solution of firefly algorithm for the economic thermal power dispatch with emission constraint in...

SOLUTION OF FIREFLY ALGORITHM FOR

THE ECONOMIC THEMAL POWER

DISPATCH WITH EMISSION CONSTRAINT

IN VARIOUS GENERATION PLANTS

THENMALAR.K 1

Assistant Professor, Dept of EEE

Vivekanandha College of Engineering for Women,

Namakkal (Dist.), India

[email protected]

Dr.A.ALLIRANI 2

Principal,

S.R.S College of Engineering & Technology,

Salem (Dist.), India

[email protected]

Abstract Economic load dispatch (ELD) is an important optimization task in power system. It is the process of

allocating generation among the committed units such that

the constraints imposed are satisfied and the energy

requirements are minimized. There are three criteria in

solving the economic load dispatch problem. They are

minimizing the total generator operating cost, total

emission cost and scheduling the generator units. In this

paper Firefly Algorithm(FA) solution to economic dispatch

problem is very useful when addressing heavily

constrained optimization problem in terms of solution

accuracy. Results obtained from this technique clearly

demonstrate that the algorithm is more efficient in terms

of number of evolution to reach the global optimum point.

The result also shows that the solution method is practical

and valid for real time applications In this paper the

Firefly Algorithm(FA) solves economic load dispatch

(ELD) power system problem of three generator system,

six generator system with emission constraints and twelve

generator system with introduced population-based

technique is utilized to solve the DED problem. A general

formulation of this algorithm is presented together with an

analytical mathematical modeling to solve this problem by

a single equivalent objective function. The results are

compared with those obtained by alternative techniques

proposed by the literature in order to show that it is

capable of yielding good optimal solutions with proper

selection of control parameters. The validity and quality of

the solution obtained Firefly Algorithm(FA) based

economic load dispatch method are checked and compared

with Artificial colony algorithm(ABC),Particle Swarm

Optimization Algorithm (PSO),simulated Annealing

Algorithm(SA)

Keywords: ABC-Artificial Bee Colony Algorithm, DED-

Dynamic Economic Dispatch, FA-Firefly Algorithm, PSO-

Particle Swarm Optimization Algorithm

I. INTRODUCTION

Economic dispatch is one of the most important

problems to be solved in the operation of power system.

The traditional method such as lambda iteration method,

the base point, the participation factors method and

gradient method are well known for the economic

thermal dispatch of generators. All the methods as well

known for the economic dispatch problem as a convex

optimization problem and it assumes the whole of the

unit operating limit (Pmin) and the maximum generation

limit (Pmax) is available for operation. This paper work

deals with the economic scheduling problem namely the

economic thermal power dispatch. In seeking the

solution for the economic dispatch problem (EDP) the

main aim is to operate a power system in such a way to

supply all the loads at the minimum cost. The solution

technique which is applied to the economic load

dispatch is Firefly Algorithm.The ultimate goal of power

plants is to meet the required load demand with the

lowest operating costs possible while taking into

consideration practical equality and inequality

constraints algorithms. Optimal operation of electric

power system networks is a challenging real-world

engineering problem. Those linked optimization

problems are the unit commitment, optimal power flow,

and economic dispatch scheduling.

The recently introduced meta-heuristic methods are

the artificial bee colony (ABC) algorithm, adapted

simulated annealing algorithm (SA),[14] firefly

algorithm (FA). These are a population-based technique

and inspired by the intelligent foraging behavior of the

honeybee swarm, birds schooling, fish behavior. The

DED problem was one of the real-world optimization

problems that has benefited from the development of the

meta-heuristic algorithms. An evolutionary programming

(EP) technique [8] in and is adopted to solve the DED

problem. Simulating annealing method (SA), quantum

evolutionary algorithm (QEA), Particle Swarm

Optimization Algorithm (PSO) and Tabu search

approach (TS) have been also designated to solve the

DED problem in [12],[13], respectively.

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4th ICCCNT 2013 July 4 - 6, 2013, Tiruchengode, India

II. OBJECTIVE FUNCTION

The objective function of the DED problem is to

minimize the operating fuels costs of committed generating units to meet the load demand, subject to

equality and inequality constraints over a predetermined

dispatch period, the results practical usefulness will be degraded if the units valve-point effects are neglected. Consequently, there are two models to represent the

units valve-point effects in the literature. The first represents the units valve in terms of prohibited operating zones which are included as inequality

constraints. The second form represents the units valve-point effects as a rectified sinusoid term which is

superimposed on the approximate quadratic fuel cost

function. The general mathematical form of the DED

problem follows: [1]

A. Minimization of Fuel Cost

The problem of an Economic Load Dispatch (ELD)

is to find the optimal combination of power generation,

which minimizes the total fuel cost, under some

constraints [14]. The ELD Problem can be,

mathematically, formulated as the following

optimization problem:

