Artificial Bee Colony Optimization For Combined ...

Chennai and Dr.MGR University Second International Conference on Sustainable Energy and Intelligent System (SEISCON 2011), Dr. M.G.R. University, Maduravoyal, Chennai, Tamil Nadu, India. July. 20-22, 2011.

Artificial Bee Colony Optimization For Combined Economic and Emission Dispatch Gaurav Prasad Dixit Department of Electrical Engineering Madhav Institute of Technology & Science Gwalior (M.P), India- 474005 E-mail:[email protected]

Hari Mohan Dubey Department of Electrical Engineering Madhav Institute of Technology & Science Gwalior (M.P), India- 474005 E-mail: [email protected]

Manjaree Pandit Department of Electrical Engineering Madhav Institute of Technology & Science Gwalior (M.P), India- 474005 E-mail:[email protected]

B. K. Panigrahi Department of Electrical Engineering Indian Institute of Technology Delhi, India-110016 E-mail: bkpanigrahi @ ee.iitd.ac.in

Keywords: Economic dispatch; Emission dispatch; Combined economic and emission dispatch; Mathematical modelling

Abstract This paper presents an application of artificial bee colony (ABC) algorithm for multi-objective optimization problem in power system. Considering the environmental impacts that grow from the emissions produced by fossil-fuelled power plant, the economic dispatch that minimizes only the total fuel cost can no longer be considered as single objective. Artificial bee colony algorithm strategy based on mathematical modeling to solve economic, emission and combined economic and emissions dispatch problems by a single equivalent objective function. Artificial bee colony algorithm has been applied to two realistic systems at different load condition. Results obtained with proposed method are compared with other techniques presented in literature. ABC algorithm is easy to implement and capable of searching near global optimum solution at fast convergence and efficiency.

1 Introduction

objective economic dispatch can no longer be considered along due to the environmental concerns that arise from the emission produced by fossil-fuelled electric power plants. Economic and environmental dispatch is a multi-objective problem. Various Multi-objective evolutionary algorithms have been applied to the economic dispatch problem [11-14] based on mathematical approximations have been developed, which directly give the solution faster. In [15-16] including emission constrains to the economic dispatch and unit commitment problems have been presented, under costminimization environment. In this paper a multi-objective optimization problem i.e. ABC algorithm inspired by foraging behavior of honey bees proposed to solve combined economic and emissions dispatch problems is presented and the effectiveness of proposed algorithm is demonstrated using three and six generating unit test systems. Result shows that the proposed methodology can reliably handles complex multi-objective optimization problem in robust and efficient manner.

2 Combined environmental economic dispatch

This paper introduce the economic dispatch problem in a power system is to determine the optimal combination of power output for all generating units which will minimize the total fuel cost while satisfying load and operational constraints. The economic dispatch problem is very complex to solve because of its colossal dimension, a non-linear objective function, and a large number of constraints.

The traditional economic dispatch problem has been defined as minimizing of an objective function i.e. the generation cost function subject to equality constraints (total power generated should be equal to total system load plus losses for all solutions) and inequality constraints (generations should lie between their respective maximum and minimum specified values)

Well known long-established techniques such as integer programming [2], dynamic programming [3], and lagarangian relaxation [4] have been used to solve the economic dispatch problem. Recently other optimization methods such as Simulated Annealing [5], Genetic Algorithm [6], Particle Swarm optimization [9], and Tabu Search Algorithm [10] are presented to solve the economic dispatch problem. This single

Minimize Φ ( x, p )

n

Φ t ( Pi ) = ∑ Φ i ( Pi ) ; i =1

(1)

Objective function n

Subject to g(x, p)

∑P − P i

L

− PD = 0 ;

i =1

Equality constraint

(2)


H ( x, P ) ≤ 0

Pi min ≤ Pi ≤ Pi max

; (3)

