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Application of Genetic Algorithm to Economic Load Dispatch

ABSTRACT:

This paper presents an approach based on genetic algorithm to solve the economic load dispatch (ELD) problem with losses for three thermal plant systems. Genetic algorithms are adaptive search methods that simulate some of the natural processes: selection, information, inheritance, random mutation and population dynamics. This approach was tested for three thermal plant systems. The performance of Genetic Algorithm - intelligent approach (GAs) is compared with the classical Kirchmayer method and it is observed that this method is accurate and may replace effectively the conventional practices presently performed in different central load dispatch centers.

INTRODUCTION:

Economic load dispatch (ELD) is a sub problem of the optimal power flow (OPF) having the objective of fuel cost minimization. The classical solutions for ELD problems have used equal incremental cost criterion for the loss-less system and use of penalty factors for considering the system losses. The lambda-iterative method has been used for ELD. Many other methods such as gradient methods, Newtonâ„¢s methods, linear and quadratic programming, etc have also been applied to the solution of ELD problems. However, all these methods are based on assumption of continuity and differentiability of cost functions. Hence, the cost functions have been approximated in the differentiable form, mostly in the quadratic form. Further, these methods also suffer on two main counts. One is their inability to provide global optimal solution and getting stuck at local optima. The second problem is handling the integer or discrete variables.

Genetic algorithms (GAs) have been proved to be effective and quite robust in solving the optimization problems. GAs can provide near global solutions and can also handle effectively the discrete control variables. GAs does not stick into local optima because GAs begins with many initial points and search for the most optimum in parallel. GAs considers only the pay-off information of objective function regardless whether it is differentiable or continuous. Consequently, the most realistic cost characteristic of power plants can be formulated. Discontinuity and non-differentiability of cost charecteristics can be effectively handled by GAs.

This paper proposes the application of GAs to solve the economic load dispatch for three thermal plant systems and the results are compared with conventional method.

CLASSIC ECONOMIC LOAD DISPATCH PROBLEM

The objective of the ELD problem is to minimize the total fuel cost at thermal plants

n

OBJ = Fi (Pi)

i=1

Subject to the constraint of equality in real power balance

n

Pi â€œ PL â€œ PD = 0

i=1

The inequality constraints of real power limits of the generation outputs are

Pi min < Pi < Pi max

Where

Fi (Pi) is the individual generation production in terms of its real power generation

Pi, Pi the output generation for unit i, n the number of generators in the system

Pd the total current system load demand, and Pl the total system transmission losses.

The thermal plant can be expressed as input-output models (cost function), where the input is the fuel cost and the output the power output of each unit, in practice, the cost function could be represented by a quadratic function.

Fi (Pi) = Ai * Pi2 + Bi * Pi + Ci

The incremental cost curve data are obtained by taking the derivative of the unit input-output equation resulting in the following equation for each generator:

dFi (Pi) / dPi = 2 Ai * Pi + Bi

Transmission losses are a function of the unit generations and are based on the system topology. Solving the ELD equations for a specified system requires an iterative approach since all unit generation allocations are embedded in the equation for each unit. In practice, the loss penalty factors are usually obtained using on line power flow software. This information is updated to ensure accuracy. They can also be calculated directly using the Bmn matrix loss formula.

PL = Pi Bij Pj

Where Bij are coefficients, constants for certain conditions.

GENETIC ALGORITHMS

GAs is inspired from phenomena found in living nature. The phenomena incorporated so far in GA models include phenomena of natural selection as there are selection and the production of variation by means of recombination and mutation, and rarely inversion, diploid and others. Most genetic algorithms work with one large panmictic population, i.e, in the recombination step each individual may potentially choose any other individual from the population as a mate. Then GA operators are performed to obtain the new child offspring.

Brief Description of GAs Operators:

There are three important GA operators which are commonly used are as follows:

(i) crossover,

(ii) Mutation, and

(iii) Selection and survival of the fittest.

