fast computation evolutionary programming algorithm for the economic dispatch problem

EUROPEAN TRANSACTIONS ON ELECTRICAL POWEREuro. Trans. Electr. Power 2006; 16:35–47Published online 14 July 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/etep.63

Fast computation evolutionary programming algorithm for theeconomic dispatch problem

P. Somasundaram*,y, K. Kuppusamy and R. P. Kumudini Devi

Power System Engineering Division, Department of Electrical and Electronics Engineering, Anna University,Chennai 600025, Tamil Nadu, India

SUMMARY

This paper essentially aims to propose a new EP based algorithm for solving the ED problem. The ED problem issolved using EP with system lambda as decision variable and power mismatch as fitness function. The algorithm ismade fast through judicious modifications in initialization of the parent population, offspring generation andselection of the normal distribution curve. The proposed modifications reduce the search region progressively andgenerate only effective offsprings. The proposed algorithm is tested on a number of sample systems with quadraticcost function and also on a 10-unit system with piecewise quadratic cost function. The computational resultsreveal that the proposed algorithm has an excellent convergence characteristic and is superior to other EP basedmethods in many respects. Copyright # 2005 John Wiley & Sons, Ltd.

key words: evolutionary programming; economic dispatch; piecewise quadratic cost function

1. INTRODUCTION

Economic dispatch (ED) is one of the most important problems to be solved in the operation of a

power system. In the literature various mathematical programming methods and optimization

techniques [1,2] are proposed to solve the ED problem. These methods include the lambda-iteration

method [3], the base point and participation factors method [3] and the gradient method [3] etc.

Recently, a global optimization technique known as evolutionary programming (EP) has become a

candidate for the ED problem. EP is a stochastic search algorithm and it searches randomly from point

to point to reach the optimum.

In References [4,5], the EP based ED problem is solved using the generator outputs as decision

variables and fuel cost as fitness function. Major drawbacks of this algorithm are inconsistent

convergence, a large number of iterations and indeterministic stopping criteria etc. Since the EP

based algorithm uses a kind of random search method, it may lead to slow convergence when the

optimization problem has a higher number of decision variables [6]. Hence it is better to formulate the

ED problem with a minimum number of decision variables when the EP based algorithm is applied.

Copyright # 2005 John Wiley & Sons, Ltd.

*Correspondence to: P. Somasundaram, Power System Engineering Division, Department of Electrical and ElectronicsEngineering, Anna University, Chennai 600025, Tamil Nadu, India.yE-mail: [email protected]

In this paper a new EP algorithm is proposed to solve the ED problem with a minimum number of

decision variables.

In the proposed Standard Evolutionary Programming Algorithm (SEPA) even though the number of

decision variables (one) is a minimum it suffers from certain drawbacks such as slow and inconsistent

convergence. The convergence is faster when compared to the other EP algorithms with unit

generations as decision variables but it is slower when compared to the conventional lambda-iteration

method. To have the convergence comparable with that of the conventional �-iteration method the

proposed SEPA, with � as decision variable, is modified and an improved approach is presented. The

improved approach is referred to as Fast Computation Evolutionary Programming Algorithm

(FCEPA).

In the improved approach modifications are proposed for the initialization of parent population,

generation of offspring and selection of normal distribution curve. The EP based algorithms, namely

the SEPA and FCEPA, are tested on sample systems with quadratic and piecewise quadratic cost

functions. Test results show that the proposed FCEPA is numerically more robust and has excellent

convergence characteristics.

2. ECONOMIC DISPATCH PROBLEM

The ED problem is stated as:

min :FT ¼Xnj¼1

FjðPjÞ ð1Þ

Subject to:

the power balance constraint (power mismatch):

Xnj¼1

Pj � PD � PL ¼ 0 ð2Þ

and the inequality constraint:

Pj;min � Pj � Pj;max ð3Þ

where the fuel cost function is usually represented by a quadratic function:

FjðPjÞ ¼ ajP2j þ bjPj þ cj ð4Þ

The ED problem is restated by considering the co-ordination equations, as follows.