Inequality constraint

Where x is a state variable, Pi is the control variable, i.e., real power setting of i th generator and n is the number of units or generators. There are several ways to include emission into the problem of economic dispatch. Reference [21] summarizes the various algorithms for solving environmental dispatch problem with different constraints. One approach is to include the reduction of emission as an objective. In this work, only NOx reduction is considered because it is a significant issue at the global level. A price penalty factor (h) is used in the objective function to combine the fuel cost, Rs/hr and emission functions, kg/hr of quadric form. The combined economic and emission dispatch problem can be formulated as to minimize n

n

ϕt = ∑ Fi ( Pi ) + h∑ Ei ( Pi ) Rs/h i =1

n

ϕi = ∑(ai Pi 2 + bi Pi + ci ) + h∑(di Pi 2 + ei Pi + f i )Rs / h i =1

i =1

(5) Subject to equality and inequality constraint defined by equations (2), (3). Once price penalty factor (h) is known, equation (5) can be rewritten as n

ϕ i = ∑ {(a i + hd i ) Pi 2 + (bi + hei ) Pi + (ci + hf i )}Rs / h i =1

(6) This has the resemblance of the familiar fuel cost equation, once h is determined. A practical way of determining h is discussed by Palanichamy and Srikrishan [7]. Consider that the system is operating with a load of PD MW, it is necessary to evaluate the maximum cost of each generator at its maximum output, i.e. (i) Evaluate the maximum cost of each generator at its maximum output, ie,

(

)

Fi ( Pi max ) = ai Pi 2max + bi Pi max + ci Rs/hr

(7)

(ii) Evaluate the maximum NOx emission of each generator at its maximum output, ie,

(

)

E i ( Pi max ) = d i Pi 2max + e i Pi max + f i kg/hr

(8)

(iii) Divide the maximum cost of each generator by its maximum NOx emission, i.e.,

( (

) )

ai Pi 2max + bi Pi max + ci Fi ( Pi max ) Rs/kg = Ei ( Pi max ) d i Pi 2max + ei Pi max + ci Recalling that

(10)

(iv) Arrange hi ( I = 1,2,......, n) in ascending order. (v) Add the maximum capacity of each unit, (Pi max ) one at a time, starting from the smallest hi unit until total demand is met as shown below. n

∑P

i max

≥ PD

(11)

i =1

(vi) At this stage, hi associated with the last unit in the process is the price penalty factor h Rs/Kg for the given load. Arrange hi in ascending order. Let ‘h’ be a vector having ‘h’ values in ascending order. (12) h = h1, h2 , h3 ,......, hn

[

]

(4)

i =1

n

Fi ( Pi max ) = hi Rs/kg Ei ( Pi max )

(9)

For a load of PD starting from the lowest hi value unit, maximum capacity of unit is added one by one and when this total equals or exceeds the load, hi associated with the last unit in the process is the price penalty factor for the given PD. Then equation (6) can be solved to obtain environmental economic dispatch using lamda iteration method [1].

3 Overview of artificial bee colony Artificial Bee Colony (ABC) is a swarm based a stochastic search algorithm which imitates the scrounging behaviour of honeybees. In ABC algorithm [17-20], the colony of artificial bees consists of three groups of bees: employed bees, onlookers and scouts. Some of the bee of colony consists of employed artificial bees and the some includes the on lookers. In other words, the number of employed bees is equal to the number of food sources around the hive. The employed bee whose food source has been abandoned becomes a scout. In ABC algorithm the position of food source determines the solution and the amount of nectar represents the fitness of the respective solution.

