Economic Dispatch Problem using Bat Algorithm

 

Yamina Ahlem GHERBI 1*, Hamid BOUZEBOUDJA 1 and Fatiha LAKDJA2

 

1 Electro technical Department, Faculty of Electrical Engineering, USTO, Oran, Algeria

2 Intelligent Control and Electrical Power System Laboratory, Djillali Liabès University of Sidi-Bel-Abbes, Algeria

E-mails: aygherbi@yahoo.fr (YAG); hbouzeboudja@yahoo.fr (HB); flakdja@yahoo.fr (FL)

* Corresponding author: Fax: 213 48 61 03 22

 

 

Abstract

A BAT algorithm to solve the economic dispatch problem is presented in this paper. The modern power system has become very complex in nature with consumption of the electric power which is in rise progression and this tendency increases with the industrialization and the growth of the population. Moreover, one loss of alimentation can cause the interruption of the various production processes, and in front of consumers who become increasingly demanding by wanting more energy and better quality, the companies of production of electrical energy must thus ensure the regular provisioning of this request, and without interruption, but these requirements compromise the cost of generation. Application of BAT algorithm in this paper is based on mathematical modeling to solve economic dispatch problems by a single equivalent objective function. The algorithm is tested on 6-unit system. BAT algorithm is easy to implement and better than other algorithms in terms of accuracy and efficiency.

Keywords

Bat Algorithm; Economic Dispatch; Optimization Methods; Metaheuristic.

 

 

Introduction

 

Economic dispatch (ED) is the operation of generation to produce energy at the lowest cost by fulfilling the demand within several limits. This is not an easy task since there are a lot of factors need to be considered especially in the large interconnected power systems. The primary objective of ED is to schedule the committed generating units output so as to meet the required load demand at minimum cost satisfying all unit and system operational constraints [1].

Various conventional methods like Bundle method, nonlinear programming, mixed integer linear programming, dynamic programming, quadratic programming, Lagrange relaxation method, network flow method, direct search method reported in the literature are used to solve such problems [1].

Practically, ED problem is nonlinear, non-convex type with multiple local optimal points due to the inclusion of valve point loading effect, multiple fuel options with diverse equality and inequality constraints.

Conventional methods have failed to solve such problems as they are sensitive to initial estimates and converge into local optimal solution and computational complexity [1].

Biology-inspired metaheuristic algorithms have recently become the forefront of the current research as an efficient way to deal with many NP-hard combinatorial optimization problems and non-linear optimization constrained problems in general [2]. These algorithms are based on a particular successful mechanism of a biological phenomenon of Mother Nature in order to achieve optimization, such as the family of honey-bee algorithms, where the finding of an optimal solution is based on the foraging and storing the maximum amount of flowers’ nectar [2]. A new algorithm that belongs in this category of the so-called nature inspired algorithms is the bat algorithm which is based on the echolocation behavior of bats [2].

In this research paper we present how the BAT algorithm can be used to solve the economic dispatch optimization problem. The effectiveness of proposed algorithm is demonstrated using six generating unit test system.

 

Formulation of the Economic Load Dispatch Problem

 

Economic Dispatch

The objective of economic load dispatch of electric power generation is to schedule the committed generating unit outputs so as to meet the load demand at minimum operating cost while satisfying all units and operational constraints of the power system. The economic dispatch problem is a constrained optimization problem and it can be mathematically expressed as follows:

(1)

where FT: total generation cost (Rs/hr) n: number of generators Pn: real power generation of nth generator (MW) Fn(Pn): generation cost for Pn Subject to a number of power systems network equality and inequality constraints [4]. These constraints include:

 

System Active Power Balance

For power balance, an equality constraint should be satisfied. The total power generated should be the same as total load demand plus the total line

(2)

where PD: total system demand (MW) PL: transmission loss of the system (MW)[4].

 

Generation Limits

Generation output of each generator should be laid between maximum and minimum limits. The corresponding inequality constraints for each generator are

Pn,min ≤ Pn ≤ Pn,max

(3)

where Pn,min: minimum power output limit of nth generator (MW) Pn,max: maximum power output limit of nth generator (MW) The generation cost function Fn(Pn) is usually expressed as a quadratic polynomial:

(4)

where an, bn and cn are fuel cost coefficients [4].

 

Network Losses

Since the power stations are usually spread out geographically, the transmission network losses must be taken into account to achieve true economic dispatch. Network loss is a function of unit generation. To calculate network losses, two methods are in general used.

One is the penalty factors method and the other is the B coefficients method. The latter is commonly used by the power utility industry. In the B coefficients method, network losses are expressed as a quadratic function:

(5)

where Bmn constants are called B coefficients or loss coefficients [4].

 

Bat Algorithm

Bats are fascinating animals. They are the only mammals with wings and they also have advanced capability of echolocation [7]. Most of bats use echolocation to a certain degree; among all the species, microbats are famous example as microbats use echolocation extensively, while megabats do not [7]. Microbats use a type of sonar, called echolocation, to detect prey, avoid obstacles, and locate their roosting crevices in the dark. If we idealize some of the echolocation characteristics of microbats, various bat-inspired algorithms or bat algorithms can be developed. For simplicity, in our approach, the following approximate or idealized rules were used:

·        All bats use echolocation to sense distance, and they also know the difference between food/prey and background barriers.

