Economic Dispatch Problem using Bat Algorithm
^{1 }Electro technical Department, Faculty of Electrical Engineering, USTO, Oran, Algeria
^{2 }Intelligent Control and Electrical Power System Laboratory, Djillali Liabès University of SidiBelAbbes, Algeria
Emails: 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 6unit 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, nonconvex 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].
Biologyinspired metaheuristic algorithms have recently become the forefront of the current research as an efficient way to deal with many NPhard combinatorial optimization problems and nonlinear 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 honeybee 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 socalled 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 F_{T}: total generation cost (Rs/hr) n: number of generators P_{n}: real power generation of n^{th} generator (MW) F_{n}(P_{n}): generation cost for P_{n} 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 P_{D}: total system demand (MW) P_{L}: 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
P_{n,min }≤ P_{n} ≤ P_{n,max} 
(3) 
where P_{n,min}: minimum power output limit of nth generator (MW) P_{n,max}: maximum power output limit of nth generator (MW) The generation cost function F_{n}(P_{n})_{ }is usually expressed as a quadratic polynomial:
(4) 
where a_{n}, b_{n} and c_{n} 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 B_{mn }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 batinspired 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 v_{i} at position x_{i} with a fixed frequency f_{min} (or wavelength λ), varying wavelength λ (or frequency f) and loudness A_{0} 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) A_{0} to a minimum value A_{min.}
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 [f_{min}, f_{max}] corresponds to a range of wavelengths [λ_{min}, λ_{max}]. 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 x_{i} and velocities v_{i} in a ddimensional search space are updated. The new solutions x_{i}^{0} and velocities v_{i}^{0} at time step t are given by:
(6) 

(7) 
where βÌ[0,1] is a random vector drawn from a uniform distribution. Here x_{0} is the current global best location (solution) which is located after comparing all the solutions among all the n bats. As the product λ_{i}f_{i} is the velocity increment, we can use either f_{i} (or λ_{i}) to adjust the velocity change while fixing the other factor λ_{i} (or f_{i}), 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 = A^{t }= < A^{t}_{j} > is the average loudness of all the bats at this time step.
Based on the above approximations and idealization, the pseudocode 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 Bcoefficient 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 
a_{n} 
b_{n} 
c_{n} 
P_{n,min} 
P_{n,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 coefficient matrix of 6Unit 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 6Unit 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.
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