The co-ordination equations for the above ED problem with transmission loss included are:

dFj

dPj

þ �@PL

@Pj

¼ �; j ¼ 1; 2; . . . ; n ð5Þ

Xnj¼1

Pj � PD � PL ¼ 0 ð6Þ

36 P. SOMASUNDARAM, K. KUPPUSAMY AND R. P. KUMUDINI DEVI

Copyright # 2005 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2006; 16:35–47

and the inequality constraint:Pj;min � Pj � Pj;max ð7Þ

The incremental cost isdFj

dPj

¼ 2ajPj þ bj ð8Þ

and the transmission loss is expressed as

PL ¼Xni¼1

Xnj¼1

PiBijPj ð9Þ

The solution for the ED problem can be obtained by solving the set of Equations (5)–(7). The problem

of solving Equations (5)–(7) may be stated as below.

Determine the decision variable � to minimize the power mismatch in Equation (6) to zero

satisfying Equations (5) and (7). Alternatively the problem may be viewed as an optimization problem

with an objective to minimize the power mismatch in Equation (6) to zero subject to the equality and

inequality constraints in Equations (5) and (7), respectively, with � as decision variable. A suitable

algorithm based on EP is developed to solve the above problem taking � as decision variable and

power mismatch as fitness function.

3. STANDARD EVOLUTIONARY PROGRAMMING ALGORITHM (SEPA)

The algorithmic steps for the proposed SEPA with � as decision variable and power mismatch as

fitness function is presented in this section.

3.1. Initialization of population

3.1.1. An initial parent population �i; i¼ 1; 2; . . . ; Np is generated by setting �i�UR(�min; �max)

where, UR(�min; �max) denotes a uniform random variable ranging over [�min; �max],

�min¼min{�1;min; �2;min; �3;min; . . .�n;min} and

�max¼max{�1;max; �2,max; �3;max; . . .�n;max}.

The �j,min and �j,max of the jth generating unit are calculated from Equation (5) by substituting Pj,min

and Pj,max for Pj.

3.1.2. The generation schedules (Pj; j¼ 1; 2, . . . ; n) corresponding to each �i; i¼ 1; 2; . . . ; Np are

obtained by solving Equation (5) iteratively subject to the inequality constraint in Equation (7).

3.1.3. The power mismatch fi (fitness function) and power loss PLi corresponding to each parent �i;i¼ 1; 2; . . . ; Np are computed using Equations (6) and (9), respectively.

3.2. Mutation

3.2.1. An offspring population �i0; i¼Npþ 1; Npþ 2; . . . ; 2Np is generated from each parent �i as

�Npþi0 ¼ �i þ Nð0; �2

i Þ i ¼ 1; 2; . . . ;Np ð10Þ

EP BASED ALGORITHM FOR SOLVING THE ED PROBLEM 37


where N(0; �i2) represents a normal random variable with zero mean and standard deviation �i, �i is

computed from

�i ¼ �ð�max � �minÞfij j

fmaxj j ð11Þ

where � is a scaling factor and fi is the value of the fitness function corresponding to �i and fmax is the

maximum fitness function value among the parent population.

3.2.2. The generation schedules (Pj; j¼ 1; 2; . . . ; n) corresponding to each �0i; i¼Npþ 1; Npþ 2; . . . ;

2Np are obtained by solving Equation (5) iteratively subject to the inequality constraint in Equation (7).

3.2.3. The power mismatch fi and power loss PLi corresponding to each offspring �i0 are computed

using Equations (6) and (9).

3.3. Competition and selection

The 2Np trial solutions, namely, Np solutions corresponding to parent population �i; i¼ 1; 2; . . . ; Np

and their offspring �i0; i¼Npþ 1; Npþ 2; . . . ; 2Np contend for survival. The weight (Wi) for each

individual in the combined population decides the survival. The weight Wi is computed as

Wi ¼Xqj¼1

wj i ¼ 1; 2; . . . ; 2Np ð12Þ

wj ¼1; if ui >

fij jfrj j þ fij j

0; otherwise

8<:

where fitness function fr is selected at random among the 2Np trial solutions, ui is a uniform random

number ranging over [0,1] and q is the arbitrarily chosen competition number which is normally taken

as Np.

The 2Np trial solutions are ranked in descending order of their weights and the first Np trial solutions

are taken as the next parent population.

3.4. Convergence criteria

The power mismatch corresponding to each member in the parent population is checked for

convergence. If power mismatch for any one of the members in the population is less than the

tolerance, then optimum solution is reached, otherwise the parent population is subjected to steps

given under sections 3.2 and 3.3. The computational steps are repeated until convergence is reached.

