a modified pso based solution approach for economic load dispatch problem in power system

8/10/2019 A MODIFIED PSO BASED SOLUTION APPROACH FOR ECONOMIC LOAD DISPATCH PROBLEM IN POWER SYSTEM

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Presented By:

-

Nishant

Chaturvedi


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CONTENTS

Introduction

Literature Review Objective

Methodology

Problem Formulation

Implementation

Results & Discussion

Conclusion

Future Scope of Work

References


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In power generation our main aim is to generate the requiredamount of power with minimum cost.

Economic load dispatch means that the generators real and

reactive power are allowed to vary within certain limits, so asto meet a particular load demand with minimum fuel cost.

This allocation of load are based on some constraints

Equality Constrain Inequality Constrain


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Literature Survey

Pluhacek michal et al (2013) a new approach for chaos driveparticle swarm optimization (PSO) algorithm is suggested. Two

different chaotic maps are alternatively used as pseudorandom

number generator and switch over during the run of chaos driven

PSO algorithm.

Rani C. et al (2013) A chotic local search operator is introduced

in the proposed algorithm to avoid premature convergence.

Park Jong-Bae et al (2010) An improved PSO framework

employing chaotic sequence combined with conventional

linearly decreasing inertia weights and adopting a cross overoperation scheme to increase both exploration and exploitation

capability of the PSO.


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Jaini et al (2010) A particle swarm optimization algorithm

(PSO) with one of the accelerating coefficient being constant are

propose to solve the economic power dispatch problem.

Tao Zhang et al (2009) A modified tent-map-based chotic PSO

(TCPSO) to solve the ELD problem. More specifically, a noveldynamic inertia weight factor was incorporated with the

modified hybrid tent-map-based chaotic PSO which balance the

global and local search better.

Literature Survey


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Literature Survey

Chaturvedi K. T. Et al (2008) A novel self organizinghierarchical particle swarm optimization ( SOH_PSO) for the

nonconvex economic dispatch to handle the problem of

premature convergence.

Araujo Ernesto et al (2008) Particle swarm optimization

approach intertwined with lozi map chaotic sequence to obtain

Takagi- Sugeno (TS) fuzzy model for representing dynaic

behavior are proposed.

Leandro dos Santos Coelho et al (2008) The use of combining

of particle swarm optimization, Gaussian probability distributionfunction and chaotic sequence.


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Park Jong- Bae et al (2006) A novel and efficient method forsolving the economic dispatch problem with valve point effect

by integrating the particle swarm optimization with the chaotic

sequences.

Chuanwen J. et al (2005) Suggested a self adaptive chaoticparticle swarm optimization is used to solve the ELD problem in

deregulated environment. Logistic map chaotic sequence to

generate the random number R1, R2 and self- adaptive inertia

weight scale in original PSO to improve the performance.

Literature Survey


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OBJECTIVE The main objective of study is to minimize generation cost using partical

swarn optimization (PSO) algorithm for the economic load dispatch (ELD)problem.

The purpose of the economic load dispatch (ELD) problem is to control the

committed generators output such that the total fuel cost is minimized,

while satisfying the power demand and other physical and operational

constraints.

To integrate PSO method with Chaotic map for solving ELD problem

having generated unit with non smooth cost function and multi-fuel.To maximize the power generation by proposing a PSO algorithm to

obtain the optimum scheduling of generator


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METHODOLOGY

Particle swarm optimization

Proposed by james kennedy & russell

eberhart in 1995

Inspired by social behavior of birds

and fishes

Combines self-experience with social

experience

Population-based optimization


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Concept of PSO

Uses a number of particles that constitute a swarm moving

around in the search space looking for the best solution.Each particle in search space adjusts its flying according to its

own flying experience as well as the flying experience of other

particles.

