bacteria foraging eld15u08

8/12/2019 BActeria Foraging ELD15U08

1/5

Bacteria Foraging Based Algorithm for Optimum

Economic Load Dispatch with Non-convex LoadsNidul Sinha1, SeniorMemberIEEE, Devaprasad Paul

2, Bir Bahadur Singh

2, Manabendra Barua

2, Yatin Chauhan

2

Abstract--An algorithm based on Bacteria Foraging (BF) was

developed to solve the problem of finding the optimum loadallocation amongst the committed units in power system with

non-convex loads. The performance of the proposed algorithm

is evaluated on a test case of 15 units. The performance of the

algorithm is compared with floating point genetic algorithm

(FPGA) with optimum parameters. In addition, the BF

algorithm is evaluated with and without swarming effect.

Results demonstrate that the performance of the BF algorithm

is far better than FPGA algorithm in terms of convergence rate

and solution quality. BF algorithm with swarming effect proves

to be more efficient as compared to that without swarming.

Index TermsBacterial Foraging, Floating point Genetic

Algorithm, Economic Load dispatch.

I. INTRODUCTIONEconomic load dispatch (ELD) in electric power

system is the optimum allocation of load amongst the

committed generating units subject to satisfaction of the

constraints. Most of the conventional classical dispatch

algorithms, like lambda-iteration method, base point and

participation factors method, and the gradient method [1], [2]

are gradient based methods and hence, cannot tackle the non-

convexity well. These algorithms usually approximate the

characteristics as quadratic ones to meet the their

requirements. However, such approximations may result into

huge loss of revenue over the time. In addition, they have the

tendency of easily getting trapped in local minima. And most

of the modern practical thermal units do have highly non-

linear input-output characteristics because of valve point

loadings prohibiting operating zones etc resulting in multi-

ple local minima in the cost function. The solution of multi-

modal optimization problems like ELD demands for solution

methods, which have no restrictions on the shape of the fuel

cost curves. Though enumerative method like dynamic

programming (DP) [1] is capable of solving ELD problems

with inherently nonlinear and discontinuous cost curves but

proves to suffer from intensive mathematical computations

and memory requirement. With nonlinear and non-

differentiable objective functions, modern heuristic search

approaches are the methods of choice. The best known ofthese are genetic algorithm (GA) [3]-[11], evolutionary

strategy (ES) [12], [13], evolutionary programming (EP)

[13]-[21], simulated annealing (SA) [3], [22], particle swarm

optimization (PSO) [23]-[25], and differential evolution (DE)

[26]-[27]. At the heart of every direct search method is a

strategy that creates new solutions and some criterion to

1Nidul Sinha is with the Department of Electrical Engineering, NIT, Silchar,

Assam, India-788010, (email:[email protected]).2All are with the Department of Electrical Engineering, NIT, Silchar, Assam.

accept or reject the new solutions. While doing this all basic

direct search methods use some greedy criteria. One of the

greedy criteria is to accept a new solution if and only if it

reduces the value of the objective function (in case of

minimization) and the other may be forcing to create more

new solutions nearer to already found better solutions.

Although the greedy decision process converges fairly fast, it

runs the risk of getting trapped in a local minimum.

Inherently all parallel search techniques like genetic and

evolutionary algorithms have some built-in safeguards like

exploration to forestall misconvergence. Though simulated

annealing [3], [22] is reported to have performed better in

solving non-linear ELD problems, the main drawback of SA

is the difficulty in determining an appropriate annealing

schedule, otherwise the solution achieved may still be a

locally optimal one. Recent trends in research, therefore,have been directed towards use of evolutionary algorithms

(EAs) i.e. GA, ES and EP, which are based on the simulated

evolutionary process of natural selection and genetics. EAs

are more flexible and robust than conventional calculus based

methods. Due to its high potential for global optimization,

GA has received great attention in solving ELD problems.

Walters and Sheble [4] reported a GA model that employed

units output as the encoded parameter of chromosome to

solve an ELD problem with valve-point discontinuities. To

enhance the performance of GA, Yalcinoz et al. [10] have

proposed the real-coded representation scheme, arithmetic

crossover, mutation, and elitism in the GA to solve more

efficiently the ELD problem, and it can obtain a high-qualitysolution with less computation time.

