters, the conjectural
variation is calculated using (9) where the forecasted load
information Q o is already available to the generator.
The value of α in the bid is equal to coefficient a of the
cost curve.
Finally, the optimization problem, with objective function
given by (12) and constraints given by (13) and (18), is
solved for determining β of the bid.
B. Application to a sample system
The above developed model has been simulated for the
system given in [18] (herein referred to as sample system 1),
using the CONOPT solver of General Algebraic Modeling
System (GAMS). The relevant data has been taken from [19].
The period for which data has been analyzed is from April
2002 to December 2002. From the data, the mean market
clearing price during the time period under consideration has
been found to be 28.55 $/MWh. The market share estimated
is presented in Table I.
The 11 generators considered are major players in an
electricity market consisting of 58 generators, who operate
among themselves 160 generating units. The 11 generators
share almost about 70% of the total load amongst them and
we concentrate on these generators for the analysis. Further
details about the analysis of the data can be obtained from [18].
Since every generator has information about its own cost
curve, the estimated cost curves have been used only for
illustration. During the actual application of the algorithm, the
generators can use their original cost curve to develop their
bids. Therefore, the risks associated with inaccurate modeling
of confidential information is eliminated. The parameters of
the estimated cost curves are as given in [18].
Table II gives the results of the simulation carried out for
each generating company. The value of the constant K is taken
to be 55.556 and the demand considered is 6000 MW. For this
market data, each generator has a CV greater than zero. This
indicates that the behavior of the generators is monopolistic
in nature [17]. The assumption of considering only the 11
prominent generators is done for simplifying the analysis. For
an elaborate analysis, all the generators in the market can be
considered. Further, instead of an average for the whole period,
each individual bidding time period can be chosen. The past
data corresponding to that particular time period will then have
to be taken for calculation of CV and this can be done for
all the time periods. However, further studies are needed for
locating a simultaneous equilibrium of the market.
The discussion on the multi-player market model with
contracts is carried out in the next section.
IV. E FFECT OF C ONTRACTS ON M ARKET
Contractual arrangements, physical or financial, play an
important role as a means of market power mitigation in
electricity markets [20]. A generating company possessing
market power needs strong motives to participate in a contract
market since the company will get higher profits by exercising
market power. However, since a contract hedges the risks of a
spot market, it will help a generating company to commit more
power at the production stage. Thus, the ability to commit a
large quantity at the production stage is a strategic benefit in
oligopolistic markets [21].
The following two sub-sections describe the impact of
contracts on the two major aspects of the market, namely,
generator bids and market power.
A. Effect on Bidding
To understand the effect of contracts on bidding, the CV
based model is applied to a multi-generator system. The
estimate of CV for this case is done using (8). The parameters
of the generator cost curves for this system (labeled as sample
system 2) are given in Table III. The objective function, which
now includes contracts, is,
p − αi
p − αi
− Ci
+ xi (f − p) (19)
max
p
βi
βi
TABLE III
PARAMETERS OF C OST C URVES OF G ENERATORS OF S AMPLE S YSTEM 2
Generator
1
2
3
4
5
6
Cost parameter ci
Cost parameter ai
$/(MWh-MWh)
$/MWh
0.00834
0.0175
0.02
0.025
0.025
0.0625
3.25
1.75
2
3
3
1
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5
TABLE IV
S IMULATION R ESULTS OF S AMPLE S YSTEM 2 WITH C ONTRACTS
γi
Generator
x = 75
x = 150
x = 200
Generator
x = 75
x = 150
x = 200
Generator
x = 75
x = 150
x = 200
Generator
x = 75
x = 150
x = 200
Generator
x = 75
x = 150
x = 200
Generator
x = 75
x = 150
x = 200
1 x = 50
2 x = 50
3 x = 50
4 x = 50
5 x = 50
6 x = 50
-0.225
-0.225
-0.225
-0.225
-0.241
-0.241
-0.241
-0.241
-0.826
-0.826
-0.826
-0.826
-0.239
-0.239
-0.239
-0.239
-0.213
-0.213
-0.213
-0.213
-0.324
-0.324
-0.324
-0.324
p
Qe
q−i
qi
βi
return
$/MWh
MWh
MWh
MWh
$/MWh/MW
$
14.775
14.641
14.238
14.052
14.808
14.704
14.391
14.183
14.770
14.761
14.733
14.714
14.852
14.763
14.494
14.315
14.897
14.802
14.520
14.332
14.813
14.770
14.642
14.556
2379.165
2386.627
2409.012
2419.310
2377.330
2383.117
2400.479
2412.054
2379.426
2379.946
2381.507
2382.548
2374.867
2379.841
2394.763
2404.711
2372.413
2377.643
2393.332
2403.791
2377.083
2379.454
2386.566
2391.307
1830.944
1828.782
1822.294
1819.310
1936.457
1934.623
1929.121
1925.454
1820.711
1818.246
1810.852
1805.922
2050.893
2049.332
2044.646
2041.522
2050.628
2049.216
2044.979
2042.155
2183.964
2182.830
2179.428
2177.159
548.221
557.845
586.718
600.000
440.873
448.494
471.358
486.601
558.715
561.700
570.655
576.626
323.974
330.510
350.117
363.189
321.785
328.427
348.353
361.637
193.119
196.624
207.138
214.148
0.021
0.020
0.019
0.018
0.030
0.029
0.027
0.026
0.023
0.023
0.022
0.022
0.037
0.036
0.033
0.031
0.037
0.036
0.033
0.031
0.072
0.070
0.066
0.063
8136.226
8231.698
8542.894
8720.971
6563.073
6654.320
6949.784
7164.864
8288.885
8346.661
8522.461
8641.724
4844.164
4934.552
5225.553
5436.086
4823.659
4913.831
5205.085
5416.535
2894.956
2958.867
3161.607
3305.938
The constraints and the demand-price elasticity are the same
as in section III. The total demand is taken to be 3200 MW.