PGi : the power generated by generator i (MW), and

n : the number of generators

B. Equality Constraint

Integration of a renewable source (modifies the

equality constraints function to be as follows

-------- (2)

Where pDt and pLt are the load demand and

systems loss at a time t respectively. The multiplier RS is set to a permissible amount of active power injected

by RS, PRS is the forecasted real power from RS

C. Inequality Constraint

The inequality constraints of the DED problem are

the units ramp-rate limits, i.e., upper rate (URi) and down rate (DRi), are considered as follows:

-------- (3)

Additional inequality constraints are the minimum and

maximum power output of each unit:

-------- (4)

Therefore, to incorporate the constraints of units ramp-rate limits in the real power output limit constraints. The

modified units real power outputs are evaluated as follows:

-------- (5)

Where Fcost : the total fuel cost ($/hr) ai,bi,ci : the fuel

cost coefficients of generator i

D. Problem Formulation

1) Economic Dispatch Problem The economic dispatch problem is defined as to

minimize

Fi = (aiPi2 + biPi+ci). Rs / hr.

n

Ft = (aiPi2 + biPi+ci). Rs / hr. i = 1 --------- (6)

Subject to

1. Pi,min Pi Pi,max

n

2. Pi = PD + PL --------- (7) i = 1 (The system demand constraint)

2) Minimum NOX Emission dispatch

Ei = di Pi2 + ei Pi + fi , kg / hr.

The minimum NOx emission dispatch problem is

defined as to minimize

n

Ei = (di Pi2 + ei Pi + fi), kg / hr. i = 1 --------- (8)

Subject to

1. The operating constraints - Pi,min Pi Pi,max

n

2. System demand constraint - Pi = PD + PL i = 1

Minimum NOx algorithm is similar to the minimum cost

algorithm.

------ (1)

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3) Combined Economic and Emission Dispatch

The Combined economic and emission dispatch

problem is defined as to minimize.

n n

i = Fi + h Ei , Rs / hr. --------- (9) i = 1 i = 1

n

i= (aiPi2 + biPi+ci)+h (diPi2 + eiPi + fi) i = 1 --------- (10)

Subject to

The operating constraints Pi,min Pi Pi,max n

System demand constraint Pi = PD + PL i = 1

Once the value of price penalty factor is known

the above equation can be rewritten in terms of known

coefficients and the unknown outputs of the generators.

n

i = [(ai + hdi) Pi2 + (bi + hei) Pi + (bi + hfi) Rs / hr. i = 1 --------- (11)

h - Price penalty factor

III. FIREFLY ALGORITHM

The development of FA is based on flashing behavior

of fireflies. There are about two thousand firefly species

where the flashes often unique for a particular species.

The flashing light is produced by a process of

bioluminescence where the exact functions of such

signaling systems are still on debating. Nevertheless,

two fundamental functions of such flashes are to attract

mating partners (communication) and to attract

potential. [9]

For simplicity, the following three ideal rules are

introduced in FA development. [12]

1) all fireflies are unisex so that one firefly will be attracted to other fireflies regardless of their

sex.

2) attractiveness is proportional to their brightness, thus for any two flashing fireflies,

the less brighter one will move towards the

brighter one.

3) the brightness of a firefly is affected by the landscape of the objective function.

For maximization problem, the brightness can

simply be proportional to the value of the objective or

fitness function.

A. Flow Chart of FIREFLY ALGORITHM

No

Yes

IV. PARTICLE SWARM OPTIMIZATION

PSO simulates the behaviours of bird flocking.

Suppose the following scenario: a group of birds are

randomly searching food in an area. There is only one

piece of food in the area being searched.[2] All the birds

do not know where the food is. But they know how far

the food is in each iteration. So what's the best strategy to

find the food? The effective one is to follow the bird,

which is nearest to the food. PSO learned from the

scenario and used it to solve the optimization problems.

In PSO, each single solution is a "bird" in the search

space. We call it "particle". All of particles have fitness

values, which are evaluated by the fitness function to be

optimized, and have velocities, which direct the flying of

the particles. The particles fly through the problem space

by following the current optimum particles.

PSO is initialized with a group of random

particles (solutions) and then searches for optima by

Start

Initialize location of fireflies

Insert variable x into load flow

Run load flow

Objective function evaluation

Ranking fireflies by their light

Find the current best solution

Run load flow program for final result

of optimal generation

Print results of the optimal dispatch, total

system loss and total generation cost

End

Move all fireflies to the better

locations

Iteration Maximum

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updating generations. In every iteration, each particle is

updated by following two "best" values. The first one is

the best solution (fitness) it has achieved so far. (The

fitness value is also stored.) This value is called pbest.

