International Journal of Computer and Electrical Engineering, Vol. 2, No. 3, June, 2010

1793-8163

560

Abstract—An Evolutionary Programming (EP) and

Efficient Particle Swarm Optimization (EPSO) techniques are

employed to solve Economic Dispatch (ED) problems including

transmission losses in power system is presented in this paper.

This paper is clearly justified with the results separately

obtained for the above two techniques and also provided with

the results by applying both the algorithms separately. With

practical consideration, ED will have non-smooth cost

functions with equality and inequality constraints that make

the problem, a large-scale highly constrained nonlinear

optimization problem. The proposed method expands the

original PSO to handle a different approach for satisfying

those constraints. In this paper, an Efficient Particle Swarm

Optimization (EPSO) technique is employed so that optimized

results are obtained, and by applying EP, faster convergence is

obtained . To demonstrate the effectiveness of the proposed

method it is being applied to test ED problems, one with

smooth and other with non smooth cost functions considering

valve-point loading effects. Comparison with other

optimization and hybrid algorithm techniques showed the

superiority of the proposed EP-EPSO approach and confirmed

its potential for solving nonlinear economic load dispatch

problems with losses.

Index Terms—Economic load dispatch , Evolutionary

Programming , Efficient Particle swarm optimization , Valve

point loading effect

I. I

NTRODUCTION

Economic Dispatch (ED) problem is one of the

fundamental issues in power system operation. In essence, it

is an optimization problem and its main objective is to

reduce the total generation cost of units, while satisfying

constraints.[1] Previous efforts on solving ED problems

have employed various mathematical programming

methods and optimization techniques excluding losses.

Recently, Eberhart and Kennedy suggested a Particle

Swarm Optimization (PSO) based on the analogy of swarm

Dr.K.Thauskkodi is with Director of Akshaya College of Engineering

and Technology , Coimbatore , Tamilnadu , 642 109 , India .email :

thanush_dr@rediffmail.com .

Manuscript received October 30 , 2009.

S.Muthu Vijaya Pandian is with Department of Electrical and

Electronics Engineering , V.L.B.Janakiammal College of Engineering and

Technology , Coimbatore , Tamil Nadu , 640 042 , India. (Phone : +91

98652 59633 ); Fax : +91 0422 2607152(email:

ajay_vijay@rediffmail.com).

of bird and school of fish. In PSO, each individual makes its

decision based on its own experience together with other

individual’s experiences. In artificial intelligence, an

evolutionary algorithm (EA) is a subset of evolutionary

computation, a generic population-based metaheuristic

optimization algorithm. An EA uses some mechanisms

inspired by biological evolution: reproduction, mutation,

recombination, and selection. Candidate solutions to the

optimization problem play the role of individuals in a

population, and the fitness function determines the

environment within which the solutions "live" (see also cost

function). Evolution of the population then takes place after

the repeated application of the above operators. Artificial

evolution (AE) describes a process involving individual

evolutionary algorithms; EAs are individual components

that participate in an AE. The main advantages of the PSO

algorithm are summarized as: simple concept, easy

implementation, and computational efficiency when

compared with mathematical algorithm and other heuristic

optimization techniques[7]. The practical ED problems with

valve-point loading effects are represented as a non smooth

optimization problem with equality and inequality

constraints. To solve this problem, many salient methods

have been proposed such as dynamic programming,

evolutionary programming, neural network approaches, and

genetic algorithm. In this paper, an alternative approach is

proposed to the non smooth ED problem using an Efficient

PSO (EPSO), which focuses on the treatment of the equality

and inequality constraints when modifying each

individual’s search. The equality constraint (i.e., the

supply/demand balance) is easily satisfied by specifying a

variable (i.e., a generator output) at random in each iteration

as a slag generator whose value is determined by the

difference between the total system demand (including

losses) and the total generation excluding the slag generator.

However, the inequality constraints in the next position of

an individual produced by the PSO algorithm can violate

the inequality constraints. In this case, the position of any

individual violating the constraints is set to maximum or

minimum depending on velocity evaluated.