PSO ALGORITHM

Basic algorithm of PSO

1. Initialize the swarm form the solution space

2.Evaluate the fitness of each particle

3. Update individual and global bests

4. Update velocity and position of each particle

5. Go to step 2, and repeat until termination condition


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FLOW CHART OF BASIC PSO

Start

Define the parameter of PSO Constants

C1, C2 Particle (P), and Dimension (D)

Initialize particles with random

Position (P) and Velocity vector (V)

Calculate fitness for each Population

Update the Population local best

Update best of local bests as gbest

Upadate Particle velocity using

eq. (1) and Postion using eq. (2)

If iteration

Completed

Stop

No

Yes


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CHOTIC THEORY

Chaos: a state of disorder and irregularity.

It describes many physical phenomena with complex behaviorby simple laws.

Dynamical systems: systems that develop in time in a non-trivial manner.

Determini stic chaos: irregular motion generated by nonlinear

dynamical systems whose laws determine the time evolution of astate of the system from a knowledge of its previous history.

i) Logistic Map

11 1.. kkk fff


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Where is a control parameter and has a real value

between [0, 4]. Despite the apparent simplicity of the

equation, the solution shows a rich variety of behaviours.

The behaviour of the system represented by above equation

is greatly changed with the change of . The value of

determines whetherfstabilizes at a constant size, oscillates

between bounded sequences of sizes, or react chaotically in

an unpredictable pattern. fk-1, is a number between Zero andOne.


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ii) Lozi map

Lozi introduced in a short note, a two-dimensional map the

equations and attractors of which resemble those of the

celebrated henon map. Simply, a quadratic term in the

latter is replaced with a piecewise linear contribution in the

former. This allows one to rigorously prove the chaotic

character of some attractors. The lozi map is depicted in fig.The map equations are given below. The parameters used in

this work are: a=1.7 and b=0.5.

nnn

bYXaX

||11

nn XY 1


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Where a and b are the real non-vanishing parameters. Inside

the region where the orbits remain bounded, the lozi map

may present both regular and chaotic behaviours.The new proposed algorithm utilizes lozi map for the first

part of the optimization process. When pre-defined number

of iterations is achieved, the lozi map is switched over to

logistic map.


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Problem formulationAn objective function expresses the main aim of the model

which is either to be minimized or maximized. It is expressed interm of design variable and other problem parameter. In presentwork the goal is to minimize the generation cost of committedgenerating unit i.e three, forty, and ten which are given below

Where,

FT: Total Generating Cost

Fi: Cost Function of ithGenerating Unitai,bi,ci: Cost Function of Generator i

Pi: Output Power of Generator i

N: Number of Generator

N

iiiT PFF

1

iiiii cPbPa 2


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Equality and inequality constraints

Active power balance equation: for power balance, an

equality constraint should be satisfied. The total generatedpower should be the same as the total load demand plus

the total line loss.

Where Pload is the total system load. The total

transmission network loss, Ploss is a function of the unit

power outputs that can be represented using B coefficients

as follows:

n

i

lossloadi PPP1

n

i

n

j

n

i

iijijiloss BPBPBPP1 1 1

000


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Here, Ploss is the total line loss. However, the transmission

loss is not considered in this research work for simplicity

(i.e., Ploss = 0).

2) Minimum and maximum power limits: power output ofeach generator should be within its minimum and maximum

limits. Corresponding inequality constraints for each

generator is.

Where Pi,min and Pi,max are the minimum and

maximum output of generator i, respectively.

m ax,m in, iii PPP


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Non-Smooth Cost Function with Valve-Point effectsThe generating units with multi valve steam turbine exhibit agreater variation in the fuel cost function. Since the valve point

result in the ripples, a cost function constraints higher order non-linearity. Here the sinusoidal functions are thus added to thequadratic cost function as follows:

Where ei and fi are the coefficients of generator i reflectingvalve-point effects.

Non-Smooth Cost function with Multi Point FuelSince the dispatching unit are practically supplied with multi

fuel sources, each unit should be representing with severalpiecewise quadratic function reflecting the effect of fuel typechange. In general, a piecewise quadratic function is used torepresent the input output curve of a generator with multi fueland described as.

|sin| min,2

iiiiiiiiiii PPfePcPbaPF


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Where, ain,bin,cinare the cost coefficients of

generator i for the p-th power level.

