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CHAPTER 4

IMPLEMENTATION OF GENETIC ALGORITHM

4.1 INTRODUCTION

Genetic Algorithm (GA) is one of the evolutionary techniques,

which follow the principles of natural genetics and natural selection to

constitute an optimization search procedure. GA is a stochastic search method

that mimics the metaphor of natural biological evolution and it has the ability

to solve highly non-linear, non-convex and non-smooth problems. The GA

works with a population of possible solutions to a problem by applying the

principle of ‘survival of the fittest’. Different population members are

assigned for reproduction in proportion to their fitness, where the fitness

function is derived from the objective function of the problem. The fitness

function should be able to provide a good measure of the quality of the

solution (Goldberg 2002). This chapter aims to solve the Gencos PBUC

problem using GA to overcome the huge computational time, solution

infeasibility and convergence problems of LR method.

The Gencos PBUC problem primarily consists of many binary

decision variables (0, 1) and is difficult to solve by deterministic methods to

obtain the optimal solution. Hence, GA is chosen to solve such a complex

combinatorial PBUC problem. GA uses a combined set of genetic operators to

search for the best combination of generating units over the entire scheduling

period. Unlike LR method, GA has an ability to add and remove a wide

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variety of equality and inequality constraints to the PBUC problem. Also, the

quality of final LR solution depends on the sensitivity of the commitment

with respect to Lagrange multipliers. Slow and unsteady convergence of LR

has always been a problem in finding the optimum solution as reported in

chapter 3. Therefore, GA technique has been implemented in this work for

solving the thermal PBUC problem and wind-thermal PBUC problem with

different constraints. The work flow diagram of GA approach to solve the

Gencos PBUC problem is shown in Figure 4.1.

Figure 4.1 Work flow diagram of GA implementation for PBUC problem

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4.2 PRINCIPLES OF GENETIC ALGORITHM

Genetic Algorithm was inspired by Darwin’s evolutionary theory

and the basic theory of GA can be found in Goldberg (2002). The algorithm

operates on a population of candidate solutions and each individual of

population is termed as ‘chromosome’. At each generation, a new set of

population is created by the process of selecting individuals according to their

level of fitness in the problem domain and breeding them together using

operators borrowed from natural genetics. This process leads to the evolution

of population of individuals that are better suited to their environment than the

individuals that they were created from, just as in natural adaptation. In this

way, GA search from many points in the search space at once and continually

narrow the focus of the search to achieve the best performance (Hobbs et al

2001). GA differs from other traditional optimization techniques such as LR

method and dynamic programming method in the following ways:

GA uses objective function information to guide the search,

not derivative or other auxiliary information.

GA uses coding of the parameters to calculate the objective

function in guiding the search, not the parameter themselves.

GA searches through many points in the solution space at one

time, not through a single point.

GA uses probabilistic rules, not deterministic rules, in moving

from one set of solutions (a population) to the next set.

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4.2.1 Basic Genetic Operators

The basic structure of a Genetic Algorithm as illustrated in

Goldberg (2002) and Azadeh et al (2012) is shown in Figure 4.2, where the

three significant genetic operators are:

(i) Selection - equates to survival of the fittest.

(ii) Crossover – represents mating between individuals.

(iii) Mutation – introduces random modifications in the

individuals.

Selection is the first genetic operator, which is applied to an existing

population. This operator is quite similar to natural selection in biological

systems, where the higher the fitness, the higher the chances of an organism

propagating its characteristics to future generations. This segment of GA

chooses the mating pairs from the population set according to the fitness

function values, the better the function value, the higher the chance of

selection. Selection is the main driving force in GA and used to choose the

suitable individuals for mating and entering to the next generation.. There are

several ways of implementing selection, the most widely used selection

techniques are roulette wheel selection, stochastic remainder selection and

tournament selection. All these selection methods allocate the population in

some proportion to the fitness of an individual with respect to the population

average.