Another "best" value that is tracked by the particle

swarm optimizer is the best value, obtained so far by any

particle in the population. This best value is a global best

and called g-best. When a particle takes part of the

population as its topological neighbours, the best value is

a local best and is called p-best. After finding the two

best values, the particle updates its velocity and positions

with following equations .

Vi(u+1) = w * Vi(u) + C1 * rand( )* (pbesti - Pi(u)) +

C2 * rand( ) * (gbesti - Pi(u)) -------(12)

Pi(u+1) = Pi(u) + Vi(u+1) ------ (13)

In the above equation,

The term rand( )* (pbesti - Pi(u)) is called

particle memory influence.

The term rand( ) * (gbesti - Pi(u)) is called

swarm influence.

Vi(u) is the velocity of ith particle at iteration

u must lie in the range

Vmin Vi Vmax

The parameter Vmax determines the resolution, or fitness, with which regions are to be searched between the present position and the target position.

If Vmax is too high, particles may fly past good solutions. If Vmin is too small, particles may not explore sufficiently beyond local solutions.

In many experiences with PSO, Vmax was often set at 10-20% of the dynamic range on each dimension.

The constants C1and C2 pull each particle towards pbest and gbest positions.

Low values allow particles to roam far from the target regions before being tugged back. On the other hand, high values result in abrupt movement towards, or past, target regions.

The acceleration constants C1 and C2 are often set to be 2.0 according to past experiences.

Suitable selection of inertia weight provides a balance between global and local explorations, thus requiring less iteration on average to find a sufficiently optimal solution.

In general, the inertia weight w is set according to the

following equation,

------ (14)

Where w -is the inertia weighting factor

Wmax - maximum value of weighting factor

Wmin - minimum value of weighting factor

ITERmax - maximum number of iterations ITER - current number of iteration

A. Flow Chart of PARTICLE SWARM OPTIMIZATION

No

Yes

V. EXPIREMENTS AND RESULTS

The power system economic dispatch problem

based on the concept of Firefly algorithm method has

been tested on 3-generator system, 6-generator system

and 12-generator system.Multiple generator limits and

total operating cost of the system is simulated in order to

evaluate the correctness as well as accuracy of this

method

A. Three Unit System

The three generating units considered are

having different characteristic. Their cost function

characteristics are given by following equations

F1=0.00533P12+11.669P1+213 Rs/Hr

F2=0.00889P22+10.333P2+ 200 Rs/Hr

F3=0.00741P32+10.833P

3+240 Rs/Hr

Initialize particles with random

position and velocity

vectors

If g best is the

optimal solution

Update particle velocity and

Position

Set best of pbest as g best

If fitness (p) is better than fitness o

(pbest) then pbest=p

For each particle position (p)

evaluate the fitness

End

Start

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According to the constraints considered in this

work among inequality constraints only active power

constraints are constraints are considered. There

operating limit of maximum and minimum power are

also different. The unit operating ranges are:

50 MW P1 200 MW 7.5 MW P2 150 MW 45 MW P3 180 MW

The transmission line losses can be calculated by

knowing the loss coefficient. The Bmn loss coefficient

matrix is given by

0.0218 0.0017 0.0028

Bmn = 0.0093 0.0228 0.0017

0.0028 0.0093 0.0179

This is the example we have taken for the

testing this novel coding scheme on the three unit

system.

Here C1 = 1.99 and C2 = 1.99 Here the

maximum value of w is chosen 0.9 and minimum value

is chosen 0.4.the velocity limits are selected as Vmax=

0.5*Pmax and the minimum velocity is selected as Vmin=

-0.5*Pmin. There are 10 no of particles selected in the

population. For different value of C1 and C2 the cost

curve converges in the different region. So, the best

value is taken for the minimum cost of the problem.

1) Simulation Results - Solution of ED Problem

The solution for ED problem of the three unit

system considered here is given below. The total

demand taken here is about 200 MW&700MW

out = 1.0e+003 *

0.050000005801763

0.079584818649999

0.074549690507549

2.977211011227124

P = 50.000005801763166

79.584818649999349

74.549690507549073

F = 2.977211011227124e+003

Table 1: Fuel cost for different Techniques (PD= 700 MW)

Objective Economic

dispatch

Emission

dispatch

Combined

economic and

emission dispatch

FA 20123.78 22234.45 32521.09

ABC 22693.16 23764.62 33987.90

PSO 23456.32 24038.33 34760.11

B. Six Unit System

The six unit generating units considered are


characteristics are given by following equations.