PiFi

2

222

2

111

iiiii

iiiii

PcPba

PcPba

2

iiniinin PcPba

21

1min

iii

iii

PPPif

PPPif

ma x1 iiin PPPif

.........

.........

.........


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IMPLEMENTATIONPseudo Code for ELD

Input required power (Pd)Initialize the coefficients a, b, c, e and f of all generators.

Select the optimization technique

Initialize the value of a_lozi = 1.7, and b_lozi = 0.5;

Provide the upper bound (UB) and lower bound (LB) constrains on generators

Initialize the PSO coefficients c1= 2, c2= 1, wmax= 0.9, wmin=0.1,

Configure the PSO running parameters population size (Psize)

= 100 and total iterations (itermax) = 50Initialize the values of fk=0.63 and mu ( ) = 4 for logistic map

Initialize the initial position and velocity matrix to zero

For iter = 1:iter_max


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For i = 1:pop_size

For j=1:nvars

If iter = = 1

Generate random number for initial positions (Pij) and velocities (Vij)

Check for upper and lower bond and modified accordingly

else

assign lastly calculated Pijand Vij

endif

Endfor

Endfor

Endfor


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Now update the variables to satisfy the Pd constrain

while (sum(init_positions (i,:)~=Pd))

temp_Pij = Pd (sum(init_positions(i,:)) init_positions(i,j));

temp_vij = init_velocity (i,j);

end

w = w_max((w_max

w_min)/iter_max) * iter;

update the value of w

If (the technique is standard)

calculate w normally

elseif (the technique is previous)calculate the next value from logistic map and use it to modify the w

fk= * fkpre * (1-fkpre);


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wnew= w* fk ;

else (the technique is proposed)

if (iter is odd the)

calculate the next value from logistic map and use it to modify the w

fk= * fkpre* (1-fkpre);

wnew= w * fk ;

else

calculate the next value from lozi map and use it to modify the w

lozi_X = 1 a_lozi * abs (lozi_X_pre) + b_lozi * lozi _Y_pre;

wnew= w * lozi _ X;

end


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for i = 1: pop_size

calculate the fitness values for all the population

x = init_positions (i, :);

fit_val(i) = obj_fun (x);

end

if P_val < G_val

G_val = P_val;

G_best = P_best;

check for the Pbest

and compare it with previous gbest

end


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init_velocity = w_new * init_velocity + c1 *rand * (Pbestinit_positions)+c2

* rand * (G_bestinit_positions);

calculate the new velocity and positions for all the population and repeat

FLOW CHART


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Start

If mod (Iter,2) ==1

End

Take Initialization

Parameters

Define Objective Function

Define Objective Constrains

Set Iter = 1

Generate Initial Population

Evaluate Objective Function

Use Lozi MapUse Logistic Map

Update Velocity and Positions

Select Best Solution

If iter == ter_max

FLOW CHART

RESULTS & DISCUSSION


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RESULTS & DISCUSSION

Test System 1: This system comprises of 3 generating unit and the

input data of 3-generating system are given in Here, the total

demand for the system is set to 850MW.

The standard PSO

0 5 10 15 20 25 30 35 40 45 508200

8250

8300

8350

8400

8450

8500

8550

8600

8650

8700

Iterations

Cost

Figure 1: Operating Cost of 3 generating unit using standard PSO


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1 2 30

50

100

150

200

250

300

350

400

Generator Number

OperatingPower

Figure 2: Operating Power of 3 generating

unit using standard PSO

Figure 3: Result window of 3 generating unit

using standard PSO

Table 1: Minimum cost of 3 generating unit using standard PSO

Technique Minimum Cost

PSO 8.2422e3


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The PSO with single chaotic operation