The maximum generation limit for all the generators is taken
as 600 MW and the contract price is taken to be 15.5 $/MWh.
Using the above data, the bids obtained for each generator
are given in Table IV.
The generators are numbered in the increasing order of the
cost coefficients based on the coefficient c i . Generator 1 is,
thus, the cheapest and generator 6, the costliest. It can be seen
that the CVs obtained for the generators have values less than
zero. This implies that we almost have a perfectly competitive
market [17].
The MCP reduces as the contract quantity for the generators
increase, irrespective of their cost coefficients. This can be
seen from Table IV, wherein the relative reduction in MCP
with contracts is more for the cheaper generators (which have
a lower value of β) than the costlier ones (with a larger β).
Furthermore, with contracts, it can be seen that the relative
reduction in β is less for the cheaper generators as compared
with the costlier generators.
The generation dispatch and the revenue of all the generators
increases with contracts. Costlier generators will prefer larger
contracts to reduce their bidding coefficients and thereby,
increase their share in the spot market. The cheaper generators
already have a higher dispatch in the spot market and contracts
do not have much effect on the bidding coefficients.
In our model, a generator which calculates its bids does
not require any information about the contracts that other
generators have entered into, which is usually the case as most
of the markets have bilateral contracts.
B. Effect on Market Power
The impact of contracts on market power is analyzed for the
power market modeled using CV. The following algorithm is
used to obtain optimal dispatch in the proposed model.
1) Algorithm: This algorithm is applicable for estimating
the optimal dispatch of a particular generator for a particular
period of bidding:
• From the previous day’s data, the power dispatched by
all generating companies, price of electricity and contracts offered by the generator itself are obtained by the
generator.
• The parameters of the demand function are then determined from the past data.
• The marginal cost of power dispatched by the generator
is next calculated and also the CV using (8).
• The contracts are now offered based on the generator’s
strategy.
• Using the contracts offered and the calculated CV, the
optimal dispatch is obtained using (7).
The effectiveness of the proposed approach and the behavior
of the market participants are studied for different cases of
competition. In all cases, we consider a system consisting of
three generators.
TABLE V
PARAMETERS OF G ENERATORS FOR S AMPLE S YSTEM 3
Generator
1
2
3
Cost parameter ci
Cost parameter ai
$/(MWh-MWh)
$/MWh
0.001
0.0015
0.002
12
10
8
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6
TABLE VI
R ESULTS OF 3 G ENERATOR C ASE FOR S AMPLE S YSTEM 3
Events
Case 1
CVi = −1
Case 2
CVi =1
Case 3
CVi = 0
Case
CV1
CV2
CV3
4
= −0.6
=0
=0
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
Generator 1
x(GWh)
q(GWh)
0
13.38
2.5
13.38
5
13.38
5
13.38
10
13.38
0
7.24
2.5
8.86
5
9.82
5
9.15
10
11.06
0
9.11
2.5
10.37
5
11.29
5
10.63
10
12.16
0
13.07
2.5
13.79
5
13.58
5
12.64
10
12.22
Generator 2
x(GWh) q(GWh)
0
10.25
0
10.25
2.5
10.25
5
10.25
10
10.25
0
6.95
0
6.6
2.5
7.47
5
8.68
10
10.42
0
8.38
0
8.03
2.5
8.82
5
9.69
10
10.99
0
7.29
0
7.09
2.5
7.84
5
8.79
10
10.28
2) Application to Sample System Consisting of 3 Generators: A three generator system (labeled as sample system 3)
is used to illustrate the relevance of the proposed CV based
model for three different classical and Game Theory based
Bidding Strategies (GTBS) market structures. The market
structures considered are perfect competition, monopoly, and
Cournot and Stackelberg [22]. In our analysis, we vary the
contracts, in an increasing order, to show their impact on market power. We consider a demand function with the following
parameters: r = 45 GWh and s = 0.5 GWh/($/MWh). The
cost parameters of the generators are shown in Table V [9].