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Figure 4.2 Basic structure of GA

The crossover operator attempts to produce new strings of superior

fitness by carrying out large changes in a strings structure. This is similar to

large jumps towards the optimum search in the solution space. In crossover,

the gene information between two selected chromosomes (individuals) is

exchanged with the aim to improve the diversity of the solution vectors. The

pair of chromosomes, taken from the mating pool, becomes the parents of two

offspring chromosomes for the new generation. The crossover operation may

be of one point crossover, multi-point crossover and uniform crossover. All

these crossover processes need a local search around a current solution and is

accomplished by another GA operator known as Mutation.

Mutation gives emphasis to maintain diversity in the population by

creating spontaneous random changes in various chromosomes. The main

purpose of using the mutation operator is to prevent early convergence and

Generate initial Population

Best individuals

Mutation

Evaluate fitness function

Selection

Is optimization criteria satisfied?

ResultStart

Generate new population Crossover

Yes

No

Mutation

Selection

Generate new population Crossover

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the permanent loss of any particular bit values. Without mutation operator,

there is no possibility of re-introducing a bit value which is missing from the

population. The choice of appropriate mutation technique depends on the

coding and problem itself. The robustness of GA has been applied in a wide

range of applications as follows:

Pattern recognition and image processing

Telecommunications networks and Chemical process

optimization

Biotechnology and medical systems

Computer aided design in all engineering branches

Power system optimization

Production planning and scheduling

One such application broadly used in finding the best possible

solution to the complex optimization problems, especially power generation

scheduling problems, which are considered with great dimensions and

constraints.

4.3 GA APPROACH FOR GENCOS PBUC PROBLEM

The stochastic searching GA technique has been implemented to

schedule the thermal and wind power generation in an isolated power system.

The GA approach solves the PBUC problem for maximizing the profit of

Gencos in deregulated environment. Each Genco decided that which of its

generating unit to be committed over the scheduling time period, while

satisfying the operating constraints. The uncertainty associated with wind

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power generation requires the scheduling of additional generation reserves to

compensate the output power fluctuations. The Gencos PBUC problem is

further complicated due to the presence of time-dependent constraints such as

minimum up/down time constraints, ramp-rate constraints and wind

generation fluctuation constraints.

In GA representation, the ON/OFF status of each generating unit

has been characterized in the form of binary strings. Also, the ramp rate

constraints limit the maximum possible loading of a generating unit as a

function of number of hours that unit has been ON and number of hours to the

next shut down interval. With the help of genetic operators, the GA approach

avoids premature convergence and helps to find the feasible solutions in the

early stages of the stochastic search process. Hence, GA provides a near

global optimum solution with less computational time. The overall procedure

for implementing Genetic Algorithm to solve the Gencos PBUC problem is

shown in Figure 4.3.

4.3.1 Choice of Genetic Algorithm Parameters

From various literatures, it has been found that the parameters

which control the GA can have a significant influence on its performance.

The GA parameters such as the population size (L), crossover probability (pc)

and mutation probability (pm) are among the major control parameters of the

genetic algorithm. Other factors such as binary coding, fitness function

evaluation and selection mechanism are also equally important and must be

decided to achieve the best optimal solution.

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Figure 4.3 Flowchart for GA approach to Gencos PBUC

4.3.2 Initial Population of Chromosomes

The size of the population is one of the major GA control

parameters. Population size represents the number of numerous chromosomes

in one iteration. If there are too few chromosomes, then GA performance is

Yes

No

Evaluate objective function and Calculate profit of PBUC problem

Is stopping criterion satisfied?

Stop

Start

Initialize GA population and iter_no.

Apply two point Crossover

Perform standard Mutation

Print optimum results

Select fittest parents by stochastic Roulette wheel selection

iter_no = iter_no + 1

Input forecasted value of demand, spot price and system data

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dominated by a single chromosome. It causes lack in diversity due to the fact

that a small part of search space was explored. On the other hand, if there are

too many chromosomes, then the solution using GA slows down the

convergence due to proportional increase in computational time. Also the

performance of good quality chromosomes has been prevented from

propagation of its features to future iterations. Therefore, moderate size of

population is best suited for many practical problems as described in Azadeh

et al (2012) and Richter et al (2000).