Solving Economic Load Dispatch Problem

Considering Transmission Losses by a Hybrid

EP-EPSO Algorithm for Solving both Smooth

and Non-Smooth Cost Function

S. Muthu Vijaya Pandian and K. Thanushkodi

International Journal of Computer and Electrical Engineering, Vol. 2, No. 3, June, 2010

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561

II. F

ORMULATION OF

E

CONOMIC

D

IAPATCH

P

ROBLEM

A. ED Problem with Smooth Cost Functions

Economic load dispatch (ELD) pertains to optimum

generation scheduling of available generators in an

interconnected power system to minimize the cost of

generation subject to relevant system constraints. Cost

equations are obtained from the heat rate characteristics

of the generating machine. Smooth costs functions are

linear, differentiable and convex functions. The most

simplified cost function of each generator can be

represented as a quadratic function as given in whose

solution can be obtained by the conventional

mathematical methods [2] :

C=∑F

j

P

j

(1)

F

j

P

j

=a

j

+b

j

P

j

+c

j

P

j

2

(2)

where C total generation cost

Fj cost function of generator j

aj bj cj cost coefficients of generator j

Fj cost function of generator j

aj bj cj cost coefficients of generator j

While minimizing the total generation cost, the total

generation should be equal to the total system demand

plus the transmission network loss.

The transmission loss is given by the equation,

PL= ∑ BojPj (3)

where B

oj

is the loss co-efficient matrix.

The equality constraint for the ED problem can be

given by,

∑ Pj = D +∑PL (4)

where D is the total demand needed by the load or

consumer. The generation output of each unit should be

between its minimum and maximum limits. That is, the

following inequality constraint for each generator should

be satisfied

Pjmin<Pj<Pjmax (5)

where P

jmin

, P

jmax

are the minimum and maximum output

of individual generators.

B. ED Problem with Non-smooth Cost Functions

In reality, the objective function of an ED problem has

non differentiable points according to valve-point effects.

Therefore, the objective function should be composed of

a set of non-smooth cost functions. In this paper, one case

of non-smooth cost function is considered i.e. the valve-

point loading problem where the objective function is

generally described as the superposition of sinusoidal

functions and quadratic functions [7].

C. Non-smooth Cost Function with Valve-Point

Effects

The generator with multi-valve steam turbines has very

different input-output curve compared with the smooth

cost function[6]. Typically, the valve point results in, as

each steam valve starts to open, the ripples like in to take

account for the valve-point effects, sinusoidal functions

are added to the quadratic cost functions as follows:

FjPj=aj+bjPj+cjPj2+ejxsin(fjx(Pjmin-Pj)) (6)

III. E

VOLUTIONARY

P

ROGRAMMING

EP is a near global search stochastic optimization

method starting from multiple points, which placed

emphasis on the behavioural linkage between the parents

and their offsprings, rather than seeking to emulate

specific operators as observed in nature to find a solution.

However EP takes a long computation time to find a

solution and sometime EP suffers from the convergence

problem. On the other hand, EPSO is a gradient based

optimization method starting from a single point and

using gradient information to obtain a solution. The

solution obtained from EPSO is a local optimal solution.

In order to obtain a high quality solution , the first part,

EP is applied to obtain a near global solution. After the

specified termination criteria for EP is reached, EPSO is

applied in the second part by using the solution from EP

as an initial starting point and searches by using a

gradient information to obtain the final optimal solution

[2].

A. Evolutionary Programming Subproblem

1) Representation

For ‘m ,particles in the swarm and ‘n’ generators of the

systems, the array of control variable vectors (S) can be

shown as

P

11

P

12

…….. P

1m

P

21

P

22

…….. P

2m

P

n1

P

n2

………. P

nm

2) Initialization

To begin the population of chromosomes is uniform

randomly initialized within the operation range of the

generator [5].

