ant colony search algorithm (acsa)

8/13/2019 Ant Colony Search Algorithm (Acsa)

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CHAPTER-1

INTRODUCTION

1.1 Introduction

The subject of minimizing distribution power losses has gained a great deal

of attention due to the high cost of electrical energy and therefore, much of current

research on distribution automation has been focusing on the minimum loss

configuration. Besides economic consideration, the effect of electric power loss is

the heat energy dissipation, which increases the temperatures of the associated

electric components and can result in insulation failure. By minimizing the power

losses, the system may acquire longer life span, and has greater reliability.

Therefore, loss minimization in distribution systems has become the subject of

intensive research.

In Practical systems, the methods employed for reduction of losses are

!etwor" reconfiguration, which is the selection of the proper

topological structure of the networ" for minimum losses.

Installation of capacitors, when this is economically justified.

#ost electric distribution feeders are configured radially for effective

coordination of their protection systems. By changing the state of networ" switches,

the radiality can always be preserved. The optimal operating condition of

distribution networ"s is obtained when line losses are minimized without any

violations of branch loading and voltage limits.

There are two types of switches in the system one is normally $closed

switch% connecting the line sections called &sectionalizing switch' and the other is

normally $open switch% on the tie(lines connecting either two primary feeders or two

substations, or loop(type laterals called &tie switch'. The change in networ"

configuration is achieved by closing or opening of these two types of switches in

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such a way that the $radiality% of the networ" is maintained. *istribution lines or

line sections show different characteristics as each has a different mi+ture of

residential, commercial and industrial type loads and their corresponding pea" times

are not coincident. This is due to the fact that some parts of the distribution system

becomes heavily loaded at certain times of the day and less loaded at other times.

Therefore, by shifting the loads in the system, the radial structure of the distribution

feeders can be modified from time to time in order to reschedule the load currents

more efficiently for loss minimization.

*uring normal operating condition, networ"s are reconfigured for two

purposes i- to minimize the system real power losses in the networ" and to

increase networ" reliability, ii- to relieve the over loads in the feeders. The former

is referred to as feeder reconfiguration for loss reduction and the latter as load

balancing. In this thesis nt /olony 0ptimization /0- is used to solve

distribution networ" reconfiguration and load balancing problem.

1.2 Literature survey

/ivanlar et al. 1)2 conducted the early wor" on feeder reconfiguration for

loss reduction. In 132, Baran et al. defined the problem of loss reduction and load

balancing as an integer(programming problem. !ara et al. 142 presented an

implementation that used a genetic algorithm to loo" for the minimum loss

configuration. In 15672, the authors suggested the use of the power flow method

based on a heuristic algorithm to determine the minimum loss configuration of

radial distribution networ"s. In 182, 192 the authors proposed a solution procedure

that employed simulated annealing :- to search for an acceptable non(inferior

solution.

In 192 the authors have formulated the load balancing and service restoration

problems by considering the capacity and voltage constraints as a mi+ed integer

nonlinear optimization problem. Baran and ;u 172 have devised the problem of loss

minimization and load balancing as an integer programming problem . correlation

e+isted between load balancing and loss reduction has been described in 1


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objective functions for load balancing and loss reduction are very similar, the

calculations for load balancing are similar to that of loss(reduction case, and

therefore the search for loss( reduction can also be applied to improve load

balancing in distribution networ"s. constrained multi(objective and non(

differentiable optimization problem with equality and inequality constraints for both

loss(reduction and load balancing has been proposed in 172. =.Peponis et al. 1)>2

have developed an improved switch(e+change method for load balancing problem,

using switch e+change operations. In the method of 1))2,1)32 some networ" branch

data are eliminated, while others are replaced by equivalents. ccurate voltage

values changes of energy losses and load balancing inde+ are calculated using the

reduced size networ" model. #.. ?ashem et al. 1))2 have proposed a load

balancing inde+ and shown that improvement in load balancing can be achieved by

networ" reconfiguration ;hei(#in @in et al. 1)32 have presented a current(inde+

based load balancing algorithm for the three 6phase unbalanced distribution

systems.

Aecently researchers have paid much attention in obtaining the solution of

distribution networ"s. Baran and ;u 1)42 have developed load flow solution in a

distribution system by the iterative solution of three fundamental equations "nown

as *ist low Branch Cquations representing real power, reactive power and voltage

magnitude similar to static load flow equations of transmission system. They have

computed the system Dacobian #atri+ using a chain rule. In their method the

mismatches and the Dacobin #atri+ involve only the evaluation of simple algebraic

e+pressions and no trigonometric functions. They have also proposed decoupled and

ast(decoupled distribution load flow algorithms.

/hiang 1)52 developed three solution algorithms for distribution system

based on Baran and ;u *ist low Branch Cquations. !ewton(Aapson !A- method

requires heavy computation in finding Dacobian #atri+. /hiang%s decoupled

algorithm modifies !A method by e+ploiting the numerical properties of the system

Dacobian to improve its computational efficiency. urther improvement can be

achieved by assuming diagonal elements of Dacobian as constants. This leads to the

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development of /hiang%s second algorithm namely ast(decoupled algorithm, which

has the same spirit as the ast(decoupled load flow methods for Transmission

systems. In this algorithm the system Dacobian is constructed only once at the first

iteration and is used for the remaining iterations. It is possible to further reduce the

computational burden associated with second algorithm based on the assumption

that system diagonal Dacobian matrices are close to identity matri+. This is

implemented in /hiang%s third algorithm namely very ast(decoupled algorithm,

which does not require any Dacobin matri+ construction and factorization.

