1. October 3-5, 2011 • Fairmont Hotel • Winnipeg, Manitoba • www.ieee.ca/epec11
A.M. (Ani) Gole
General Chair EPEC 2011
University of Manitoba
66 Chancellors Circle
Winnipeg, MB R3T 2N2
Canada
August 24, 2011
Mr. Masoud Yadollahi Zadeh
Student, Azad University-South Tehrab Branch – Iran
No. 377B – Golestan 12 Street – Braim
Abadan, YT IRAN
06316944494
Dear Mr. Yadollahi;
Re: Invitation to IEEE EPEC 2011 Conference, Winnipeg Manitoba, Canada
On behalf of the 2011 IEEE EPEC Organizing Committee, it is my pleasure to extend this invitation to you as a
registered attendee, to participate in the Conference on October 3-5, 2011 held in Winnipeg, Canada. We are pleased
that you will be presenting your paper (#1569473665) titled “Nash Equilibrium In competitive Electricity Markets”
as part of the conference technical program.
The objective of this annual three day conference is to provide a forum for experts in Electrical Power and Energy to
disseminate their recent research outcomes and exchange views on the future research directions of these fields.
We acknowledge that all your expenses related to and incurred during the aforementioned conference, as well as for
travel to and from Winnipeg, will be covered by you or your company.
If you have any questions, please contact our Registration Chair, Ms. Kimberly Laing at +1 204 989–1266.
Sincerely,
A.M. (Ani) Gole
General Chair EPEC 2011

2. Nash Equilibrium in Competitive Electricity
Markets
Masoud.Yadollahi zadeh*, Hasan Monsef**
*Electrical Engineering Department- Azad University -South Tehran Branch-Iran.
Email:masoud.yadollahi@gmail.com
**Electrical and Computer Engineering Department - Tehran University-Iran.
Email:hmonsef@ut.ac.ir
Abstract – Electricity market participants (generators)
will choose their bids in order to maximize their profit
in a competitive environment. This paper presents an
efficient mathematical technique, considering
transmission congestion and losses, to determine
generator profit maximization. By the algorithm
presented in this paper some converged bidding
coefficients has been resulted so that each supplier can
bid higher than its marginal price in the market and
get the maximum benefit. The paper further
demonstrates the establishment of Nash equilibrium
when all suppliers are trying to maximize their profit in
this manner. Finally at the end of the paper, this
algorithm is applied to a typical system and results are
presented.
Keywords –Bidding Strategy, Transmission Congestion,
Transmission Losses, Nash Equilibrium, Spot Pricing.
I. INTRODUCTION
The economic operation of a utility in a
competitive environment brings about optimization
problems such as generation costs, bidding strategies,
system constraints and many other problems. In a fully
competitive environment, power producers use various
manners to keep continuity in the market. Thus many
methods have been presented in papers and researches
to show how a market participant can solve power
system problems to gain maximum benefit.Operation
of electricity market and spot trading is discussed in
[2]. The interaction of long term contracting and spot
market transactions between Gencos and Discos is
modeled in [3, 4]. In [5], it is assumed that power
suppliers are to bid a linear supply function and paid
the market clearing price. A stochastic optimization
model is established and two methods to estimate of
bidding coefficients of rivals are developed. Imperfect
knowledge of rivals is modeled, too. First method is to
estimate bidding coefficients of rivals by normal
distribution. Second method, is to estimate by mean
value vectors. In [1], a continuous bid curve for
suppliers and consumers is assumed. However the
variation in bidding will be limited to the variation of a
single parameter k for each supplier and consumer.
This parameter will vary the bid around the true
marginal curve to get the maximum welfare by
choosing a bid which is a best response.
The aim of this paper is to simplify the applied
method in [1].According to [1], to obtain k coefficients
a Newton- step method is used to get the maximum
benefit and establish Nash Equilibrium . This needs too
much calculation . Specially in large networks using
this method makes the problem more complicated. A
simple way is presented in this paper based on Newton
itteration formula.
II. MATHEMATICAL FORMULATION
When performing market analysis of the power
system, a market participant is interested in what its profit
will be for various bidding strategies. This profit will
depend not only on its bid, but also on the bids of the
other participants in the market. In general, suppliers
follow a linear curve for their marginal cost bidding.
Fig.1, shows a linear bid curve.
In a perfect electricity market, any power supplier is a
price taker. Microeconomic theory hold the optimal
bidding strategy for a supplier is simply to bid marginal
cost. When a generator bids other than marginal cost, in
an effort exploit imperfections in the market to increase
profits, this behavior is called strategic bidding. If the
generator can successfully increase its profits by strategic
bidding or by any means other than lowering its costs, it
is said to have market power. The real electricity markets
are not perfectly competitive, and as a result, a supplier
can increase profits through strategic bidding, or in other
words, through exercising market power.
As it is seen in fig.1, each supplier submits a minimum
price πmin at which it will sell power along with a slope ms
defining the slope of the linear curve. Using these bids,
the pool operator (such as a power exchange or possibly
an ISO.) solves the OPF under the assumption that the
participants are submitting their true marginal. The
amount of dispatch received is then awarded according to
the solution of this OPF. With these bids as a base, to
test the market model, bids are chosen as some
percentage over or under true marginal cost. In order to
bid k times higher than the true marginal cost, the
supplier must submit a new bid ( min,πsm ) which

