DocumeOPTIMAL POWER DISPATCH IN MULTI NODE ELECTRICITY MARKET USING GENETIC ALGORITHM

7/29/2019 DocumeOPTIMAL POWER DISPATCH IN MULTI NODE ELECTRICITY MARKET USING GENETIC ALGORITHM

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Chapter 1. Introduction

In 1988 almost all electric power utilities throughout the world operated with an

organizational model in which one controlling authority-the utility-operated the

generation, transmission, and distribution systems located in a fixed geographic area.

Economists for some time had questioned whether this monopoly organization was

efficient. With the example of the economic benefits to society resulting from the

deregulation of other industries such as telecommunications and airlines, and in a

political climate friendly to the notion of deregulation, the United Kingdom was the first

to restructure its nationally owned power system, creating privately owned companies to

compete with each other to sell electric energy. Deregulation followed in Norway,

Australia, and New Zealand, and then, in the 1992 National Energy Policy Act (NEPA),

in the United States.

The electricity industries in number of countries have recently been deregulated to

introduce competition. In a centralized power industry, the planning is done to minimise

the production cost. In a competitive electricity market, generation resources are,

scheduled based on offers and bids of the suppliers and consumers. Many approacheshave been proposed in literature to solve the optimal power dispatch problem for

electricity markets [1,3,4].

One of the competitive electricity market models is the auction market model, in which

participants place their bids to sell or buy electricity. In an electricity auction market, the

two main participants are distribution companies and generation companies. These

participants will submit their bids to an independent system operation (ISO) company. A

supply bid is given as a cost per MW and a quantity in MW which a generation company

is willing to generate in a particular period. Each generation company may place several

bids. A demand bid is given as a cost per MW and a quantity in MW which a distribution

company is willing to consume in a particular period. Several demand bids may be placed

by each distribution company.

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A strong motive for considering auctions for the pricing of electric power is given by the

assumption that the electric power industry will move from regulated rate of return

pricing to market-based pricing in the near future. This requires consideration of various

pricing mechanisms. An additional reason is that the natural gas industry spent much time

and effort in researching auction mechanisms for the pricing of natural gas when their

industry underwent deregulation. The electric power industry is quite similar to the

natural gas industry in that both industries produce, transport, and sell their respective

commodities. The need for a pricing mechanism coupled with the example of the natural

gas industry is sufficient reason for considering auctions in the electric power pricing

arena.

The optimal power dispatch models proposed by several researchers [1,3,4] have the

objective to maximize the total benefit to the participants in the multinode auction

market. This thesis demonstrates the application of a genetic algorithm to solve the

optimal power dispatch problem for a multi-node auction market. The model used in this

thesis, like most of the models available in literature, does not directly consider the

reactive power market and the transmission cost. The advantage of the proposed genetic

algorithm is the simplicity of handling non-linear constraints, without having to simplify

the power flow constraints. In addition, the algorithm is easy to implement and additional

features such as security constraints can be easily incorporated in the algorithm.

A new model using genetic algorithm is developed to solve the optimal power dispatch

problem for a multi-node auction market. The above methods are tested on 17-node 26-

line system and compared to demonstrate their performance.

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Chapter 2. Power System Deregulation

2.1 Introduction

Electrical power industry has been dominated by large utilities that have overall

authorities overall activities in generation, transmission and distribution of power refer to

as vertically integrated utilities. During the nineties many electrical utilities and power

network companies world wide have been forced to changed their ways of doing business

from vertically integrated mechanism to open market system. This kind of process is

called as deregulation or restructuring.

Deregulation word refers to un-bundling of electrical utility or restructuring of electrical

utility and allowing private companies to participate. The aim of deregulation is to

introduce an element of competition into electrical energy delivery and thereby allow

market forces to price energy at low rates for the customer and higher efficiency for the

suppliers.

2.2 Vertically Integrated Electrical Utility (VIEU)

VIEU is referred as Regulated Electrical Power Industry.

Regulation means that the Government has set down laws and rules that put limits on

and define how a particular industry or company can operate.

2.2.1 Need for regulation

1. Risk free way to finance the creation of electric industry

2. Recognition and support from local government to utilities

3. Assured return on investments

4. Establishment of local monopoly

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In Fig. 2-1 shows the basic structure of regulated power system, in which one controlling

authority-the utility-operated the generation, transmission and distribution systems

located in a fixed geographic area.

Fig. 2-1: Basic structure of VIEU

2.2.2 Features of VIEU

1. Overall authority, overall activities in generation transmission distribution of

power utility lie within its domain of operation.

2. VIEU will be the only electricity provider in the region and it has obligation to

provide electricity to every one in the region.

3. Information flow is a bilateral one between generation and transmission system

but money flow was unidirectional.

2.2.3 Demerits of VIEU

1. It was often difficult to regulate the cost incurred in generation transmission and

losses occurred in distribution.

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2. Losses occurred in distribution is accounted by spreading the cost over all three

components. Hence utilities often charged their customers at an average tariff

depending upon their aggregated cost during the particular period.

3. The prices setting is done by an external regulator agency often involved

considerations other than economics. (Political party interferences or government

policies on new issues etc.)

4. The main objective of VIEU is to minimize the total cost while satisfying all the

associated problems and constraints, but this leads to complex operation issues

because of the big size VIEU. Further VIEU needs centralized planning for long-

term generation, transmission expansion, midterm planning activities such as

maintenance, production scheduling, fuel scheduling for optimal cost.

In spite of all the above demerits VIEU have performed satisfactorily over the long years

with respect to operation, control and planning. But after 1990 there has been very big

mismatch between the growth of the load and the generation expansion. This has led to

ineffective operation of the system. Hence the concept of deregulation has been mooted.

When the generation, transmission and distribution system control are separated in terms

of management and ownership, the power system is said to be deregulated.

2.3 Deregulated electrical power industry

Deregulation in power industry is a restructuring of the rules and economic incentives

that governments set up to control and drive the electric power industry.

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Competitive

Generation

Market

Multiple sellers

Competitive

Retail market

Multiple Buyers

Transco & Disco

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Fig. 2-2: Unbundling the system

Fig. 2-3: Typical configuration of restructured or deregulated power system

Competition Regulated monopoly Competition

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Fig. 2-4: The competition

ISO

ISO was appointed for the whole system and its main responsibility is to keep the system

in balance. i.e.,

Imports + productions = Exports + Consumption + losses.

Thus ISO must be an independent authority without any involvement in market

competition. But it validates all the transactions before the actual operation takes place

from the point of view of security of the systems, congestion management, real time

operation etc.

Responsibilities of Independent System Operator

1. System security and reliability

2. Power delivery

3. Transmission pricing

4. Service quality assurance

5. Promotion of economic efficiency and equity

6. Fair market

Market trader/Market operator (Retailer)

Market operator is an entity in the de-regulated environment and is responsible for the

operation of market trading of electricity. He receives the bid offers from various market

participants and determines the markets price based on certain criteria in accordance with

the market structure.

2.3.1 Need for Deregulation

1. To provide cheaper electricity.

2. To offer greater choice to the customer in purchasing the economic energy.

3. To give more choice of generation.

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4. To offer better services w.r.t power quality i.e. Constant voltage, constant freq.

and uninterrupted power supply.

2.3.2 Benefits of deregulated power system

1. Cheaper electricity.

2. Efficient capacity expansion planning at GENCO level, TRANSCO level and

DISCO level.

3. Pricing is cost effective rather than a set tariff.

4. More choice of generation.

5. Better service is possible.

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Chapter 3. Genetic Algorithms

3.1 Introduction

Genetic Algorithms (GAs) were invented and developed by John Holland. He invented

genetic algorithm with decision theory for discrete domains. Holland emphasized the

importance of recombination in large populations.

Genetic algorithms are search algorithms based on the mechanics of natural selection and

natural genetics, inspired from the biological evolution, survival of the fittest among

string structures with a structured yet, randomized information exchange with in the

population to form a search algorithm with some of the innovative flair of human search.

In every generation a new set of artificial creatures (strings) created using bits and piece

of the old, an occasional new part is tried for good measure. Being randomized GAs

exploit historical information to speculate on new search points with expecting improved

performance. The current literature identifies three main types of search methods or

optimization techniques. They are:

i. Calculus based method

ii. Enumerate method

iii. Random search techniques

Calculus based and enumerative methods are comfortable in their ability to deliver

solutions in applications involving search spaces of limited problem domain. Both

methods are local in scope, the optima they seek are the best in a neighborhood of the

current search point. But in their application to real world of search, which is fraught withdiscontinuities of functions and their derivatives and vast multi-modal noisy search

spaces, they break down on problems of even moderate size and complexity. Their

inability and inefficiency to overcome the local optima and reach the global optimum

make them insufficiently robust, precluding their application to complex problems as

search method.

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On the other hand, random search algorithms managed to overcome the inherent

disabilities of the calculus and enumerative methods. Yet, random schemes that searches

and save the best must also be discounted because of the efficiency requirement. Random

searches, in the long run can be expected to do no better than enumerative schemes. In

our haste to discount strictly random search methods, we must be careful to separate them

from randomized techniques.