Dasmon E @ee 1)F2 further developed the distribution power flow equations

such that the loss terms in two of the fundamental equations are grouped and

represented in a single line equivalent. This process represents the actual

distribution networ" by a simple single line equivalent. This method e+tends the

single line equivalent networ" to be used for load flow calculations and for deriving

the conditions for voltage collapse to occur. The conventional load flow methods

can indicate the possibility of voltage collapse but are unable to predict its

occurrence in advance. *ue to the simplicity of the single line equivalent technique,

Dasmon E @ee method is most suitable for use in real(time distribution system

monitoring as stability analysis based on this method is much simplified. s this

method is much simplified for finding losses, it doesn%t provide accuracy. ll

voltage terms are eliminated from the equations for solving the load flows there by

simplifying the equations for iterative solution.

The conventional distribution load flow methods involve the formation of

Dacobin matri+ and trigonometric functions. ?ersting 1)72 presented a load flow

technique based on the ladder networ" theory and it appears to wor" very well. *as,

?otari and ?alam 1)82 developed a simple and efficient method, which involves

only the evaluation of a simple algebraic e+pression of voltage magnitudes. :o this

method is efficient and requires less computer memory.

nt /olony 0ptimization /0- is a paradigm for designing metaheuristic

algorithms for combinatorial optimization problems. The first algorithm which can

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be classified within this framewor" was presented in )


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CHAPTER-2

LOAD LO! "ETHOD OR RADIAL DI#TRI$UTION

#%#TE"#

2.1 Introduction

The conventional load flow methods of transmission systems are not suitable

for distribution systems. #any researches have suggested modified versions of the

conventional load flow methods for solving the distribution networ" by considering

it as ill(conditional power networ" and they included admittance matri+, Dacobins,

Trigonometric functions that results in large computational time.

2.2 Load &o' #o&ution

The thesis uses a new load flow technique for solving radial distribution networ"s

which involves only the evaluation of a simple algebraic e+pression of receiving end

voltage and involves no trigonometric function as apposed to the standard load flow

methods. Gsing ?irchoff%s current law and ?irchoff%s voltage law a set of iterative

equations were developed. It is very efficient and has e+cellent convergence

characteristics. The radial topology of distribution networ"s has been fully e+ploited by

this method. In solving the radial distribution networ" for load flow some assumptions

were assumed to simplify the solution.

Assu()tions

). It is assumed that the three phase radial distribution networ"s are balanced

and represented by their equivalent single line representation.

3. Half(line charging susceptances of distribution lines are negligible and

these distribution lines are represented as short lines.

4. :hunt capacitor ban"s are treated as loads.

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2.2.1 Circuit (ode&

In this section circuit model of a )7(node radial distribution system is

represented.

s it is assumed that three(phase radial distribution system is balanced, it can

be represented by its equivalent single line diagrams. ig.3.) shows single line

diagram of ICCC )7 node radial distribution system.

2.2.2 #o&ution "et*odo&o+y

In any radial distribution system, the electrical equivalent of a branch(),

which is connected between node ) and 3 having a resistance A)- and inductive

reactance )-is shown in ig.3.3

/onsider branch ). The receiving(end node voltage can be written as

-)-)-)-3 JIKK = L3.)

:ectionalizing :witch

Tie :witch

@oad /enter

eeder ) eeder 3 eeder 4

) 3 4

5

F

7

8

9

))

)3

)4

)5

)F

)7

)

3

4

5

F

7

89

))

)3

)4

)5

)F)7

i+.2.1, IEEE 1 $us #yste(

) , 3 !ode number

), 3,. . . . Branch number

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:imilarly for branch 3,

-3-3-3-4 JIKK = L3.3

s the substation voltage K)-is "nown ta"en as ) p.u-, so if I)- is "nown,

i.e., current of branch(), it is easy to calculate K3- from Cqn.3.). 0nce K 3- is

"nown, it is easy to calculate K 4-from Cqn.3.3, if the current through branch 3 is

"nown. :imilarly, voltages of nodes 5,F,L. nd number of nodes- can easily be

calculated if all the branch currents are "nown. Therefore, a generalized equation of

receiving(end voltage, sending(end voltage, branch current and branch impedance is

i )- i- j- j-

K K I J+ = L3.4i-

;here $j% is the branch number.

i M sending end node of branch $j%

iN) M receiving end node of branch $j%

Cqn.3.4 can be evaluated for j M ),3L., nb number of branches-. /urrent

through branch ) is equal to the sum of the load currents of all the nodes beyondbranch ), i.e.

=

=nd

3i

-i@-) II L3.4ii-

In general

I@3-

i+, 2.2 #in+&e &ine dia+ra( o a /ranc*

-)-)K -3,-3,K

J)-

MA)-

Nj)-

P@3-

NjO@3-

) 3I)-

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= "i -i@-j II L3.5

The current through branch 3 is equal to the sum of the load currents of all

the nodes beyond branch 3 plus the sum of the charging currents of all the nodes

beyond branch 3. Therefore, if it is possible to identify the nodes beyond all the

branches, it is possible to computer all the branch currents. Identification of the

nodes beyond all the branches is realized through an algorithm as e+plained in

:ection 3.4.

The load current of node $i% is

i-

-i@-i@

-i@KjOPI

+= L3.F

;here $i% M 3,4,L., nd

I@i-M@oad current of node $i%

iQQnodetoconnectedloadpower/omple+OjP -i@-i@ =+

@oad currents are computed iteratively. flat voltage profile for all the

nodes is assumed for the first iteration and load currents of all the loads are

computed using Cqn.3.F. The branch currents are computed using load currents in

Cqn.3.5. detailed load(flow(calculation procedure is described in :ection 3.5.