0 5 10 15 20 25 30 35 40 45 508200

8250

8300

8350

8400

8450

8500

8550

8600

8650

8700

Iterations

Cost

Figure 4: Operating Cost of 3 generating unit using PSO 1


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1 2 30

50

100

150

200

250

300

350

400

Generator Number

OperatingPower

Figure 5: Operating Power of 3 generating unit using PSO 1


PSO 1 8.2416e3

Figure 6: Result window of 3 generating unit using PSO 1

Table 2: Minimum cost of 3 generating unit using PSO 1


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The PSO with double (alternative) chaotic operation

0 5 10 15 20 25 30 35 40 45 508200

8250

8300

8350

8400

8450

8500

8550

8600

8650

8700

Iterations

Cost


400


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1 2 30

50

100

150

200

250

300

350

400

Generator Number

OperatingPower

Figure 8: Operating Power of 3 generating unit using PSO 2


PSO 2 8.2341e3

Figure 9: Result window of 3 generating unit using s PSO 2



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Minimum Operational Cost by all Three Techniques

0 5 10 15 20 25 30 35 40 45 508200

8250

8300

8350

8400

8450

8500

8550

8600

8650

8700

Iterations

Cost

PSO

PSO 1

PSO 2

Figure 10 : Comparison of cost minimization vs. iterations for PSO, PSO with chaotic map (PSO 1) and

Proposed PSO (PSO 2) with 2 chaotic maps.


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1 2 30

50

100

150

200

250

300

350

400

Generator Number

OperatingPower

PSO

PSO 1

PSO 2

Figure 11: Comparison of optimum operational condition for 3 generator units for

PSO, PSO with chaotic map (PSO 1) and Proposed PSO (PSO 2) with 2 chaotic maps.


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Figure 12: Result window for comparison of 3 generating unit for PSO, PSO with chaotic map

(PSO 1) and Proposed PSO (PSO 2) with 2 chaotic maps.

Technique Minimum cost

PSO 8.2422e3

PSO 1 8.2416e3

PSO 2 8.2341e3

Table 4: Minimum Operational Cost for 3 generating unit by all Three Techniques


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Test System 2: In this case the test system consists of 40-generating units and the input data are given. The total

demand is set to 10500 MW.The standard PSO

0 5 10 15 20 25 30 35 40 45 501.315

1.32

1.325

1.33

1.335

1.34

1.345

1.35

1.355x 10

5

Iterations

Cost



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0 5 10 15 20 25 30 35 40 450

100

200

300

400

500

600

Generator Number

OperatingPower

Figure 14: Operating Power of 40 generating

unit using standard PSO

Figure 15: Result window of 40 generating unit using

standard PSO



PSO 1.3195e5


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0 5 10 15 20 25 30 35 40 45 501.305

1.31

1.315

1.32

1.325

1.33

1.335

1.34

1.345

1.35

1.355 x 10

5

Iterations

Cos

t


600


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0 5 10 15 20 25 30 35 40 450

100

200

300

400

500

600

Generator Number

OperatingPower

Figure 17: Operating Power of 40 generating unit using PSO 1 Figure 18: Result window of 40 generating unit using PSO 1

Table 6: Minimum cost of 40 generating unit using PSO

1


PSO 1 1.3093e5


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0 5 10 15 20 25 30 35 40 45 50

1.27

1.28

1.29

1.3

1.31

1.32

1.33

1.34

1.35

1.36x 10

5

Iterations

Cost


600


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0 5 10 15 20 25 30 35 40 450

100

200

300

400

500

Generator Number




PSO 2 1.2717e5


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0 5 10 15 20 25 30 35 40 45 501.28

1.29

1.3

1.31

1.32

1.33

1.34

1.35

1.36x 10

5

Iterations

Cost

PSO

PSO 1

PSO 2

Figure 22 : Comparison of cost minimization vs. iterations for PSO, PSO with chaotic map

(PSO 1) and Proposed PSO (PSO 2) with 2 chaotic maps


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0 5 10 15 20 25 30 35 40 450

100

200

300

400

500

600

Generator Number

Ope

ratingPower

PSO

PSO 1

PSO 2




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Figure 24: Result window for comparison of 40 generating unit for PSO, PSO with chaotic map

(PSO 1) and Proposed PSO (PSO 2) with 2 chaotic maps.