The randomized search techniques incorporated the basic advantages of random search

but used it only as a tool to guide a more highly exploitative search. In these methods, the

search is carried out randomly and information gained from a search is used in guiding

the next search. Genetic algorithm is an example of such technique, which drew

inspiration from the robustness of nature.

Genetic algorithms in their quest for robustness surpassed their traditional cousins and

differ in some very fundamental ways. GAs are different in the following aspects:

i. GAs work with a coding of the parameter set, not the parameters themselves.

ii. GAs searches from a population of points, not from a single point as in

conventional search algorithms.

iii. GAs uses objective function information, not derivatives or other auxiliary

knowledge.

iv. GAs use probabilistic tradition rules but not deterministic rules.

In this chapter, Genetic algorithm and its operators have been discussed in detail.

3.2 Phases of Genetic Algorithm

The first step in Genetic Algorithm is the random generation of large number of search

points from the total search space. Each and every point in the search space corresponds

to one set of values for the parameters of the problem. Each parameter is coded with a

string of bits. The individual bit is called gene. The content of each gene is called

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allele. The total string of such genes of all parameters written in a sequence is called

chromosome. So, there exits a chromosome for each point in the search space. The set

of search points selected and used for processing is called a population. i.e., population

is a set of chromosomes. The number of chromosomes in a population is called

population size and the total number of genes in a string is called string length. The

population is evaluated through various operators of GA to generate a new population.

This process is carried out until global optimum point is reached. Typically it consist of

three phases,

i. Generation

ii. Evaluation

iii. Genetic operation

3.2.1. Generation

In this phase number of chromosomes equal to population size is generated and each is of

length equals to string length. The size of population is direct indication of effective

representation of whole search space in one population. The population size affects both

the ultimate performance and efficiency of GA. If it is too small it leads to local optimum

solution. The selection of string length depends on the accuracy and resolution

requirement of the optimization problem. The higher the string length, the better the

accuracy and resolution. But this may lead to slow convergence. Also, the number of

parameters in the problem will have a direct effect on the string length of the

chromosome, for a particular resolution and accuracy requirements the string length is

chosen appropriately. The chromosome should in some way contain the information

about solution, which it represents. After the selection of string length and population

size, the initial population is encoded. Most commonly used encoding schemes are :

a) Binary encoding

In binary encoding every chromosome is a string of bits 0 or 1. The chromosome looks

like

Chromosome 1: 110110010011

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Chromosome 2: 110111100001

Each chromosome has one binary string. Each bit in this string can represent some

characteristic of the solution or the whole string can represent a number.

b) Permutation encoding

In permutation encoding every chromosome is a string of numbers, which represent

number in a sequence. Permutation encoding is only useful for ordering problems. The

chromosomes in this encoding looks like

Chromosome 1: 1 5 3 2 6 4 7 9 8

Chromosome 2 : 8 5 6 7 2 3 1 4 9

c) Value encoding

Direct value encoding can be used in problems, where some complicated value, such as

real numbers, is used. Use of binary encoding for this type of problems would be very

difficult. In the encoding, every chromosome is a string of some values. Values can be

anything connected to problem, real numbers or characteristics to some complicated

objects. The chromosomes in this encoding looks like:

Chromosome 1: 1234 5.3243 0.4556 2.3293 2.4545

Chromosome 2: ABDJEIFJDHDIERJFDLDFLFEGT

Value encoding is very good for some special problems. On the other hand, for this

encoding is often necessary to develop some new crossover and mutation specific for the

problem.

Random generation techniques are used in accomplishing this task. Any of the encoding

techniques can be used but binary encoding is mostly used.

Now, the initial population of chromosomes is decoded and all the parameters are

calculated for each chromosome. This results in a set of solutions whose size is equal to

population size.

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3.2.2. Evaluation

In the evaluation phase, suitability of each of the solutions from the initial set as the

solution of the optimization problem is determined. For this function called fitness

function is defined. This is used as a deterministic tool to evaluate the fitness of eachchromosome. The optimization problem may be minimization or maximization type. In

the case of maximization type, the fitness function can be a function of variables that bear

direct proportionality relationship with the objective function. For minimization type

problems, fitness function can be function of variables that bear inverse proportionality

relationship with the objective function or can be reciprocal of a function of variables

with direct proportionality relation ship with the objective function. In either case, fitness

function is so selected that the most fit solution is the nearest to the global optimum

point. The programmer of GA is allowed to use any fitness function that adheres to the

above requirements. This flexibility with the GA is one of its fortes.

On the whole for a typical optimization problem, evaluation phase consists of calculation

of individual parameters, testing of any equality or inequality constraints that need to be

satisfied, evaluation of objective function, and finally evaluation of fitness from fitness

function. This evaluation is discrete in nature vis--vis some genetic operators which

operate on more than one chromosome at a time.

3.2.3. Genetic operation

In this phase, the objective is the generation of new population from the existing

population with the examination of fitness values of chromosomes and application of

genetic operators. These genetic operators are reproduction, crossover, and mutation.

This phase is carried out if we are not satisfied with the solution obtained earlier. The GA

utilizes the notion of survival of the fittest by transferring the highly fit chromosomes to

the next generation of strings and combining different strings to explore new search

points.

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Reproduction

Reproduction is simply an operator where by an old chromosome is copied into a Mating

pool according to its fitness value. Highly fit chromosomes receive higher number of

copies in the next generation. Copying chromosomes according to their fitness means that

the chromosomes with a higher fitness value have higher probability of contributing one

or more offspring in the next generation.

Crossover

It is recombination operation. Here the gene information (information in a bit) contained

in the two selected parents is utilized in certain fashion to generate two children who bear

some of the useful characteristics of parents and expected to be more fit than parents.

There are various techniques that are used for performing this crossover. But first of all

we need to pick up two parents from the existing population to perform crossover. This

selection can be done using two methods.

a) Random selection b) Roulette Wheel selection

In the random selection technique, the parents are picked up randomly from the existing

population. In roulette wheel selection technique, selection is usually implemented as a

linear search through roulette wheel with slots weighed in proportion to string fitness

values. This is achieved using the following steps.

i. Total sum of the fitness (fitsum) of all the strings is calculated.

ii. A random real number (rand-sum) between 0 and fitsum is generated.

iii. Starting with the first member of existing population, for each member n the

fitness sum of members 1 to n is compared with the randomly generated

number.

iv. If (fitness of member n) > rand-sum, n is selected as parent. Otherwise the

process is continued by incrementing n.

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All the above steps are useful in selecting a parent. Therefore, before performing each

crossover, we have to execute the above steps twice. Obviously, through this Roulette

wheel selection we are giving more reproductive chances to those population members

that are the fit. Thus, we are ensuring that the picking of chromosomes as parents is

according to their objective function values. It is important to note that the convergence

rates and efficiency of GA with roulette wheel selection techniques far vis--vis random

selection technique. In the roulette wheel selection technique, still faster rate of

convergence can be achieved by sorting the members of the population in the descending

order of their fitness before selecting parents.

Now crossover is carried out using any of the following three methods.

a) Simple or Single Point Crossover

b) Multi-point Crossover

c) Uniform Crossover

a) Single Point Crossover

In this method crossover is carried out at a single point. This is illustrated in the

following example. Let Par1 and Par2 be the two parents selected for crossover. Assume

the strings par1 and par2 as below.

Par 1: 1 1 0 0 0 1 0 1 Par 2: 1 0 1 1 0 1 1 1

Now, a crossover site is selected randomly as an integer between 1 and string length. For

illustration the string length is taken as 8, but in the project work we used 10 as string

length. Let this crossover site is 4. Then children Child 1 and Child 2 are generated as

below.

Child 1: 1 2 3 4 5 6 7 8 = 1 1 0 0 0 1 1 1

|

Child 2: 1 2 3 4 5 6 7 8 = 1 0 1 1 0 1 0 1

|

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b) Multi Point Crossover

Two or more crossover points are selected, binary string from the beginning of the

chromosome to the first crossover point is copied from the first parent, the part from the

first to the second crossover point is copied from the other parent and the rest is copied

from the first parent again.

Par1 = 111 010 10

Par2 = 110 011 11

If two crossover points (3 & 6) are selected,

Child1 = 111 011 10

Child2 = 110 010 11

c) Uniform Crossover

In this method, crossover is performed over the entire length of the string of bits. For this,

a mask is generated randomly. This mask is nothing but a string of bits of value 0 or 1

and sizes same as string length. With the information in the mask, we generate the

children as below.

Par1 : 1 1 0 0 1 0 1 1

Par2 : 0 1 0 0 0 1 0 0

Mask : 0 0 1 0 1 1 0 1

Child 1: 1 1 0 0 0 1 1 0 (If mask=0, Child 1= Par 1 & Child 2= Par 2)

Child 2: 01 0 0 1 0 0 1 (If mask=1, Child 1= Par 2 & Child 2= Par 1)

Here we need to generate a mask for each crossover but we dont need to store them, so

number of masks needed is equal to the no of crossover need to be performed. We

generate them as and when required and discard them thereafter.

Thus we have seen that each crossover resulted in two children. So the number of

crossovers required to be performed for next generation depends on the number of

children we need. Usually it is a general practice to copy some of the best parents as it is

into the next generation the required strings as children. This phenomenon of copying

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best parents into the next generation is called Elitism and the number of parents so

copied is indicated by a parameter of GA called Percentage of Elitism (Pe). This is

nothing but the % of parents so copied of the total number of parents. This Elitism is

basically carried out to not to loose the best strings obtained so far which otherwise may

be lost.

