CHAPTER - 8
Optimization of Strategic Bidding to maximize Profit
through Double side Action in Electricity Market using
Firefly Algorithm
8.1 Introduction
The reconstruction of the electric power industry creates a significant effect on
issues pertaining to the economic and reliable operation of the electric power system.
The generation companies and large consumers experience a different task of
designing the bidding methodologies. Therefore the systematic development of
optimal bidding strategy becomes a major concern of Generation companies and
power consumers. An innovative approach for defining optimal bidding strategy is
presented as a multi objective stochastic optimization problem and solved by Firefly
Algorithm (FA).
The Firefly Algorithm is introduced for the bidding problem for both
Independent Power Producers (IPPs) and large consumers (Double side action) to
obtain the global optimal solution. It effectively maximizes the IPPs profit for the
benefit of large consumers. A numerical example with six suppliers and two large
consumers is considered to illustrate the prominent features of the proposed method
and the test results presented. The simulation result shows that this approach
effectively maximize the profit and benefit of power suppliers and large consumers,
converge much faster and more reliable when compared with available methods
[133].
118
8.2
Problem Formulatio
Formulation
8.2.1 Mathematical Model
In the competitive Electricity market, the Independent Power Producers (IPPs)
and large consumers participat
participate in bidding methodologies for their own benefits. The
mathematical model of Pool based Electricity Market is presented in Fig 8.1.
Fig. 8.1. Mathematical model of electricity market for Double side action
Let the system consist of ‘ m ’ generators in the power network managed by an
ISO, an aggregated consumer load which may not participate in the demand side
bidding but remain inaccessible to the price elasticity, and ‘ n ’ large consumers who
participate in demand side bidding. The power suppliers and large consumers need to
bid a linear non- decreasing supply and non
non-increasing
increasing demand from PX, in the form
of a linear supply curve.
by
The m Independent Power Producers bid a Linear supply curve mentioned
mentio
R = ai + bi Pi where i=1,
=1, 2….. m and n large consumers bid a linear demand curve
denoted by R = c j − d j L j where j=1, 2….. n . In the above equation Pi represents the
real power output of i th generator, ai , bi the bidding co-efficient
efficient of i th supplier, L j the
load demand of j th consumer, and c j , d j the bidding co-efficient
efficient of j th large consumer.
The co-efficient ai , bi , c j and d j are non-negative numbers.
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The power exchange perform
performs the generation scheduling and demand dispatch
management while satisfying the security and reliability constraints using unique
dispatch procedure, with the objective of maximizing pay
pay-off.
off. Hence power exchange
manipulates the system output and demand of large consumers so as to improve the
profit of GENCOs. This process is graphically explained using Fig 8.2.
Fig. 8.2.
2. Market Equilibrium Point for Double side action
The auction based electricity market, data for the successive bidding period is
unknown and therefore the suppliers/large consumers may not have substantial
information to evolve a solution for this optimization problem. However, the prior
bidding allows the
he chance for estimating the bidding co-efficient
efficient of the rival as a
possibility.
The objective of independent power producers is to maximize their profit. The
cost function of ith power supplier is given by the equation (8.1)
2
Ci (Pi ) = ei Pi + fi Pi
(8.1)
The objective function of independent power producer can be defined as in equation
(8.2)
m
m
i =1
i =1
Max : F ( a i , b i ) = ∑ RPi − ∑ C i ( Pi )
(8.2)
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Similarly, the objective of large consumer is to maximize their benefit. The Revenue
function jth large consumer is expressed as in equation (8.3)
B j (L j ) = g j L j − h j L j
2
(8.3)
The objective function of large consumer can be written as in equation (8.4)
n
n
j =1
j =1
Max : G (c j , d j ) = ∑ B j ( L j ) − ∑ R j L j
(8.4)
The Market Clearing Price (R) is defined as ratio of bidding co-efficient of
power suppliers (ai , bi ) and large consumers (c j , d j ) with aggregated load demand and
represented by the equation (8.5)
n c
ai
+∑ j
i =1 bi
j =1 d j
R=
m
n
1
1
K +∑ +∑
i =1 bi
j =1 d j
m
Q0 + ∑
(8.5)
The aggregated load demand is formulated as in equation (8.6)
Q( R) = Qo − KR
(8.6)
Where
Q o = Constant number.
K
= Coefficient of the price elasticity of the aggregate demand.
Constraints
2. Power balance constraints:
m
n
i =1
j =1
∑ Pi = Q( R) + ∑ Li
pi =
R − ai
bi
(8.7)
i =1,2..........
...m
121
(8.8)
Lj =
cj − R
j = 1,2.......... ...n
dj
(8.9)
2. Power generation limit constraints:
pi min ≤ pi ≤ pi max
i = 1,2.......... ...m
(8.10)
3. Power consumption limit constraints:
L j min ≤ L j ≤ L j max
j = 1,2.......... ...n
(8.11)
Once the problem is analyzed from the pth (p=1,2,……,n + m) participant, the
bidding coefficients of the jth (j=1,2,…..,n and j≠p supplier, aj and bj ,follows a joint
normal distribution with the following probability density function ( pdf ).
pdf ( a j , b j ) =
1
2Π σ (j a )σ (jb ) 1 − ρ 2j
×