The comple+ power loss of a branch $j% between node $i% and node $iN)% is

computed as follows

Power fed into the branch $j% between bus $i% and $iN)% at bus $i% is ( )

i- j-K I -

:imilarly power fed into the branch $j% at bus $iN)% is ( )i )- j-K I -+

Therefore the power loss in the branch $j% is written as


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=+-j-j @Oj@P ( )i- m-K I - N ( )

i )- m-K I -+

The real and reactive power losses of branch $m% are given by

( )

m- i- j- i )- j-@P real K I K I+=

( ) m- i- j- i )- j-@O imag K I K I+= L3.7

2.0 I&&ustration o Node Identiication

/onsider ICCC()7 node radial distribution system shown in ig.3.). The

formation of various vectors used in sparsity technique for node identification is

given below

2.0.1 A&+orit*( or Node Identiication

ollowing algorithm e+plains the methodology of identifying the nodes and

branches connected to a particular node in detail, which will help in finding the

e+act load feeding through that particular node.

2.0.1.1 A&+orit*( or or(ation o ectors Adn3 Ad/3 " and "T4

:tep ) Aead system branch data

:tep 3 Initialize vector # with ) E :M>

:tep 4 Initialize the count for node iM)

:tep 5 Initialize count for branch count jM)

:tep F if iM M :C 1j2- go to step 8 else go to step 7

:tep 7 if iM M AC1j2 go to step 9 else go to step


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db 1:2 Mj

:tep

:tep )> #T1i2 M:

# 1iN)2 M#T1i2 N)

:tep )) if iRMnd-

iMiN) go to step 5 else go to step )3

:tep )3 stop

Ta/&e 2.2, " and "T vectors o i+ 14

!ode no. # 1i2 #T 1i2

) ) )

3 3 3

4 4 4

5 5 7

F 8 8

7 9 )>

9 )) )4

< )5 )7

)> )8 )8

)) )9 )9

)3 )< ) 33

)5 34 34

)F 35 3F

)7 37 37

;here,

#1i2 M#emory location from

#T1i2 M#emory location to for a particular node $i%.

;here, iM) to nd

Ta/&e 2.0 Ad5acent /ranc* Ad/4 6 node Adn4 vectors o i+. 2.1

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The details of the implementing vectors in sparsity technique are given in

Tables 3.) and 3.3. $#% and $#T% govern the reservation allocation of memory

location for each node. ;ith the help $dn% and $db%, vectors constructed $#%

and $#T% vectors it is very simple to calculate the effective branch currents and

voltages at any particular node.

2.7 Load &o' Ca&cu&ation

The loadflow used in this thesis is forward(bac"ward distribution loadflow.

Initially flat voltage profile is assumed for all the nodes i.e., voltage is set to )p.u.

2.7.1 $ac8'ard Pro)a+ation

The purpose of the bac"ward propagation computation is to obtain updated

branch currents in each section, by considering the previous iteration voltages at

:.no. !ode dn db :.no. !ode dn db

) ) 5 ) )5

)7 )3 9 8

F F 3 )9 )) < 9

7 7 4 )< )3 <

)4

4 )>

9 7 5 4 3) )5 ))

< 8 5 33 )F )3

)> 8 7 5 34 )5 )4 ))

))

9

3 F 35 )F )4 )3

)3 < 7 3F )7 )4

)4 )> 8 37 )7 )F )4

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each node. *uring bac"ward propagation, voltage values are held constant at the

values obtained in the forward path and updated branch currents are transmitted

bac"ward along the feeder using bac"ward path. Bac"ward propagation starts at the

e+treme end branch and proceeds towards source node.

2.7.2 or'ard Pro)a+ation

The purpose of the forward propagation is to calculate the voltages at each

node starting from the feeder source branch. The feeder substation voltage is set at

its actual value. *uring forward propagation the current in each branch is held

constant to the value obtained in bac"ward wal". The node voltages are calculated

using Cqn.3.4.

2.7.0 Test or Conver+ence

The convergence criterion is the voltage mismatch between voltages obtained

in the current iteration and the previous iteration. @oadflow iterations are stopped

when the iterations reach a ma+imum iteration count or when the ma+imum

deviation in the node voltages for a successive iterations is less than a pre(specified

tolerance value.

2.7.7 A&+orit*( or Load &o' Ca&cu&ations

:tep ) Aead the line and load data

or jM) to nd()-

iM3 to nd

Initialize K1i2MKK1i2M).>

TP@MTO@M>.>

Crr M >.>>>>)

:tep 3 IT M )

:tep 4 Aead #, #T, dn, db vectors

:tep 5 /alculate load current at each node starting from the last load.

Gsing the load currents obtain branch currents using Cqn.3.5

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:tep F or bus iM3 to nd

or jM #1i2, nMdn1j2, "Mdb1j2

/alculate the node voltages using Cqn.3.4

:tep 7 or iM3 to nd

If K1i2(KK1i2- R Crr- go to step 9 else go to step 5

:tep 8 ITNN

:tep 9 or jM) to nb-

/alculate real and reactive power loss using Cqn.3.7

:tep stop

CHAPTER-0

ANT COLON% OPTI"I9ATION

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0.1 Introduction

Insects that live in colonies, such as ants, bees, wasps and termites, follow

their own agenda of tas"s independent from one another. However, when these

insects act as a whole community, they are capable of solving comple+ problems in

their daily lives, through mutual cooperation. Problems such as selecting and

pic"ing up materials, and finding and storing foods, which require sophisticated

planning, are solved by insect colonies without any "ind of supervisor or controller.

This collective behavior which emerges from a group of social insects has been

called &swarm intelligence'. nts are capable of finding the shortest route between

a food source and the nest without the use of visual information, and they are also

capable of adapting to changes in the environment

The natural metaphor on which ant algorithms are based is that of ant

colonies. Aeal ants are capable of finding the shortest path from a food source to

their nest without using visual cues by e+ploiting pheromone information. ;hile

wal"ing, ants deposit pheromone on the ground and follow, in probability,

pheromone previously deposited by other ants. In ig.4.), we show a way ants

e+ploit pheromone to find a shortest path between two points. /onsider ig.4.)a-

ants arrive at a decision point in which they have to decide whether to turn left or

right. :ince they have no clue about which is the best choice, they choose randomly.

