![International Journal of Electronic and Electrical Engineering.
ISSN 0974-2174, Volume 7, Number 5 (2014), pp. 523-528
© International Research Publication House
http://www.irphouse.com
Genetic Algorithm for Solving the Economic Load Dispatch
Satyendra Pratap Singh1
, Rachna Tyagi2
and Anubhav Goel3
1
Research Scholar, Electrical Engineering Department, IIT (BHU) Varanasi, UP
2
Asst. Prof. Electrical Engineering Department, AITM Varanasi, UP
3
Lecturer, Electronics and Communication Engineering, JPIET Meerut, UP
E-mail: 1
spsingh.rs.eee@iitbhu.ac.in, 2
er.rachnatyagi@gmail.com,
3
anubhavgoel207@gmail.com
Abstract
In this paper, comparative study of two approaches, Genetic Algorithm
(GA) and Lambda Iteration method (LIM) have been used to provide
the solution of the economic load dispatch (ELD) problem. The ELD
problem is defined as to minimize the total operating cost of a power
system while meeting the total load plus transmission losses within
generation limits. GA and LIM have been used individually for solving
two cases, first is three generator test system and second is ten
generator test system. The results are compared which reveals that GA
can provide more accurate results with fast convergence characteristics
and is superior to LIM.
Keywords: Economic load dispatch, genetic algorithm, lambda
iteration method, generator systems.
1. Introduction
Economic Load Dispatch is the very important issues in the area of Power System.
Load demands are increasing day by day. With the development of integrated power
system, it becomes necessary to operate the plant units economically. An important
objective in the operation of such a power system is to generate and transmit power to
meet the system load demand at minimum fuel cost by an optimal mix of various types
of plants [1]. Thus ELD occupies an important position in the electric power system.
For any specified load condition, ELD determines the power output of each plant (and
each generating unit within the plant) which will minimize the overall cost of fuel
needed to serve the system load taking in consideration all practical constraints [2].
ELD is the very huge topic and lots of research works have been done in this area.
In [3], an arithmetic crossover GA has been proposed to solve the ELD problem. In](https://image.slidesharecdn.com/ijeeev7n5132-150627074939-lva1-app6891/85/Genetic-Algorithm-for-Solving-the-Economic-Load-Dispatch-1-320.jpg)
![Satyendra Pratap Singh et al524
[4], a hybrid method which is the combination of GA and fuzzy logic is used to
optimize the cost of generation.
2. Economic Load Dispatch
The minimization of objective function is the primary concern of an ELD problem.
The objective function meets the demand of generation and satisfies all other
constraints. Mathematically objective function of ELD problem with constrained
optimization problem is
= ∑ ( ) (1)
is the total generation cost; N is the total number of generating units; is the
power generation cost function of the unit. The total cost of operation includes the
fuel cost, costs of labour, maintenance and supplies. Mostly, costs of labour, supplies
and maintenance are fixed percentages of incoming fuel costs. Now assume that the
variation of fuel cost of each generator with the active power output is given by a
quadratic polynomial
= ∑ ( + + ) (2)
Where, is power output of generator i; , , and are cost coefficients.
The ELD problem is defined as to minimize the total operating cost of a power
system while meeting the total load plus transmission losses within generator limits.
Subject to (1) the energy balance equation
∑ = + (3)
(2) the inequality constraints
( ) ≤ ≤ ( ) (4)
Where is the power transmission loss.
3. LIM for the Solution of the ELD Problem
The LIM is the most popular method for the solution of the economic load dispatch
problem. It gives a decentralized solution to the ELD problem by equating the
marginal cost of generation of each thermal unit to the price of electricity, or,
equivalently, the marginal revenue of each unit under perfect competition conditions,
known as system lambda [5].The minimum and maximum lambda values are initially
computed,
= ,
,
(5)
= ,
,
(6)
The initial value chosen for lambda is the mid-point of the interval ( , ),
i.e,
= (7)](https://image.slidesharecdn.com/ijeeev7n5132-150627074939-lva1-app6891/85/Genetic-Algorithm-for-Solving-the-Economic-Load-Dispatch-2-320.jpg)
![Genetic Algorithm for Solving the Economic Load Dispatch 525
4. Genetic Algorithm
The GA is a stochastic global search method that mimics the metaphor of natural
biological evolution such as selection, crossover, and mutation [6-7]. GA’s work on
string structures where string is binary digits which represent a coding of control
parameters for a given problem. All parameters of the given problem are coded with
strings of bits. The individual bit is called ‘gene’ and the content of the each gene is
called ‘allele’. Typically, the genetic algorithms have three phases initialization,
evaluation and genetic operation. The fitness function for the maximization problem is
( ) = ( ) (8)
and for the minimization problem is
( ) = ( )
(9)
Where f(x) is fitness function and F(x) is objective function.