3) Fitness Evaluation

The active power generations at all the buses except the

first bus in all intervals are control variable, which are

itself constrained. Equality constraint can be handled as

2

n

ij Dj ij

i

P P P

=

= −

∑

4) Creation of Offspring

A new population of solutions is produced from the

existing population by adding a guassian random number

with zero mean and pre-defined standard deviation as

follows:

International Journal of Computer and Electrical Engineering, Vol. 2, No. 3, June, 2010

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Pit’ = Pit + N(0,σit2) (7)

Where σ

it

can be calculated from the following equation:

σit = ? * fs/fmax(Pitmax- Pitmin) (8)

Where ϐ is a scaling factor, which can be tuned during the

process of search for optimum.

5) Selection And Competition

The selection technique used here is the stochastic

tournament method. The 2P individuals compete with

each other for selection. A weight value ωs is assigned to

each individual as follows

1

p

j

s jω ω

=

=

∑

(9)

ωj = 1 if fs<fr

ωj =0 otherwise

In biological DNA systems, the basic units are the

adenine (A), thymine (T), guanine (G) and cytosine (C)

nucleotides that join the helical strands. In genetic

algorithms, the basic unit is called a symbol. The nature

of symbols depends on the particular genetic algorithm.

In gene expression programming, the symbols consist of

functions, variables and constants. Symbols for variables

and constants are called terminals, because they have no

arguments.

An ordered set of symbols form a gene, and an ordered

set of genes form a chromosome. In GEP programs, genes

typically have 4 to 20 symbols, and chromosomes are

typically built from 2 to 10 genes; chromosomes may

consist of only a single gene. The DNA strand for a

mammal typically contains about 5x10

9

nucleotides.

B. Gene Expression Programming

Gene Expression Programming is a procedure that

mimics biological evolution to create a computer program

to model some phenomenon. Gene expression

programming can be used to create many different types

of models including decision trees, neural networks and

polynomial constructs. The type of gene expression

programming implemented in DTREG is Symbolic

Regression so named because it creates a symbolic

mathematical or logical function [4].

Fig.1 Biological evolution of EP

DTREG provides a full implementation of the Gene

Expression Programming algorithm developed by

Cândida Ferreira. Here are some of the features of

DTREG’s implementation :

• Continuous and categorical target variables

• Automatic handling of categorical predictor

variables

• A large library of functions that you can select

for inclusion in the model

• Mathematical and logical (AND, OR, NOT, etc.)

function generation

• Choice of many fitness functions

• Both static linking functions and evolving

homeotic genes

• Fixed and random constants

• Nonlinear regression to optimize constants

• Parsimony pressure to optimize the size of

functions

• Automatic algebraic simplification of the

combined function

• Several forms of validation including cross-

validation and hold-out

C. Expression Trees and Karva

The key to GEP’s ability to quickly mutate valid

expressions is the way it encodes symbols in genes. This

notation is called the Karva Language. Expressions

encoded using Karva are called K-expressions. Consider

the simple mathematical expression:

a*b+c (10)

This can be encoded as an expression tree of the form

Fig 2: Karva-expressions encoding model 1

An expression tree is an excellent way to represent an

expression in a computer, because the tree can be

arbitrarily complex, and expression trees can be evaluated

quickly.

To convert an expression tree to the Karva notation,

start at the left-most symbol in the top line of the tree and

scan symbols left-to-right and top-to-bottom. Each time a

symbol is encountered, add it to the K-expression in left-

to-right order. When there are no more symbols on a line,

advance to the left end of the following line. Using this

method, the tree shown above is converted to the K-

expression:

+*cab (11)

Note that + is the first symbol found on the first line, at

the end of that line scanning begins on the second line

and finds * followed by c. It then starts with the third line

International Journal of Computer and Electrical Engineering, Vol. 2, No. 3, June, 2010

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563

and finds a and b.

As a second example, consider the expression

a*b+sqrt(c*d) (12)

The corresponding expression tree is

Fig 3: Karva-expressions encoding model 2

Where ‘Q’ represents square root. This can be translated

to the K-expression

+*Qab*cd (13)

The process of converting an expression tree to a K-

expression can be carried out quickly by a computer. A

reverse process can quickly convert a K-expression back

to an expression tree.