It can be e+pected that, on average, half of the ants decide to turn left and the other

half to turn right. This happens both to ants moving from left to right and to those

moving from right to left. igs.4.)b- and 4.)c- shows what happens in the

immediately following instants, supposing that all ants wal" at appro+imately the

same speed. The number of dashed lines is roughly proportional to the amount of

pheromone that the ants have deposited on the ground. :ince the lower path is

shorter than the upper one, more ants will visit it on average, and therefore

pheromone accumulates faster. fter a short transitory period the difference in the

amount of pheromone on the two paths is sufficiently large so as to influence the

decision of new ants coming into the system 1this is shown by ig.4.)d-2. rom

now on, new ants will prefer in probability to choose the lower path, since at the

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decision point they perceive a greater amount of pheromone on the lower path. This

in turn increases, with a positive feedbac" effect, the number of ants choosing the

lower, and shorter, path. Kery soon all ants will be using the shorter path.

i+. 0.1 $e*avior o Rea& Ants

The above behavior of real ants has inspired Ant system, an algorithm in

which a set of artificial ants cooperate to the solution of a problem by e+changing

information via pheromone deposited on graph edges. The ant system has been

applied to combinatorial optimization problems such as the traveling salesman

problem T:P- and the quadratic assignment problem. The ant colony system

/:-, the algorithm presented in this thesis, builds on the previous ant system in

the direction of improving efficiency when applied to hard combinatorial problems

such as traveling salesmen problem quadratic assignment problem and even to

networ" reconfiguration problem.

The concept of nt colony system can be better e+plained by applying it to

Traveling salesmen problem. The main idea is that of having a set of agents, called

Ants, search in parallel for good solutions to the T:P and cooperate through

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pheromone(mediated indirect and global communication. Informally, each ant

constructs a T:P solution in an iterative way it adds new cities to a partial solution

by e+ploiting both information gained from past e+perience and a greedy heuristic.

#emory ta"es the form of pheromone deposited by ants on T:P edges, while

heuristic information is simply given by the edge%s length.

0.2 Ant syste(

Ant system 1)92 is the progenitor of all the research efforts with ant

algorithms and was first applied to the T:P. nt system utilizes a graph

representation which is augmented as follows in addition to the cost measure

-s,r , each edge -s,r has also a desirability measure -s,r , called

pheromone, which is updated at run time by artificial ants ants for short-. ;hen ant

system is applied to symmetric instances of the T:P, -r,s-s,r = but when it is

applied to asymmetric instances it is possible that -r,s-s,r .

Informally, ant system wor"s as follows. Cach ant generates a complete tour

by choosing the cities according to a probabilistic state transition rule S ants prefer to

move to cities which are connected by short edges with a high amount of

pheromone. 0nce all ants have completed their tours a global pheromone updatingrule global updating rule, for short- is appliedS a fraction of the pheromone

evaporates on all edges edges that are not refreshed become less desirable-, and

then each ant deposits an amount of pheromone on edges which belong to its tour in

proportion to how short its tour was in other words, edges which belong to many

short tours are the edges which receive the greater amount of pheromone-. The

process is then iterated. The state transition rule used by ant system, called a

random-proportional rule, is given by Cqn.4.), which gives the probability with

which ant $"% in city $r% chooses to move to the city $s%

[ ] [ ]

[ ] [ ]

=

otherwise,>

r-Dsif-u,r-u,r

-s,r-s,r

-s,rp"

-rDu"

"

L4.)

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;here is the pheromone, )= is the inverse of the distance -s,r ,

-rD" is the set of cities that remain to be visited by ant " positioned on city r to

ma"e the solution feasible-, and is a parameter which determines the relative

importance of pheromone versus distance -> > .

In Cqn.4.) we multiply the pheromone on edge -s,r by the corresponding

heuristic value -s,r . In this way we favor the choice of edges which are shorter

and which have a greater amount of pheromone.

In ant system, the global updating rule is implemented as follows. 0nce all

ants have built their tours, pheromone is updated on all edges according to

m

"

" )

r,s- ) - r,s- r,s- =

+ L4.3

;here,

=otherwise>

"antbydonetours-ifr,,@

)

-s,r""

)>


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distributed on the edges of the graph. This allows an indirect form of

communication called stigmergy.

lthough ant system was useful for discovering good or optimal solutions for

small T:P%s, the time required to find such results made it infeasible for larger

problems. #arco *origo and @uca #aria =ambardella devised three main changes to

improve nt system performance which led to the definition of the nt colony

system /:-.

0.0 Ant Co&ony #yste(

The nt /olony :ystem /:- differs from the previous ant system because

of three main aspects

i- The state transition rule provides a direct way to balance between

e+ploration of new edges and e+ploitation of a priori and accumulated

"nowledge about the problem

ii- The global updating rule is applied only to edges which belong to the

best ant tour,

iii- ;hile ants construct a solution a local pheromone updating rule local

updating rule, for short- is applied.

Informally, the /: wor"s as follows $m% ants are initially positioned on $n%

cities chosen according to some initialization rule e.g., randomly-. Cach ant builds

a tour i.e., a feasible solution to the T:P- by repeatedly applying a stochastic

greedy rule the state transition rule-. ;hile constructing its tour, an ant also

modifies the amount of pheromone on the visited edges by applying the local

updating rule. 0nce all ants have terminated their tour, the amount of pheromone on

edges is modified again by applying the global updating rule-. s was the case in

ant system, ants are guided, in building their tours, by both heuristic information

they prefer to choose short edges- and by pheromone information. n edge with a

high amount of pheromone is a very desirable choice. The pheromone updating rules

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are designed so that they tend to give more pheromone to edges which should be

visited by ants.

