genetic algorithm to economic load dispatch

8/9/2019 Genetic Algorithm to Economic Load Dispatch

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Application of Genetic

Algorithm to Economic

Load Dispatch


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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.


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INDEX

Contents

1. Introduction

2. Economic Load Dispatch (ELD)

3. Factors To Consider In The EDC

3.1 Cost of Generation

3.2 Price

3.3 Quantity Supplied

4. Objective of EDC

5. Classic Economic Load Dispatch Problem

6. Genetic Algorithm (GA)

7. Brief Description of Gas Operators

7.1 Crossover

7.2 Mutation

7.3 Selection

8. Basic Structure of GA

9. Application of Gas To Economic Load Dispatch Problem

9.1 Encoding And Decoding

9.2 Objective And Fitness Function Formulation

10. Algorithm For ELD Using GA

11. Simulation Results And Performance

12. Conclusion


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1. 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 forconsidering the system losses. The lambda-iterative method has been used for ELD. Many other

methods such as gradient methods, Newtons 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.

2. Economic Load Dispatch:-

An economic dispatch calculation (EDC) is performed to dispatch, or schedule, a set of

online generating units to collectively produce electricity at a level that satisfies a specified

demand in an economical manner. Each online generating unit may have many characteristics

that make it unique, and which must be considered in the calculation. The amount of electricity

demanded can vary quickly and schedule produced by an EDC should leave units able to

respond and adapt without major implications to cost or profit. The electric system may have

limits (e.g., voltage, transmission, etc.) that impact the EDC and hence should be considered.


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Generating units may have prohibited generation levels at which resonant frequencies may cause

damage or other problems to the system. The impact of transmission losses, congestion, and

limits that may inhibit the ability to serve the load in a particular region from a particular

generator (e.g., a low-cost generator) should be considered. The market structure within an

operating region and its associated regulations must be considered in determining the specified

demand, and in determining what constitutes economical operation. An independent system

operator (ISO) tasked with maximizing social welfare would likely have a different definition of

economical than does a generation company (GENCO) wishing to maximize its profit in a

competitive environment. The EDC must consider all of these factors and develop a schedule

that sets the generation levels in accordance with an economic objective function.

3. Factors to Consider in the EDC:-

3.1 The Cost of Generation:-

Cost is one of the primary characteristics of a generating unit that must be

considered when dispatching units economically. The EDC is concerned with the short-term

operating cost, which is primarily determined by fuel cost and usage. Fuel usage is closely

related to generation level. Very often, the relationship between power level and fuel cost is

approximated by a quadratic curve:

F=aP2 + bP + c.

c is a constant term that represents the cost of operating the plant, b is a linear term that

varies directly with the level of generation, and a is the term that accounts for efficiency changes

over the range of the plant output.Aquadratic relationship is often used in the research

literature.However, due to varying conditions at certain levels of production (e.g., the opening or

closing of large valves may affect the generation cost [Walters and Sheble, 1992]), the actual

relationship between power level and fuel costmay be more complex than a quadratic equation.

Many of the long-term generating unit costs (e.g., costs attributed directly to starting and

stopping the unit, capital costs associated with financing the construction) can be ignored for the

EDC, since the decision to switch on, or commit, the units has already been made. Other

characteristics of generating units that affect theEDCare theminimum andmaximumgeneration


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levels at which theymay operate.When binding, these constraints will directly impact the EDC

schedule.

3.2 The Price:-

The price at which an electric supplier will be compensated is another important

factor in determining an optimal economic dispatch. In many areas of the world, electric power

systems have been, or still are, treated as a natural monopoly. Regulations allow the utilities to

charge rates that guarantee them a nominal profit. In competitive markets, which come in a

variety of flavors, price is determined through the forces of supply and demand. Economic

theory and common sense tell us that if the total supply is high and the demand is low, the price

is likely to be low, and vice versa. If the price is consistently below a GENCOs average total

costs, the company may soon be bankrupt.

3.3 The Quantity Supplied:-

The amount of electric energy to be supplied is another fundamental input for the

EDC. Regions of the world having regulations that limit competition often require electric

utilities to serve all electric demand within a designated service territory. If a consumer switches

on a motor, the electric supplier must provide the electric energy needed to operate the motor. In

competitive markets, this obligation to serve is limited to those with whom the GENCO has a

contract. Beyond its contractual obligations, the GENCO may be willing (if the opportunity

arises) to supply additional consumer demand. Since the

consumers have a choice of electric supplier, a GENCO determining the schedule of its own

online generating units may choose to supply all, none, or only a portion of that additional

consumer demand.

The decision is dependent on the objective of the entity performing the EDC (e.g., profit

maximization, improving reliability, etc.). _ 2006 by Taylor & Francis Group, LLC.

EDC and System Limitations A complex network of transmission and distribution lines

and equipment are required to move the electric energy from the generating units to the

consumer loads. The secure operation of this network depends on bus voltage magnitudes and


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angles being within certain tolerances. Excessive transmission line loading can also affect the

security of the power system network. Since superconductivity is a relatively new field, lossless

transmission lines are expensive and are not commonly used. Therefore, some of the energy

being transmitted over the system is converted into heat and is consequently lost. The schedule

produced by the EDC directly affects losses and security; hence, constraints ensuring proper

system operation must be considered when solving the EDC problem.