Figure 4.4 Population of PBUC schedules

The population of chromosomes (PBUC schedules) for the Gencos

PBUC problem is shown in Figure 4.4 and this population of chromosome is

called as parent. In this research work, the population size is selected based on

the total number of generating units and scheduling time period. The

population size of 60 is considered for standard 10 thermal generating unit

test system and 30 has been considered for standard 3 unit test system.

PBUC schedule L Hour 1 2 3 4 5 6 7 ……………24

PBUC schedule 1 Hour 1 2 3 4 5 6 7 ……………24 T1: 1 1 1 1 1 1 1 …………… 1 T2: 1 1 1 1 1 1 1 …………… 1 T3: 0 0 0 0 0 1 1 …………… 0 T4: 0 0 1 1 1 1 1 …………… 0 ……T10: 0 0 0 0 0 0 0 …………… 0

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4.3.3 Binary Coding

The binary coding must be careful ly designed to transfer the

information between chromosome strings and objective function (profit

maximization function) of the problem. The most commonly used way of

coding the generating unit status is binary string. Here, each chromosome

(PBUC schedule) is represented by a binary string and each bit in the string

represents the significant characteristics of the solution. Therefore, a

chromosome must contain binary information about the solution that

represents. By considering the generating unit ON/OFF status as the main

decision variable, the power generation limits of each generating unit are

represented by a 12-bit binary number. The binary string representation is

given in the following manner:

Chromosome 1: 110110010011

Chromosome 2: 100111100001

Chromosome 3: 001000110010

This binary representation implicitly takes care of the minimum and

maximum limits of the generating unit as given in the Equation (4.1), because

the binary representation is made to assure the ratings between the limits.

)()()( maxmin iPiPiP TGTGTG (4.1)

Once the required accuracy of the output of a generating unit is

decided, then the number of binary bits required to represent each unit's

output can be calculated. Since each unit must be loaded within the limits of

)(min iPTG and )(max iPTG , the value of ),( tiPTG in each interval has been represented

by a string length (lsi) as given (Kothari & Dhillon 2004) in the

Equation (4.2).

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PPiPiP

l TGTGsi

})()({log

minmax

2 (4.2)

where P is the variation in power output of generating unit ‘i’ and the length

of kth chromosome (lki) is given by the Equation (4.3).

12

1............2,1

ssiki Nill (4.3)

The total string length is obtained by concatenating the binary bits

that represent the generating unit’s output. Finally the unit status of each

generator is evaluated in the form of binary values.

4.3.4 Fitness Function Evaluation

The fitness function is derived from the problem objective function

of profit maximization function as given in the Equation (4.4).

TCTRPFmax (4.4)

The profit maximization function for the thermal PBUC problem is

given by the Equation (4.5).

TG

TGTG

N

i

T

tTGTGTGTGTG

N

i

T

tTG

N

i

T

tTG

tiUtiRtiPFtiSDtiUStiPF

tiUtPRtiRtiUtSPtiPPF

1 1

1 11 1

),()),(),((),(),()),((

),()(),(),()(),(max

(4.5)

Similarly, the profit maximization function for the wind-thermal

PBUC problem is given by the Equation (4.6).

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WG

TG

WGTGTG

N

j

T

tWG

N

i

T

tTGTGTGTGTG

N

j

T

tGW

N

i

T

tTG

N

i

T

tTG

jFOCWGjVOCWGtjP

tiUtiRtiPFtiSDtiUStiPF

tjVtSPtjPtiUtPRtiRtiUtSPtiPPF

1 1

1 1

1 11 11 1

)())(),((

),()),(),((),(),()),((

),()(),(),()(),(),()(),(max

(4.6)

Based on the objective function of Gencos PBUC problem, the

fitness function is formed as follows:

tfuelthermal

profitGencoffittness i cosmin

max(4.7)

The genetic algorithm searches for the optimal solution by

maximizing the fitness function. After evaluating the fitness function, fitness

scaling was applied to the fitness values to prevent the domination of super

individual chromosome which often leads to premature convergence in

subsequent generations. The fitness scaling (FIT) adopted in this work is

given by the Equation (4.8).