0.0.1 AC# #tate Transition Ru&e

In the /: the state transition rule is as follows an ant positioned on node

$r% chooses the city $s% to move to by applying the rule given by

[ ] [ ]{ }

= S

ururts rJu k

-,-,ma+arg -

n-e+ploratioBiasedotherwise

ion-e+ploitatqqif > L4.4

where q is a random number uniformly distributed in 1> )2, q> is a parameter

( ))q> > , and $:% is a random variable selected according to the probability

distribution given by Cqn.4.).

The state transition rule resulting from Cqns.4.) and 4.4 is called pseudo-

random-proportional rule. This state transition rule, as with the previous random(

proportional rule, favors transitions toward nodes connected by short edges and with

a large amount of pheromone. The parameter q >determines the relative importance

of e+ploitation versus e+ploration ;hen a particular ant is positioned in node r, a

random number q ( ))q> is generated. If ( )>qq , then the best branch is selected,

this means that e+ploitation was the decisive factor, while in the opposite case, the

selection of the route is performed according to the probabilistic transition rule Cqn.4.)

0.0.2 AC# :&o/a& U)datin+ Ru&e

In /: only the globally best ant i.e., the ant which constructed the shortest

tour from the beginning of the trial- is allowed to deposit pheromone. This choice,

together with the use of the pseudo(random(proportional rule, is intended to ma"e

the search more directed. nts search in a neighborhood of the best tour found up to

the current iteration of the algorithm. =lobal updating is performed after all ants

have completed their tours. The pheromone level is updated by applying the global

updating rule of Cqn.4.5.

3>


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r,s- ) - r,s- r,s- + L4.5

where

( ) =

otherwise>

bestglobals-r,if@-s,r)

gb

( ))>


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s real ant colonies, ant algorithms are composed of a population, or colony,

of concurrent and asynchronous entities globally cooperating to find a good

&solution' to the tas" under consideration. lthough the comple+ity of each

artificial ant is such that it can build a feasible solution as a real ant can somehow

find a path between the nest and the food-, high quality solutions are the result of

the cooperation among the individuals of the whole colony. nts cooperate by

means of the information they concurrently readwrite on the problem%s states they

visit.

0.0.7.2 P*ero(one trai& and sti+(er+y

rtificial ants modify some aspects of their environment as the real ants do.

;hile real ants deposit on the world%s state they visit a chemical substance, the

pheromone, artificial ants change some numeric information locally stored in the

problem%s state they visit. This information ta"es into account the ant%s current

history or performance and can be readwritten by any ant accessing the state. By

analogy, we call this numeric information artificial pheromone trail, pheromone

trail for short. In /0 algorithms local pheromone trails are the only

communication channels among the ants. This stigmergetic form of communication

plays a major role in the utilization of collective "nowledge. Its main effect is to

change the way the environment the problem landscape- is locally perceived by the

ants as a function of all the past history of the whole ant colony. Gsually, in /0

algorithms, an evaporation mechanism similar to real pheromone evaporation

modifies pheromone information over time. Pheromone evaporation allows the ant

colony slowly to forget its past history so that it can direct its search toward new

directions without being over(constrained by past decisions.

0.0.7.0 #*ortest )at* searc*in+ and &oca& (oves

rtificial and real ants share a common tas" to find a shortest minimum

cost- path joining an origin nest- to destination food- sites. Aeal ants do not jumpS

they just wal" through adjacent terrain%s states, and so do artificial ants, moving

33


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step(by(step through &adjacent states' of the problem. The e+act definitions of state

and adjacency are problem specific.

0.0.7.0 #toc*astic and (yo)ic state transition )o&icy

rtificial ants, as real ones, build solutions applying a probabilistic decision

policy to move through adjacent states. s for real ants, the artificial ants% policy

ma"es use of local information only and it does not ma"e use of loo"(ahead to

predict future states. Therefore, the applied policy is completely local, in space and

time. The policy is a function of both the a priori information represented by the

problem specifications equivalent to the terrain%s structure for real ants-, and of the

local modifications in the environment pheromone trails- induced by past ants.

rtificial ants also have some characteristics that do not find their

counterpart i.e. in real ants.

rtificial ants live in a discrete world and their moves consist of transitions from

discrete states to discrete states.

rtificial ants have an internal state. This private state contains the memory of

the ants% past actions.

rtificial ants deposit an amount of pheromone that is a function of the quality

of the solution found.

rtificial ant%s timing in pheromone laying is problem dependent and often does

not reflect real ant%s behavior. or e+ample, in many cases artificial ants update

pheromone trails only after having generated a solution.

To improve overall system efficiency, /0 algorithms can be enriched with

extra capabilities such as loo"(ahead, local optimization, bac"trac"ing, and so

on that cannot be found in real ants. In many implementations ants have been

hybridized with local optimization procedures.

34


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0.7 Case study,

Traveling salesmen problem T:P- for )F cities

The position of each city is represented by the co(ordinates in two dimension plane./onsidering the following parameters for nt /olony :earch algorithm

!umber of antsF

3= , ).>= , ).>= , >>>).>> = , q>M>.>

Table 4.) *ata for )F /ities position for traveling salesmen problem

/itiesPosition of /ities in 3*

coordinate system

measured in ?m from

reference point-

) 49,


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0 10 20 30 40 50 60 70 80 90 10010

20

30

40

50

60

70

80

90

100

City-1

City-2

City-3City-4

City-5

City-6

City-7

City-8

City-9

City-10

City-11

City-12

City-13

City-14

City-15

i+, 0.2 Route (a) o t*e sa&es(an

The )F cities problem ant colony search algorithm is applied for hundred

iterations and the convergence of the algorithm for best fitness is shown in fig.4.4.