BFA has been recently proposed [28] and further

applied to: harmonic estimation problem in power systems

[29], optimize both real power loss and voltage stability

Limit [30] and optimize active power filter for load

compensation [31]. The algorithm is based on the foraging

behavior of E. coli bacteria present in human intestine. The

objective is the minimization of the total production cost over

the scheduling horizon while the constraints must be satisfied

during the optimization process.

BFA includes most of the features like chamotaxis,

swarming, reproduction, elimination, and dispersal of

improved modern heuristic search methods, which make the

algorithm very promising.

Very few works are reported on the performance of BF

algorithm on highly nonlinear power system problems. And

the recent reported impressive performance of BFA on

benchmark mathematical functions have induced us in

applying this method on highly nonlinear ELD problems.

In view of the above, the main objectives of the present

work are:

(i) To develop a program based on BF algorithm to solvethe non-convex ELD problem.


2/5

(ii) To compare the performance of the algorithm on thesame problem with that of recent FPGA method.

(iii)To investigate into the performance of the algorithmwith swarming as well as without swarming on the

same problem.

II. BACTERIAL FORAGING ALGORITHM (BFA)The idea of foraging under BFA is based on the fact

that natural selection tends to eliminate animals with poor

foraging strategies and favour those having successful

foraging strategies. After many generations, poor foraging

strategies are either eliminated or reshaped into good ones.

The E. coli bacteria that are present in our intestines have a

foraging strategy governed by four processes, namely,

chemotaxis, swarming, reproduction, and elimination and

dispersal [28].

A. Chemotaxis:

This process is achieved through swimming and tumbling.

Depending upon the rotation of the flagella, the bacterium

decides what direction it should move (tumbling) and if the

new location of bacterium after movement is better, the

bacterium will continue to swim in the same previous

direction (swimming) for a specific number of steps.

Suppose that we want to find the minimum of J(), Rp.

Assume that is the position of a bacterium and J ( )

represents the amount of the food at the position J()0 representing that the bacterium at location

is in nutrient-rich, neutral, and noxious environments,

respectively. To represent a tumble, a unit length random

direction, say (i) , is generated. This will be used to define

the direction of movement after a tumble. In particular

i(j+1, k, l ) =

i( j, k, l ) + C(i) (i) (1)

where i(j, k, l) represents the i

thbacterium at j

th chemotactic,

kth

reproductive, and lth

elimination and dispersal step. C(i) is

the step-size taken in random direction specified by the

tumble. If at i(j+1, k, l), the cost of J (i, j, k, l) is lower thanat

i(j, k, l), then another step of size C(i) in the direction (i)

will be taken and bacterium will begin to swim in the

direction (i) . This swim is continued as long as it continues

to reduce the cost, but only up to a maximum number of

steps, Ns. This represents that the cell will tend to keep

moving if it is headed in the direction of increasingly

favorable environments.

B. Swarming:

It is always desired that the bacterium that has searched the

optimum path of food should try to attract other bacteria so

that they reach the desired place more rapidly. Swarming

makes the bacteria congregate into groups and hence move asconcentric patterns of groups with high bacterial density. Let

P( j k l ) = { i(j, k, l)|i = 1,2,..., S}.

Mathematically, swarming can be represented by:

)])-(wexp([h

)])-(wexp([-dJ)(J

2

repellent

s

1i

repellent

2

attract

s

1i

attract

s

1i

cci

cc

+

==

=

==

i

jj

i

jj

(2)

where Jcc(,P(I,j,l)), due to the movements of all the cells, is a

time varying function that is added to J(i,j,k,l ) so that the

cells will try to find nutrients, avoid noxious substances, and

at the same time try to move toward other cells, but not too

close to them. S is the total number of bacteria. p is the

number of parameters to be optimized that are presented in

each bacterium position. dattract, wattract, hrepelent, and wrepelentare

different coefficients that are to be chosen judiciously.

C. Reproduction:

Half of the total bacteria i.e. Sr= S/2 with least health die,

and the comparatively remaining healthier bacteria each split

into two bacteria, which is placed in the same location. This

makes the population of bacteria constant and follows the

natural principle of preferring better fit bacteria to survive

and produce.

D. Elimination and Dispersal:

It is possible that in the local environment, the life of a

population of bacteria changes either gradually by

consumption of nutrients or suddenly due to some other

influence. Events can kill or disperse all the bacteria in a

region. They have the effect of possibly destroying

chemotactic progress, at the same time they also have the

possibility of assisting in chemotaxis, since dispersal may

result into bacteria at better locations i.e. solutions..