·        Bats fly randomly with velocity vi at position xi with a fixed frequency fmin (or wavelength λ), varying wavelength λ (or frequency f) and loudness A0 to search for prey. They can automatically adjust the wavelength (or frequency) of their emitted pulses and adjust the rate of pulse emission rÌ[0,1], depending on the proximity of their targets;

·        Although the loudness can vary in many ways, we assume that the loudness varies from a large (positive) A0 to a minimum value Amin.

Another obvious simplification is that no ray tracing is used in estimating the time delay and three dimensional topographies. In addition to these simplified assumptions, the following approximations have been used, for simplicity. In general the frequency f in a range [fmin, fmax] corresponds to a range of wavelengths [λmin, λmax]. For example, a frequency range of [20 kHz, 500 kHz] corresponds to a range of wavelengths from 0.7 mm to 17 mm.

In simulations, we use virtual bats naturally. We have to define the rules how their positions xi and velocities vi in a d-dimensional search space are updated. The new solutions xi0  and velocities vi0 at time step t are given by:

(6)

(7)

where βÌ[0,1] is a random vector drawn from a uniform distribution. Here x0 is the current global best location (solution) which is located after comparing all the solutions among all the n bats. As the product λifi is the velocity increment, we can use either fi (or λi) to adjust the velocity change while fixing the other factor λi (or fi), depending on the type of the problem of interest. For the local search part, once a solution is selected among the current best solutions, a new solution for each bat is generated locally using random walk:

(8)

where EÌ[0,1] is a random number, while = At = < Atj > is the average loudness of all the bats at this time step.

Based on the above approximations and idealization, the pseudo-code of the Bat Algorithm (BA) can be summarized below.

The loudness usually decreases once a bat found its prey and rate of pulse emission increases. In our experiments, the loudness and pulse emission rate are varied once a solution is improved. The bat is moving towards optimal solution according to:

(9)

where α and γ are constants [7]. Figure 1 represents flowchart of the Bat Algorithm.

 

Results and Discussion

 

This paper presents the development of Bat Algorithm for solving an economic dispatch problem with six units. The programs are written in MATLAB software package.

The generator cost coefficients; generation limits and B-coefficient matrix of six unit system are given below. Economic Load Dispatch solution for six unit system is solved using BAT algorithm.

The cost coefficients and power limits of our 6 unit system are given in Table1.

 

Table 1.Cost coefficients and power limits of 6 unit system.

Unit

an

bn

cn

Pn,min

Pn,max

1

756.79886

38.53

0.15240

10

125

2

451.32513

46.15916

0.10587

10

150

3

1049.9977

40.39655

0.02803

35

225

4

1243.5311

38.30553

0.03546

35

210

5

1658.5696

36.32782

0.02111

130

325

6

1356.6592

38.27041

0.01799

125

315

 

Figure 1. Flowchart of the Bat Algorithm

 

The loss co-efficient matrix of 6-Unit system:

(9)

 

Table 2 shows the powers generated with the method of bat algorithm by varying demand.

 

Table 2.Test results of BAT algorithm for 6-Unit System: Power dispatch (MW)

Power

demand

(MW)

BA

P1

P2

P3

P4

P5

P6

600

23.8587

10.0000

95.6376

100.7030

202.8428

181.1953

650

26.0690

10.0000

107.2678

109.6703

216.7651

196.9558

700

28.2911

10.0000

118.9568

118.6753

230.7668

212.7419

750

30.4787

11.2272

130.4517

127.5111

244.4575

228.1844

800

32.5877

14.4827

141.5491

136.0426

257.6603

243.0085

850

34.7084

17.7685

152.7123

144.6200

270.8849

257.8618

900

36.8470

21.0780

163.9343

153.2216

284.1710

272.7362

950

39.0000

24.4149

175.2179

161.8868

297.4793

287.6303

 

In table 2 is seen that the results obtained by the method of Bat algorithm are satisfactory.

To verify the effectiveness of our method (BA), we fixed power demand at 700MW and the results obtained by bat algorithm are compared with other methods such as Genetic Algorithm (GA) [8], PSO [9] and Simulated Annealing (SA) [10], and the results of this comparison are shown in Table 3.

 

Table 3.Comparison results between BA, GA, PSO and SA

 

AG[8]

PSO[9]

SA[10]

BA

P1

21.4220

29.2400

28.2942

28.2911

P2

23.6700

12.5700

10.0001

10.0000

P3

118.0564

118.9667

118.9561

118.9568

P4

118.3270

118.3270

118.6795

118.6753

P5

230.5502

230.7553

230.7623

230.7668

P6

230.7945

212.7338

212.7395

212.7419

The results of comparison of table 3 are represented by the graph of the Figure 2.

Figure 2. Graph showing the result of comparison between BA, GA, PSO and SA

 

The result of comparison between BA, GA, PSO and SA obtained from Table 2 shows that the method bat algorithm gives better results.

Additionally, the advantages of BAT algorithm are that BAT algorithm is easier to implement and there are fewer parameters to adjust.

 

 

Conclusions

 

The results obtained by applying the proposed algorithm were compared to those obtained by AG, PSO and SA. The BAT algorithm has superior features, including quality of solution, stable convergence characteristics and good computational efficiency. The comparison shows that BAT algorithm performs better than the mentioned methods.

Therefore, BAT optimization is a promising technique for solving complicated problems in power system.

 

 

References

 

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