3F


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0 10 20 30 40 50 60 70 80 90 100360

380

400

420

440

460

480

Iterations

BestTourLength

Convergence characteristics for Travelling Salesmen Problem

i+, 0.0 conver+ence c*aracteristics or 1< cities trave&in+ sa&es(an )ro/&e(

0.7.1 actors eectin+ Peror(ance o Ant co&ony searc* a&+orit*(

). !umber of ants /onvergence can be accelerated by increasing the number of

cooperative agents $nts%-. The ma+imum number of ants which can be used

for /: algorithm is limited by the number of cities. The optimal number of

ants is close to the number of cities. /onvergence performance of /: algorithm

for different number of ants is shown in ig.4.5.

3. $Beta% value represents the relative importance between pheromone and

distance between cities. or >= , /: algorithm wor"s solely on the

pheromone amount deposited on the path and the effect of path distance is not

ta"en into account, and the convergence is delayed. ;ith >> convergence

performance is improved as shown in ig.4.F

37


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4. > Initial pheromone value > is set to a small value. #ore precisely initial

pheromone value should be set to a value near to the inverse of the fitness value to

improve the convergence performance.

5. , , values are determined e+perimentally with different values for the

specified problem and the values are ta"en which gives best performance.

0 10 20 30 40 50 60 70 80 90 10036 0

38 0

40 0

42 0

44 0

46 0

48 0

Iterations

B

estTourLength

Convergence characteristics for Different Number of Ants

3 Ants

6 Ants

9 Ants

i+, 0.7, Conver+ence o AC# a&+orit*( or t*ree3 si@ and nine Ants

38


28/63

0 10 20 30 40 50 60 70 80 90 100

350

400

450

500

550

600

650

700

750

800

Iterations

B

estTourLength

Convergence characterist ics for Different "Beta" Values

Beta=0

Beta=1

Beta=2

i+.0.


29/63

remains firmly embedded in biology, and so it is common to discuss &parents,'

&children', &alleles' and so on.

0.


30/63

3. second method which may require fewer fitness evaluations is

tournament selection. In this method, solutions are randomly selected to

participate in a &tournament'S the solution with the highest fitness is

selected, and the process repeats until enough parents are chosen.

#ost selection methods are stochastic, and so may allow a small number of

less(fit solutions to reproduce. This has the advantage of maintaining diversity in

the population.

0. to ) or visa versa in a binary string, or adding a random

value to an allele in a string of real numbers.

0. T*e a&+orit*(

4>

Parent ) > > > v> > > >v >

Parent 3 ) ) ) v) ) ) )v )/rossover

/hild ) > > > ) ) ) ) >

/hild 3 ) ) ) > > > > )


31/63

The most basic = simply runs through these processes in order, and repeats

until either an adequate solution has been found or a certain amount of time has

passed. The canonical = therefore proceeds as follows

=enerate an initial population

*0

:elect a set of parents by some fitness(based method

Perform crossover on parents to produce children

Perform mutation on children

G!TI@ a terminating condition has been reached

CHAPTER-7

Net'or8 Reconi+uration

7.1 Introduction

4)


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Between 4> and 5> U of total investments in the electrical sector goes to

distribution systems, but nevertheless, they have not received the technological

impact in the same manner as the generation and transmission systems. *istribution

networ"s wor" mostly with minimum monitoring. The manual control of capacitors,

sectionalizing switches and voltage regulators are operated manually without

adequate computation support for the systemQs operators. !evertheless, there is an

increasing trend to automate distribution systems to improve their reliability,

efficiency and service quality. utomation is possible due to the advance

microprocessor control technology, to its increasing cost reduction and due to its

joint use with telecommunications technologies. It is possible to install distribution

operation centers where the networ" is constantly monitored and control actions can

be made remotely. ;ith the aid of these technologies it is possible to monitor

substations and feeders to reconfigure feeders and to control voltage and reactive

power.

If the networ" reconfiguration and voltage control and reactive power

adjustments become routine operations, the operators will not trust only on their

criteria and e+perience to operate the system. It will be necessary to have dedicated

software that assists the operator in selecting appropriate control actions. 0ne of

these actions is the networ" reconfiguration that can be oriented to different

objectives. Gnder normal operating conditions, the networ" is reconfigured to

reduce the systemQs losses andor to balance load in the feeders. Gnder conditions of

permanent failure, the networ" is reconfigured to restore the service, minimizing the

zones without power.

*istribution systems consist of groups of interconnected radial circuits. The

configuration may be varied via switching operations to transfer loads among the

feeders. Two types of switches are used in primary distribution systems. They are

normally closed switches sectionalizing switches- or normally open switches tie

switches-. Both types are designed for both protection and configuration

management. !etwor" reconfiguration is the process of changing the topology of

distribution systems by altering the openclosed status of switches.

43


33/63

!etwor" reconfiguration is a complicated combinatorial, non(differentiable,

constrained optimization problem because the distribution system involves many

candidate(switching combinations.

In this thesis an ant colony search algorithm /:- 1)


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whereT,@ossis the total real power loss of the system. Parameters ! and " are

the penalty constants, S/Kthe squared sum of the violated voltage constraints, and

S/I is the squared sum of the violated current constraints. #oreover, the penalty

constants are determined as follows

)- /onstant ! " - is given a value of $>%, if the associated voltage current-

constraint is not violated.

3- significant value is given to ! " - if the associated voltage current-

constraint is violatedS this ma"es the objective function to move away from the

undesirable solution.

or secure operation, the voltage magnitude at each bus must be maintained

within its limits. The current in each branch must satisfy the branch%s capacity.