Table 2: Fuel Cost Coefficient For Six Unit

System (PD=700mw)

Fuel

Cost

function

a(p2) b(p) c

(Rs/hr)

pmin (MW)

pmax (MW)

F1 0.1524 38.539 756 10 125

F2 0.1058 46.159 451 10 150

F3 0.0280 40.396 1049 35 225

F4 0.0354 38.305 1243. 35 210

F5 0.0211 36.327 1658 130 325

F6 0.0179 38.270 1356 125 315

Table 3: Total Generation using different Techniques

(PD= 700 MW)

OBJE

CTIV

E

P1 P2 P3 P4 P5 P6

Econo

mic

Dispa

tch

41.37

485

19.56

447

122.0

292

95.87

92

230.2

597

215.0

321

Emiss

ion

Dispa

tch

86.20

00

84.83

09

110.9

136

111.0

101

161.7

126

162.6

996

Comb

ined

Econo

mic

67.02

26

64.07

49

115.9

380

119.4

393

174.9

152

177.2

328

Table 4: Fuel cost for different Techniques (PD= 700 MW)

Objective Economic

dispatch

Emission

dispatch

Combined economic

and emission dispatch

FA 39299.56 40135.56 6123.45

ABC 40103.23 41533.65 6476.53

PSO 40392.66 42998.72 66239.72

C. Ten Unit System

The ten unit generating units considered are


characteristics are given by following equations.

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Table 5: Fuel Cost Coefficient for 12 Unit

System(PD=1200MW)

Fuel

cost a(p2) b(p)

C

(Rs/

hr)

pmin (MW)

pmax (MW)

F1 0.0051 2.2034 15 12 73

F2 0.0040 1.9104 25 26 93

F3 0.0039 1.8518 40 42 143

F4 0.0038 1.6966 32 18 700

F5 0.0021 1.8015 29 30 93

F6 0.0026 1.5354 72 100 350

F7 0.0029 1.2643 49 100 248

F8 0.0015 1.2130 82 40 190

F9 0.0013 1.1954 105 70 590

F10 0.0014 1.1285 100 40 113

F11 0.0013 1.1954 105 70 190

F12 0.0014 1.1285 100 40 113

Table 6: Total Generation using different Techniques

(PD= 1200 MW)

Algorithm FA ABC PSO

P1 12 15 10

P2 26 16 19.30

P3 42 53 20

P4 42.34 45.64 50

P5 51.64 70.12 53.6

P6 100 120 70

P7 130 110 95

P8 190 175 95

P9 190 192 222

P10 113 103 175

P11 190 175 200

P12 113 125 190

Total

power 1199.98 1199.76

1199.99

Fuel cost 2.61E+003 2.7537

E+003

2.652E+003

Table 4: Fuel cost for different Techniques

(PD= 300 MW,700 MW and 1200 MW)

Uni

t

Power

Demand

(mw)

Algorithm Fuel cost

(RS/HR)

Total

Power

(MW)

3 300

FA 1.32 E+003 329

ABC 1.47 E+003 327

PSO 1.57 E+003 325

6 700

FA 3.215E+003 725.8721

ABC 3.5623 E+003 723.33

PSO 3.691E+003 719.4354

12 1200

FA 2.61E+003 1199.98

ABC 2.7537 E+003 1199.76

PSO 2.652E+003 1199.98

VI. SUMMARY AND CONCLUSION

An accurate solution is available on comparison

with other technique results. The validity of the proposed

method is demonstrated with the help of three standard

test systems. The Emission Constrained Economic

Dispatch (ECED) provides lower cost in emission. This

property is revealed for various load conditions.

The computational results of table (IV) consist

of twelve generator test system. It shows that FA fuel

cost function is less than ABC and PSO. The claims of

some of the recent reports provide near optimal solution

for large computationally intensive problem. Many of the

hybrid algorithms only help to improve the solution

accuracy. But this algorithm is very well defined and the

solution accuracy is excellent and converges to near

global minimum with less search account. It is high

efficiency than other methods. Thus, it obtains the

solution with high accuracy. An Firefly algorithm has

been developed for the economic thermal power dispatch

problem of electric generating units. The algorithm was

implemented and tested on 3-generator test system, 6-

generator test system and 12- generator test system.

Firefly Algorithm is an efficient tool for the economic

scheduling for generating units with the given generator

constraints and other data. The quality of the solution

shows that an improved Firefly Algorithm Search offers

a promising viable approach for solves the economic

thermal power dispatch problem.

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[19] Economic Load Dispatch Using Firefly Algorithm K.

Sudhakara Reddy1, Dr. M. Damodar Reddy2 1-(M.tech

Student, Department of EEE, SV University, Tirupati 2- (Associate professor, Department ofEEE, SV University,

Tirupati) July-August 2012

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solution of firefly algorithm for the economic thermal power dispatch with emission constraint in...

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