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1

max

FF

wFIT (4.8)

where w is a scaling coefficient and is the smallest fitness function value in

the population, thresholdfF /1 and the value of fthresold =0.0001; Fmax is the

maximum of F within the population.

The fitness scaling is used to speed up the convergence of the

evolutionary process. It mainly fixes the relative spread between the highest

objective function values and the average objective function values occurring

in a population. Hence, the convergence problems can be avoided by properly

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scaling or spreading out the objective function values for all the population

members before selection.

4.3.5 Stochastic Roulette Wheel Selection

After calculating the fitness function of each individual, the

stochastic Roulette wheel selection technique (Kothari & Dhillon 2004) is

used to select the best parents according to their fitness values. In Roulette

wheel selection, by repeatedly spinning the wheel, individuals are chosen

using a stochastic sampling with replacement until the population is filled.

The individuals are selected probabilistically according to the cumulative

probability of each string and no new strings are formed in the selection

process. Using the fitness value fi of all strings, the probability pi for selecting

the ith string is given below:

L

kk

ii

f

fp

1

(4.9)

where L is the population size; fi is the fitness of ith string and fk is the sum of

fitness values of all the strings in the population.

The Roulette wheel is spun ‘L’ times, each time the pointer of the

Roulette wheel selects the string. The ‘L’ random numbers between 0 and 1

are generated at random in order to choose ‘l’ strings. If the chosen random

number for a string is in the cumulative probability range, then that string is

copied for the mating pool. Each time an individual is selected, the size of

selection segment is reduced by1 and while the segment size becomes

negative, then the selection probability is set to zero.

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4.3.6 Crossover Operation

The crossover operator basically combines the substructures of two

parent chromosomes to produce new chromosomes. In crossover operation,

the information is exchanged among the strings of mating pool to create new

strings. If the good quality substrings from the parents get combined by

crossover, then the children (offspring) are likely to have improved fitness.

Therefore, it is necessary to preserve some of the high-quality strings in the

mating pool, not all the strings used in crossover.

Figure 4.5 Two point crossover

The two point crossover has been considered to create candidate

offspring in this research work as shown in Figure 4.5. For two point or multi-

point crossover operations, only the string bits lying between the randomly

chosen crossover points are exchanged among the two parents to create the

offspring. Also, a crossover probability (pc) is used to decide whether a given

part of the mating pool will be crossed. The probability for crossover

operation is Pc =0.8 in this work. This pc controls the rate at which the

crossover occurs for every chromosome in the search process. The higher the

value, more quickly the new chromosomes are introduced into the population.

If the crossover rate is too large, high performance chromosomes are

discarded faster, while if the rate is too low, the search may stagnate. Hence,

the crossover operator is mainly responsible for the global search property of

the GA approach.

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4.3.7 Mutation Operation

In this research work, the standard flip mutation is applied to the

new offsprings which are created from crossover process. In this mutation

process, flip the bits of chromosome means changing the bits from 1 to 0 or 0

to 1. The flip mutation as shown in Figure 4.6 involves turning a randomly

selected generating unit ON or OFF within a given PBUC schedule in such a

way that to achieve the optimal unit commitment schedule. The mutation

operator has been applied with a small probability of 0.002. Because, the

mutation probability (pm) decides how often the strings of chromosome get

mutated. If the pm = 100%, the entire length of chromosome is changed, while

if the pm = 0%, there is no mutation and the offspring are generated

immediately after crossover (or directly copied) without any change.

Figure 4.6 Flip mutation process

Mutation process leads an offspring to have its own uniqueness.

Usually a very low mutation rate is selected to decrease the amount of

randomness introduced into the solution. Selecting a proper value of mutation

operator is the key, to avoid GA getting captured from local minima.