In genetic operation phase, we generate a new population from the previous
population using genetic operators. They are reproduction, crossover and mutation.
Reproduction is the operator used to copy the old chromosome into matting pool
according to its fittest value. Higher the fitness of the chromosome more is number of
the copies in the next generation chromosome.
The various methods of selecting chromosomes for parents to crossover are
roulette-wheel selection, boltzmann selection, tournament selection, rank selection,
steady state selection etc. The commonly used reproduction operator is the roulette-
wheel selection method where a string is selected from the mating pool with a
probability proportional to the fitness [10]. The roulette-wheel mechanism is expected
to make / copies of string of the mating pool. The average fitness is
= ∑
̅
(10)
The basic operator for producing new chromosome is crossover. In this operator,
information is exchanged among strings of matting pool to create new strings. The
final genetic operator in the algorithm is mutation. In general evolution, mutation is a
random process where one allele of a gene is replaced by another to produce a new
genetic structure. Mutation is an important operation, because newly created
individuals have no new inheritance information and the number of alleles is
constantly decreasing.
5. Results and Discussions
The GA and classical method (lambda iteration) are used to solve ELD problems and
results are discussed and compared. The algorithms are implemented in MATLAB to
solve ELD problem. The main objective is to minimize the cost of generation of
thermal plants using GA and classical Lambda Iteration Method. The performance is
evaluated with losses for two set generator data, which are referred as Problem I and
Problem II.
Problem I: Three generator test systems [9]
Problem II: Ten generator test systems [8]
For GA problem assume the length of the string, l is 16, population of string, pop is
20, crossover probability, pc is 0.8 and mutation probability, pm is 0.01.](https://image.slidesharecdn.com/ijeeev7n5132-150627074939-lva1-app6891/85/Genetic-Algorithm-for-Solving-the-Economic-Load-Dispatch-3-320.jpg)
![Satyendra Pratap Singh et al526
6. Problem I: Three generator test systems
The coefficients of fuel cost are given below in Table 1. The power demand is
considered to be 300MW. The results corresponding to LIM and GA for problem I are
detailed in Table 2.
Table 1: Coefficients of Fuel Cost for Three generator test systems
Unit No.
1 0.00525 8.66 328.13
2 0.00609 10.040 136.91
3 0.00592 9.760 59.16
Table 2: Three Generator Test Results ( = 300 MW)
LIM GA
P1 202.49 202.464
P2 81.0267 80.9787
P3 27.0149 27.0799
Fitness - 0.999957
Losses 10.5311 10.5354
Error 0.000652 0.0129291
Total cost 3615.11 3614.95
Developed program returns the generated power, total cost, total losses and error.
7. Problem II: Ten generator test systems
Again the proposed technique has been performed on a sample system which consists
of ten generator system. The power demand is considered to be 1440MW.
Transmission loss coefficients are given in Table 3 [8]. The results corresponding to
LIM and GA for problem II is detailed in table 4.
Table 3: Coefficients of Fuel Cost for Ten generator test systems.
Unit no. ai bi ci
1 0.001220 7.92 630
2 0.004700 7.91 190
3 0.001320 7.93 625
4 0.001153 7.92 723
5 0.001154 7.93 717
6 0.001562 7.92 561
7 0.001153 7.92 723
8 0.001321 7.91 618
9 0.001319 7.00 561
10 0.001530 7.00 561](https://image.slidesharecdn.com/ijeeev7n5132-150627074939-lva1-app6891/85/Genetic-Algorithm-for-Solving-the-Economic-Load-Dispatch-4-320.jpg)
![Genetic Algorithm for Solving the Economic Load Dispatch 527
Table 4: Ten Generator Test Results ( = 1440 MW).
LIM GA
P1 160 160
P2 65 65
P3 150 150
P4 170 170
P5 160 160
P6 130 130
P7 170 170
P8 145 145
P9 140 140
P10 163.926 163.981
Fitness - 0.999976
Losses 13.9357 13.9261
Error 0.026341 0.0345486
Total cost 17608.4 17607.7
8. Conclusion
In this paper, Genetic Algorithm and Lambda Iteration method have been successfully
implemented to obtain the optimum solution of ELD. Due to the large variation in load
from time to time and it is not possible to have the load dispatch for every possible
load demand. Since there is no general procedure for find out the optimum solution of
economic load dispatches. This is where GA plays an important role to find out the
optimum solution in a fraction of second.