The results obtained are tabulated in Table VI.
Case 1 – Perfect Competition: In this case, all generators
have CVi equal to −1 in the case of a perfectly competitive
market. The results, thus, show that the prices remain constant
for any value of contracts offered by the generators. This
implies that no generator can exercise market power in a
perfectly competitive environment, hence, contracts do not
have any impact on the bidding behavior.
Case 2 – Monopoly: In the case of monopoly, generators
have CVi equal to 1. Hence, they can exercise market power
and this gets manifested in the higher values of market clearing
price. The effect of contracts offered by individual generators
is, therefore, significant in lowering the market price.
Case 3 – Cournot Oligopoly: In an oligopoly market, with
a Cournot type bidding strategy, the CVs of all generators
will be zero. This means that each generator believes that its
pseudo-competitors (q −i ) will show no response to its own
generation changes. Hence, in this case, contracts will lead to
a lower market price, thereby, resulting in the mitigation of
market power.
Case 4 – Stackelberg Oligopoly: In the Stackelberg model,
generator 1 acts as a leader and has a CV equal to −0.6,
while the other generators act as followers with their CVs
being equal to zero. Thus, the other generators will benefit by
the price set by generator 1. Hence, in this case, contracts have
Generator 3
x(GWh)
q(GWh)
0
8.69
0
8.69
0
8.69
2.5
8.69
5
8.69
0
6.7
0
6.38
0
7.18
2.5
8.29
5
9.88
0
7.83
0
7.53
0
6.97
2.5
7.73
5
7.62
0
6.88
0
6.71
0
7.36
2.5
8.19
5
9.5
Price
($/MWh)
25.38
25.38
25.38
25.38
25.38
48.21
46.31
41.08
37.75
27.29
39.34
38.12
35.86
33.9
28.47
35.52
34.82
32.44
30.76
26
a significant effect in reducing the market power exercised by
the leader. An increase in the contracts of the leader will lead
to reduction in the market price.
The above case studies establish that the classical GTBS
with contracts are simply special cases of CVBS with contracts.
V. C ONCLUSIONS
Until now, the CV based methods have been employed to
obtain the optimal dispatch of generators in the absence of
contracts. In this work, we have developed a multi-market
model based on CV considering contracts. Using this model
the generator bids are obtained as parameters of linear supply
function. The model has been examined on sample systems to
study their bidding behavior in the presence of contracts. The
results have shown that, in the absence of contracts, generators
exhibit monopolistic behavior (indicated by the positive values
of CVs) to gain more profits by bidding higher. However, with
contracts in place, the generators have to bid lower, resulting
in reduced price of electricity but with higher profits from
the contracts. Hence, the behavior of generating companies is
competitive in nature as indicated by the negative values of
CV.
Another application of the proposed model is for market
power mitigation using contracts. The model is applied to a
pool based market with forward contracts. The resulting lower
prices are indicative of market power mitigation. Further, the
contractual behavior of a generator, that is the extent to which
a generator is encouraged to cover its output, in the forward
market is decided by its cost parameters and CV alone. It has
also been shown that, in the presence of contracts, the classical
GTBS is only a special case of CVBS.
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N. Sasi Kiran received his B.E. degree from Andhra University College
of Engineering, Vishakhapatnam, India in 2002 and his M.Tech. from IIT
Bombay, Mumbai in 2005. His research interests include risk management,
power system analysis and deregulation.
S. A. Khaparde (M’88-SM’91) is a Professor with the Department of
Electrical Engineering, IIT Bombay, Mumbai, India. His research interests
include power system computations, analysis and deregulation in the power
systems.
Sangamesh G. Sakri received his B.E. degree from Gulbarga University,
Gulbarga, India in 1989 and his M.Tech. from IIT Bombay, Mumbai in 2005
He is currently with P.D.A. College of Engineering, Gulbarga, Karnataka,
India. His research interests are energy conservation, energy management and
deregulation in power systems.
Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY BOMBAY. Downloaded on January 29, 2009 at 03:55 from IEEE Xplore. Restrictions apply.