4. EXAMPLE AND RESULTS

The proposed SEPA is tested on a number of sample systems. The results of the 20-unit system alone

are presented. The data for the 20-unit system is given in Reference [7]. The parameters used for the

proposed EP methods are:



Scaling factor : � ¼ 0:01

Population size : Np ¼ 5

Convergence criteria ¼ 1 MW

The results obtained using the proposed SEPA are given in column 3 of Table I. The results in

column 3 exactly match with those in column 2 obtained using the lambda-iteration method; but the

computation time is very high compared to that required for the lambda-iteration method. It is found

that the number of iterations for convergence are different for different trial studies for the same

problem. This is due to randomness involved in the algorithm. The number of iterations is found to

vary between 5 and 50. The computation times corresponding to the maximum number of iterations

are given in the bottom row of Table I.

5. DRAWBACKS AND MODIFICATIONS OF THE SEPA

The drawbacks as well as the modifications of the SEPA are discussed in the following sections.

5.1. Drawbacks of the algorithm

1. In step 3.1.1 the parent population �i; i¼ 1; 2, . . . ; Np is generated by setting �i�UR(�min; �max)which is random in nature. This may lead to a different parent population for different trial

Table I. Generation schedule for 20-unit system.

Unit generation (MW) Lambda-iteration method Proposed methods

SEPA FCEPA

P1 512.83 512.83 512.85P2 169.13 169.13 169.15P3 126.91 126.91 126.92P4 102.88 102.88 102.89P5 113.69 113.69 113.69P6 73.58 73.58 73.59P7 115.29 115.29 115.29P8 116.41 116.41 116.42P9 100.41 100.42 100.42P10 106.04 106.04 106.04P11 150.24 150.24 150.25P12 292.77 292.78 292.78P13 119.12 119.12 119.12P14 30.83 30.83 30.83P15 115.82 115.82 115.82P16 36.26 36.26 36.26P17 66.87 66.87 66.88P18 87.98 87.98 87.99P19 100.81 100.81 100.82P20 54.32 54.32 54.32Loss, PL (MW) 91.98 91.98 91.99Fuel cost in $/h 62461.13 62461.48 62463.73�optimum 20.95 20.95 20.95Range of iterations 8 5–50 1–4CPU time in ms (1 GHz, IBM PC) 4.5 50 5



studies. In step 3.2.1 the offspring population �i0; i¼Npþ 1; Npþ 2; . . . ; 2Np is generated using a

separate normal distribution curve for each parent individual. On account of this the offspring �i0;

i¼Npþ 1; Npþ 2; . . . ; 2Np may differ widely and may not be closer to each other. This may leadto slow convergence to optimum.

2. In step 3.2.1, the offspring is generated randomly using Equation (10) without any check toassess whether the offspring to be generated is better than its parent or not with respect to fitnessfunction. An offspring is effective and useful only if its fitness function converges with respect toits parent’s fitness function value or else the offspring is ineffective. Hence a selective process isneeded to generate only the effective offsprings. An offspring is an ineffective member comparedto its parent in the following cases:

(i) If the fitness function value fi of a parent is positive and the normal random number N(0; �i2) is

also positive, then the offspring, �iþNP0 ¼�iþN(0; �i

2) is ineffective since the fitness valuefiþNP0 of �iþNP

0 will be more positive than fi.(ii) If the fitness function value fi of the parent is negative and the normal random number N(0;

�i2) is also negative, then the offspring, �iþNP

0 ¼�iþN(0; �i2) is ineffective since the fitness

value fiþNP0 of �iþNP

0 will be more negative than fi.3. The third drawback also arises in step 3.2.1. In this step �i is obtained from Equation (11) to

generate a normal distribution curve. The patterns of normal curves obtained during successiveiterations of a sample system study for a specific parent (decision variable) are shown in Figure 1.From this figure it is observed that the width of the curves does not decrease progressively asiteration count progresses. The width of the curve corresponding to the second iteration is morethan that of the first iteration, but for reliable and non-oscillatory convergence, the width of thenormal distribution curve should decrease progressively as the iteration count progresses, sincethe width of the curve decides the maximum range of random number N(0; �i

2) which in turndecides the change in decision variable.

It may be recalled that in the conventional gradient optimization methods, the step size to move

along the negative gradient direction is fixed arbitrarily to start with and reduced progressively during

subsequent iterations to achieve faster and non-oscillatory convergence. On similar grounds the width

of the normal distribution curve may be generated by fixing arbitrary width (maximum range of ��) as

displayed in Figure 2 and the width may be reduced successively during subsequent iterations to get

faster non-oscillatory convergence.

Figure 1. Actual curves.