4.4 GA SIMULATION PARAMETERS

Simulations were carried out to find out the accuracy of GA method

for solving the Gencos PBUC problem with the following simulation

parameters:

Population size (L1) =60 (for Genco-I and III)

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Population size (L2) =30 (for Genco-II only)

Crossover probability (pc) =0.8

Mutation probability (pm) =0.002 and

Maximum no. of iterations =200.

4.5 SIMULATION RESULTS OF GA TECHNIQUE FOR

GENCOS PBUC PROBLEM

The performance of GA method to solve the Gencos PBUC problem

was tested by MATLAB simulation. Three different Gencos which are

described in chapter 3 has been taken as case study. In GA approach, the

objective function and the scheduling time period of the Gencos PBUC

problem are represented based on the 12-bit binary coding. The numerical

results of thermal and wind-thermal PBUC problem are also been discussed in

this section.

4.5.1 Case Study-Genco I: Results of 10 Thermal Units with 24 Hours

System

For Genco-I, the scheduling time period was segregated as 6 cycles

and each cycle consists of 4 hours. The summary of optimal output of the GA

method for Genco-I is given in Table 4.1, where the thermal generating units

T9 and T10 are continuously OFF during the scheduling time period.

Because, the Genco-I produces the power below forecasted level in some of

the operating time period as given in the Equation (2.9). When L1 = 60, the

total cost is 24539390 and the profit is 4457610. As stated in section 4.3.2,

the size of population controls the optimal performance of the system. The

effect of population size on total cost and profit is shown in Figure 4.7, where

the population size is varied from 20 to 80.

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When L1 = 20, the results are not satisfactory, because there are

fewer chromosomes and hence lack of diversity in the population. Although,

the total cost is reduced and profit is increased when L1=80, the convergence

time is too high as shown in Figure 4.8. The effect of population size (varied

from 20 to 80) on computational time is also simulated and it produces the

optimal computational time in 60 population size. When L1 = 120, the GA

method get diverged and produced sub-optimal results with high

computational time of 210 seconds.

Table 4.1 Optimal output of Genco-I for the population size of 60

HourPower scheduling of Thermal units (MW) Total

Revenue ()

Total Cost ()

Profit ()T1 T2 T3 T4 T5 T6 T7 T8 T9 T10

1 455 245 0 0 0 0 0 0 0 0 749000 615589.8 133410.22 455 295 0 0 0 0 0 0 0 0 802500 662900 139600 3 455 395 0 0 0 0 0 0 0 0 909500 747900 161600 4 455 455 0 0 0 0 0 0 0 0 1016500 799000 217500 5 455 455 0 70 0 20 0 0 0 0 1070000 885700 184300 6 455 455 0 130 0 20 0 0 0 0 1177000 936400 240600 7 455 455 90 130 0 20 0 0 0 0 1230500 973900 256600 8 455 455 130 130 0 20 0 0 0 0 1284000 1072200 211800 9 455 455 130 130 110 20 0 0 0 0 1391000 1120700 270300

10 455 455 130 130 150 70 10 0 0 0 1498000 1261800 236200 11 455 455 130 130 162 80 28 10 0 0 1551500 1347300 204200 12 455 455 130 130 162 80 78 10 0 0 1605000 1357000 248000 13 455 455 130 130 162 20 38 10 0 0 1498000 1287500 210500 14 455 455 130 130 85 20 25 0 0 0 1391000 1198000 193000 15 455 455 90 130 25 20 25 0 0 0 1284000 1119500 164500 16 455 455 0 130 25 20 0 0 0 0 1123500 983500 140000 17 455 370 0 130 25 20 0 0 0 0 1070000 947500 122500 18 455 455 0 130 25 20 0 0 0 0 1177000 1018300 158700 19 455 455 0 130 125 20 0 10 0 0 1284000 1101800 182200 20 455 455 0 130 160 20 0 10 0 0 1498000 1309800 188200 21 455 455 0 130 160 20 0 10 0 0 1391000 1170500 220500 22 455 455 0 130 60 0 0 0 0 0 1177000 988100 188900 23 455 445 0 0 0 0 0 0 0 0 963000 851100 111900 24 455 345 0 0 0 0 0 0 0 0 856000 783400 72600

Total 28997000 24539390 4457610Computational time 125 seconds

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Figure 4.7 Effect of population size on total cost and profit for Genco-I

Figure 4.8 Effect of population size on computational time for Genco-I

While comparing the performance of LR and GA methods for

Genco-I as shown in Figure 4.9, the GA approach produces 4.16% higher

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profit than Lagrange relaxation method. Also, the savings in thermal fuel cost

is about 6.37% with the computational time of 125seconds.