Technique Minimum cost

PSO 1.3017e5

PSO 1 1.2932e5

PSO 2 1.2839e5

Table 8: Minimum Operational Cost for 40 generating unit by all Three Techniques

T S M l i F l i h V l P i Eff Th


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Test System 3: Multi-Fuels with Valve-Point Effect The testsystem consists of 10-generating units considering multi-fuels

with valve-point effects. The total system demand is set to 2700

MW.The standard PSO

0 5 10 15 20 25 30 35 40 45 50300

400

500

600

700

800

900

1000

Iterations

Cost


450

500


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1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

350

400

Generator Number

OperatingPower

Figure 26: Operating Power of 10 generating unit using

standard PSOFigure 27: Result window of 10 generating unit using standard PSO



PSO 318.4248


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0 5 10 15 20 25 30 35 40 45 50200

300

400

500

600

700

800

900

1000

Iterations

Cost


450

500


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1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

350

400

450

Generator Number

OperatingPowe

r




PSO 1 294.1963


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0 5 10 15 20 25 30 35 40 45 50

200

300

400

500

600

700

800

900

1000

Iterations

Cost


300

350


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1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

Generator Number

OperatingPow

er




PSO 2 239.8838


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0 5 10 15 20 25 30 35 40 45 50200

300

400

500

600

700

800

900

1000

Iterations

C

ost

PSO

PSO 1

PSO 2

Figure 34: Comparison of cost minimization vs. iterations for PSO, PSO with chaotic map

(PSO 1) and Proposed PSO (PSO 2) with 2 chaotic maps


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1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

350

400

450

500

Generator Number

Op

eratingPower

PSO

PSO 1

PSO 2




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Figure 36: Result window for comparison of 10 generating unit for


Table 12: Minimum Operational Cost for 10 generating

unit by all Three Techniques


PSO 317.5348

PSO 1 302.1667

PSO 2 247.6402

CONCLUSION AND FUTURE SCOPE


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This work presents an efficient approach for enhancing the

performance of standard PSO algorithm by alternative use of twodifferent chaotic maps for velocity updation and applied to the ELD

problem and tested for three different systems and objectives. The

simulation results shows the superiority of the proposed algorithm

over the previously proposed single chaotic map based PSO algorithm

and support the idea that switching over of chaotic pseudorandom

number generators in the PSO algorithm improves its performance

and the optimization process.



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The results for three different experiments are collected with different

settings and results compared with other methods which show that the

proposed algorithm improves the results by at least 10% for all three

cases. Although the result has improved we can further develop the

algorithm by utilizing multiple maps and optimizing the chaotic mapsparameters however these considerations are leaved for future

enhancements.


REFERENCES


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REFERENCES

[8] L d d S C lh d Ch Sh L S l i E i L d Di h


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[8] Leandro dos Santos Coelho and Chu-Sheng Lee, SolvingEconomic Load Dispatch

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[11] K M H G W Zh Y D d Kit P W Q t


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List of PublicationsNishant Chaturvedi, A. S. Walkey and N. P. Patidar, A Modified PSO Based


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, y ,

Solution Approach for Economic Load Dispatch Problem in Power System.

International Journal of scientific and engineering Research. ISSN : 2229-5518,

Vol. 5, Issue 4,pp. 292-300, April 2014.

Nishant Chaturvedi and A. S. Walkey, A Survey on Economic Load Dispatch

Problem using Particle Swarm Optimization technique International Journal of

emerging Technology and Advanced Engineering. ISSN: 2250-2459, ISO 9001:

2008, Vol. 4, Issue 3, pp. 188-193, March 2014.

Nishant Chaturvedi and A. S. Walkey, A Noval Approach for Economic Load

Dispatch Problem Based on GA and PSO, International Journal of Engineering

Research and Application. ISSN: 2248-9622, Vol. 4, Issue 3 (Version 2), March

2014, pp. 24-31.


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a modified pso based solution approach for economic load dispatch problem in power system

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