5.2. Modifications to the proposed algorithm

5.2.1. Selection of single parent. The first drawback is contained by generating a single parent

deterministically instead of generating many parents randomly for generating the offspring popula-

tion. The initial single parent is generated as

�p ¼ �1 þ �2 þ � � � þ �n

nð13Þ

where

�j ¼dFjðPjÞ

dPj

� 1

1 � dPLdPj

� � j ¼ 1; 2; . . . ; n

Pj ¼Pj;maxPPj;max

� PD j ¼ 1; 2; . . . ; n ð14Þ

The system MW demand PD is distributed among the units in proportion to their capacity as stated

in Equation (14) for fixing initial generation schedule.

The offspring population is generated from the single parent as:

�i ¼ �p þ Ni

�0; �2

p

�i ¼ 1; 2; . . . ;Np ð15Þ

where the normal random numbers for i¼ 1; 2; . . . ; Np are generated from a single normal distribution

curve with standard deviation �p. The procedure for generating the single normal distribution curve is

given in section 5.2.3.

5.2.2. Generation of effective offsprings. The offsprings are generated using a selective procedure as

follows.

Consider that the parent be �p and its fitness value is fp. Let the normal random number generated

to create the i-th offspring be Ni(0; �p2). If fp is negative and Ni(0; �p

2) is positive or if fp is positive

and Ni(0; �p2) is negative, then generate the i-th offspring as �i¼�pþNi(0; �p

2). In the above two cases,

the fitness function of the offspring fi converges with respect to its parent’s fitness function value fp

Figure 2. Ideal curves.



while in all other cases its fitness function diverges. Hence the selectivity ensures effective offsprings

and non-oscillatory faster convergence. The above selective process is used to generate Np offsprings

from the single parent.

5.2.3. Selection of �p for distribution curve. The maximum permissible range of initial variation in the

decision variable �p to generate the first offspring population is fixed judiciously based on experience.

For example, it may be fixed as t¼� � (�max ��min), � being an arbitrary constant and the maximum

range of the random variation in the decision variable, �, may be taken as ��¼ � t as depicted in

Figure 3. The normal distribution curve is constructed such that the probability of occurrence of

�� t is zero. This will ensure that the random number N(0; �p2) generated would be less than t.

Mathematically,

1ffiffiffiffiffiffi2�

p� �

Z 1

t

e�12

��ð Þ2

¼ 0 ð16Þ

By setting the value for t, we can compute � from Equation (16) and generate the normal

distribution curve. It may be difficult to solve the above equation with the right-hand side equal to zero.

Hence by setting the right-hand side value to a small number, say 0.00003, the probability of

occurrence of �� t is 0.00003. Then Equation (16) becomes

1ffiffiffiffiffiffi2�

p��

Z 1

t

e�12

��ð Þ2

¼ 0:00003 ð17Þ

The solution of the above equation is obtained by using the standard normal tables [8]. The width of

�� is progressively decreased during successive iterations and the respective normal distribution

curve is generated to ensure smoother convergence. The value of maximum width of �� may be

decreased by 50% during each successive iteration. This procedure ensures ideal normal distribution

curves similar to that shown in Figure 2.

5.3. FCEP algorithm

The SEPA with the above modifications is referred to as Fast Computation Evolutionary Programming

Algorithm (FCEPA). The main computational steps with the proposed modifications are briefly

summarized as follows.

Figure 3. Search range.



Step 1: Choose the initial permissible variation in decision variable � and select the population size for

offspring.

Step 2: Compute the initial parent �p using Equation (13) and the corresponding generation schedule

and power mismatch using Equations (5) and (6). Check for power mismatch convergence, if

the convergence criterion is satisfied go to step 7; otherwise, go to next step.

Step 3: Generate the normal distribution curve and offspring population as explained in previous

sections 5.2.3 and 5.2.2, respectively.

Step 4: Compute the generation schedule to each offspring by solving Equation (5) and the power

mismatch using Equation (6).

Step 5: Select the best offspring individual having the minimum fitness function value among the

offspring population. Check for power mismatch convergence for this best offspring, if the

convergence criterion is satisfied go to step 7; otherwise, go to next step.

Step 6: Replace the parent individual by the best offspring individual and go to step 3.

Step 7: Terminate the computation.