Figure 4.9 Profit comparison of LR and GA methods for Genco-I

4.5.2 Case Study-Genco II: Results of 3 Thermal Units with 12 Hours

System

The optimal results of GA approach for Genco-II is given in

Table 4.2. From the results, it can be observed that the optimal solution is

produced at the population size of 30. Not only the total cost and

computational times are optimized, the profit of Genco-II are also optimized

to best possible solution in the population size of 30. It has been indicated in

Figure 4.10. The profit comparison between LR method and GA approach for

Genco-II is shown in Figure 4.11. The average profit for 12 hours achieved by

the GA method is about 1.16 times greater than that of LR technique

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Table 4.2 Optimal results of GA method for Genco-II with 30 population size

Hour Demand(MW)

T1 T2 T3 Revenue()

Total cost ()

Profit ()

1 170 0 0 1 95985.00 69550.00 26435.00 2 250 0 0 1 110455.00 82500.00 27955.00 3 400 0 0 1 94600.00 79750.00 14850.00 4 520 0 0 1 96275.00 77000.00 19275.00 5 700 0 1 1 301465.00 291500.00 9965.00 6 1050 0 1 1 360355.50 292600.00 67755.50 7 1100 0 1 1 362625.00 294250.00 68375.00 8 800 0 1 1 345350.00 296450.00 48900.00 9 650 0 1 1 336265.00 297000.00 39265.00 10 330 0 1 1 200930.00 158680.00 42250.00 11 400 0 1 1 232340.00 192500.00 39840.00 12 550 0 1 1 318135.00 269860.00 48275.00

Total 2854780.50 2401640.00 453140.50Computational time 75 seconds

Figure 4.10 Effect of population size on computational time for Genco-II

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Figure 4.11 Profit comparison of LR and GA methods for Genco-II

However with the GA technique, there is a significant performance

improvement over the LR method with reduced computational time of 75

seconds. Moreover, while running the GA program beyond the maximum of

200 iterations, has not resulted in any considerable improvements in solution

quality.

4.5.3 Case Study-Genco III: Results of 10 Thermal + 2 Wind Units

with 24 Hours System

The commitment schedule of both thermal and wind-thermal

generating units by GA method is given in Table 4.3. During PBUC

scheduling process, only thermal units are committed from 1st hour to 7th hour

of 24hours scheduled time period and then wind generating units are taken

into the commitment schedule from 8th hour onwards, due to the fact that the

cut-in speed of both the wind power units W1 and W2 is 3.5m/sec.

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Table 4.3 Optimal PBUC schedule of GA method for Genco-I and Genco-III

HourLoad

Demand (MW)

Commitment status of Thermal and Wind units Thermal PBUC schedule Wind-Thermal PBUC schedule

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 W1 W21 700 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 2 750 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 3 850 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 4 950 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 5 1000 1 1 0 1 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 6 1100 1 1 0 1 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 7 1150 1 1 1 1 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 8 1200 1 1 1 1 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 9 1300 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1

10 1400 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 0 0 1 1 11 1450 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 0 0 1 1 12 1500 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 0 0 1 1 13 1400 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 0 0 0 1 1 14 1300 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 15 1200 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 16 1050 1 1 0 1 1 1 0 0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 17 1000 1 1 0 1 1 1 0 0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 18 1100 1 1 0 1 1 1 0 0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 19 1200 1 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 0 0 0 0 1 0 20 1400 1 1 0 1 1 1 0 1 0 0 1 1 0 1 1 1 0 0 0 0 1 1 21 1300 1 1 0 1 1 1 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 22 1100 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 23 900 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 24 800 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0