3.1 Step in algorithm Step1: Initialization A randomly distributed initial population solutions (Xi = 1, 2... D) is being dispersed over the D dimensional problem space. Step 2: Reproduction An artificial onlooker bee chooses a food source depending on probability value associated with food source, Pi, calculated by following expression:

Pi = f i t i

(13)

N

∑ n =1

fitn


Where fiti is the fitness value of the solution i which is proportional to the nectar amount of the food source in the position i and N is the number of food sources which is equal to the number of employed bees. In order to produce a candidate food position from the old one in memory, the ABC used the following expression: (14) Vij = X Kj + ϕ ij ( X ij − X Kj ) Where k = {1, 2… D} and j = {1, 2… N} are randomly chosen indexes. ϕ ij , is a random number between [-1, 1]

The applicability and validity of the ABC algorithm for practical applications has been tested on two test cases. The programs are developed using MATLAB 7.9 and the system configuration is Pentium IV processor with 3.2 GHz speed and 1 GB RAM. The Parameter for ABC algorithm considered here are: N=20; Food-Number=N/2; and limit=500. Test case 1: The system consists of three thermal units. The parameters of all thermal units are presented in table: 1 and table: 2.

Step 3: Replacement of bee and selection In ABC, Providing that a position can not be improved further through a predetermined number of cycles, then that food source is assumed to be abandoned. The value of predetermined number of cycles is an important control parameter of the ABC algorithm, which is called “limit” for abandonment. Assume that the abandoned source is Xi and j= {1, 2… N} then the scout discovers a new food source to be replaced with Xi. This operation can be defined as: j j j X i j = X min + rand (0,1) * ( X max − X min )

4 Simulation result and discussions

Table: 1 Fuel cost coefficient unit

Fuel cost coefficient ai

bi

PGmin (MW)

PGmax (MW)

ci

G1

0.03546

38.30553

1243.53110

35

210

G2

0.02111

36.32782

1658.56960

130

325

G3

0.01799

38.27041

1356.65920

125

315

Table: 2 NOX emission equation in kg/h are given below in (15)

After each candidate source position Vij is produced and then evaluated by the artificial bee, its performance is compared with that of its old one. If the new food has equal or better fitness than the old source, it is replaced with the old one in the memory. Otherwise, the old one is retained in the memory.

unit

Emission coefficient

3.2 Pseudo-code

The Bmn loss coefficient matrix is

di

ei

PGmin (MW) fi

PGmax (MW)

G1

0.00683

– 0.5455

40.26690

35

210

G2

0.00461

– 0.5116

42.89553

130

325

G3

0.00461

– 0.5116

42.89553

125

315

B mn =

0.0000 70 0.000030 0.000025

0.000025 0.000069 0.000032

0.000030 0.000032 0.000080

Table: 3 Comparison of test results for Three Generating units system


Form Table: 3, it is clear that ABC algorithm gives optimum result in terms of minimum fuel cost, emission level and the total operating cost. Table: 4 gives the best optimum power output of generators for CEED problem using ABC algorithm for load demand 400MW, 500MW and 700MW. Table: 4 Optimum Power dispatch Results by Proposed method for three units system PD MW

P1 MW

P2 MW

P3 MW

Escape Time (second)

400

102.5422

153.7333

151.1370

1.895377

500

128.8241

192.5837

190.2859

2.340603

700

182.6011

271.2813

269.4840

3.770030

4

6.8

Total operating Cost

Table: 3 shows the summarized result of CEED problem for load demand of 400MW, 500MW and 700MW are obtained by the proposed ABC algorithm with stopping criteria based on maximum-generation=100.

x 10

6.78

PD=700MW

6.76 6.74 6.72 6.7 6.68 6.66

0

10

20

30

40

50

60

70

80

90

100

genetation

Figure: 3 convergence of Three generating units system for PD=700MW Test case II: The system consists of six thermal units. The parameters of all thermal units are presented in table: 5 and table: 6, the summarized result of CEED problem for load demand of 500MW and 900MW are obtained by the proposed ABC algorithm with stopping criteria based on maximumgeneration=200 is presented in Table: 7. Table: 5 Fuel cost coefficient unit

The convergence tendency of proposed ABC based strategy for power demand of 400MW, 500MW and 700 MW is plotted in figure: 1, figure: 2 and figure: 3. it shows that the technique converges in relatively fewer cycles thereby possessing good convergence property and resulting in low operating cost.