Elimination and dispersal prevents bacteria from gettingtrapped in local optima. For each elimination-dispersal event

each bacterium in the population is subjected to elimination-

dispersal with probability ped. To keep the number of bacteria

constant, if we eliminate a bacterium, simply disperse one to

a random location in the optimization domain. The flowchart

of the bacterial foraging algorithm is shown in

Fig.1.

Fig.1 Flowchart of the Bacteria Foraging Algorithm.

E. BFA Algorithm in brief:

Step 1Initialization

First following variables must be chosen.

1) S: Number of bacteria to be used in the search.

2) p: Number of parameters to be optimized.

3) Ns: Swimming length.

4) Nc: Number of chemotactic steps.

5) Nre: Number of reproduction steps.


3/5

6) Ned: Number of elimination and dispersal events.

7) ped: Probability of elimination and dispersal.

8) C(i), i= 1,2,,S: unit length run for every bacterium

9) The values of dattract, attract, hrepelent and repelent

10) Initial values for the i, i= 1,2,,S

Step 2Iterative algorithm for optimization

1) Elimination-dispersal loop: l=l+1

2) Reproduction loop: k=k+1

3) Chemotaxis loop: j=j+1

a) For i =1,2,,S, take a chemotactic step for bacterium i

as follows.

b) Compute J (i, j, k, l ).

Let J(i j k l ) = J (i j k l )+ Jsw(i j k l )+Jcc(i( j k l ),P( j k l ))

(i.e., add on the cell-to-cell attractant effect for swarming

behavior).

c) Let Jlast= Jsw(i j k l ) to save this value since we may find

a better cost via a run.

d) Tumble: Generate a random vector (i) Rp with each

element m(i), m= 1,2,..., p, a random number on [1,1].

e) Move:

Let: (i) =(i)/ (T(i)(i))

1/2

i( j+1, k ,l )=

i( j, k ,l )+ C(i) (i)

This results in a step of size C(i) in the direction ofthe tumblefor bacterium i.

f) Compute J (i, j +1, k, l ), and then

Let Jsw (i j+1, k, l) =J (I, j+1, k, l,)+Jcc (i( j+1,k, l),P( j+1,

k,l))

g) Swim

i) Let m = 0 (counter for swim length).

ii) While m


4/5

Parameters for BFA:

Population Size = 60

Maximum chemotactic steps = 50

Penalty multiplier = 100.

Nc= 50

Nre= 10

Ns= 6

ped= 0.25

Ned= 4

Table-1: Units data for the test case (15 units case)

with load 2650MW (with valve point loadings)

PiOutput Limits Fuel Coefficients

Min Max a b c e f

1 15 55 323.79 12.41 0.004447 120 0.077

2 150 455 574.54 10.22 0.000183 300 0.035

3 20 130 374.59 8.8 0.001126 120 0.077

4 20 130 374.59 8.8 0.001126 120 0.077

5 150 470 461.37 10.4 0.000205 300 0.035

6 135 460 630.14 10.1 0.000301 300 0.035

7 135 465 548.2 9.87 0.000364 300 0.0358 60 300 227.09 11.5 0.000338 200 0.042

9 25 162 173.72 11.21 0.000807 120 0.077

10 20 160 175.95 10.72 0.001203 120 0.077

11 20 80 186.86 11.21 0.003586 120 0.077

12 20 80 230.27 9.9 0.005513 120 0.077

13 25 85 225.28 13.12 0.000371 120 0.077

14 15 55 309.03 12.12 0.001929 120 0.077

15 150 455 671.03 10.07 0.000299 300 0.035

The convergence characteristics of the FPGA and BFA

algorithms are shown in Fig.2 & Fig.3 for the test case.

Investigation of the figures 2 & 3 reveals that BFA

algorithms both with swarming and without swarming

converges faster than FPGA. BFA with swarming converges

faster than that without swarming.

To investigate the effects of initial trial solutions all three

algorithms were run with 10 different initial trial solutions

and the performance is reported in table-2. The average cost

achieved in all the runs with each of the algorithm shows the

capability of the algorithm in escaping local minima and find

the better global solutions. Also, the lower value in the

difference between maximum and minimum values further

demonstrate better performance. It can be observed from the

table that BFA with swarming has the least average cost

amongst three and the least difference between maximum

and minimum values. The performance of BFA algorithms in

both forms is better than FPGA.