The scheduled power of thermal and wind power generating units

over 24 hours time period is shown in Figure 4.12. The peak load demand of

1500MW is supplied by 1390MW from thermal units and 110MW from wind

generating units. Hence, the profit of Genco-III has increased by about 0.12

times the Genco-I and the savings in thermal fuel cost is about 4.83%. By

varying the population size from 20 to 80, the performance of GA method for

Genco-III is observed and is shown in Figure 4.13. Though the results of 60

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and 80 population size are nearer, the convergence time for the population 80

is more than that of 60 population.

Figure 4.12 Scheduled power of thermal and wind power generating units

Figure 4.13 Revenue, total cost and profit of Genco-III with different population size

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Table 4.4 Results of 20 test trials by GA method for Genco-I and Genco-III

Genco-I: 10 Thermal units only

Genco-III: 10 Thermal +2 Wind units

Total cost () Profit () Total cost () Profit ()

1st trial 24540975 4454525 23004195 51838552nd trial 24596320 4446025 23001088 51846573rd trial 24563985 4452650 23086892 51713254th trial 24566745 4450530 23011091 51836005th trial 24539550 4457255 23001115 51843256th trial 24592730 4448183 23069268 51721737th trial 24560865 4453155 23003326 51839648th trial 24628970 4444761 23089700 51696259th trial 24649218 4443925 23077894 5172082

10th trial 24543985 4457105 23002055 518417611th trial 24579896 4449629 23092658 516821712th trial 24627025 4445875 23012340 518121513th trial 24669680 4440115 23004224 518364714th trial 24541836 4454270 23058997 517520815th trial 24540135 4455892 23002070 518409516th trial 24539086 4456975 23087995 517045017th trial 24539390 4457610 23048456 517634518th trial 24650724 4441268 23005178 518360819th trial 24587615 4448265 23013673 517845620th trial 24553525 4453248 23003082 5183981Average 24580612.80 4450563.05 23033765.00 5178750.20

Best 24539390 4457610 23001088 5184657% deviation 0.175% 0.124% 0.162% 0.120%

Parameters Population size=60;

No. of iterations=200 standard deviation ‘1’

Population size=60; No. of iterations=200 standard deviation ‘1’

Average computational

time 127.25 seconds 117.50 seconds

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Due to the stochastic nature of GA, 20 independent test trials are

made for each population set, with each run starting with different random

initial population. The test results of Genco-I and Genco-III are given in

Table 4.4 and it can be observed that the best and average values of total cost

and profit over the 20 test trials are close to each other. Each run of PBUC

program was terminated after a set number of iterations and the final solution

obtained is printed.

4.6 SUMMARY

The Genetic Algorithm is a stochastic search technique for obtaining

appropriate solution in the large discrete solution spaces of the given problem.

GA has been described with its significant characteristics to solve the

complex optimization problem in the first part of this chapter. Then the GA

method was employed to solve the thermal PBUC and wind-thermal PBUC

problems in deregulated environment. While implementing GA approach to

the Gencos PBUC problem, some issues have been observed. Primarily, GA

does not result in the same solution with every run of the algorithm due to the

stochastic nature of GA and it leads to less guaranteed convergence for the

Gencos PBUC problem. Even though several if-loops are used to cut down

the computational time with respect to infeasible states, GA has taken more

amount of time to solve the PBUC problem for a particular hour. As the load

demand varies, the number of possible states increases exponentially and this

exponential growth is directly proportional to the GA computation time and

memory requirement.

However, GA is a powerful tool for solving the difficult, high-

dimensional optimization problem; it has the major limitations of premature

convergence and lack of good local search ability. Therefore, clonal selection

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based Artificial Immune System (AIS) algorithm inspired by the

immunological principles are considered to overcome the drawbacks of GA

method in the next chapter. The AIS method is found to be accurate and time

efficient for a given system, greatly speeding up the convergence calculation

with lesser memory.