 a j − µ j ( a )
1


exp −
2
 σ (j a )
ρ
−
2
(
1
)
j 


2
(b )

2 ρ j ( a j − µ (j a ) )(b j − µ j )  b j − µ (jb )
 −
+
( a ) (b )
 σ (b )

σ
σ
j
j
j






2

 


(8.12)
(b )
Where ρ j is the correlation co-efficient between aj and bj .The mean µ (j a ) , µ j and
standard deviation σ (j a ) , σ (bj ) are the parameters of the joint distribution. The marginal
(b )
distribution of a j , bj are normal with mean values µ (j a ) ), µ j and standard deviations
σ (j a ) , σ (bj ) respectively. Similarly, the above probability density function (
pdf ) is also
used for finding the bidding coefficients of the large consumers. Based on historical
bidding data these distributions can be determined. Using probability density function
(8.12) for suppliers as well as large consumers the joint distribution between aj , bj ,
and between c j , d j the optimal bidding problem with objective functions given in
equation (8.2)and (8.4) and constraints (8.7) to (8.11) becomes
optimization problem.
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a stochastic
8.3
Implementation
of
Firefly
algorithm
to
solve
Bidding
Strategies for Double side action
By the way of its global searching behavior, the power producers and
consumers need exclusive bidding strategies that require to consider constraints such
as Power balance, Generator limits and Load consumption limits of market
participants. The Firefly Algorithm can directly solve the optimal bidding problem by
way of which the electric power can be sold at optimal prices and enable to have
maximized profit. The flow chart of proposed method is shown in fig. 8.3. The Firefly
Algorithm entails four essential parameters, Population size (n), Attractiveness ( β ),
randomization parameter ( a ) and Absorption coefficient ( γ ).The feasible parameters
obtained by iterative processes are as follows. α = 0.2 – 0.9, β = 0.2 – 1.0, γ = 0.1–
10 and n = 25 – 50.
The solution of the optimal bidding problem of six independent power
producers and two large consumers is obtained using the following parameter setting.
Where n = 30, β = 0.20, α = 0.25, γ = 1 and maximum number of iterations = 5000.
Owing to the random nature of the FA, their performance cannot be judged by the
result of a single run. Many trials with independent population initializations are made
to obtain a useful conclusion of the performance of the approach. The superiority of
the proposed FA is established by comparing the results with those already reported
by the most recently published methods such as PSO, GA and Monte Carlo method
for solving the bidding problem. The study is programmed in MATLAB 9.0 and
simulation carried out on a computer with a Pentium IV, Intel Dual core 2.2 GHz, 2
GB RAM.
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Start
Read system data, Power suppliers data (Cost
Coefficients, Generator limits), Consumer data
(Revenue coefficients, load limits), Aggregated
load and Price Elasticity
Initialize the FA parameters: Population size (n).
Attractiveness ( β ), randomization parameter ( α ),
Absorption coefficient ( γ ) and number of iterations
Create the initial random population of bidding