0 50 100 150 200 250 300 350 4003.446

3.448

3.45

3.452

3.454

3.456

3.458

3.46

3.462

3.464

3.466x 10

4

Generations

Cost($)

Fig. 2 The convergence nature of FPGA on the test case.

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

5

Chamotectic steps

Cost($)

BFA with swarming

BFA without swarming

Fig.3: The convergence nature of BFA with swarming and

without swarming.

Table-2. Statistical test results of 10 runs with different

initial solutions (with non-smooth cost curves) for the test

case.

Method Average cost

(Rs.)

Maximum cost

(Rs,)

Minimum cost

(Rs.)

FPGA

BFA

(withoutswarming)

BFA (withswarming)

34131

33757

33373

34540

34084

33557

33523

33430

33237

.


5/5

V. CONCLUSIONAlgorithms based on FPGA, BFA (swarming) and BFA

(without swarming) are developed in Matlab and their

performances are tested on a test case of 15 units for non-

convex economic load dispatch problems with valve point

loading effects. Experimental results reveal that all the

algorithms are competent to provide better quality solutions.

BFA with swarming the three exhibits the highest capability

of converging to better quality solutions with higher

convergence rate. BFA in both forms is superior to FPGA in

terms of convergence rate and better quality solutions. In

between BFAs with swarming and without swarming, the

earlier one demonstrate to be more efficient in finding better

quality solutions and converging to the global optimal at a

faster rate.

Hence, BFA with swarming is recommended for solution of

highly nonlinear ELD problems in power system. However,

lot more scope is there for further works in improvement of

BFA algorithm like adaptive tuning of chemotactic steps,

step size etc.

VI. REFERENCES

1. Liang, Zi-Xiong and Glover, J. Duncan, A zoom feature for adynamic programming solution to economic dispatch including

transmission losses, IEEE Trans. on Power Systems, Vol.7, No.2,

May 1992, pp.544-549.2. Wood, A.J., Wollenberg, B.F., Power Generation, Operation and

Control, second edition, Wiley, New York, USA, 1996.

3. Wong, K.P. and Wong, Y.W., Thermal generator scheduling usinghybrid genetic/simulated annealing approach, IEE Proc. Part-C,

Vol.142, No.4, July 1995, pp.372-380.

4. Walter, D.C. and Sheble, G.B., Genetic algorithm solution ofeconomic dispatch with valve point loading, IEEE Trans. on Power

Systems, Vol.8, No.3, August, 1993, pp.1325-1332.

5. Bakirtzis, A. Petridis, V. and Kazarlis, S., Genetic algorithm solutionto the economic dispatch problem,IEE Proc. Part-D, Vol. 141, No. 4,July 1994, pp. 377382.

6. Sheble, G.B. and Brittig, K., Refined genetic algorithm Economicdispatch example, IEEE Trans. on. Power Systems, Vol. 10, Feb.

1995, pp. 117124.

7. Chen, P.H. and Chang, H.C., Large-scale economic dispatch bygenetic algorithm, IEEE Trans. on Power Systems, Vol.10, No.4,

November 1995, pp.1919-1926.

8. Fogel, D.B., A comparison of evolutionary programming and geneticalgorithms on selected constrained optimization problems, Simulation,

June 1995, pp.397-404.

9. Goldberg, D.E., Genetic Algorithms in Search, Optimization andMachine Learning, Addison Wesley, MA, 1989.

10. Yalcionoz, T., Altun, H., and Uzam, M., Economic dispatch solutionusing a genetic algorithm based on arithmetic crossover, Proc. PowerTech. Conf., IEEE, Portugal, Sept. 2001.

11. Houck, C.R., Joines, J.A., and Kay, M.G., A genetic algorithm forfunction optimization: A Matlab implementation, Technical Report

NCSU-IE TR 95-09, North Carolina State University, 1995.

12. Bck, Th. and Schwefel, H.P., An overview of evolutionaryalgorithms for parameter optimization, Evolutionary Computation,Vol.1, No.1, 1993, pp.1-23.

13. Fogel, D.B.,Evolutionary Computation: Toward a New Philosophy ofMachine Intelligence, IEEE Press, Piscataway, NJ, 1995.

14. Fogel, D.B., An introduction to simulated evolutionary optimization,IEEE Trans. on Neural Networks, Vol.5, No.1, 1994, pp.3-14.