For the testing of GA and LIM, three generators and ten generators test systems are
used. The results obtained from both methods are compared with each other. It is
found that GA is giving better results than LIM. i.e. GA proves itself as fast algorithm
and yields true optimum generations of both operating costs and transmission line
losses of the power system.
References
[1] Sharma A., Tyagi R., and Singh S. P., “Sort Term Hydrothermal Scheduling
using Evolutionary Programming”, Int. J. of Inventions in Research,
Engineering Science and Technology (IJIREST),vol.1,no.1, April 2014.
(ISSN(Online):2348-8077)
[2] A. J. Wood and B. F. Wollenberg, “Power Generation, Operation and
Control”, 2nd Edition, New York: John Wiley & Sons, 1996.
[3] Yalcinoz. T, Altun. H, and Uzam. M, “Economic dispatch solution using
genetic algorithm based on arithmetic crossover, “in Proc. IEEE Porto Power
Tech. Conf., Porto, Portugal, Sep. 2001](https://image.slidesharecdn.com/ijeeev7n5132-150627074939-lva1-app6891/85/Genetic-Algorithm-for-Solving-the-Economic-Load-Dispatch-5-320.jpg)
![Satyendra Pratap Singh et al528
[4] Singh S. P., Bhullar S., “Hybrid Approach to Economic Load Dispatch”,
National Conference on Artificial Intelligence and Agents: Theory &
Applications, IIT (BHU) Varanasi, Dec. 2011.
[5] Chowdhury B.H., Rahman Saifur, “A Review of Recent Advances in
Economic Dispatch”, IEEE Transactions on Power Systems, Vol. 5, No. 4,
November 1990
[6] http:// en.wikipedia.org/wiki/Genetic_algorithm
[7] Mitchell M., “An Introduction to Genetic Algorithm”, MIT Press, 1998.
[8] Roa C.A-Sepulveda, Herrera M., Pavez-Lazo. B, Knight U.G., Coonick A.H.,
“Economic Dispatch using fuzzy decision trees”, Electric Power Systems
Research, vol. 66, no. 2, pp. 115-122, Aug. 2003.
[9] Kothari D.P., Dhillon J.S., “Power System Optimization”, Prentice-Hall of
India Vijayalakshmi. G.A., Rajsekaran. S, “Neural Networks, Fuzzy Logic,
and Genetic Algorithms” synthesis and application, PHI Learning Pvt. Ltd.](https://image.slidesharecdn.com/ijeeev7n5132-150627074939-lva1-app6891/85/Genetic-Algorithm-for-Solving-the-Economic-Load-Dispatch-6-320.jpg)
Genetic Algorithm for Solving the Economic Load Dispatch
- 1. International Journal of Electronic and Electrical Engineering. ISSN 0974-2174, Volume 7, Number 5 (2014), pp. 523-528 © International Research Publication House http://www.irphouse.com Genetic Algorithm for Solving the Economic Load Dispatch Satyendra Pratap Singh1 , Rachna Tyagi2 and Anubhav Goel3 1 Research Scholar, Electrical Engineering Department, IIT (BHU) Varanasi, UP 2 Asst. Prof. Electrical Engineering Department, AITM Varanasi, UP 3 Lecturer, Electronics and Communication Engineering, JPIET Meerut, UP E-mail: 1 spsingh.rs.eee@iitbhu.ac.in, 2 er.rachnatyagi@gmail.com, 3 anubhavgoel207@gmail.com Abstract In this paper, comparative study of two approaches, Genetic Algorithm (GA) and Lambda Iteration method (LIM) have been used to provide the solution of the economic load dispatch (ELD) problem. The ELD problem is defined as to minimize the total operating cost of a power system while meeting the total load plus transmission losses within generation limits. GA and LIM have been used individually for solving two cases, first is three generator test system and second is ten generator test system. The results are compared which reveals that GA can provide more accurate results with fast convergence characteristics and is superior to LIM. Keywords: Economic load dispatch, genetic algorithm, lambda iteration method, generator systems. 1. Introduction Economic Load Dispatch is the very important issues in the area of Power System. Load demands are increasing day by day. With the development of integrated power system, it becomes necessary to operate the plant units economically. An important objective in the operation of such a power system is to generate and transmit power to meet the system load demand at minimum fuel cost by an optimal mix of various types of plants [1]. Thus ELD occupies an important position in the electric power system. For any specified load condition, ELD determines the power output of each plant (and each generating unit within the plant) which will minimize the overall cost of fuel needed to serve the system load taking in consideration all practical constraints [2]. ELD is the very huge topic and lots of research works have been done in this area. In [3], an arithmetic crossover GA has been proposed to solve the ELD problem. In