Crossover

The task of crossover is the creation of new individuals of the current population. An individual can be viewed on chromosome level as Cj = (cj1,cj2,cj3.......cjn). The child chromosome Ck = ((c1,c1â„¢),(c2,c2â„¢),Â¦..(cn,cnâ„¢)) is created by recombination of its parent chromosomes pi = (c1,c2,c3,Â¦..,cn) and Pj = (c1â„¢,c2â„¢,Â¦..,cnâ„¢).

The recombination operation (ci,ciâ„¢) is the projection to the first on second component of the parameter list, namely,

P1 = ( 0 0 1 0 : 1 1 0 ) and

P2 = (1 0 1 1 : 0 0 1),

|__________________ xsite

The child strings can be obtained after the recombination or crossover are

C1 = ( 1 0 1 1 : 1 1 0) and

C2 = (0 0 1 0 : 0 1 1).

Hence, it can be concluded that the crossover operator has three distinct sub steps, namely,

i. Slice each of the parent strings in the sub strings,

ii. Exchange a pair of corresponding sub strings of the parents, and

iii. Merge the two respective sub strings to form offspring.

Basic Structure of GA

Mutation

Mutation is the important operator, because newly created individuals have no new inheritance information and the number of alleles is constantly decreasing. This process results in contraction of the population to one point, which is only wished at the end of the population to one point, which is only wished at the end of the convergence process, after the population works in a very promising part of the search space. Diversity is necessary to search a big part of the search space. It is on goal of the learning algorithm to search always in regions not viewed before. Therefore, it is necessary to enlarge the information contained in the population. One way to achieve this goal is mutation. The mutation operator M (chromosome) selects a gene of that chromosome and changes the allele by an amount m, the mutation variance. This happens with a mutation frequency m. the parameter m and m have major influence of the quality of the learning algorithm. Mutation can be illustrated with the help of an example,

Let a string is P1 (0 0 1 0 1 0 0)

|__________________ msite

after the mutation at the second position

p1=(0110100)

selection

In the implementation of genetic algorithm, the best individuals using roulette wheel with slot sized according to fitness is selected, so that the probability of selection of best strings are more. Further more , one only accept an offspring as a new member of the population, if it differs enough from the other individuals , at least by some significant amount . After accepting a new individual, one of the worst individuals is removed,i.e. its fitness value is quite low from the population in order to hold the population size constant. In the present implementation the worst fit individual is removed because the algorithm is not sensible against this selection. The complete genetic algorithm is represented with the help of the flow chart as shown in Fig.2

To maximize the efficiency of GAs, the three inherent parameters of GAs are to be optimized, the mutation probability Pm,crossover probability Pc and the population size POPSIZE.

APPLICATION OF GAs TO ECONOMIC LOAD DISPATCH PROBLEM

ENCODING AND DECODING

Encoding is a process of coding a problem as a number of finite strings. It typically utilizes the binary alphabet {0,1}. The types of encoding schemes have been developed by researches, which are called series encoding and embedded encoding. The series encoding simply stacks each unitâ„¢s output value structure in series with each other in the string. Each unitâ„¢s output gene structure is assigned the same number of loci with in the string. The embedded encoding scheme uses the same systems for representation and decoding as the first, except the assigned gene structures are embedded with in each other through out the string. The string is made up of a series of smaller gene structures, each containing one gene locus for each unit. It has been reported that series encoding can provide a better ED solution. In this paper a binary series coding is used through out all the GAs.

Decoding a binary string into an unsigned integer can play very important roles in GA implementation. The inequality power limit constraint is performed in such a way that the individual string is normalized over the unitâ„¢s operating region. The inequality constraints are handled in this manner, which efficiently reduces the searching space, and thus enhances the performance of the system.

The decoding method is formulated in Eq. (8).

Value = bit 0 X x 20 + bit 1 X x 21+Â¦Â¦Â¦+ bit i X x 2 i + Â¦Â¦+

Bit chrom-length X x 2 chrom-length

If the optimized parameter belongs to (Pimax â€œ Pimin) decoding value of the parameter in computer by Eq. (9).