coefficients (bi , d j )
By using bidding co-efficient calculate Market
Clearing Price ( R ) using the equation
n c
ai
j
+∑
b
d
j =1
i =1
j
i
R=
m
n
1
1
K +∑ +∑
i =1 bi
j =1 d j
m
Q0 + ∑
By using MCP calculate fitness of each
population using the equation
Maximize: F (ai , bi ) = RPi − Ci ( Pi ) &
Maximize : B (c j , d j ) = B j ( L j ) − RL j
Apply FA parameters to obtain the optimal
solution (Max profit and benefit)
No
Whether optimal
solution is reached
Yes
Print the profits of power suppliers and benefits
of large consumers.
Stop
Fig.8.3. Flow chart for Profit and Benefit Maximization of Market
Participants by Proposed method
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8.4
Case study and Results
The proposed Firefly approach is applied to a test system given in reference
[103] which consists of Six Independent power Producers (GENCOs) and two large
consumers. The cost coefficients of power generation and maximum/ minimum limits
of six Independent power Producers are given in appendix A.11 (Table A.11.1).
Similarly the revenue cost coefficients and load consumption limits of two large
consumers are listed in Appendix A.11 (Table A.11.2). The fuel cost function of each
generator and revenue cost function of consumers is expressed in the form of a
quadratic equation. The parameters associated with the load characteristics are
considered from the same reference [103] where in the aggravated load Q0 equals to
300 MW and the price elasticity K equals to 5.
Table 8.1 Simulation Results for Six Power suppliers
1
2
3
Bidding
Strategy ($/MW)
0.0656
0.1214
0.2647
Bidding
Power (MW)
160.00
122.50
51.00
Revenue
($)
2637.120
1524.585
840.502
Fuel Cost
($)
1248.00
934.828
510.637
Profit
($)
1389.120
589.756
329.945
4
5
0.0834
0.1716
112.50
41.25
1384.488
679.880
987.668
496.867
396.820
183.013
6
0.1716
41.25
679.880
496.867
183.013
GENCOs
Total Profit
Market clearing price ($/MWh)
3071.667
16.482
Table 8.2 Simulation Results for Two Large Consumers
Large
Bidding Strategy Bidding Load
Consumers
($/MW)
(MW)
1
0.0876
169.00
2
0.0706
141.95
Total Benefit ($)
125
Revenue
($)
3927.56
2944.42
Marginal
Cost ($)
2775.458
2339.780
Benefit
($)
1152.102
604.636
1756.738
Table 8.3 Comparison of Bidding Strategies of Sis Power suppliers
GENCOs
1
2
3
4
5
6
FA
(Proposed)
PSO
[118]
GA
[118]
Monte
Carlo [103]
bi
bi
bi
bi
0.0656
0.1214
0.2647
0.0834
0.1716
0.1716
0.062
0.079
0.243
0.046
0.124
0.124
0.058
0.101
0.221
0.035
0.116
0.116
0.0297
0.124
0.092
0.074
0.170
0.170
Table 8.4 Comparison of Bidding Strategies of Two Large Consumers
Large
Consumers
FA
(Proposed)
dj
PSO
[118]
dj
GA
[118]
dj
Monte
Carlo [103]
dj
1
2
0.0876
0.0706
0.072
0.0514
0.064
0.049
0.097
0.077
The simulation results of Independent power Producers and large consumers
are presented in Table 8.1 and Table 8.2. The graphical representation of revenue, fuel
cost and profit of independent power Producers are shown in Fig.8.4. The
comparative studies with Particle Swarm Optimization [118], Genetic Algorithm
[118] and Monte Carlo method [103] are part of the investigation to analyze the
bidding coefficients of market participants and displayed in Tables 8.3 and 8.4. The