In order to control crossover also there is a parameter called Crossover Probability

(Pc). This probability is used as a decision variable before performing the crossover.

This is done as follows. A random number between 0 and 1 is generated and if that

number is less then Pc, crossover is performed. The randomly generated number is

greater than Pc, Child1 and Child2 are directly selected as Par1 and Par2. This is

equivalent to the case of crossover where crossover site is equal to the string length.

There are various other techniques too for implementing the Pc and the programmer of

GA is given freedom to choose any one. But the above technique is mostly used.

Mutation

This operator is capable of creating new genetic material in the population to maintain the

population diversity. It is nothing but random alteration of a bit value at a particular bit

position in the chromosome. The following example illustrates the mutation operation.

Original String: 1011001 Mutation site: 4 (assumption)

String after mutation: 1010001

Some programmers prefer to choose random mutation or alternate bit mutation.

Mutation Probability (Pm) is a parameter used to control the mutation. For each string

a random number between 0 and 1 is generated and compared with the Pm. if it is less

than Pm mutation is performed on the string. Some times mutation is performed bit-by-

bit also instead of strings. These results in substantial increase in CPU time but

performance of GA will not increase to the recognizable extent. So this is usually not

preferred.

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Thus obviously mutation brings in some points from the regions of search space which

otherwise may not be explored. Generally mutation probability will be in the range of

0.001 to 0.01. This concludes the description of Genetic Operators.

3.3 Standard genetic algorithm

Begin

Initialize

chromosomes in the population

evaluate fitness of all chromosomes

do until

number of generations is large enough

do until

the new population is formed

select parents from the old population

produce offsprings via reproduction, crossover or mutation process

evaluate fitness of offsprings

enddo

enddo

end

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Chapter 4. Application of Genetic Algorithm to Optimal

power dispatch

4.1 Problem description for single node electricity market

For a single node auction market, the supply and demand curves at each single node can

be illustrated as shown in Fig. 1. The supply curve is obtained by ordering selling bids in

increasing order of price, where as the demand curve is obtained by ordering buying bids

in decreasing order of price. In this figure, the x-axis gives the cumulative value of the

bidding quantity and the y-axis gives the bidding price. The spot price at a single node is

the price which matches the supply and demand bids, i.e. the point at which the supply

and demand curves intersect each other. At the spot price, the benefit of participants is

maximised and this is illustrated by the shaded area in Fig. 4-1. This single node auction

model can be mathematically described as follows:

Fig. 4-1:An example of the supply and demand curves

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Assuming that there areMksupply bids andNkdemand bids at the kth node. Let Sikbe the

ith supply bid at node kand is given by Sik= {xs

ik, psik}, wherex

sikis the selling price and

psikis the selling quantity. In addition, letBikbe the ith demand bid at node kand is given

byBik = {xdik, p

dik}, wherex

dikis the buying price andpdikis the buying quantity. If kx

denotes the spot price and kp denotes the spot quantity, then the maximum participants

benefit, which is the sum of suppliers benefit and consumers benefit, can be given as

( ) ( )s dk k

s s d d

k k ik ik jk k jk

i M j N

B x x p x x p

= + % % ----------- (4.1)

whered

ikp% ands

ikp% are consumers and suppliers dispatched quantity, respectively,s

kM

andd

kN are the sets of all dispatched suppliers and dispatched consumers, respectively.

The following table 4.1 shows the participants benefit and spot prices for a single node

electricity market of the 17-node, 26-line system.

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Table 4.1

Node Spot price($/MW) Participants benefit($)

1 1.140000 14.4400002 0.700000 376.500000

3 ----- -----

4 1.400000 2.700000

5 1.100000 8.300000

6 1.000000 2.000000

7 0.010000 30.760000

8 ----- -----

9 ----- -----

10 1.000000 12.900000

11 1.400000 34.200000

12 1.000000 4.80000013 1.300000 35.500000

14 ----- -----

15 1.000000 65.500000

16 1.000000 63.800000

17 ----- -----

Total 651.56

4.2 Problem description for multi node electricity market

For a multi-node electricity auction market, there are transmission lines connected

between bidding nodes. The connections result in real power pk and reactive power qk

injection to the network at each node. The real power injection to a node can be modelled

as an additional demand bid (or a supply bid if the real power injection is negative) by the

network for the quantity pkat the selling (or buying) price kx , which is equal to the spot

price. This network effect is described in detail in [1]. As an example, Fig. 4-2 illustrates

the dispatch of the bids when the real power injection is considered. In Fig. 4-2a , the

injection ofPk to the node is supplied by the partly dispatched generator bid. The spot

quantity has increased and the price has not changed. If the injected power is greater than

the undispatched amount of the partly dispatched supply bid then the additional amount

cannot be supplied at the same price. Therefore, the spot price will increase as shown in

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Fig. 4-2b. This will result in displacing some consumers as shown by dc in Fig. 4-2b. It

can be seen in Fig. 4-2 that the spot price and spot quantity may be changed due to the

effect of the real power injection. This may result in changing the sets Bikand Sikof all

dispatched suppliers and dispatched consumers. Consequently, the participants benefit at

node kis now given by [1]

' ( ) ( )s dk k

s s d d

k k ik ik jk k jk k k

i M j N

B x x p x x p x p

= + & &

& % & % &-------------(4.2)

wheres

kM& and

d

kN& are the new sets of all dispatched suppliers and dispatched

consumers respectively, kx& is the new spot price and the last term is the amount paid by

the transmission line. In addition, the total participants benefit at all nodes can be

expressedas

'

1

( ) ( )s dk k

Ks s d d

k k ik ik jk k jk k k

k i M j N

B x x p x x p x p=

= +

& &

& % & % & ------------(4.3)

whereKis the number of nodes.

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Fig. 4-2a

Fig. 4-2b

Fig. 4-2: Examples of the network effects.

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It can be easily seen that the participants benefit at each node ('

kB ) is a function of the

real power injection. Therefore, the optimization problem of the total participants benefit

at all nodes is similar to the conventional optimal load flow problem, with the exception

that the objective is to maximize the participants benefit, rather than minimize thegeneration cost. This optimization problem can be described as

Maximize

1

( ) ( )s dk k

Ks s d d

k ik ik jk k jk k k

k i M j N

x x p x x p x p=

+

& &

& % & % & -------------- (4.4)

subject to the following constraints:

The capacity constraints which provide the limits on real power (pk) and reactive power

(qk) injection to the network by any node, i.e.

k k kp p p ------------------ (4.5)

k k kq q q ------------------ (4.6)

where kp , kp are the minimum and maximum real power injection limit associate with

node k and kq , kq are the minimum and maximum reactive power output limits of

generators associate with node k.

Constraints on the limit of power flow along lines which are given by

kl kl p p --------------------- (4.7)

where klp is the maximum limit of a power flow in a line connecting node kand node l.

In addition, the real and reactive power injection at each node can be determined as a

summation of the real and reactive power flows along lines which are connected to that

node. These are given by

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1

K

k kl

ll k

p p=

= -------------------- (4.8)

1

K

k kl

ll k

q q=

= -------------------- (4.9)

wherepkland qklare the real power and reactive power flow along the transmission line

connecting node k and node l, respectively. Furthermore, the real power and reactive

power flow are given by the following equations

2

( cos( )) ( sin( ))kl kl k k l k l kl k l k l p G v v v B v v = ----------------- (4.10)

2( cos( )) ( sin( ))kl kl k k l k l kl k l k l q B v v v G v v = + ----------------- (4.11)

where Gkl and Bkl are real and imaginary component of the admittance of the line

connecting node k and node l, k and l are angles at node k and l and vk and vlare

voltages at node kand node l.

This optimisation problem has non-linear constraints which is difficult to solve using the

linear programming technique. A genetic algorithm is proposed in the following section

to solve the above problem. The genetic algorithms are simple to implement and it is easy

to incorporate additional constraints into the problem.

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Fig. 4.3 One line diagram of the 17 bus test system

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Table: 1.Transmission line data

Line

data

From

node

To

node

X(pu) B(pu) Capacity(mw)

L1 1 16 0.015 0.06045 960

L2 2 4 0.00115 0.0073 2470

L3 3 1 0.000733 0.002967 858

L4 3 4 0.00065 0.00535 1494

L5 4 5 0.0164 0.0966 286

L6 4 9 0.0678 0.1912 69

L7 5 7 0.0107 0.0631 286

L8 6 4 0.01525 0.07235 488

L9 7 12 0.0014 0.0082 286

L10 8 7 0.00125 0.00925 1144

L11 8 10 0.0099 0.0239 207

L12 9 1 0.1595 0.4272 69

L13 9 11 0.02535 0.06695 138

L14 11 12 0.0008 0.0045 1492

L15 11 14 0.1951 0.3683 61

L16 11 15 0.1467 0.3999 69

L17 12 6 0.0063 0.02995 488

L18 13 11 0.043 0.0823 122

L19 13 12 0.0084 0.0543 488

L20 13 14 0.053167 0.0108 183

L21 14 15 0.0111 0.02405 152

L22 15 12 0.000967 0.008633 975

L23 16 13 0.0046 0.0323 488

L24 16 15 0.00395 0.0271 976

L25 16 17 0.0068 0.0645 747

L26 17 15 0.0023 0.0191 716

Table:2.Reactive power constraints at nodes

Node No Qk(min)MW Qk(max)MW

1 187.6 -119

2 400 -400

3 0 0

4 32.2 -21.68

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5 58.1 -92.4

6 96 -140

7 66 -71.6

8 50 -50

9 0 0

10 64 -72.811 47 -62.6

12 403 -531

13 43.5 -63

14 0 0

15 140 -124

16 468 -432

17 0 0

5. Implementation of genetic algorithm

Several essential schemes need to be designed in order to apply a genetic algorithm to a

multi-node electricity market. These are the encoding scheme, fitness function, crossover

method and control parameters.