D. Implementation of PSO for ED Problems

The PSO algorithm searches in parallel using a group

of individuals, in a physical dimensional search space, the

position and velocity of individual i are represented as the

vectors Xi = (xib……….xin) and Vi = (vib……….vin) )

respectively, be the position of the individual i and its

neighbors’ best position so far. Using the information, the

updated velocity of individual i is modified under the

following equation in the PSO algorithm[3]:

V

i

k+1

=ωV

i

k

+c

1

rand1 x (P

besti

k

- x

i

k

)+c

2

rand

2

x (G

best

k

– x

i

k

)

Where

V

i

k+1

velocity of individual of iteration at k

ω weight parameter

C

1

, C

2

acceleration factors

rand

1

rand

2

random numbers between 0 and 1.

X

i

k

Position of individual i at iteration k

Pbest best position of group throughout iteration k

Each individual moves from the current position to the

next one by the using the following equation:

X

ik+1

=X

ik

+X

ik+1

The search mechanism of the PSO using the modified

velocity and position of the individual i based on (7)

and (8) is illustrated in fig (4)

Fig. 4 Search mechanism of PSO

IV. E

FFICIENT

PSO

FOR

ED

P

ROBLEMS

In this section, a new approach to implement the PSO

algorithm will be described while solving the ED

problems considering losses [3]. The main process of the

efficient PSO algorithm can be summarized as follows:

Step1) Initialization of a group at random while

satisfying constraints.

Step2) Velocity and position updates while satisfying

constraints

Step3) Update of Pbest and Gbest.

Step4) Calculate transmission losses for the obtained

Pbest and Gbest

Step5) Increment the demand with the transmission

losses

Step6) Go to Step 2 until satisfying stopping criteria.

In the subsequent sections, the detailed implementation

strategies of the EPSO are described.

A. Initialization of Individuals

In the initialization process, a set of individuals (i.e.,

generation outputs) is created at random. Therefore,

individual i position at iteration 0 can be represented as

the vector of n is the number of generators.[3] The

velocity of individual i is given by corresponds to the

generation update quantity covering all generators. The

following procedure is suggested for satisfying

constraints for each individual in the group:

Step1) Set j=1, i=1 element (i.e., generator) of an

individual i.

Step2) Select the jth element of the individual i.

Step3) Create the value of the element (i.e., generation

output) at random satisfying its inequality constraint.

Step4) If j=n-1 then go to step 5; otherwise j=j+1 and go

to Step 2.

Step5) The value of the last element of an individual is

determined by subtracting ∑P from the total demand

Step6) If i=no of individuals go to step 7; otherwise put

i=i+1 and go to step 2.

Step7) Stop the initialization process.

After creating the initial position of each individual, the

velocity of each individual is also created at random. The

following strategy is used in creating the initial velocity:

(P

min

- €)-P

ij

0

<v

ij

0

<P

max

-€-P

ij

0

(14)

Where e is a small positive real number

The velocity of element j of individual i is generated at

random within the boundary [8].

B. Velocity Update

To modify the position of each individual, it is

necessary to calculate the velocity of each individual in

the next stage, which is obtained from (7). In this velocity

updating process, the values of parameters such as w, c1

and c2 should be determined in advance. The weighting

function is defined as follows

w=w

max

-(w

max

-w

min

/iter

max

)*iter (15)

International Journal of Computer and Electrical Engineering, Vol. 2, No. 3, June, 2010

1793-8163

564

Where

w

max

final weight

w

min

initial weight

iter

max

maximum number of iteration

iter current iteration number

C. Position Modification Considering Constraints

The position of each individual is modified by (8). The

resulting position of an individual is not always

guaranteed to satisfy the inequality constraints due to

over/under velocity [4]. If any element of an individual

violates its inequality constraint due to over/under speed

then the position of the individual is fixed to its maximum

or minimum operating point. Therefore, this can be

formulated as follows:

(16)

To resolve the equality constraint problem without

intervening the dynamic process inherent in the PSO

algorithm, we propose the following procedures:

Step1) Set j=1, i=1. Let the present iteration be k.