3. satisfies
k
sm
sm = and minmin *ππ k= . Fig.2, Shows a
bid that is k times higher than the true marginal cost bid.
Initially, the optimal bid for each supplier is found under
the assumption that the other suppliers bid their true
marginal cost. Only the individual supplier is allowed to
change its bid. In a perfectly competitive market, the best
response for each supplier is to bid its marginal cost. This
is a well-known economic principle which can be
proven very simply. Define supplier profit as Revenue
minus Costs, which is written:
[ ])( GiPiCGiPiiR −= β (14)
At which:
Ri: The ith generator profit
βi: The ith generator bid
PGi: The ith generator power generated
Ci(PGi): The ith generator generation cost
Thus the objective function for each supplier in the
market is:
{ }
max
max
1
2
.
)]([
jTjT
GiPGiP
l
j
jTK.DiPGiP
tosubject
GiPiCGiPiMaxiMaxR
≤
≤
=
+=
−=
∑
β
(15)
WhereTj is transmitted power through line j.
On the other hand, the general form of the cost function is
as follows:
2)( iPiciPibiaiPiC ++= (16)
If we put (16) into (14) then profit function is expanded
as follows:
2
Gi
PicGiPibiaGiPiiR −−−= β (17)
Generation
Bid
[MW]
ms
Price=π
πmin [$/MWh]
Figure 1. Linear bid curve
Generation
Bid
[MW] ms
k
ms
Price=π
πmin kπmin [$/MWh]
Figure 2. Bidding k times higher than the marginal cost
Considering (16) the true marginal cost bid for a
generator is of the form :( indices can be ignored for the
time being)
)min()()(
2
1
)( πππππ −=−=−= smbsmb
c
GP 1
(18)
At which, π is the spot price.
So, for new bidding





=
=
minmin .ππ k
k
m
m s
s
new true marginal
cost bid is as follows:
)(
2
1
)min()( kb
kc
smGP −=−= ππππ (19)
πβ k= (20)
Putting (19) and (20) into (17) will result (21):
a
c
b
kc
b
ckc
b
kc
b
c
kb
c
R
akb
ck
kb
kc
b
kb
kc
kR
−−+−+−−=⇒
−−−−−−=
4
2
224
2
2
2
222
2
)2(
24
1
)(
2
))(
2
1
(
πππππ
ππππ
(21)
The supplier's profit sensitivity to variations in its bid can
be used to determine a Newton-step that improves
profit[1]. This Newton-step is defined as shown in (22):
oldk
k
R
oldk
k
R
oldknewk ||
1
2
2
∂
∂
−








∂
∂
−= (22)
According to [1],obtaining knew , needs to form large
matrixes and then transposing them and obtain inverse
matrixes, this will complicate the problem ,specially
when we work on a large network.To avoid complication,
simply we derive from R in relation to k. Thus:
ck
bk
k
R
3
32
2
ππ −
=
∂
∂
(23)
1 - See figure 1

4. ckk
R
4
2
2
2
2
3π
−=
∂
∂ (24)
Therefore, using (22), the ith generator k coefficients are
obtained:
iodk
ik
iR
ioldk
ik
iR
oldik
newik ||
1
2
2
∂
∂
−








