A Meta-Heuristic Approach of Bat Algorithm to Evaluate the Combined EED Problem

A Meta-Heuristic Approach of Bat Algorithm to Evaluate the Combined EED Problem

1 M. SaiCharan Reddy,

PG Student (M.Tech Electrical Power Systems), VBIT Engineering College, Hyderabad,India,

2Dr. P. S. Subramanyam.

Dept. of EEE, VBIT Engineering College, Hyderabad, India,

Abstract: Electrical Power Systems are designed and operated to meet continuous variation of power demand. In power system, minimization of operation cost is very important. This paper presents an application of BAT 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. Application of BAT algorithm in this paper is based on mathematical modelling to solve economic, emission and combined economic and emissions dispatch problems by a single equivalent objective function. BAT 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. BAT algorithm is easy to implement and much superior to other algorithms in terms of accuracy and efficiency.

Index Terms: Economic dispatch, BAT algorithm, Artificial Bee Colony algorithm, Combined economic and emission dispatch, Mathematical modelling.

1 INTRODUCTION

Economic load dispatch can be defined as process of allocating generation levels to generating units so that the system load is supplied entirely and most economically. This paper introduces the economic dispatch problem in a power system 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.

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 [7] and Tabu Search Algorithm [8] are presented to solve the economic dispatch problem. This single 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.

Recently, various modern heuristics multi- objective evolutionary algorithms such as Non-dominated Sorting Genetic Algorithm- II (NSGA-II) [9], Evolutionary Programming algorithm (EP) [10], Strength Pareto Evolutionary Algorithm (SPEA) [11] and Multi-Objective Particle Swam Optimization algorithm (MOPSO) [12] may prove to be efficient in solving EED problem by tackling both two objectives of EED problem simultaneously as competing objectives. But all these methods seem to be lack of ability to find the Pareto- optimal front due to their drawbacks: NSGA-II and SPEA may obtain only near Pareto-optimal front with long simulation time when applied to solve EED problem because of the premature convergence of Genetic Algorithm (GA). EP suffers from the oscillation of the solution and computational time may be too long when applying EP to solve EED problem. The premature convergence of PSO may lead optimization progresses of MOPSO methods to the local Pareto-optimum front, which would degrade their performance in solving EED problem. Including emission constrains to the economic dispatch and unit commitment problems have been analysed under cost- minimization environment.

In this paper a multi-objective optimization problem i.e., BAT algorithm is proposed to solve combined economic and emissions dispatch problems and the effectiveness of proposed algorithm is demonstrated using three and six generating unit test systems.

2 COMBINED ENVIRONMENTAL ECONOMIC DISPATCH (CEED)

The traditional economic dispatch problem has been defined as minimizing of an objective function i.e., the generation cost function subject to equality constraints i.e., total power generated should be equal to total system load plus losses for all solutions and inequality constraints

i.e. generations should lie between their respective maximum and minimum specified values.

The objective function equation (1) is minimised subjected to equality constraint equation (2) and inequality constraints equation (3).

………. (1)

………. (2)

…….. (3)

Where x is a state variable, Pi is the control variable, i.e., real power setting of generator and n is the number of units or generators.

There are several ways to include emission into the problem of economic dispatch. There are various algorithms for solving environmental dispatch problem with different constraints [19]. One approach is to include the reduction of emission as an objective. In this work, only 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

……………. (4)

..(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

..(6)

This has the resemblance of the familiar fuel cost equation, once h is determined. A practical way of determining is discussed by Palanichamy and Srikrishna [6]. 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.

1. Evaluate the maximum cost of each generator at its maximum output, i.e.,

   ..(7)

2. Evaluate the maximum emission of each generator at its maximum output, ie,

   (8)

3. Divide the maximum cost of each generator by its maximum emission, i.e.,

   ……….. (9)

   Recalling that

   Rs/kg ………………. (10)

4. Arrange hi (i = 1, 2,……,n) in ascending order.

5. Add the maximum capacity of each unit, one at a time, starting from the smallest hi unit until total demand is met as shown below.

……………… (11)

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)

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 lambda iteration method.

3 BAT ALGORITHM

Bats are fascinating animals. They are the only mammals with wings and they also have advanced

capability of echolocation. Most of bats uses echolocation to a certain degree, among all the species, microbats are famous example as microbats use echolocation extensively, while mega bats do not. Microbats use a type of sonar, called echolocation, to detect prey, avoid obstacles, and locate their roosting revices in the dark.