Step2) Select the jth element (i.e., generator) of an

individual i.

Step3) Modify the value of element j using (7), (8), and

(11).And satisfy inequality constraint.

Step4) If j=n-1 then go to Step 5, otherwise j=j+ 1and go

to Step 2.

Step5) The value of the last element of an individual is

obtained by subtracting

∑P

ij

0

from the total system

demand.

Step6) If i=no. of individuals then go to step 7; otherwise

i=i+1 and go to Step2

Step7) Stop the modification procedure

D. Update of Pbest and Gbest

The Pbest of each individual at iteration k+1 is updated

as follows:

P

besti

k+1

=X

i

k+1

if TC

i

k+1

< TC

i

k

(17)

P

besti

k+1

=P

besti

k

if TC

i

k+1

> TC

i

k

(18)

Where

TC

i

– object function evaluated at the position of the

individual i.

Additionally, Gbest at iteration k+1 is set as the best

evaluated position among Pbest

k+1

V. S

IMULATED

R

ESULT

A

NALYSES

A. ED Problem with Non Smooth Cost Functions with

Valve point effect

TABLE

1:

I

NPUT DATA FOR

40

U

NIT

S

YSTEM

Generator P

jmin

P

jmax

a

i

b

i

c

i

e

i

f

i

1 36 114 0.00690 6.73 94.705 100 0.084

2 36 114 0.00690 6.73 94.705 100 0.084

3 60 120 0.02028 7.07 309.54 100 0.084

4 80 190 0.00942 8.18 369.54 150 0.063

5 47 97 0.01140 5.35 369.03 120 0.077

6 68 140 0.01142 8.05 148.89 100 0.084

7 110 300 0.01142 8.03 222.33 200 0.042

8 135 300 0.00357 6.99 287.71 200 0.042

9 135 300 0.00492 6.60 391.88 200 0.042

10 130 300 0.00573 12.9 455.76 200 0.042

11 94 375 0.00605 12.9 722.82 200 0.042

12 94 375 0.00515 12.8 635.20 200 0.042

13 125 500 0.00569 12.5 654.69 300 0.035

14 125 500 0.00421 8.84 913.40 300 0.035

15 125 500 0.00752 9.15 1760.4 300 0.035

16 125 500 0.00708 9.15 1728.3 300 0.035

International Journal of Computer and Electrical Engineering, Vol. 2, No. 3, June, 2010

1793-8163

565

17 220 500 0.00708 7.97 1728.3 300 0.035

18 220 500 0.00313 7.95 647.83 300 0.035

19 242 550 0.00313 7.97 647.81 300 0.035

20 242 550 0.00313 7.97 647.85 300 0.035

21 254 550 0.00313 6.63 785.96 300 0.035

22 254 550 0.00218 6.63 785.96 300 0.035

23 254 550 0.00284 6.66 794.53 300 0.035

24 254 550 0.00284 6.66 794.53 300 0.035

25 254 550 0.00277 7.10 801.32 300 0.035

26 254 550 0.00277 7.10 801.32 300 0.077

27 10 150 0.52124 3.33 1055.1 120 0.077

28 10 150 0.52124 3.33 1055.1 120 0.077

29 10 150 0.52124 6.43 1055.1 120 0.077

30 47 97 0.01140 6.43 148.89 120 0.063

31 60 190 0.00160 6.43 222.92 150 0.063

32 60 190 0.00160 8.95 222.92 150 0.063

33 60 190 0.00160 8.62 222.92 150 0.042

34 90 200 0.00010 8.62 107.87 200 0.042

35 90 200 0.00010 5.88 116.58 200 0.042

36 90 200 0.00010 5.88 116.58 200 0.098

37 25 110 0.0161 5.88 307.45 80 0.098

38 25 110 0.0161 3.33 307.45 80 0.098

39 25 110 0.0161 3.33 307.45 80 0.098

40 242 550 0.00313 7.97 647.83 300 0.035

TABLE

2:

H

YBRID

R

ESULTS FOR

(NN-EPSO)

40

U

NIT

S

YSTEMS

Unit Output (MW)

1 114

2 114

3

120

4

190

5

97

6

140

7

300

8

300

9

300

10

300

11

375

12

375

13

500

14

500

15

500

16

500

17

409.273

International Journal of Computer and Electrical Engineering, Vol. 2, No. 3, June, 2010

1793-8163

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18

225

19

508

20

458

21

356

22

394

23

355

24

525

25

310

26

448

27

72

28

131

29

75

30

67

31

151

32

112

33

139

34

90

35

129

36

104

37

36

38

89

39

104

40

550

Demand = 10500 MW

Losses = 62.27 MW

Optimal cost = 130328.3256 $/hr

Elapsed Time = 8.3590 sec

The above table 2 given the output results for forty unit

systems hybrid Neural network and Efficient particle

swarm optimization method [8]

Fig 4: Convergence plot for 40 unit systems in

(NN-EPSO) Method

B. Simulated result sfor forty Units hybrid EP-EPSO

TABLE

3:

O

UTPUT

R

ESULTS FOR

F

ORTY

U

NITS

H

YBRID

(EP-EPSO

)

M

ETHOD

Generator

Output (MW)

1

114

2

114

3

120

4

190

5

97

6

140

7

300

8

300

9

300

10

300

11

375

12

375

13

500

14

500

15

500

16

500

17

409.273

18

225

19

510

20

458

International Journal of Computer and Electrical Engineering, Vol. 2, No. 3, June, 2010

1793-8163

567

21

354

22

394

23

355

24

525

25

310

26

450

27

72

28

129

29

75

30

67

31

151

32

112

33

139

34

90

35

129

36

104

37

36

38

89

39

104

40

550

Demand = 10500 MW

Losses = 61.57 MW

Optimal cost = 130227.3256 $/hr

Elapsed Time = 7.7590 sec

1

2

3

4

5

6

7

8

9

10

1.26

1.27

1.28

1.29

1.3

1.31

1.32

x 10

5

ITERATIONS

TOTAL COST ($/hr)

ECONOMIC LOAD DISPATCH USING HYBRID EP-EPSO

Fig 5: Convergence plot for 40 unit systems in

(EP-EPSO) Algorithm

TABLE

4:

C

OMPARISON BETWEEN

H

YBRID

EP-EPSO

AND

NEURO-EPSO

M

ETHOD

Generator

Output Of Hybrid

NEURO-EPSO (MW)

Output Of Hybrid EP-

EPSO (MW)

1

114 114

2

114 114

3

120 120

4

190 190

5

97 97

6

140 140

7

300 300

8

300 300

9

300 300

10

300 300

11

375 375

12

375 375

13

500 500

14

500 500

15

500 500

16

500 500

17

409.273 409.273

18

225 225

19

508 510

20

458 458

21

356 354

22

394 394

23

355 355

24

525 525

25

310 310

26

448 450

27

72 72

28

131 129

29

75 75

30

67 67

31

151 151

32

112 112

33

139 139

International Journal of Computer and Electrical Engineering, Vol. 2, No. 3, June, 2010

1793-8163

568

34

90 90

35

129 129

36

104 104

37

36 36

38

89 89

39

104 104

40

550 550

TABLE

5

C

OMPARISION OF

T

OTAL

P

RODUCTION

C

OST AND

S

IMULATION

T

IME AMONG

NN-EPSO

AND

EP-EPSO

VI. C

ONCLUSION

In this paper, a new methodology for solving non-

smooth ED problem including valve point loading using

EP combined with EPSO. The proposed algorithm

consists of two parts. The first part employs the property

of EP, which can provide a near global search region at

the beginning. When the specified termination criteria of

EP is reached, the local search EPSO is applied to tune

the control variables in order to obtain the final optimal

solution. It is clear from the Table 5 mean cost value and

simulation obtained by EP-EPSO is comparatively less

compared to all other methods. Simulation results

demonstrate that the proposed method can give a cheaper

total production cost than those obtained from EPSO, EP,

NN-EPSO and EP-SQP. The resultant EP-EPSO has been

suggested as a powerful optimization tool for non-convex

ED problem.

