This document is a letter informing Dr. Masoud Yadollahi Zadeh that their paper titled "A New Approach To Obtain Maximum Benefit In Electricity Markets Based On The Newton Iteration Equation" has been accepted for presentation at the 13th IASME/WSEAS International Conference on Mathematical Methods and Computational Techniques in Electrical Engineering in Angers, France from November 17-19, 2011. The letter provides information on registering for the conference, guidelines for preparing papers for the proceedings, and notes that the best papers may be invited for journal publication. It encourages Dr. Yadollahi Zadeh's attendance and participation in the conference.

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Date (27/9/2011)
Dear Dr. (Masoud Yadollahi Zadeh),
We are pleased to inform you that your paper submitted to 13th IASME/WSEAS International Conf. on
MATHEMATICAL METHODS AND COMPUTATIONAL TECHNIQUES IN ELECTRICAL ENGINEERING is
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Paper ID number: 673-106
Title: A New Approach To Obtain Maximum Benefit In Electricity Markets Based On The Newton Iteration Equation
Authors: MASOUD YADOLLAHI ZADEH - HASAN MONSEF
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2. A New Approach To Obtain Maximum Benefit In Electricity Markets
Based On The Newton Iteration Equation
MASOUD YADOLLAHI ZADEH - HASAN MONSEF
Electrical Engineering Department
Azad University -South Tehran Branch
AHANG Boulevard -Tehran
Iran
Email:masoud.yadollahi@gmail.com
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. 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.
1 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.
2 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.2, 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
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3. 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 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:
[ ])( GiiGiii PCPR −= 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
K.
.
)]([
jj
GiGi
l
j
jDiGi
GiiGiii
TT
PP
TPP
tosubject
PCPMaxMaxR
≤
≤
+=
−=
∑
=
β
(15)
WhereTj is transmitted power through line j.
On the other hand, the general form of the cost
function for generators is as follows:
2)( iiiiiii PcPbaPC ++= (16)
If we put (16) into (14) then profit function is
expanded as follows:
2
GiiGiiiGiii PcPbaPR −−−= 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
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ISBN: 978-1-61804-051-0 51

4. Considering (16) the true marginal cost bid for a
generator is of the form :( indices can be ignored
for the time being)
)()()(
2
1
)( minπππππ −=−=−= ssG mbmb
c
P 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
mP sG −=−= ππππ (19)
πβ k= (20)
Putting (19) and (20) into (17) will result (21):
c
b
kc
b
ckc
b
kc
b
c
kb
c
R
kb
ck
kb
kc
b
kb
kc
kR
4242222
)(
4
1
)(
2
))(
2
1
(
2
2
222
2
2
−+−+−−=⇒
−−−−−=
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):
oldkoldkoldnew
k
R
k
R
kk ||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)
ckk
R
4
2
2
2
2
3π
−=
∂
∂ (24)
Therefore, using (22), the ith generator k
coefficients are obtained:
iodk
i
i
ioldk
i
i
oldinewi
k
R
k
R
kk ||1
2
2
∂
∂
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∂
∂
−=
−
(25)
1 - See figure 1
3 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.
Table 1. Market participants information
Market
Participant
Power(M
W)
Cost Function
G1 10-250 C1(P1)=150+5P1+0.1
1P1
2
G2 10-300 C2(P2)=600+8P2+0.0
85P2
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
Figure 3. Typical 6-bus system
Recent Researches in Mathematical Methods in Electrical Engineering and Computer Science
ISBN: 978-1-61804-051-0 52

5. By analysing the network (using Power World
software) no considering losses and congestion ,we
obtain the following results(fig.4):
Figure 4.The 6 bus system power flow assuming
zero losses and infinity capacity of lines
In fig 4 all spot prices are equal. But if we consider
losses and congestion in the system we see that
some lines would be overloaded.
Figure 5. The 6 bus system considering losses
and congestion of lines
In order to release the lines from overload, we
need to do optimal power flow on this system, so :
Figure 6.The OPF of the 6 bus system
Now k coefficient for each generator to obtain
the maximum benefit should be found . Using (25)
k coefficients obtained for each generator are
shown in table 3:
Table 3. k coefficients
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
4 Conclusion
A method to obtain k coefficients for power
suppliers in a competitive electricity market in
order to gain maximum profit is presented in this
paper. 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.
Recent Researches in Mathematical Methods in Electrical Engineering and Computer Science
ISBN: 978-1-61804-051-0 53

6. Refrences:
[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,
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[3] D.J.Wu, Paul Kleindorfer, Jin E.Zhang,
"Optimal bidding and contracting
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Annual Hawaii International Conference
on System Sciences Publication Year:
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[4] D.J.Wu, P.Kleindorfer, J.E.Zhang,
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[6] Song. H, C. C.Liu, J.Lawarree," Nash
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[7] Tengshun peng, K.Tomsovic,"Congestion
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[8] Y.P.Molina, R.B.Prada, O.R. Saavedra,
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Recent Researches in Mathematical Methods in Electrical Engineering and Computer Science
ISBN: 978-1-61804-051-0 54