- 2. Satyendra Pratap Singh et al524 [4], a hybrid method which is the combination of GA and fuzzy logic is used to optimize the cost of generation. 2. Economic Load Dispatch The minimization of objective function is the primary concern of an ELD problem. The objective function meets the demand of generation and satisfies all other constraints. Mathematically objective function of ELD problem with constrained optimization problem is = ∑ ( ) (1) is the total generation cost; N is the total number of generating units; is the power generation cost function of the unit. The total cost of operation includes the fuel cost, costs of labour, maintenance and supplies. Mostly, costs of labour, supplies and maintenance are fixed percentages of incoming fuel costs. Now assume that the variation of fuel cost of each generator with the active power output is given by a quadratic polynomial = ∑ ( + + ) (2) Where, is power output of generator i; , , and are cost coefficients. The ELD problem is defined as to minimize the total operating cost of a power system while meeting the total load plus transmission losses within generator limits. Subject to (1) the energy balance equation ∑ = + (3) (2) the inequality constraints ( ) ≤ ≤ ( ) (4) Where is the power transmission loss. 3. LIM for the Solution of the ELD Problem The LIM is the most popular method for the solution of the economic load dispatch problem. It gives a decentralized solution to the ELD problem by equating the marginal cost of generation of each thermal unit to the price of electricity, or, equivalently, the marginal revenue of each unit under perfect competition conditions, known as system lambda [5].The minimum and maximum lambda values are initially computed, = , , (5) = , , (6) The initial value chosen for lambda is the mid-point of the interval ( , ), i.e, = (7)
- 3. Genetic Algorithm for Solving the Economic Load Dispatch 525 4. Genetic Algorithm The GA is a stochastic global search method that mimics the metaphor of natural biological evolution such as selection, crossover, and mutation [6-7]. GA’s work on string structures where string is binary digits which represent a coding of control parameters for a given problem. All parameters of the given problem are coded with strings of bits. The individual bit is called ‘gene’ and the content of the each gene is called ‘allele’. Typically, the genetic algorithms have three phases initialization, evaluation and genetic operation. The fitness function for the maximization problem is ( ) = ( ) (8) and for the minimization problem is ( ) = ( ) (9) Where f(x) is fitness function and F(x) is objective function. In genetic operation phase, we generate a new population from the previous population using genetic operators. They are reproduction, crossover and mutation. Reproduction is the operator used to copy the old chromosome into matting pool according to its fittest value. Higher the fitness of the chromosome more is number of the copies in the next generation chromosome. The various methods of selecting chromosomes for parents to crossover are roulette-wheel selection, boltzmann selection, tournament selection, rank selection, steady state selection etc. The commonly used reproduction operator is the roulette- wheel selection method where a string is selected from the mating pool with a probability proportional to the fitness [10]. The roulette-wheel mechanism is expected to make / copies of string of the mating pool. The average fitness is = ∑ ̅ (10) The basic operator for producing new chromosome is crossover. In this operator, information is exchanged among strings of matting pool to create new strings. The final genetic operator in the algorithm is mutation. In general evolution, mutation is a random process where one allele of a gene is replaced by another to produce a new genetic structure. Mutation is an important operation, because newly created individuals have no new inheritance information and the number of alleles is constantly decreasing. 5. Results and Discussions The GA and classical method (lambda iteration) are used to solve ELD problems and results are discussed and compared. The algorithms are implemented in MATLAB to solve ELD problem. The main objective is to minimize the cost of generation of thermal plants using GA and classical Lambda Iteration Method. The performance is evaluated with losses for two set generator data, which are referred as Problem I and Problem II. Problem I: Three generator test systems [9] Problem II: Ten generator test systems [8] For GA problem assume the length of the string, l is 16, population of string, pop is 20, crossover probability, pc is 0.8 and mutation probability, pm is 0.01.