Bidding power of independent power producers and large consumers are shown in
Fig. 8.5. It is seen from Fig.8. 6, the total profits and benefits of the proposed method
is improved than the other available methods.
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Table 8.5 Comparison of Bidding Power and Profits of Six Power suppliers
GENCOs
FA
(Proposed)
PSO
[118]
GA
[118]
Monte
Carlo [103]
P(MW)
Profit($)
P(MW)
Profit($)
P(MW)
Profit($)
P(MW)
Profit($)
1
160.00
1389.120
156.00
1320.3
152.00
1310.1
160.00
1368
2
122.50
589..756
104.38
574.1
102.83
504
89.4
572.7
3
51.00
329.945
47.271
316.2
41.921
291.8
45.7
322.9
4
112.50
396.82
119.380
416.1
116.28
384.7
88.8
386.4
5
41.25
183.010
48.762
178.4
46.025
165.8
43.1
177.5
6
41.25
183.010
48.762
178.4
46.025
165.8
43.1
177.5
3000
Revenue ($)
Fuel cost ($)
Profit ($)
!"#$%&'(
2500
2000
1500
1000
500
0
1
2
3
4
Power suppliers
5
6
Fig. 8.4. Revenue, Fuel cost and Profit of Six Power suppliers
Table 8.6 Comparison of Bidding load and Benefits of Two large Consumers
Large
Consumers
FA
(Proposed)
PSO
GA
[118]
[118]
Monte
Carlo [103]
L
(MW)
Benefit
($)
L
(MW)
Benefit
($)
L
(MW)
Benefit
($)
L
(MW)
Benefit
($)
1
169.00
1152.102
168.97
1146
162.61
1135.6
139.70
1126.3
2
141.95
604.636
140.92
611.8
139.95
596.4
112.10
592.6
127
Table 8.7 Comparison of MCP, Total profits and Computational
time of Market participants
MCP ($/hr)
Total
Profits ($)
Computational
time (sec)
FA
(Proposed)
16.482
PSO
[118]
16.47
GA
[118]
15.81
Carlo [103]
4828.405
4741.3
4554.2
4723.9
3.65
6.24
12.28
--
Monte
16.35
Fig 8.5. Comparison of Bidding Power of Market participants
It is because the FA algorithm plays a vital role to obtain of the global optimal
solution. The Table 8.5 and 8.6 throw bright on the Bidding power, Bidding load,
Profit and Benefit of six Independent power producers and two large consumers.
Fig.8.6. Comparison of Profit of Market participants
128
Fig.8.7.
.7. Comparison of Total profits of Proposed with Existing methods
for Market participants
The comparison
omparison of market clearing price (MCP), total profits and
computational time of market participants are presented in Table 8.7. The total profits
of market participants are compared with proposed and existing methods in Fig .8.7.
It is clear that the proposed
oposed method provides improved profits and benefits compared
to existing methods. The computational time is also seems to be lower than the other
available methods.
8.5
Summary
The Firefly Algorithm has been applied to solve bidding strategy problem in
orderr to improve the profit and benefit of Independent power Producers and two large
consumers in an open Electricity market. The simulation result ha
have been compared
with Particle Swarm Optimization (PSO), Genetic Algorithm (GA) and Monte Carlo
method. The performance has been able to confirm the feasibility and reliability of FA
algorithm as an efficient methodology in analyzing the optimal bidding strategy of
market participants. The studies have been able to showcase promising nature of the
Firefly algorithm
thm for solving complicated power system optimization problem under
deregulated environment.
129