5.4. Sample system study and results

The proposed FCEPA was tested on a number of sample systems. The results of the 20-unit system are

given in column 4 of Table I. The solution exactly matches with that in columns 2 and 3. The optimum

result is almost the same for several trial runs and the range of iterations varies from 1 to 4. The

computational time is reduced drastically by 90% compared to the standard EP algorithm. The

convergence pattern of power mismatch for the proposed SEPA and FCEPA are shown in Figures 4

and 5, respectively. The modifications proposed are very effective in achieving faster, reliable and non-

oscillatory convergence.

6. ED PROBLEM WITH PIECEWISE QUADRATIC COST FUNCTION

Traditionally, the cost function of each generator has been approximately represented by a single

quadratic cost function. It is more realistic, however, to represent the generation cost function for fossil

fired plants as a segmented piecewise quadratic function [5,9–12] for generators supplied with

multiple fuel sources (gas and oil).

Figure 4. Convergence characteristics for 20-unit system with proposed standard EP.



The proposed FCEP algorithm is tested on a 10-unit system with piecewise quadratic cost function

in order to test the robustness of the algorithm. In this algorithm initially the cost coefficients ajk; bjkand cjk of the j-th unit, k¼ 1; 2; . . . ; m are decided by the initial assumed value of Pj’s as per

Equation (14) and in the successive iteration the values of ajk, bjk and cjk are selected depending on

the value of Pj’s.

The data for the 10-unit system is taken from Reference [5]. Each generator has two or three

segmented piecewise quadratic cost functions and four operating conditions of the system with load

levels, namely 2400, 2500, 2600 and 2700 MW, being considered. The results of the proposed

algorithm are compared with that obtained using the algorithm proposed in Reference [5] wherein the

ED problem is solved using the EP algorithm by taking generator outputs as variables and the cost

function as fitness function. The parameters used for the EP based algorithm given in Reference [5]

and the proposed algorithm are given in Table II. The results of the three methods (namely EP with unit

generation as variables, proposed standard EP with system lambda as variable, and FCEP) are given in

Table III.

From Table III it can be seen that the optimal fuel costs with the proposed methods closely matches

with that given in Reference [5]. The number of iterations of the proposed FCEPA algorithm is greatly

reduced compared to the method proposed in Reference [5]. The population size for the proposed

approaches is taken as 5, which is one eighth of the population size of the method proposed in

Reference [5]. A comparative study of the computation time is given in the bottom row of Table III. It

may be observed that on average there is a 90% reduction in computation time for the proposed

methods when compared to the method proposed in Reference [5] which shows the computational

efficiency of the proposed methods.

Figure 5. Convergence characteristics for 20-unit system with proposed FCEP.

Table II. EP method—parameters.

Parameters Method given in Proposed methodsReference [5]

SEPA FCEPA

Scaling factor (�) 0.01 0.01 —Population size (Np) 40 5 5Convergence criteria 400 iterations 1 MW—power mismatch 1 MW—power mismatch



Tab

leII

I.G

ener

atio

nsc

hed

ule

for

10

-un

itsy

stem

wit

hu

nit

gen

erat

ion

asvar

iab

lean

dth

ep

rop

ose

dm

eth

od

s.

Un

itP

rop

ose

dm

eth

od

sg

ener

atio

n(M

W)