[ value X (Pimax â€œ Pimin) ]

Pi = Pimin +-------------------------------

2 chrom-length â€œ 1

Objective and Fitness Function Formulation

In the ED problem, the goal is to minimize the objective function

n

Ft = Fi (Pi)

i=1

with the constraint of equality

n

Pi â€œ PL â€œ PD = 0

i=1

is changed to constraint optimization problem and thus forming fitness function.

n

Fct = Ft + PF [ Pi â€œ PL â€œ PD ]

i=1

Where PF is penalty factor. The penalty function is placed into the objective function in such a way that it penalizes any violation of constraints and forces that unconstrained optima towards the feasible region. In the ELD problem the goal is to minimize the objective function FCT,while the objective when using GAs is to maximize a fitness function. It is therefore necessary to map the fitness function FCT in the given form.

Ftt = EXP [ - (K1* Fct) K2 ]

K1 and K2 are constants and the value is problem dependent. Considerin the evolutionary process of the GAs, the solution is improved through the generations and also to decrease the penalty function over the successive iterations can be adapted with the penalty function varying directly with the number of generations. This ensures that only the objective function is ultimately minimized with a feasible solution.

SIMULATION RESULTS AND PERFORMANCE

Three thermal plant system

To focus on the evaluation of the proposed GA, a three-unit power system is used. The data used in this paper are obtained from Sheble and Britting are as follows:

F1 = 0.00156 X P12 + 7.92 X P1 + 561 Rs/hr

F2 = 0.00194 X P22 + 7.85 X P2 + 310 Rs/hr

F3 = 0.00482 X P32 + 7.97 X P3 + 78 Rs/hr

0.0000050 0.000005 0.0000075

Bmn = 0.0000050 0.000015 0.0000100

0.0000075 0.000010 0.0000450

The total operating ranges for this example are

100 MW < P1 < 600 MW

100 MW < P2 < 400 MW

50 MW < P3 < 200 MW

The parameters used in GA are as follows

Population size 10 Chromosome length 36 Sub-Chromosome lengths 13,12,11 Crossover probability 0.5

Mutation Probability 0.01

Total load classical Kirchmayer Method

Pd PG1 PG2 PG3 PL Cost(Rs/hr)

812.57 325.116 371.012 130.00 13.558 7986.093

585.33 233.258 268.106 90.933 6.962 5890.063

869.00 345.120 400.660 138.610 15.420 8522.450

Total load GA

Pd PG1 PG2 PG3 PL Cost(Rs/hr)

812.57 314.381 383.003 128.334 13.146 7986.069 585.33 243.450 257.655 91.475 7.250 5890.0947

869.00 355.524 395.091 134.196 15.812 8122.852

Genetic algorithm claims to provide near optimal or optimal solution for computationally intensive problems. Therefore the effectiveness of genetic algorithm solutions should always be evaluated by C Language was tested for three thermal plant systems. The performance of Genetic Algorithmic approach (GAs) is compared with the classical Kirchmayer method and as given in table1. It is observed that this method is accurate and may replace effectively in the conventional practices presently performed in different central load dispatch centres.

CONCLUSION

This paper has attempted to solve economic load dispatch problem of the power system networks The results are obtained for three thermal plant systems.

Future Scope

This method can be extended to one plant as combined cycle cogeneration plant in multi thermal plant system.

REFERENCES

1. A J Wood and B F Wollenburg power generation operation and control. John wiley and sons, 1984.

2. D E Goldberg and J H Holland , Genetic Algorithms in search optimization and Machine Learning Addison Wesley,1989.

3. Z Michealewicz Genetic Algorithms + Data structure=Evolution Programs Springer verlag,Berlin,Heidelberg,Newyork,1992.

4.Y H Song and C S V Chov.â„¢Advanced Engineered conditioning Genetic Approach to Power Economic Dispatch.â„¢ IEE Proceedingsâ€Generation Transmission and Distribution,vol 144, no 3,May1997,p285.