- 4. Satyendra Pratap Singh et al526 6. Problem I: Three generator test systems The coefficients of fuel cost are given below in Table 1. The power demand is considered to be 300MW. The results corresponding to LIM and GA for problem I are detailed in Table 2. Table 1: Coefficients of Fuel Cost for Three generator test systems Unit No. 1 0.00525 8.66 328.13 2 0.00609 10.040 136.91 3 0.00592 9.760 59.16 Table 2: Three Generator Test Results ( = 300 MW) LIM GA P1 202.49 202.464 P2 81.0267 80.9787 P3 27.0149 27.0799 Fitness - 0.999957 Losses 10.5311 10.5354 Error 0.000652 0.0129291 Total cost 3615.11 3614.95 Developed program returns the generated power, total cost, total losses and error. 7. Problem II: Ten generator test systems Again the proposed technique has been performed on a sample system which consists of ten generator system. The power demand is considered to be 1440MW. Transmission loss coefficients are given in Table 3 [8]. The results corresponding to LIM and GA for problem II is detailed in table 4. Table 3: Coefficients of Fuel Cost for Ten generator test systems. Unit no. ai bi ci 1 0.001220 7.92 630 2 0.004700 7.91 190 3 0.001320 7.93 625 4 0.001153 7.92 723 5 0.001154 7.93 717 6 0.001562 7.92 561 7 0.001153 7.92 723 8 0.001321 7.91 618 9 0.001319 7.00 561 10 0.001530 7.00 561
- 5. Genetic Algorithm for Solving the Economic Load Dispatch 527 Table 4: Ten Generator Test Results ( = 1440 MW). LIM GA P1 160 160 P2 65 65 P3 150 150 P4 170 170 P5 160 160 P6 130 130 P7 170 170 P8 145 145 P9 140 140 P10 163.926 163.981 Fitness - 0.999976 Losses 13.9357 13.9261 Error 0.026341 0.0345486 Total cost 17608.4 17607.7 8. Conclusion In this paper, Genetic Algorithm and Lambda Iteration method have been successfully implemented to obtain the optimum solution of ELD. Due to the large variation in load from time to time and it is not possible to have the load dispatch for every possible load demand. Since there is no general procedure for find out the optimum solution of economic load dispatches. This is where GA plays an important role to find out the optimum solution in a fraction of second. For the testing of GA and LIM, three generators and ten generators test systems are used. The results obtained from both methods are compared with each other. It is found that GA is giving better results than LIM. i.e. GA proves itself as fast algorithm and yields true optimum generations of both operating costs and transmission line losses of the power system. References [1] Sharma A., Tyagi R., and Singh S. P., “Sort Term Hydrothermal Scheduling using Evolutionary Programming”, Int. J. of Inventions in Research, Engineering Science and Technology (IJIREST),vol.1,no.1, April 2014. (ISSN(Online):2348-8077) [2] A. J. Wood and B. F. Wollenberg, “Power Generation, Operation and Control”, 2nd Edition, New York: John Wiley & Sons, 1996. [3] Yalcinoz. T, Altun. H, and Uzam. M, “Economic dispatch solution using genetic algorithm based on arithmetic crossover, “in Proc. IEEE Porto Power Tech. Conf., Porto, Portugal, Sep. 2001
- 6. Satyendra Pratap Singh et al528 [4] Singh S. P., Bhullar S., “Hybrid Approach to Economic Load Dispatch”, National Conference on Artificial Intelligence and Agents: Theory & Applications, IIT (BHU) Varanasi, Dec. 2011. [5] Chowdhury B.H., Rahman Saifur, “A Review of Recent Advances in Economic Dispatch”, IEEE Transactions on Power Systems, Vol. 5, No. 4, November 1990 [6] http:// en.wikipedia.org/wiki/Genetic_algorithm [7] Mitchell M., “An Introduction to Genetic Algorithm”, MIT Press, 1998. [8] Roa C.A-Sepulveda, Herrera M., Pavez-Lazo. B, Knight U.G., Coonick A.H., “Economic Dispatch using fuzzy decision trees”, Electric Power Systems Research, vol. 66, no. 2, pp. 115-122, Aug. 2003. [9] Kothari D.P., Dhillon J.S., “Power System Optimization”, Prentice-Hall of India Vijayalakshmi. G.A., Rajsekaran. S, “Neural Networks, Fuzzy Logic, and Genetic Algorithms” synthesis and application, PHI Learning Pvt. Ltd.