These constraints are e+pressed below

ma+imin KKK L5.3

ma+,ii II L5.4

where iK

is voltage magnitude of bus i, Kminand Kma+ are minimum and ma+imum

bus voltage limits, respectively. iI and Ii,ma+are current magnitude and ma+imum

current limit of branch i, respectively.

The proposed method uses a set of simplified feeder(line flow formulations for

power flow analysis to prevent complicated computation.

7.0 A))&ication o Ant Co&ony #earc* A&+orit*( AC#A4 to

reconi+uration )ro/&e(

7.0.1 #tate transition ru&e and &oca&+&o/a& u)datin+ ru&e

s illustrated in chapter 4, by the guidance of the pheromone intensity, the

ants select preferable path. inally, the favorite path rich of pheromone become the

best tour, the solution to the problem. This concept develops the emergence of the

45


35/63

/: method. t first, each ant is placed on a starting state. Cach will build a full

path, from the beginning to the end state, through the repetitive application of state

transition rule. ;hile constructing its tour, an ant also modifies the amount of

pheromone on the visited path by applying the local updating rule. 0nce all ants

have terminated their tour, the amount of pheromone on edge is modified again

through the global updating rule. In other words, the pheromone(updating rules are

designed so that they tend to give more pheromone to paths which should be visited

by ants. In the following, the state transition rule, the local updating rule, and the

global updating rule are briefly introduced.

7.0.1.1 #tate transition ru&e

The state transition rule used by the ant system, called a random(proportional

rule, is given by Cqn.5.5, which gives the probability with which ant k in node i

chooses to move to node#.

[ ] [ ]

[ ] [ ]

=

otherwise,>

i-Dsif-m,i-m,i

-j,i-j,i

-j,ip"

-iDm"

"

L5.5

where is the pheromone which deposited on the edge between nodes i and#,

the inverse of the edge distance, Jki- the set of nodes that remain to be visited by

ant k positioned on node i, and is a parameter which determines the relative

importance of pheromone versus distance. Cqn.5.5 indicates that the state transition

rule favors transitions toward nodes connected by shorter edges and with greater

large amount of pheromone.

7.0.1.2 Loca& u)datin+ ru&e

;hile constructing its tour, each ant modifies the pheromone by the local

updating rule. This can be written below

>-j,i-)-j,i += L5.F

4F


36/63

where > is the initial pheromone value and is a heuristically defined parameter.

The local updating rule is intended to shuffle the search process. Hence, the

desirability of paths can be dynamically changed. The nodes visited earlier by a

certain ant can be also e+plored later by other ants. The search space can be

therefore e+tended. urthermore, in so doing, ants will ma"e a better use of

pheromone information. ;ithout local updating, all ants would search in a narrow

neighborhood of the best previous tour.

7.0.1.0 :&o/a& u)datin+ ru&e

;hen tours are completed, the global updating rule is applied to edges

belonging to the best ant tour. This rule is intended to provide a greater amount of

pheromone to shorter tours, which can be e+pressed below

)-j,i-)-j,i

+= L5.7

where is the distance of the globally best tour from the beginning of the trial and

2)>1 is the pheromone decay parameter. This rule is intended to ma"e the

search more directedS therefore, the capability of finding the optimal solution can be

enhanced through this rule in the problem solving process.

7.7. Co()utationa& )rocedures o t*e )ro)osed (et*od

The solution process begins with encoding parameters. tie(switch T:- and

some sectionalizing switches with the feeders form a loop. particular switch of

each loop is selected to open to ma"e the loop radial such that the selected switch

naturally becomes a tie switch. The networ" reconfiguration problem is identical tothe problem of selecting an appropriate tie switch for each loop to minimize the

power loss. coding scheme 13>2 that recognizes the positions of the tie switches is

proposed. The total number of tie switches is "ept constant, regardless of any

change in the system%s topology or the tie switches% positions. ig.5.) shows an

individual that is composed of tie switches% positions. *ifferent switches from a

47


37/63

loop are, respectively, selected for cutting the loop circuit and trying to become a tie

switch. fter each loop is made radial, a configuration is proposed. The fitness

value defined as the system loss- associated with this proposed configurations is

determined using Cqn.3.7. inally, the best one among the proposed configurations

is selected, and which is a feasible solution radial configuration- with minimum

loss.

The fitness function Total real power loss- to be minimized is as follows

=

==nb

)j

@osslossT, -jPminPminmin f L5.8

where $nb%is the total number of branches in the system. ig.5.3 shows a flowchartof the main computational procedures. The proposed method mainly involves power

loss computation using Cqn.3.7, bus voltage determination using Cqn.3.4 and /:

application. The computation finds configurations with various states of switches so

that the value of the objective function is successively reduced.

Tie switch no.) Tie switch no.3 . . . . Tie switch no. $n%

i+ure.7.1 co()osition o an individua&

The main computational processes are briefly stated below.

#te) 1 Initiation- t first, the colonies of ants are randomly selected which

estimated the initial fitness in the different permutations. random number

generator can be employed to generate the number of ants within the

feasible search space. In addition, these ants are positioned on different

nodes while the initial pheromone value of > is also given at this step.

#te) 2 Cstimation of the fitness- In this step, the fitness of the ants, which is

defined as the objective function, is estimated. Then, the pheromone can be

added to the particular direction in which the ants have chosen. t this

stage, a roulette selection algorithm can be employed based on the

48


38/63

computed fitness. Then, by spinning this designated roulette, a new

permutation of pheromone associated with different paths is formed. In

other words, based on a roulette selection method, a path fitness- with

higher amount of pheromone will be easy to find a new path. Hence, it

would be more suitable for guiding the ants to the direction.