If we idealize some of the echolocation characteristics of microbats, we can develop various bat- inspired algorithms or bat algorithms. For simplicity, we now use the following approximate or idealized rules:

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

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

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

Another obvious simplification is that no ray tracing is used in estimating the time delay and three dimensional topography. In addition to these simplified assumptions, we also use the following approximations, for simplicity. In general the frequency f in a range

500 kHz] corresponds to a range of wavelengths from 0.7mm to 17mm.

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

…………. (13)

……………… (14)

where [0, 1] is a random vector drawn from a uniform distribution. Here is the current global best location (solution) which is located after comparing all the solutions among all the n bats. As the product is the velocity increment, we can use either (or ) to adjust the velocity change while fixing the other factor (or ), 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

………………….. (15)

Where [1, 1] is a random number, while

=< > 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.

3.1 PSEUDO-CODE OF THE BAT ALGORITHM

Objective function f(x), x =

Initialize the bat population (i = 1, 2, .., n) and Define pulse frequency at

Initialize pulse rates and the loudness

while (t <Max number of iterations)

Generate new solutions by adjusting frequency,

and updating velocities and locations/solutions [equations (13) to (15)]

if (rand > )

Select a solution among the best solutions

Generate a local solution around the selected best solution

end if

Generate a new solution by flying randomly

if (rand < & ) < )) Accept the new solutions Increase and reduce

end if

Rank the bats and find the current best

end while

Post process results and visualization

4 SIMULATION RESULTS AND DISCUSSIONS

The applicability and efficiency of BAT algorithm has been applied to two test cases. The programs are developed using MATLAB 7.14.

The Parameters for BAT algorithm considered

here are:

n=20, A=0.9, r=0.1, , .

Test case 1: The system consists of three thermal units. The parameters of all thermal units are adapted from [1].

Table: 1 Comparison of test results for Three Generating units

Table: 1 shows the summarized results of CEED problem for load demand of 400MW, 500MW and 700MW are obtained by the proposed BAT algorithm with stopping criteria based on maximum-generation=100.

Form Table: 1, it is clear that BAT algorithm gives optimum result in terms of minimum fuel cost,

emission level and the total operating cost compared to other algorithms.

Table: 2 gives the best optimum power output of generators for CEED problem using BAT & ABC algorithm for load demand 400MW, 500MW and 700MW.

Table: 2 Optimum Power dispatch Results by ABC, Proposed BAT method for three units system

The convergence tendency of proposed BAT algorithm based strategy for power demand of 400MW, 500MW and 700 MW is plotted in figure: 1. It shows that

the technique converges in relatively fewer cycles thereby possessing good convergence property.

Figure: 1 convergence of three generating units system for load demand values 400MW, 500MW & 700MW.

Test case II: The system consists of six thermal units. The parameters of all thermal units are adapted from [1]. The summarized result of CEED problem for load demand of

500MW an 900MW are obtained by the proposed BAT algorithm with stopping criteria based on maximum- generation=100 is presented in Table: 3.

Table: 3 comparison of test Results for six generating unit system

Form Table: 3, it is clear that BAT algorithm gives the optimum result in terms of minimum fuel cost, emission level and the total operating cost compared to other algorithms.

Table: 4 gives the best optimum power output of generators for CEED problem using BAT & ABC algorithms for load demand 500MW and 900MW.

Table: 4 Optimum Power dispatch results by ABC Approach for six unit system

The convergence tendency of proposed BAT algorithm based strategy for power demand of 500MW and 900 MW is plotted in figure:2. It shows that the technique

converges in relatively fewer cycles there by possessing good convergence property.

Figure: 2 convergence of six generating unit system for load demand values of 500MW and 900MW.

5 CONCLUSION

In this paper, a new optimization of BAT 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 and ABC. The comparison shows that BAT algorithm performs better than above mentioned methods. The BAT algorithm has superior features, including quality of solution, stable convergence characteristics and good computational efficiency. Bat algorithm gives optimum dispatch evaluation with less number of iterations. Therefore from the results it is concluded that BAT optimization is a promising technique for solving complicated problems occurring in power systems.

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*h values are considered based on literature.