5.1. Encoding scheme

The optimisation problem considered in this case is to find the spot price and spot

quantity at all nodes which maximise the participants benefit (given in Eq. (4.4)). As

mentioned earlier, the spot price and quantity at each node depend on the real power (pk)

injection which is in turn depending on the voltage (vk) and the angle ( k ). Therefore, a

candidate solution at each node can be either an array of the real power and reactive

power injection or an array of the voltage and angle. Although an array of random

voltages and angles at all nodes may lead to easy evaluation of power flows (using Eqs.

(4.10) and (4.11)) and real power injections (using Eq. (4.8)) at all nodes, the evaluated

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results are unlikely to satisfy the power capacity limit constraints at all nodes and the line

capacity constraints at all transmission lines.

Fig. 5.1: Encoding Scheme.

On the other hand, with an array of real power and reactive power injections (as shown in

Fig. 4-3), power flows in the network can only be determined via an iterative load flow

solution [6], but the power capacity limit constraints can be incorporated into the

encoding scheme. Both representations were tried during the early part of our work and it

was found that the choice of real power and reactive power is better than voltage and

phase angle. The encoding chromosome consists of 2*10*(K- 1) bits, in which each 10-

bit binary string is used to represent a range between the maximum and minimum real

power (or reactive power) limit at each node. In addition, the real power and reactive

power injection at the reference node can be obtained from the load flow solution.

5.2. Fitness value

The objective function in this optimisation problem is given by Eq. (4.4) and this can be

used as a fitness function in the genetic algorithm. Therefore, the fitness value of each

chromosome can be determined by

1

( ) ( )s dk k

Ks s d d

k ik ik jk k jk k k

k i M j N

F x x p x x p x p=

= +

& &

& % & % & ------------------

(5.12)

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In addition, the load flow problem need to be evaluated for each chromosome to ensure

that none of the power flows along transmission lines violates the line capacity constraint.

In this thesis, the fast-decoupledload flow method is used to solve the load flow problem

[7]. If a chromosome has violated the power flow limit, a penalty value will be assigned

to its fitness. This will result in a small fitness value and the violated chromosome is

unlikely to be selected as a parent in the next reproduction process.

5.3. Crossover and mutation schemes

Several crossover methods have been proposed in the literature, these include one point

crossover, two point crossoverand uniform crossover. The one point crossover method

selects a random crossover point along the parents and swaps binary bits of the parent

chromosomes beyond the selected point. The two point crossover method is similar to

one point crossover except two random crossover positions are selected and binary bits

between two selected points are swapped. In the uniform crossover method, crossover

positions are randomly selected and a binary bit at each selected point is swapped. There

is no simple way of choosing the best crossover method; the success or failure of a

particular crossover method also depends on the selection of the fitness function and

control parameters. A simple mutation method is to randomly toggle the content of each

binary bit position in a chromosome. As an example, if mutation occurring at the third bit

position of the string 1001011 would give 1011011.

5.4. Control parameters

The performance of the genetic algorithm also depends on control parameters, such as

population size, crossover probability and mutation probability. The population size is the

number of chromosomes in each generation, typically the size increasing according to the

problem difficulty. The crossover probability is a probability that crossover occurs after

the reproduction process. Typical value for the crossover probability ranges from 0.5 to

0.95. The mutation probability is the probability of the mutation operator in each bit

position. The mutation probability is typically very small (0.0010.01).

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5.5. Algorithm for multi node electricity market using GA:

1. Read the system data such as no.of buses, no.of lines, slack bus no., line data, bus

data.

2. Read the genetic algorithm data such as Pe, Pc, Pm, maximum no.of generations,

chromosome length, population size.

3. Read the suppliers bidding data and consumers bidding data. A bidding data is

comprised of quantity of power injection, price.

4. Form Ybus with given system data.

5. Form B1, B2 matrices in the fast decoupled algorithm.

6. Generate random population of given population size each having given

chromosome length.

7. Decode the chromosome into decimal value and apply the maximum and

minimum limits.

8. The decoded values gives the real power and reactive power injections at all the

buses except slack bus.

9. Supply these power injections as inputs to fast decoupled load flow.

10. The output of fast decoupled load flow will be voltage magnitudes and phase

angles at all the buses.

11. After the convergence of the load flow compute power flows through all the lines,

real and reactive power injections of slack bus.

12. Evaluate fitness value (total benefit) using Eq.(4.12).

13. Generate a new population from the present population using the following steps:

14. Copy chromosomes with the best fit 10% to the new population.

15. The remaining offsprings can be generated by repeating the following steps until

the new population is filled.

16. Using Roulette wheel technique select two parents.

17. Generate a random number and if it is greater than the crossover probability then

generate two offsprings via the crossover process else the two parents becomes

offsprings.

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18. With the mutation probability, apply the mutation process to the offspring.

19. Check for convergence of genetic algorithm by calculating the error in each

chromosome.

20. If error < tolerance go to step 22 else go to step15.

21. If total no.of offsprings equal to population size update old population with new

population and go to step 7.

22. Print the power injections, voltages, phase angles, power flows, spot price,

participants benefit, lines benefit and total benefit at all the nodes.

23. End.

5.5 Case studies

The genetic algorithm was implemented on a Pentium-133

microcomputer using a C__ programming language and it was applied

to a test system with 17 nodes and 26 lines shown in Fig. 4.3. The

transmission line data of the network are given in Table 1 and the

reactive power capacity limits are given in Table 2. The real power

injection at a given node is maximum when all selling bids are

dispatched. Therefore the maximum possible injection is equal to thetotal amount of power offered by suppliers at that node. Similarly, the

minimum power injection (i.e. maximum negative injection) is when no

selling bids are dispatched and all buying bids are dispatched. In this

case, it is equal to the total amount of power bid by the consumers.

One difficulty in using a genetic algorithm is the selection of the control

parameters and the crossover methods. In this study, genetic

algorithms were executed with different combinations of the control

parameters which were varied from the following list:

. population size: 100, 200, 400

. crossover probability: 0.5, 0.7, 0.9

. mutation probability: 0.01, 0.001

. crossover methods: two-point, uniform

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Each genetic algorithm was run 20 times and each run was carried out

over 200 generations. The average final objective function value of all

runs in each genetic algorithm is used as a measure of the algorithms

performance. The genetic algorithms with the population size of 400

requires a computational time of approximately 20 times that of 200

population for the test system. The results have shown that the

uniform crossover method performs better than the two point

crossover method. In addition, the genetic algorithms performed well

with the population size of 400, the crossover probability of 0.7 and the

mutation probability of 0.01. Nevertheless, most of the genetic

algorithms converged to good solutions for this test system. The result

obtained from the genetic algorithm associated with the chosen control

parameters are given in Tables 57. Table 5 gives the results of the

real power and reactive power injection to the system and the

associated voltages and angles at the nodes. Table 6 gives the results

of the power flows along transmission lines. Table 7 gives the

participants benefits and spot prices at all nodes. It can be seen that

the total participants benefit is 658.96 when trading within the node.

(i.e. without transmission network) When the trading among the nodestake place through the network, the genetic algorithm gives a total

participants benefit of 1278.14. In addition, the spot price differences

among nodes have been decreased due to trading on the network, in

which the spot price at each node is equal or close to 1.20. Results

show that the transaction across the network has resulted in

decreasing the price differences between nodes. Further, the total

benefit to the participants has increased as expected. According to

basic concepts, there cannot be a price difference between two ends of

an a line if the line has not reached the capacity limit. Further, if there

is a price difference due to line capacity limits, the power flow must be

in the direction of the low price node to the high price node. The

results obtained are consistent with the above concepts excepts for

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some lines.These exceptions are lines L3, L4 and L5. Inspection of

dispatched bids at node 3 reveals that, although the price at node 3 is

1.1, it is at the corner of 1.1 and 1.2; and theoretically the price can be

anywhere between 1.1 and 1.2. If the spot price at node 3 is 1.2

instead of 1.1, the direction of power flow in lines L3 and L4 are not

unusual. Further the network topology is such that the route to channel

power from node 2 to 8 is through node 5 (the generation at node 7 is

fully utilised). This explains why the power in line L5 is from node 4 to

5. The network earns a surplus due to price differences at the ends of

lines, while it lose revenue due to power losses in the lines. In this test

system the price differences are very small. As a result the benefit to

the network is very small.