R

EFERENCES

[1] A.J. Wood and B.F.Wollenberg, “ Power Generation ,Operation

and Control ” , New York: John Wiley & Sons , Inc., 1984.

[2] Pathom Attaviriyanupap and Hiroyuki Kita , “ A Hybrid EP and

SQP for Dynamic Economic Dispatch with Non-smooth Fuel Cost

Function ” IEEE Transactions on Power Systems , vol 17 ,no.2 ,

May 2002.

[3] J.B.Park , K.S.Lee , K.Y.Lee , “ A Patticle Swarm Optimization

for Economic Dispatch with Non-smooth Cost Function ,” IEEE

Trans. Power Systems , vol 20 , Feb 2005.

[4] H.T.Yang , P.C.Yang , and C.L.Huang , “ Evolutionary

Programming Based Economic Dispatch for Units with Non-

Smooth Fuel Cost Functions , ” IEEE Trans. Power Systems , vol

11, no.1 , pp.112-118 ,Feb1996.

[5] N.Sinha , R. Chakrabarti and P.K.chattopadhyay , “Evolutionary

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[6] J.Nanda,A.Sachan,.L.Pradhan M.L.Kothari,,A.Koteswara Rao ,

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Dispatch , ” IEEE Trans. Power Systems , vol 22 , 1997.

[7] B.H.Chowdhury and S.Rahman , “ A Review of Recent Advances

in Economic Dispatch , ” IEEE Trans. Power Systems , vol

5 ,no.4 , pp.1248-1259 , Nov 1990.

[8] S.Muthu Vijaya Pandian and Dr.K.Thanushkodi , “ A Hybrid

(Neural Network – Efficient Particle Swarm Optimization (NN-

EPSO) for Economic Dispatch Problems Considering

Transmission Losses with Non-smooth Cost Functions ”

International Journal of Computer Science and Electrical

Engineering (IJCEE) no.3, vol1 June 2009 ,pg 207-214.

S.Muthu Vijaya Pandian was born in Karaikudi , Tamil Nadu , India

on the 19th of June 1978. He received his M.B.A from Madurai

Kamaraj University , Madurai ,Tamil Nadu , India , in 2002 . He then

received his M.E in Power Systems from Government College of

Technology , Coimbatore , Tamil Nadu , India., in 2004 and his

currently pursuing his Ph.D in Anna university , Chennai , India . He is

currently Lecturer at the Department of Electrical and Electronics

Engineering ,V.L.B.Janakiammal College of Engineering and

Technology , Coimbatore , Tamil Nadu , India. He has published Three

International Journal , One International conference and his current

research interests include areas of Power Systems.

Dr.K.Thanushkodi was born in Theni District , Tamil Nadu , India in

1948. He received the B.E degree in Electrical and Electronics

Engineering and the M.sc.(Engg) degree from Madras University ,

Chennai , India in 1972 and 1974 , respectively , and the Ph.D degree in

Electrical and Electronics Engineering from Bharathiar University ,

Coimbatore , India , in 1994. He served as thirty three years of teaching

experience and guided 4 Ph.D . Now he guiding 12 Research scholars

in Anna University , Chennai and 30 Research scholar in Anna

University , Coimbatore , India. He published 75 papers in International ,

National Journals and Conferences. He is currently Director of Akshaya

College of Enggineering and Technology , Coimbatore and also

Syndicate member Anna University ,Chennai , India. His research

include computer modeling and simulation , computer networking and

power systems. .

Method Cost $/hr Time(s)

NN 146069.7350 28.07

EPSO 130330.3647 7.232

EP 143799.0000 9.242

NN-EPSO 130328.3246 8.3529

EP-EPSO 130227.3256 7.7590

EP-SQP 1035748.0000 1251.0

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