EP

wit

hu

nit

gen

erat

ion

asvar

iab

leS

EPA

FC

EPA

24

00

MW

25

00

MW

26

00

MW

27

00

MW

24

00

MW

25

00

MW

26

00

MW

27

00

MW

24

00

MW

25

00

MW

26

00

MW

27

00

MW

P1

18

9.7

20

6.2

21

1.1

22

6.3

18

9.7

20

6.5

20

9.8

21

8.2

18

9.7

20

6.5

20

9.8

21

8.2

P2

20

2.9

20

6.7

20

7.9

21

4.3

20

2.3

20

6.5

20

7.9

21

1.7

20

2.3

20

6.5

20

7.9

21

1.7

P3

25

3.2

26

8.3

26

9.7

28

9.1

25

3.9

26

5.7

26

9.9

28

0.7

25

3.9

26

5.7

26

9.9

28

0.7

P4

23

2.4

23

6.3

23

6.4

24

2.5

23

3.1

23

5.9

23

7.0

23

9.6

23

3.1

23

5.9

23

7.0

23

9.6

P5

24

0.1

25

7.2

26

2.6

29

4.2

24

1.8

25

8.0

26

3.7

27

8.5

24

1.8

25

8.0

26

3.7

27

8.5

P6

23

2.9

23

5.8

23

7.0

24

2.7

23

3.1

23

5.9

23

7.0

23

9.6

23

3.1

23

5.9

23

7.0

23

9.6

P7

25

4.4

26

8.7

27

4.4

30

3.8

25

3.3

26

8.9

27

4.4

28

8.6

25

3.3

26

8.9

27

4.4

28

8.6

P8

23

2.8

23

5.8

23

7.4

24

2.2

23

3.1

23

5.9

23

7.0

23

9.6

23

3.1

23

5.9

23

7.0

23

9.6

P9

32

0.9

33

0.9

40

3.2

35

6.4

32

0.3

33

1.5

40

2.8

42

8.5

32

0.3

33

1.5

40

2.8

42

8.5

P1

02

40

.72

54

.12

60

.32

88

.42

39

.42

55

.12

60

.62

74

.82

39

.42

55

.12

60

.62

74

.9�

op

tim

um

——

——

0.4

28

20

.46

27

0.4

74

90

.50

63

0.4

28

20

.46

27

0.4

74

90

.50

64

Fu

el4

81

.73

52

6.2

55

74

.74

62

6.2

74

81

.68

52

6.2

65

74

.73

62

3.7

44

81

.74

52

6.2

45

74

.74

62

3.7

8co

st($

/h)

No

.o

f4

00

40

04

00

40

05

–5

05

–5

05

–5

05

–5

02

–4

2–

42

–4

2–

4it

erat

ion

sC

PU

tim

e1

67

51

67

51

67

51

67

51

51

51

51

54

44

4in

ms



7. CONCLUSION

This paper presents a computationally very efficient FCEPA for solving the ED problem. The

formulation of the ED problem is modified so as to make the EP algorithm very efficient.

Modifications are made in the SEPA in order to achieve faster and reliable convergence. The proposed

algorithm is computationally efficient because of the modifications proposed for the generation of

parent and offspring population, construction of normal distribution curve, etc. The effectiveness of

the proposed algorithm has been tested on a number of sample systems. The proposed method is

relatively simple, reliable and efficient.

8. LIST OF SYMBOLS AND ABBREVIATIONS

n Total number of units

m Total number of fuel types

Fj(Pj) Fuel cost of j-th generator

FT Total fuel cost

fi Fitness value of i-th individual

Pj Power output of j-th generator

PD Total load

PL Total system transmission loss

aj, bj, cj Cost coefficients of the j-th generator

Bij Loss coefficients

Np Population size

t Maximum value of normal random number

� Mutation scale

�� Correction required for �Wi Weight value of i-th individual

ui Uniform random number of i-th individual

wj Weight value corresponding to j-th competition

q Competition number

ED Economic Dispatch

EP Evolutionary Programming

SEPA Standard Evolutionary Programming Algorithm

FCEPA Fast Computation Evolutionary Programming Algorithm

REFERENCES

1. IEEE Committee Report. Present practices in the economic operation of power systems. IEEE Transactions on PowerApparatus and Systems 1971; 90:1768–1775.

2. Chowdhury BH, Rahman S. A review of recent advances in economic dispatch. IEEE Transactions on Power Systems 1990;5:1248–1259.

3. Wood AJ, Wollenberg BF. Power Generation, Operation and Control, 2nd edn. Wiley: New york, 1996.4. Jayabarathi T, Sadasivam G, Ramachandran V. Evolutionary programming based economic dispatch of generators with

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AUTHORS’ BIOGRAPHIES

P. Somasundaram (born in 1977) obtained his BE degree in Electrical and ElectronicsEngineering and his ME in Power Systems from Madras University and AnnamalaiUniversity, India, in 1998 and 2000, respectively. He has been engaged in carrying outresearch work in Anna University, in the area of power system generation and operation.

K. Kuppusamy (born in 1944) obtained his BE degree in Electrical Engineering and hisMSc (Engg.) in Power Systems from Madras University, India. In 1981, he obtained his Ph.Dfrom Madras University, India. Presently he is a Professor of Electrical Engineering in AnnaUniversity, India. His areas of interest include power system analysis, power systemoptimization and AI techniques to power system problems.

R. P. Kumudini Devi (born in 1968) obtained her BE degree in Electrical and Electronics Engineering and herME in Power Systems from SV University and Anna University, India, in 1990 and 1992, respectively. In 2000,she obtained her Ph.D. from Anna University, India. Presently she is an Assistant Professor of ElectricalEngineering in Anna University. Her areas of interest include power system stability, optimization and AItechniques to power system problems.



fast computation evolutionary programming algorithm for the economic dispatch problem

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