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Table 5: Power injection, voltage and phase angles

Node P(MW) Q(MVAR) Voltage(V) Angle(degree)

1 162.00 139.51 1.05 7.00

2 281.34 11.72 1.05 8.993 -51.16 -38.19 1.049 6.92

4 -123.55 -39.37 1.05 6.90

5 46.40 -36.98 1.05 5.18

6 15.23 30.54 1.05 3.47

7 234.84 81.92 1.05 2.50

8 -156.52 -85.05 1.05 1.72

9 -35.0 -2.17 1.042 2.42

10 -3.26 12.99 1.05 1.67

11 -234.14 -160.73 1.05 1.32

12 231.23 363.40 1.05 1.82

13 -113.96 -60.97 1.05 1.76

14 -30.10 -5.26 1.041 0.751

15 -437.96 -158.88 1.05 0.58

16 443.20 126.14 1.05 4.24

17 -213.00 -310.80 1.0 0.0

Table 6: Power flows

Line no. Pkl (MW) Plk(MW) Line no. Pkl(MW) Plk(MW)

L1 85.64 -84.93 L14 -209.06 209.38

L2 281.34 -279.66 L15 3.35 -3.33

L3 -57.50 57.53 L16 3.14 -3.12

L4 6.33 -6.33 L17 -101.09 101.71

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L5 33.35 -33.18 L18 8.14 -8.11

L6 41.51 -40.40 L19 -2.15 2.15

L7 79.59 -78.96 L20 24.02 -23.26

L8 -86.47 87.57 L21 -3.49 3.63

L9 153.72 -153.41 L22 -273.54 274.20

L10 -159.78 160.08 L23 144.87 -143.99L11 3.26 -3.26 L24 255.23 -252.87

L12 -18.28 18.82 L25 128.01 -126.68

L13 23.68 -23.46 L26 -86.31 87.94

Table 7: Participants benefits and spot prices

Node With Network Effect

Spot

price($/MW)

Total

benefit($)

Participants

benefit($)

Lines

benefit($)

1 1.14 -170.08 14.60 -184.68

2 1.20 118.98 456.60 -337.61

3 1.10 72.48 16.20 56.28

4 1.20 161.66 13.40 148.265 1.10 -42.74 8.30 -51.04

6 1.20 -7.68 10.60 -18.28

7 1.10 2.42 260.75 -258.32

8 1.20 204.72 16.90 187.82

9 1.20 44.7 2.70 42.00

10 1.20 21.71 17.8 3.91

11 1.20 333.16 52.20 280.96

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12 1.20 -245.67 31.80 -277.47

13 1.20 177.25 40.50 136.75

14 1.20 40.82 4.70 36.12

15 1.20 640.16 114.60 525.56

16 1.20 -378.94 157.90 -531.84

17 1.20 295.90 40.30 255.60Total 1278.88 1259.85 14.03

Table 8: Participants benefits and spot prices comparison

Node With Network Effect Without

network

Without

network

Spot

price($/MW)

Total

benefit($)

Participants

benefit($)

Lines

benefit($)

Spot price Participants

benefit

1 1.14 -170.08 14.60 -184.68 1.14 14.60

2 1.20 118.98 456.60 -337.61 0.70 376.50

3 1.10 72.48 16.20 56.28 0.69 15.09

4 1.20 161.66 13.40 148.26 1.40 2.70

5 1.10 -42.74 8.30 -51.04 1.10 8.30

6 1.20 -7.68 10.60 -18.28 1.00 2.00

7 1.10 2.42 260.75 -258.32 0.01 30.76

8 1.20 204.72 16.90 187.82 0.25 35.79

9 1.20 44.7 2.70 42.00 1.00 2.45

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10 1.20 21.71 17.8 3.91 1.00 12.90

11 1.20 333.16 52.20 280.96 1.30 38.20

12 1.20 -245.67 31.80 -277.47 1.00 4.80

13 1.20 177.25 40.50 136.75 1.30 38.90

14 1.20 40.82 4.70 36.12 1.20 35.27

15 1.20 640.16 114.60 525.56 1.00 65.5016 1.20 -378.94 157.90 -531.84 1.00 63.80

17 1.20 295.90 40.30 255.60 1.00 2.24

Total 1278.88 1259.85 14.03 658.96

The following graph shows the error Vs generation .

Fig 6: Graph of Error Vs Generation

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Chapter 6: source code

clc;

clear;

ip=fopen('new1105.in','r+');

op=fopen('new1105.out','w+');

n=fscanf(ip,'%d',1);

fprintf(op,'THE NUMBER OF BUSES ARE %d\n',n);

nline=fscanf(ip,'%d',1);

fprintf(op,'THE NUMBER OF LINES ARE %d\n',nline);

nslack=fscanf(ip,'%d',1);

fprintf(op,'THE SLACK BUS %d\n',nslack);

itermax=fscanf(ip,'%d',1);fprintf(op,'THE MAXIMUM NUMBER OF ITERATIONS ARE

%d\n',itermax);

Linedata=fscanf(ip,'%f',[8,nline]);

Linedata=Linedata';

lp=Linedata(:,1); % ASSIGNING COLUMN 1 OF

DATA TO Fm

lq=Linedata(:,2); % ASSIGNING COLUMN 2 OF

DATA TO To

R=Linedata(:,3); % ASSIGNING COLUMN 3 OF

DATA TO RX=Linedata(:,4); % ASSIGNING COLUMN 4 OF

DATA TO X

ycp=complex(0,Linedata(:,5)); %ASSIGNING

COLUMN 5 OF DATA TO Ycharg

ycq=complex(0,Linedata(:,6));

tap=Linedata(:,7); %ASSIGNING COLUMN 7 OF DATA TO

tap ratios

cap=Linedata(:,8);

fprintf(op,'\nLINE DATA OF THE SYSTEM \n');

fprintf(op,'\nNo Fm TO R(k) X(k) Ycp(k)Ycq(k) tap(k)\n');

for k=1:nline

fprintf(op,'%d\t%d\t%d\t%f\t%f\t%f\t%f\t

%f\n',k,lp(k),lq(k),R(k),X(k),imag(ycp(k)),imag(ycq(k)),tap

(k));

end

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ycap=fscanf(ip,'%f',[1,n]);

ycap=ycap';

fprintf(op,'\nTHE SHUNT ADMITTANCES ARE\n');

for i=1:n

Yshunt(i)=complex(0,ycap(i));

fprintf(op,'%d\t%f\n',i,ycap(i));end

pop_size=40;

chro_size=320;

maxgen=100;

pop=round(rand(pop_size,chro_size));

Busvltg=fscanf(ip,'%f',[6,n]);

Busvltg=Busvltg';

Itype=Busvltg(:,1);

Vspec=Busvltg(:,2);

pmax=Busvltg(:,3);

pmin=Busvltg(:,4);

Qmax=Busvltg(:,5);

Qmin=Busvltg(:,6);

fprintf(op,'\nTHE BUS VOLTAGES ARE\n');

fprintf(op,'bus itype Vspec Qmax

Qmin\n');

for i=1:n

fprintf(op,'%d\t%f\t%f\t%f\t

%f\n',i,Itype(i),Vspec(i),Qmax(i),Qmin(i));end

sell=fscanf(ip,'%f',[14,n]);

sell=sell';

sum_piks=fscanf(ip,'%f',[1,17]);

sum_piks=sum_piks';

xjkd=fscanf(ip,'%f',[1,8]);

pjkd=fscanf(ip,'%f',[8,17]);

pjkd=pjkd';

sum_pjkd=fscanf(ip,'%f',[1,17]);

sum_pjkd=sum_pjkd';j=1;

for i=1:7

xiks(:,i)=sell(:,j);

j=j+1;

piks(:,i)=sell(:,j);

j=j+1;

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end

xk=[1.14

1.2

1.1

1.2

1.11.2

1.1

1.2

1.2

1.2

1.2

1.2

1.2

1.2

1.2

1.2

1.2];

%****************FLAT VOLTAGE

START****************************

for i=1:n

if Itype(i)==1

Ebus(i)=complex(1,0);

else

Ebus(i)=complex( Vspec(i),0); end

Vmag(i)=abs(Ebus(i));

delA(i)=0;

end

%***********************READING BUS DATA

OVER*****************************

%FORMING RESERVATION CHART

for k=1:n

nlcont(k)=0;

endfor k=1:nline

p=lp(k);

q=lq(k);

nlcont(p)=nlcont(p)+1;

nlcont(q)=nlcont(q)+1;

end

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%FORMATION OF ITAGF AND ITAGTO VECTORS

itagf(1)=1;

itagto(1)=nlcont(1);

for i=2:n

itagf(i)=itagto(i-1)+1;

itagto(i)=itagto(i-1)+nlcont(i);end

%FORMATION OF ADJQ AND ADJL VECTORS

for k=1:n

nlcont(k)=0 ; %reintialisation

end

for k=1:nline

p=lp(k);

q=lq(k);

lpq=itagf(p)+nlcont(p);

lqp=itagf(q)+nlcont(q);

nlcont(p)=nlcont(p)+1; %UPDATE NLCOUNT

nlcont(q)=nlcont(q)+1;

adjq(lpq)=q; %FORMING ADJQ VECTOR

adjl(lpq)=k; %FORMING ADJL VECTOR

adjq(lqp)=p;

adjl(lqp)=k;

end

for k=1:nline

Z(k)=complex(R(k),X(k));yline(k)=1/Z(k);

end

%MODELLING OF OFF NOMINAL TAP CHANGING TRANSFORMER

for k=1:nline

a=tap(k);

if a~=1

a1=1-1/a;

a2=-a1/a;

ycp(k)=a2*yline(k);

yline(k)=yline(k)/a;ycq(k)=a1*yline(k);

else,end

end

% FORMATION OF DIAGONAL ELEMENTS OF YBUS

for i=1:n

ypp(i)=complex(0,0);