Fuel cost coefficient ci

PGmin

PGmax

(MW)

(MW)

ai

bi

G1

0.15247

38.53973

756.79886

10

125

G2

0.10587

46.15916

451.32513

20

150

G3

0.02803

40.39655

1049.99770

35

225

G4

0.03546

38.30552

1243.53110

35

210

G5

0.02111

36.32782

1658.55960

130

325

G6

0.01799

38.27041

1356.65920

125

325

4


3.06

x 10

Table: 6 NOX emission equation in kg/h are given in

3.05

PD=400MW

3.04

unit

3.03

Emission coefficient fi

PGmin

PGmax

(MW)

(MW)

di

ei

G1

0.00419

0.32767

13.85932

10

125

G2

0.00419

0.32767

13.85932

20

150

G3

0.00683

– 0.54551

40.26690

35

225

G4

0.00683

– 0.54551

40.26690

35

210

G5

0.00461

– 0.51116

42.89553

130

325

G6

0.00461

– 0.51116

42.89553

125

325

3.02 3.01 3 2.99 2.98

0

10

20

30

40

50

60

70

80

90

100

genetation

Figure: 1 convergence of Three generating units system for PD=400MW

4


4.05

x 10

PD=500 MW 4

Bmn loss coefficient matrix in the order of 10-4 is given as:

3.95

3.9

0

10

20

30

40

50

60

70

80

90

100

genetation

Figure: 2 convergence of Three generating units system for PD=500MW


Form Table: 7, it is clear that ABC algorithm gives the optimum result in terms of minimum fuel cost, emission level and the total operating cost.

Table: 8 Optimum Power dispatch results by ABC Approach for six unit system

Table: 8 gives the best optimum power output of generators for CEED problem using ABC algorithm for load demand 500MW and 900MW.

Table: 7 comparison of test Results for six generating unit system PD, MW

Performance

h Rs/kg

Conventional Method [10]

RGA [10]

Hybrid GA[10]

Hybrid GTA[10]

Proposed ABC

Fuel cost, Rs/hr

27638.300

27692.1

27695

27613.4

27613

Emission , kg/hr

262.454

263.472

263.37

263.000

263.012

8.830

10.172

10.135

8.930

8.9343

Total cost, Rs/hr

39159.500

39258.100

39257.5

39158.9

39156.9

Fuel cost, Rs/hr

48892.900

48567.7

48567.5

48360.9

47045.3

Emission , kg/hr

701.428

694.169

694.172

693.570

693.791

PL, MW

35.230

29.725

29.718

28.004

28.0087

Total cost, Rs/hr

82436.580

81764.5

81764.4

81529.100

81527.6

Computational time for ABC

4.111050

500 43.898

PL, MW

3.948641

900 47.822

The convergence tendency of proposed ABC based strategy for power demand of 500MW and 900 MW is plotted in figure: 4 and figure: 5. it shows that the technique converges in relatively fewer cycles thereby possessing good convergence property and resulting in low operating cost. 5


4

x 10

3.5

PD=500MW

3 2.5 2 1.5 1 0.5 0

0

20

40

60

80

100

120

140

160

180

200

genetation

Figure: 4 convergence of six generating unit system for PD=500MW

5 Conclusion In this paper, a new optimization of artificial bee colony (ABC) algorithm has been proposed. In order to prove the effectiveness of algorithm it is applied to CEED problem with three and six generating unit. The results obtained by proposed method were compared to those obtained conventional method, RGA and SGA and Hybrid GA. The comparison shows that ABC algorithm performs better then above mentioned methods. The ABC algorithm has superior features, including quality of solution, stable convergence characteristics and good computational efficiency. Therefore, this results shows that ABC optimization is a promising technique for solving complicated problems in power system

4


8.45

x 10

Acknowledgements

PD=900MW

8.4

The authors are thankful to Director, Madhav Institute of Technology & Science, Gwalior (M.P) India for support and facilities to carry out this research work

8.35 8.3 8.25 8.2 8.15

0

20

40

60

80

100

120

140

160

180

200

References

genetation

Figure: 5 convergence of six generating unit system for PD=900MW

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