4. The Objective of EDC:-

In a regulated, vertically integrated, monopolistic environment, the obligated-to-

serve electric utility performs the EDC for the entire service area by itself. In such an

environment, providing electricity in an economical manner means minimizing the cost of

generating electricity, subject to meeting all demand and other system operating constraints. In a

competitive environment, the way an EDC is done can vary from one market structure to

another. For instance, in a decentralized market, the EDC may be performed by a single GENCO

wishing to maximize its expected profit given the prices, demands, costs, and other constraints

described above. In a power pool, a central oordinating entity may perform an EDC to centrally

dispatch generation for many GENCOs. Depending on the market rules, the generation owners

may be able to mask the cost information of their generators. In this case, bids would be

submitted for various price levels and used in the EDC.

5. 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 balanc

n

Pi PL PD = 0

i=1

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


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Pi min < Pi < Pi max

Where

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


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6. GENETIC ALGORITHM:-

Quadratic Representation of Unit 1 Fuel Costs

4000

3000

2000

1000

0

100150 200 250 300 350 400 450 500

MWs generated

Corresponding Average Fuel Costs for Unit 1

8

7

6

5

100 150 200 250 300 350 400 450 500

Relationship between fuel input and power output.


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facilitates simple crossover operations. Variable length representations may also be used, but

crossover implementation is more complex in this case. Tree-like representations are explored

ingenetic programming and graph-form representations are explored inevolutionary

programming.

The fitness function is defined over the genetic representation and measures the quality of

the represented solution. The fitness function is always problem dependent. For instance, in

the knapsack problem one wants to maximize the total value of objects that can be put in a

knapsack of some fixed capacity. A representation of a solution might be an array of bits, where

each bit represents a different object, and the value of the bit (0 or 1) represents whether or not

the object is in the knapsack. Not every such representation is valid, as the size of objects may

exceed the capacity of the knapsack. Thefitness of the solution is the sum of values of all objects

in the knapsack if the representation is valid, or 0 otherwise. In some problems, it is hard or even

impossible to define the fitness expression; in these cases,interactive genetic algorithms are

used.

Once we have the genetic representation and the fitness function defined, GA proceeds to

initialize a population of solutions randomly, then improve it through repetitive application of

mutation, crossover, inversion and selection operators.

7. Brief Description of GAs Operators:-

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

crossover,

Mutation, and

(i) Selection and survival of the fittest.
http://en.wikipedia.org/wiki/Genetic_programminghttp://en.wikipedia.org/wiki/Genetic_programminghttp://en.wikipedia.org/wiki/Evolutionary_programminghttp://en.wikipedia.org/wiki/Evolutionary_programminghttp://en.wikipedia.org/wiki/Evolutionary_programminghttp://en.wikipedia.org/wiki/Knapsack_problemhttp://en.wikipedia.org/wiki/Interactive_evolutionary_computationhttp://en.wikipedia.org/wiki/Interactive_evolutionary_computationhttp://en.wikipedia.org/wiki/Genetic_programminghttp://en.wikipedia.org/wiki/Evolutionary_programminghttp://en.wikipedia.org/wiki/Evolutionary_programminghttp://en.wikipedia.org/wiki/Knapsack_problemhttp://en.wikipedia.org/wiki/Interactive_evolutionary_computation


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7.1 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,c1),(c2,c2),..(cn,cn)) is created by recombination of its parentchromosomes pi = (c1,c2,c3,..,cn) and Pj = (c1,c2,..,cn).

The recombination operation (ci,ci) 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.

7.2 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


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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)

7.3 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 a flow chart.

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.


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8. BASIC STRUCTURE OF GA:-


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9.2 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 PL PD = 0

i=1

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

n

Fct = Ft + PF [ Pi PL 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.


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ALGORITHM FOR ELD USING GA:-

For solution of ELD using SGA, incremental fuel cost of the generators i.e. lambda is

encoded in the chromosome. The algorithm for implementing ELD without losses using Simple

GA is as follows.

1. Read population size, chromosome length, unit data, Pdemand, Probability of Elitism,

Crossover and mutation.

2. Randomly generate population of chromosomes.

3. Decode the chromosomes using (6).

4. Lambda_act = Lambda_min + (Lambda_max - Lambda_min)* Decoded lambda

5. Use the Lambda_act and cost coefficients of the generators, and calculate real power output of

the generators (Pgen).

6. Calculate the error of each chromosome as (Sum of Pgen) Pdemand.

7. Fitness(i) of each chromosome is calculated as 1/(1+error(i)/Pdemand).

8. Arrange the chromosomes in the descending order of their fitness.

9. Check if error(1) 0.0001*Pdemand

10. If yes STOP and calculate Optimal fuel cost and Pgen of units

Else

11. Check if Fitness(1)=Fitness(last chromosome)

12. If yes print All chromosomes have equal values, calculate Optimal fuel cost and Pgen ofunits and STOP.

Else

13. Apply elitism, Reproduction (RWS), crossover and mutation and generate new population

from old one.

14. Update generation count.

15. Check if Generation count > maximum generations?

16. If yes, print Problem not converged in maximum number of generations, STOP.

Else

17. Repeat from step 3.


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SIMULATION RESULTSAND 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)


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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.


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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.Advanced Engineered conditioning Genetic Approach to PowerEconomic Dispatch. IEE ProceedingsGeneration Transmission and Distribution,vol 144, no

3,May1997,p285.

genetic algorithm to economic load dispatch

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