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end

for k=1:nline

p=lp(k);

q=lq(k);

ypp(p)=ypp(p)+yline(k)+ycp(k);

ypp(q)=ypp(q)+yline(k)+ycq(k);end

for i=1:n

ypp(i)=ypp(i)+Yshunt(i);

end

fprintf(op,'\nTHE DIAGONAL ELEMENTS ARE\n');

for i=1:n

fprintf(op,'ypp(%d)= %f

%fi\n',i,real(ypp(i)),imag(ypp(i)));

end

%FORMATION OF OFF DIAGONAL ELEMENTS OF YBUS

for i=1:2*nline

k=adjl(i);

ypq(i)=-yline(k);

end

fprintf(op, '\nTHE OFF DIAGONAL ELEMENTS ARE\n');

for i=1:2*nline

fprintf(op,'ypq(%d)=%f +

%fi\n',i,real(ypq(i)),imag(ypq(i)));

end%******************FORMING YBUS HAS BEEN

COMPLETED*************************/

%FORMATION OF B1 MATRIX

for i=1:n

for j=1:n

B1(i,j)=0;

B2(i,j)=0;

end

end

for k=1:nlinep=lp(k);

q=lq(k);

temp=1/(X(k));

B1(p,q)=-temp;

B1(q,p)=B1(p,q);

B1(p,p)=B1(p,p)+temp;

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B1(q,q)=B1(q,q)+temp;

end

B1(nslack,nslack)= 10^20;

%Formation of B1 matrices over

for i=1:n

for j=1:n if B1(i,j)~=0

fprintf( op,'B1(%d,%d)=%f+

%f\n',i,j,real(B1(i,j)),imag(B1(i,j)));

end

end

end

%DECOMPOSITION OF B1 MATRIX BY CHOLESKY METHOD

B1(1,1)=sqrt(B1(1,1));

for j=2:n

B1(1,j)=B1(1,j)/B1(1,1);

B1(j,1)=B1(1,j);

end

for i=2:n

for j=i:n

if i==j

sum=0;

for k=1:i-1

sum=sum+B1(i,k)^2;

endB1(i,i)=sqrt(B1(i,i)-sum);

else

sum=0;

for k=1:i-1

sum=sum+B1(i,k)*B1(k,j);

end

B1(i,j)=(B1(i,j)-sum)/B1(i,i);

B1(j,i)=B1(i,j);

end

endend

%Formation of B1 matrices over

%FORMATION OF B2 MATRIX

for i=1:n %diagonal elements

B2(i,i)=-imag(ypp(i));

end

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for i=1:n

Jstart=itagf(i);

Jstop=itagto(i);

for j=Jstart:Jstop

q=adjq(j);

k=adjl(j);B2(i,q)=imag(yline(k)); %off diagonal elements

end

end

for i=1:n

if(Itype(i)==2)

B2(i,i)=10^20;

end

B2(nslack,nslack)=10^20;

end

for i=1:n

for j=1:n

if B2(i,j)~=0

fprintf( op,'B2(%d,%d)=%f\n',i,j,B2(i,j));

end

end

end

%formation of B2 matrix over

%DECOMPOSITION OF B2 MATRIX BY CHOLESKY METHOD

B2(1,1)=sqrt(B2(1,1));for j=2:n

B2(1,j)=B2(1,j)/B2(1,1);

B2(j,1)=B2(1,j);

end

for i=2:n

for j=i:n

if i==j

sum=0;

for k=1:i-1

sum=sum+B2(i,k)^2; end

B2(i,i)=sqrt(B2(i,i)-sum);

else

sum=0;

for k=1:i-1

sum=sum+B2(i,k)*B2(k,j);

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end

B2(i,j)=(B2(i,j)-sum)/B2(i,i);

B2(j,i)=B2(i,j);

end

end

end%**********************************************************

******%

tic

t=0;

generation=1;

while(t==0)

%calculation of coded values

for c=1:pop_size

failure(c)=0;

j1=0;

for i=1:n-1

psp(i)=0;

for j=1:10

psp(i)=psp(i)+(2^-j)*pop(c,j+j1);

end

j1=j1+j;

Pspec(i)=pmin(i)+(pmax(i)-pmin(i))*psp(i);

end

next=i+1;nextnext=2*(n-1);

for i=next:nextnext

l=i+1-next;

x1(l)=0;

for j=1:10

x1(l)=x1(l)+(2^-j)*pop(c,j+j1);

end

j1=j1+j;

Qspec(l)=Qmin(l)+(Qmax(l)-Qmin(l))*x1(l);

endfor iter=0:itermax

dPmax=0;

dQmax=0;

%CALCULATION OF INJECTED POWERS AT ALL BUSES

for i=1:n

if i~=nslack

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Inew=ypp(i)*Ebus(i);

Jstart=itagf(i);

Jstop=itagto(i);

for j=Jstart:Jstop

q=adjq(j);

Inew=Inew+ypq(j)*Ebus(q); end

s=Ebus(i)*conj(Inew);

Pcal(i)=real(s);

Qcal(i)=imag(s);

else, end %if closing

end

% CALCULATION OF INJECTED POWERS OVER

%CALCULATION OF MISMATCHES

for i=1:n

if i~=nslack

dP(i)=Pspec(i)-Pcal(i);

if Itype(i)==1

dQ(i)=Qspec(i)-Qcal(i);

else

dQ(i)=0.0; %for PV bus

end

else

dP(i)=0.0; %for slack bus

dQ(i)=0.0;end

end

dPmax=max(abs(dP));

dQmax=max(abs(dQ)) ;

for i=1:n

dP(i)=dP(i)/Vmag(i);

end

%FORWARD SUBSTITUTION

if(dPmax> 0.0001||dQmax>0.0001)

Y(1)=dP(1)/B1(1,1);for i=2:n

temp=0.0;

for j=1:i-1

temp=temp+B1(i,j)*Y(j);

end

Y(i)=(dP(i)-temp)/B1(i,i);

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end

%BACKWARD SUBSTITUTION

X1(n)=Y(n)/B1(n,n);

for i=n-1:-1:1

temp=0.0;

for j=i+1:ntemp=temp+B1(i,j)*X1(j);

X1(i)=(Y(i)-temp)/B1(i,i);

end

end

%UPDATING PHASE ANGLES

for i=1:n

delA(i)=delA(i)+X1(i);

e(i)= Vmag(i)*cos(delA(i));

f(i)= Vmag(i)*sin(delA(i));

Ebus(i)=complex(e(i),f(i));

end

iter=iter+.5;

else

converged=1;

break;

end

%HALF ITERATION OVER**************************

dPmax=0;

dQmax=0;%CALCULATION OF INJECTED POWERS AT ALL BUSES

for i=1:n

if i~=nslack

Inew=ypp(i)*Ebus(i);

Jstart=itagf(i);

Jstop=itagto(i);

for j=Jstart:Jstop

q=adjq(j);

Inew=Inew+ypq(j)*Ebus(q);

ends=Ebus(i)*conj(Inew);

Pcal(i)=real(s);

Qcal(i)=imag(s);

else, end %if closing

end %end of ith loop

% CALCULATION OF INJECTED POWERS OVER

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for i=1:n

if i~=nslack

dP(i)=Pspec(i)-Pcal(i);

if Itype(i)==1

dQ(i)=Qspec(i)-Qcal(i);

elsedQ(i)=0.0; %for PV bus

end

else

dP(i)=0.0; %for slack bus

dQ(i)=0.0;

end

end

dPmax=max(abs(dP));

dQmax=max(abs(dQ)) ;

for i=1:n

dQ(i)=dQ(i)/Vmag(i);

end

%FORWARD SUBSTITUTION

if(dPmax>0.0001||dQmax>0.0001)

Y(1)=dQ(1)/B2(1,1);

for i=2:n

temp=0.0;

for j=1:i-1

temp=temp+B2(i,j)*Y(j); end

Y(i)=(dQ(i)-temp)/B2(i,i);

end

%BACKWARD SUBSTITUTION

dV(n)=Y(n)/B2(n,n);

for i=n-1:-1:1

temp=0.0;

for j=i+1:n

temp=temp+B2(i,j)*dV(j);

enddV(i)=(Y(i)-temp)/B2(i,i);

end

%UPDATING THE VOLTAGE MAGNITUDES

for i=1:n

Vmag(i)=Vmag(i)+dV(i);

e(i)= Vmag(i)*cos(delA(i));

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f(i)= Vmag(i)*sin(delA(i));