15. Chellapilla, K., Combining mutation operators in evolutionaryprogramming, IEEE Trans. on Evolutionary Computation, Vol.2,

No.3, 1998, pp.91-96.

16. Wolpert, D.H. and Macready, W.G., No free lunch theorems foroptimization, IEEE Trans. on Evolutionary Computation, Vol.1,No.1, 1997, pp.67-82.

17. Yang, H.T., Yang, P.C. and Huang, C.L., Evolutionary programmingbased economic dispatch for units with non-smooth fuel costfunctions, IEEE Trans. on Power Systems, Vol.11, No.1, February

1996, pp.112-118.

18. Yao, X. , Liu, Y. and Lin, G., Evolutionary programming madefaster,IEEE Trans. Evolutionary Computation, Vol. 3, July 1999, pp.

82102.

19. Sinha, Nidul, Chakrabarti, R. and Chattopadhyay, P.K., Evolutionaryprogramming techniques for economic load dispatch,IEEE Trans. on

Evolutionary Computation, Vol.7, No.1, February, 2003, pp.83-94.

20. Sinha, Nidul, Chakrabarti, R. and Chattopadhyay, P.K., FastEvolutionary programming techniques for short-term hydrothermal

scheduling,IEEE Trans. on Power Systems, Vol.18, No.1, February

2003, pp.214-220.21. Yao, X. and Liu, Y., Fast evolutionary programming, Proc. 5th

Annu.Conf. Evolutionary Programming, L. J. Fogel, T. Bck, and P. J.

Ange-line, Eds. Cambridge, MA, 1996, pp. 451460.22. Wong, K.P. and Fung, C.C., Simulated annealing based economic

dispatch algorithm, IEE Proc. Part-C, Vol.140, No.6, 1992, pp. 544-

550.

23. Kennedy, J., and Eberhart, R., Particle Swarm Optimization,Proceedings of IEEE International Conference on Neural Networks,

Vol. IV, Perth, Australia, 1995, pp. 1942-1948.24. Park, J.-B., Lee, K.-S., Shin, J.-R., and Lee, K.Y., "A particle swarm

optimization for economic dispatch with non-smooth cost Functions,

IEEE Trans. on Power Systems, Vol. 20, No. 1, February 2005, pp. 34-42.

25. El-Gallad, A., El-Hawary, M., Sallam, A., and Kalas, A., Particleswarm optimizer for constrained economic dispatch with prohibitedoperating zones, Proc. IEEE Canadian Conf. on Electrical and

Computer Engineering, 2002, pp.78-81.

26. Storn, R., System design by constraint adaptation and differentialevolution,IEEE Trans. on Evolutionary Computation, Vol. 3, No. 1,

1999, pp. 22-34.

27. Price K., Differential Evolution: A Fast and Simple NumericalOptimizer, NAFIPS 1996, Berkeley, pp.524-527.

28. K. M. Passino, Biomimicry of bacterial foraging fordistributedoptimization and control, IEEE Control Syst. Mag., vol. 22,no. 3, pp. 5267, Jun. 2002.

29. S. Mishra, A hybrid least square-fuzzy bacteria foraging strategy forharmonic estimation, IEEE Trans. Evol. Comput., vol. 9, no. 1, pp.

6173, Feb. 2005.30. M. Tripathy and S. Mishra, Bacteria foraging-based solution to

optimize both real power loss and voltage stability limit, IEEE Trans.Power Syst., vol. 22, no. 1, pp. 240-248, Feb. 2007.

31. S. Mishra, C. N. Bhende, Bacterial Foraging Technique-BasedOptimized Active Power Filter for Load Compensation, IEEE Trans.Power Syst., vol. 22, no. 1, pp. 457 565, Jan. 2007.

Nidul Sinha received his B.E. degree in electricalengineering from Calcutta University, in 1984 and

M.Tech. degree in power apparatus and systems from

Indian Institute of Technology, New Delhi, in 1989. Hereceived his Ph.D. degree in electrical engineering from

Jadavpur University. His research interests include

application of soft computing techniques to operation,

control and economics of electrical power systems,deregulation and optimization. He is a senior member of IEEE. He is also a

reviewer of the journals IEEE TPWRS, TPWRD, PESL, IEEE TEC, IEEPart-C, and EPSR.

bacteria foraging eld15u08

Documents