∂
∂
−= (25)
The determination of economic equilibrium such as
Nash Equilibrium is of interested.
To determine a Nash Equilibrium point, the k
coefficient for each supplier can be iteratively obtained
by all participants until a point is reached where each
supplier's k coefficient remains constant. At this point
Nash Equilibrium in the market will be established.
IV. NUMERICAL EXAMPLE
To illustrate the proposed approach, a sample 6-bus
network shown in fig.3 is considered. The system data is
provided in tables 1 and 2.
Figure 3. Typical 6-bus system
Table 1. Market participants information
Market
Participant
Power(MW) Cost Function
G1 10-250 C1(P1)=150+5P1+0.11P1
2
G2 10-300 C2(P2)=600+8P2+0.085P2
2
G3 10-270 C3(P3)=335+10P3+0.1225P3
2
D4 200 ------------------
D5 200 ------------------
D6 200 -------------------
Table 2. Transmission line limits
Line Capacity (MW) From bus ...To...
1 100 1-2
2 120 1-4
3 100 1-5
4 100 2-3
5 100 2-4
6 100 2-5
7 100 2-6
8 120 3-5
9 120 3-6
10 100 4-5
11 100 5-6
By analysing the network (using Power World
software) no considering losses and congestion ,we will
obtain the results shown in fig.4 (see appensix A).To
simulate a real network we need to consider losses and
lines congestion. Fig.5 shows an analysed network with
these constraints.In fig 5, all spot prices are equal. But we
see that some lines are overloaded.
In order to release the lines from overload, we need to
do optimal power flow on this system. In fig 6.The OPF
of the 6 bus system has been shown.(see appendix A)
Now k coefficient for each generator should be found
in order to obtain the maximum benefit and reach to Nash
Equilibrium.
Using (24) k coefficients obtained for each generator
are shown in table 3:
Table3. k coefficients for each generator
Iteration k1 k2 k3
0 1 1 1
1 1.2713 1.2840 1.6480
2 1.5330 1.5779 1.9702
3 1.7014 1.7981 2.1630
4 1.7487 1.8819 2.2100
5 1.7515 1.8906 2.2122
6 1.7515 1.8907 2.2122
7 1.7515 1.8907 2.2122
To illustrate Nash Equilibrium, these k coefficients are
shown as curves.Nash Equilibrium occures when k
coefficients for generators remains constant after some
itterations . in fig 7, k coefficients as curves have been
shown to show their constant rate after some iteration .
(see appendix A)
CONCLUSION
A method to obtain k coefficients to reach to Nash
Equilibrium point for power suppliers in a competitive
electricity market in order to gain maximum profit is
presented in this paper. By solving the problem, Nash
equilibrium is established after some iteration. In this
paper, real circumstances of a power system are
considered so that market participants could find actual
parameters of a market. It has been shown that market
suppliers do have some market power violating an
underlying assumption of competitive markets. In other
words, each supplier's bidding strategy has an effect on
the market price. This encourages them to bid higher
than their marginal cost. Furthermore, it has been shown
that network constraints such as losses and transmission
line congestion cause market power.
For continuation of the study, participate consumers as
competitors and analysis the system by proposed method
is recommended. The influence of reactive power as a
network constraint on the bidding strategy and market
analyzing is recommended for future study, as well.

5. REFERENCES
[1] J.D.Weber, T.J.Owerbye, "A two level
optimization problem for analysis of market
bidding strategies," 1999IEEE, Volume: 2,
Jul1999, Page(s):926-936
[2] W.Mielczarski, G.Michalik, M.Widiaia,
"Bidding Strategies in Electricity Msrkets,"
Power Industry computer
applications.1999.PICA99. Proceeding of the
21st
1999 IEEE International conference,
Publication Year: 1999, Page(s):71-76
[3] D.J.Wu, Paul Kleindorfer, Jin E.Zhang,
"Optimal bidding and contracting strategies in
the deregulated electric power market: part І,"
System Sciences, 2000. Proceedings of the 33rd
Annual Hawaii International Conference on
System Sciences Publication Year: 2000.
[4] D.J.Wu, P.Kleindorfer, J.E.Zhang, "Optimal
bidding and contracting strategies in the
deregulated electric power market: part
II,"System Sciences, 2001. Proceedings of the
34th Annual Hawaii International Conference on
System Sciences Publication Year: 2001
[5] Fushuan Wen, A.K.David, "Optimal bidding
strategies and modeling of imperfect information
among competitive generators," Power System
IEEE Transactions, Volume: 16, Issue: 1,
Publication Year: 2001, Page(s): 15 - 21
[6] Song. H, C. C.Liu, J.Lawarree," Nash
Equilibrium Bidding Strategies in a Bilateral
Electricity Market," Power Engineering Review,
IEEE, Volume: 22, Issue: 2, Publication Year:
2002, Page(s): 62 - 62
[7] Tengshun peng, K.Tomsovic,"Congestion
influence on bidding strategies in an electricity
market," Power system, IEEE Transactions,
Volume: 18, Issue: 3, Publication Year: 2003,
Page(s):1054-1061
[8] Y.P.Molina, R.B.Prada, O.R. Saavedra,
"Allocation of transmission loss cost using game
theory," Power Tech, 2007 IEEE Lausanne,
Publication Year: 2007, Page(s): 407 – 412
[9] F. Gao, G.B. Sheble "Electricity market
equilibrium model with resource constraint and
transmission congestion" Original Research
Article. Electric Power Systems Research,
Volume 80, Issue 1, January 2010, Pages 9-18

6. Appendix A:
Resulted Figures
Figure 4.The 6 bus system power flow assuming zero losses and infinity line capacity

7. Figure 5. The 6 bus system considering losses and congestion of lines
Figure 6.The OPF of the 6 bus system
Figure 7. k coefficients as curves