Ebus(i)=complex(e(i),f(i));

end

iter=iter+0.5;

else

converged=1;break;

end

end% iter loop

if(converged==1)

fprintf(op,'PROBLEM CONVERGED IN %f ITERATIONS\n',iter);

fprintf(op,'THE BUS VOLTAGES ARE\n');

for i=1:n

fprintf(op,'%f\t%f\n',Vmag(i),delA(i)*180/3.14);

end

else

fprintf(op,'THE PROBLEM FAILED TO CONVERGE IN %d

ITERATIONS\n',iter);

end

%PROBLEM CONVERGED,COMPUTE POWER FLOWS

for k=1:nline

p=lp(k);

q=lq(k);

temp1=conj(Ebus(p))*((Ebus(p)-Ebus(q))*yline(k)

+Ebus(p)*ycp(k));temp2=conj(Ebus(q))*((Ebus(q)-Ebus(p))*yline(k)

+Ebus(q)*ycq(k));

Ppq(k)=real(temp1);

Pqp(k)=real(temp2);

end

for i=1:nline

fprintf(op,'L%d\t%f\n',i,Ppq(i)*100);

end

limit=zeros(1,nline);

for k=1:nline if abs(Ppq(k))>(cap(k)/100)

limit(k)=1;

% fprintf(op,'\n%dth line limit has exceeded \n',k);

break;

end

%assigning chromosome to 1

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if limit(k)==1

failure(c)=1;

end

end

%calculation of slackbus power

Islack=ypp(nslack)*Ebus(nslack);for i=nslack

jstart=itagf(i);

jto=itagto(i);

for g1=jstart:jto

q=adjq(g1);

Islack=Islack+ypq(g1)*Ebus(q);

end

end

P1=Ebus(nslack)*conj(Islack);

Pslack=real(P1);

Qslack=imag(P1);

fprintf(op,'\nthe active power of slack bus is %f',Pslack);

fprintf(op,'\nthe reactive power of slack bus is

%f',Qslack);

Pk=[Pspec Pslack];

%***********************************************************

%CALCULATION OF FITNESS VALUE

for i=1:17

disps(i)=0; for j=1:7

if xiks(i,j)0

if xk(i)>xiks(i,j)

pbenefit(i)=pbenefit(i)+(xk(i)-

xiks(i,j))*piks(i,j);

end

sdisp=sdisp-piks(i,j);

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end

end

for j=1:8

if xjkd(j)>xk(i)

pbenefit(i)=pbenefit(i)+(xjkd(j)-

xk(i))*pjkd(i,j); end

end

end

lbenefit=0;

for i=1:17

lbenefit(i)=-(xk(i)*(Pk(i)*100));

end

tbenefit(i)=0;

for i=1:17

tbenefit(i)=pbenefit(i)+lbenefit(i);

end

benefit=tbenefit(1);

for i=2:17

benefit=benefit+tbenefit(i);

end

fprintf(op,'\nc=%d,benefit=%f',c,benefit);

benefit

sum2(c)=benefit;

if failure(c)==1sum2(c)=sum2(c)*0.1;

end

end%end of pop_size

population=[pop sum2'];

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%

%Sorting the population

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%

for i=1:pop_sizefor j=i+1:pop_size

if population(j,321)>population(i,321)

temp=population(i,:);

population(i,:)=population(j,:);

population(j,:)=temp;

end

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end

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%

%Calculation of Fitness Sum

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Fitsum=zeros(1,pop_size);

Fitsum(1)=population(1,321);

for i=2:pop_size

Fitsum(i)=Fitsum(i-1)+population(i,321);

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%

population=[population Fitsum'];

%EVALUATION OF NEXT POPULATION

nextpop=zeros(pop_size,chro_size);

for i=1:(.1*pop_size)

for j=1:320

nextpop(i,j)=population(i,j);

end

end

Tfit=population(pop_size,322);

g=(.1*pop_size)+1;

while gr && failure(i)~=1

kk=kk+1;

for h=1:320

parent(kk,h)=population(i,h); end

break;

end

end

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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a=0;

b=1;

r1=a+(b-a)*rand;

%checking for pc

if r1


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nextpop(g,:)=child(1,:);

nextpop(g+1,:)=child(2,:);

g=g+2;

end%end of population generation

%cheking for convergence

error(generation)=abs(population(1,321)-population(pop_size,321));

if(abs(population(1,321)-population(pop_size,321))maxgen)

t=1;

else

generation=generation+1;

pop=nextpop;

end

end

t=toc

k=1:generation;

plot(k,error);

fclose('all');

Chapter 7: Results

THE NUMBER OF BUSES ARE 17THE NUMBER OF LINES ARE 26

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THE SLACK BUS 17

THE MAXIMUM NUMBER OF ITERATIONS ARE 30

LINE DATA OF THE SYSTEM

No Fm TO R(k) X(k) Ycp(k) Ycq(k) tap(k)

1 1 16 0.010500 0.060450 0.000000 0.000000 1.000000

2 2 4 0.002300 0.014000 0.000000 0.000000 1.000000

3 3 1 0.000733 0.002967 0.000000 0.000000 1.000000

4 3 4 0.000650 0.005350 0.000000 0.000000 1.000000

5 4 5 0.016400 0.096600 0.000000 0.000000 1.000000

6 4 9 0.067800 0.191200 0.000000 0.000000 1.000000

7 5 7 0.010700 0.063100 0.000000 0.000000 1.000000

8 6 4 0.015250 0.072350 0.000000 0.000000 1.000000

9 7 12 0.001400 0.008200 0.000000 0.000000 1.000000

10 8 7 0.001250 0.009250 0.000000 0.000000 1.000000

11 8 10 0.009900 0.023900 0.000000 0.000000 1.000000

12 9 1 0.159500 0.427200 0.000000 0.000000 1.000000

13 9 11 0.025350 0.066950 0.000000 0.000000 1.000000

14 11 12 0.000800 0.004500 0.000000 0.000000 1.000000

15 11 14 0.195100 0.368300 0.000000 0.000000 1.000000

16 11 15 0.146700 0.399900 0.000000 0.000000 1.000000

17 12 6 0.006300 0.029950 0.000000 0.000000 1.000000

18 13 11 0.043000 0.082300 0.000000 0.000000 1.000000

19 13 12 0.008400 0.054300 0.000000 0.000000 1.000000

20 13 14 0.053167 0.010800 0.000000 0.000000 1.000000

21 14 15 0.011100 0.024050 0.000000 0.000000 1.000000

22 15 12 0.000967 0.008633 0.000000 0.000000 1.000000

23 16 13 0.004600 0.032300 0.000000 0.000000 1.000000

24 16 15 0.003950 0.027100 0.000000 0.000000 1.000000

25 16 17 0.006800 0.064500 0.000000 0.000000 1.000000

26 17 15 0.002300 0.019100 0.000000 0.000000 1.000000

THE SHUNT ADMITTANCES ARE

1 0.000000

2 0.000000

3 0.000000

4 0.000000

5 0.000000

6 0.000000

7 0.000000

8 0.000000

9 0.000000

10 0.000000

11 0.00000012 0.000000

13 0.000000

14 0.000000

15 0.000000

16 0.000000

17 0.000000

THE BUS VOLTAGES ARE

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bus itype Vspec Qmax Qmin

1 2.000000 1.050000 1.876000 -1.190000

2 2.000000 1.050000 4.000000 -4.000000

3 1.000000 1.050000 0.000000 0.000000

4 2.000000 1.050000 0.322000 -0.216800

5 2.000000 1.030000 0.581000 -0.924000

6 2.000000 1.040000 0.960000 -1.4000007 2.000000 1.030000 0.660000 -0.716000

8 2.000000 1.020000 0.500000 -0.500000

9 1.000000 1.020000 0.000000 0.000000

10 2.000000 1.020000 0.640000 -0.728000

11 2.000000 1.020000 0.470000 -0.626000

12 2.000000 1.020000 4.030000 -5.310000

13 2.000000 1.010000 0.435000 -0.630000

14 1.000000 1.000000 0.000000 0.000000

15 2.000000 1.010000 1.400000 -1.240000

16 2.000000 1.030000 4.680000 -4.320000

17 0.000000 1.000000 0.000000 0.000000

6th line limit has exceeded
















PROBLEM CONVERGED IN 73 GENERATIONS

Power injection,voltage and angle results

-----------------------------------------------

Node P(MW) Q(MVAR) Voltage(V) Angle(degree)

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-----------------------------------------------

1 5.893125 -0.357629 1.050000 4.221735

2 -2.782500 3.031250 1.050000 0.247870

3 -0.069727 0.000000 1.048874 3.533069

4 -0.872461 -0.216800 1.050000 2.334715

5 0.332471 0.144491 1.030000 2.605349

6 0.240937 -0.330625 1.040000 1.0605957 2.324707 0.501437 1.030000 1.497762

8 -0.175312 -0.047852 1.020000 1.662875

9 -0.130313 0.000000 1.027778 0.484511

10 0.351680 -0.085414 1.020000 2.204400

11 -2.590918 -0.530742 1.020000 -0.303904

12 0.213867 2.014238 1.020000 0.300300

13 0.246875 0.387158 1.010000 0.164960

14 -0.359365 0.000000 1.004618 -0.399732

15 -1.146094 1.111250 1.010000 -0.202966

16 -1.518750 2.763984 1.030000 -0.270507

Power flow results

-----------------------------------------------

Line no. Ppq(MW) Pqp(MW)

-----------------------------------------------

L1 142.738696 -140.775589

L2 -278.250000 279.919288

L3 -428.907134 430.164406

L4 421.934534 -420.854679

L5 -1.548066 1.620513L6 20.116044 -19.849499

L7 31.626554 -31.523100

L8 -34.942917 35.121282

L9 281.337198 -280.207456

L10 17.499373 -17.343396

L11 -35.030623 35.167969

L12 -16.019793 16.409369

L13 22.838045 -22.710034

L14 -235.993826 236.436445

L15 2.125563 -2.098666

L16 0.425020 -0.416676

L17 -58.642028 59.036562

L18 3.022868 -2.938628L19 -7.183938 7.213359

L20 13.575977 -13.345797

L21 -20.483598 20.548344

L22 -116.356567 116.586454

L23 -15.064500 15.263302

L24 6.420731 -6.209307

L25 -2.455677 2.604874

L26 12.244453 -12.174439

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the active power of slack bus is 0.148493

the reactive power of slack bus is -1.005658

Participants benefits and spot prices

------------------------------------------------------

Node Spot price Total benefit Participan benefit line benefit

------------------------------------------------------1 1.140000 -657.216250 14.600000 -671.816250

2 1.200000 790.500000 456.600000 333.900000

3 1.100000 23.869922 16.200000 7.669922

4 1.200000 118.095312 13.400000 104.695312

5 1.100000 -28.271777 8.300000 -36.571777

6 1.200000 -18.312500 10.600000 -28.912500

7 1.100000 5.032227 260.750000 -255.717773

8 1.200000 37.937500 16.900000 21.037500

9 1.200000 18.337500 2.700000 15.637500

10 1.200000 -24.401563 17.800000 -42.201563

11 1.200000 363.110156 52.200000 310.910156

12 1.200000 6.135937 31.800000 -25.664063

13 1.200000 10.875000 40.500000 -29.62500014 1.200000 47.823828 4.700000 43.123828

15 1.200000 252.131250 114.600000 137.531250

16 1.200000 348.150000 157.900000 182.250000

17 1.200000 22.480808 40.300000 -17.819192

------------------------------------------------------

1278.003 1259.850000 48.427351

THE EXECUTION TIME IN SEC=9.734134

benefit =1.2717e+003


benefit = 976.2094


benefit = 1.2461e+003

benefit = 1.2718e+003

benefit = 1.2403e+003

benefit = 1.2234e+003

benefit = 1.2564e+003

benefit = 1.1792e+003

benefit = 1.1752e+003

benefit = 1.2414e+003

benefit = 1.2785e+003

benefit = 1.1287e+003

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benefit = 1.2381e+003

benefit = 1.0501e+003

benefit = 1.2595e+003

benefit = 1.2654e+003

benefit = 1.0235e+003

benefit = 1.2377e+003

benefit = 1.2573e+003

benefit = 1.2306e+003

benefit = 1.1140e+003

benefit = 1.1241e+003

benefit = 1.1508e+003

benefit = 1.2557e+003

benefit = 1.1563e+003

benefit = 1.2443e+003

benefit = 1.2784e+003

benefit = 1.2332e+003

benefit = 1.2355e+003

benefit = 1.1013e+003

benefit = 1.1821e+003

benefit = 1.1010e+003

benefit = 1.1176e+003

benefit = 1.2389e+003

benefit = 1.1565e+003

benefit = 1.0173e+003

benefit = 1.0964e+003

benefit = 1.2785e+003

benefit = 1.2784e+003

benefit = 1.2718e+003

benefit = 1.2717e+003

benefit = 1.2414e+003

benefit = 932.5910

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benefit =976.2076

benefit = 1.2785e+003

benefit = 1.1563e+003

benefit = 1.2595e+003

benefit = 1.1792e+003

benefit = 1.2377e+003

benefit = 1.2403e+003

benefit = 1.2573e+003

benefit = 1.2785e+003

benefit = 1.1010e+003

benefit = 932.5909

benefit = 1.1176e+003

benefit = 1.1563e+003

benefit = 1.2654e+003

benefit = 1.2785e+003

benefit = 1.2381e+003

benefit = 1.1010e+003

benefit = 1.2557e+003

benefit = 1.2332e+003

benefit = 1.2414e+003

benefit = 1.0612e+003

benefit = 1.2654e+003

benefit = 1.1792e+003

benefit = 1.0173e+003

benefit = 1.1140e+003

benefit = 1.0964e+003

benefit = 1.0964e+003

benefit = 1.1508e+003

benefit = 1.1324e+003

benefit = 956.3551

benefit = 1.2654e+003

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benefit = 1.2785e+003

benefit = 1.2304e+003

benefit = 1.2403e+003

benefit = 1.2785e+003

benefit = 1.2785e+003

benefit = 1.2785e+003

benefit = 1.2785e+003

benefit = 1.2332e+003

benefit = 1.2573e+003

benefit = 1.1140e+003

benefit = 1.2654e+003

benefit = 1.2654e+003

benefit = 1.2377e+003


benefit = 1.1792e+003

benefit = 1.2403e+003

benefit = 1.1563e+003

benefit = 1.2717e+003

benefit = 1.0964e+003

benefit = 1.0173e+003

benefit = 1.2654e+003

benefit = 1.2785e+003

benefit = 1.2332e+003

benefit = 1.1792e+003

benefit = 976.2074

benefit = 1.1253e+003

benefit = 1.2515e+003

benefit = 1.2414e+003

benefit = 1.1140e+003

benefit = 1.1324e+003

benefit = 1.1563e+003

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benefit = 1.2381e+003

benefit = 932.5911

benefit = 1.2381e+003

benefit = 1.0173e+003


benefit = 1.2654e+003

benefit = 1.2595e+003

benefit = 1.1563e+003

benefit = 1.0964e+003

benefit = 1.1508e+003

benefit = 1.2414e+003

benefit = 1.2573e+003

benefit = 1.2785e+003

benefit = 1.2785e+003

benefit = 1.2785e+003

benefit = 1.2785e+003

benefit = 1.1792e+003

benefit = 1.1253e+003

benefit = 932.5909

benefit = 1.1324e+003

benefit = 1.2381e+003

benefit = 1.2785e+003

benefit = 1.2403e+003

benefit = 1.2785e+003

benefit = 1.2785e+003

benefit = 1.2785e+003

benefit = 1.2785e+003

benefit = 1.2785e+003

benefit = 1.2785e+003

benefit = 1.2785e+003

benefit = 1.2785e+003

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benefit = 1.2785e+003

t = 9.0496

error vs generation

0 5 10 15 20 250

50

100

150

200

250

300

350

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Chapter 8. Conclusion& Future Scope

In this thesis the genetic algorithm approach for solving the optimal power dispatch in a

multi-node electricity market has been proposed. The objective of the algorithm is to

maximise the total participants benefit at all nodes in the system, which in turn depends

on the real power injection to the system.

The algorithm has been implemented by using the real power and reactive power

injection at all nodes as a candidate (chromosome). The total participants benefit is given

as a chromosomes fitness and it hasbeen determined by solving the load flow problem.

The genetic algorithm has been implemented with various control parameters and tested

on a 17-node, 26-line system. The results have shown that the proposed algorithm

provides a good solution

In future days micro genetic algorithm is developed ,which will become even more

efficient when specialists knowledge (Eg. Fuzzy Logic) about the problem is included.

In this way, it is possible to reduce search space and, consequently, decrease the

execution time, increasing the chances to reach global optimal solution.

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Bibliography

[1] D.L. Post, S.S. Coppinger, G.B. Sheble, Application of auctions

as a pricing mechanism for the interchange of electric power, IEEE

Trans. Power Syst. 10 (1995) 15801584.

[2] U.D. Annakkage, R.A.S.K. Ranatunga, Optimal power dispatch

of multi-node electricity markets, in: VI SEPOPE, Salvador,

Brazil, 1998.

[3] R.W. Ferrero, S.M. Shahidehpour, Optimality conditions in

power transactions in deregulated power pools, Elect. Power Syst.

Res. 42 (1997) 209214.

[4] N. Pamudji, R.J. Kaye, H.R. Outhred, Network effects in a

competitive electricity industry: No linear and linear nodal auction

models, in: Stockholm Power Tech Conference, 1995.

[5] H.R. Outhred, R.J. Kaye, Incorporating network effects in a

competitive electricity industry: an Australian perspective, Electricity

Transmission Pricing and Technology, Kluwer, Dordrecht,

1996.

[6] M. Mitchell, An Introduction to Genetic Algorithms, MIT Press,

Cambridge, MA, 1996.

[7] D.E. Goldberg, Genetic Algorithms in Search, Optimisation and

Machine Learning, Addison-Wesley, Reading, MA, 1989.

[8] T.T. Maifeld, G.B. Sheble, Genetic-based unit commitment

algorithm,

IEEE Trans. Power Syst. 11 (1996) 13591370.

[9] S.O. Orero, M.R. Irving, Economic dispatch of generators withprohibited operating zones:: A genetic algorithm approach, IEE

Proc. Gener. Trans. Distrib. 143 (1996)

Documents

DocumeOPTIMAL POWER DISPATCH IN MULTI NODE ELECTRICITY MARKET USING GENETIC ALGORITHM