#te) 0 nt reconfiguration- The ant%s reconfigurations are based on the level of

pheromone and distance. s Cqn.5.5 shows, each ant selects the ne+t node

to move ta"ing into account -j,i and -j,i values. greater -j,i

means that there has been a lot of traffic on this edgeS hence, it is more

desirable to approach the optimal solution. 0n the other hand, a greater

-j,i indicates that the closer node should be chosen with a higher

probability. In the networ" reconfiguration study, this can be seen as the

difference between the original total power loss and the new total power

loss. Therefore, in this step, the ant reconfiguration process helps convey

ants by selecting directions based on these two parameters.

#te) 7 @ocalglobal updating rule- ;hile constructing a solution of the networ"

reconfiguration problem, ants visit edges and change their pheromone level

by applying the local updating rule of Cqn.5.F. Its purpose is not only

broadening the search space, but dynamically increasing the diversity of ant

colony. fter n iterations, all ants have completed a tourS the pheromone

level is updated by applying the global updating rule of Cqn.5.7 for the trail

which belongs to the best selected path. Therefore, according to this rule,

the shortest path found by the ants is allowed to update its pheromone, also

this shortest path will be saved as a record for the later comparison with the

succeeding iteration.

#te) < Termination of the algorithm- Cnd the process if &the ma+imum iteration

number is reached' or &all ants have selected the same tour' is satisfiedS

otherwise, repeat the outer loop. In addition, the number of ants and the

number of iterations were e+perimentally determined. ll the tours visited

49


39/63

by ants in each iteration should be evaluated. If a better path found in the

process, it will be saved for later reference.

4


40/63

7.< &o' c*art o Ant co&ony searc* a&+orit*(

:tart

Aead line and load

data

:olve feeder flow for the system, compute itsfitness to determine its initial power loss

Aandom selection of ant initiation

pplying state transition rule and using roulette

wheel a new permutation of pheromone associatedwith different paths is formed

pply local pheromone updating rule

:olve feeder flow and determine system

power loss

Gpdate pheromone using globalupdating rule

Termination of

the algorithm

:tartProceed to ne+t generation

=enM=enN)

Ves

!o

i+.7.2 &o'c*art o Ant co&ony searc* a&+orit*(

5>


41/63

7. I()rove(ent o conver+ence c*aracteristics o net'or8

reconi+uration usin+ &ine contro& strate+y

The simplicity of the proposed methodology ma"es it suitable for an on(line control

strategy for feeder loss reduction. The strategy for selecting the best switching

option is further e+plained via the e+ample system of ig.5.3.

To maintain continuous power supply to all the loads and to improve convergence

characteristics, the following set of rules to be adopted for selection of switches.

Aule) ll switches those do not belong to any loop are to be closed.

Aule3 ll switches connected to the sources are to be closed.

Aule4 :ectionalizing switches, those lie on @K side from load flow- of the tie

switches, are ta"en as opening options of the initial configuration.

;hen closing the tie switch F, five options for opening sectionalizing

switches ), 3,

))

)3

)4

)5

)F

)7

)

3

4

5

F

7

89

))

)3

)4

)5

)F)7

i+.7.2, IEEE 1 $us #yste(

) , 3 !ode number

), 3,. . . . Branch number

K R K)) F 5)


42/63

transferring loads on eeder() to eeder(3 is e+pected to increase

losses. /onsequently, from rule 3 opening the sectionalizing switch ) or 3 are

regarded as undesirable and need not be considered. Therefore, associated with

closing the tie switch F are three candidate options, viz., opening the sectionalizing

switches )5

53


43/63

inde+ is minimized. In other words, all the branch load balancing indices are set to

be more or less the same value and are also nearly equal to the system load

balancing inde+.

The load(balancing problem is formulated in the form of branch load

balancing and system load balancing indices are

The branch load balancing inde+ ma+-i

-i

-i:

:@B = L5.9

The system load balancing inde+

=

=)nb

)ima+

-i

-i

sys:

:

nb

)@B L5..>F).>F) )5 >.>F).>F).)FF.)FF


61/63

$r.

No.

#endin+

end node

Receivin+

end node

Resistance

4Reactance

4Rea& )o'er

B!4

Reactive

)o'er

BAr4

) ) 3 >.>.>58> )>>.>> 7>.>>

3 3 4 >.5 >.3F)) .>> 5>.>>

4 4 5 >.477> >.)975 )3>.>> 9>.>>5 5 F >.49)) >.).>> 4>.>>

F F 7 >.9) >.8>8> 7>.>> 3>.>>

7 7 8 >.)983 >.7)99 3>>.>> )>>.>>

8 8 9 >.8))5 >.34F) 3>>.>> )>>.>>

9 9 < ).>4>> >.85>> 7>.>> 3>.>>

< < )> ).>55> >.85>> 7>.>> 3>.>>

)> )> )) >.).>7F> 5F.>> 4>.>>

)) )) )3 >.4855 >.)349 7>.>> 4F.>>

)3 )3 )4 ).579> ).)FF> 7>.>> 4F.>>

)4 )4 )5 >.F5)7 >.8)3< )3>.>> 9>.>>

)5 )5 )F >.F >.F37> 7>.>> )>.>>)F )F )7 >.8574 >.F5F> 7>.>> 3>.>>

)7 )7 )8 ).39 ).83)> 7>.>> 3>.>>

)8 )8 )9 >.843> >.F85> .>> 5>.>>

)9 3 )< >.)75> >.)F7F .>> 5>.>>

)< )< 3> ).F>53 ).4FF5 .>> 5>.>>

3> 3> 3) >.5>.5895 .>> 5>.>>

3) 3) 33 >.8>9< >.> 5>.>>

33 4 34 >.5F)3 >.4>94 .>> F>.>>

34 34 35 >.9 >.8>.>> 3>>.>>

35 35 3F >.9 >.8>)) 53>.>> 3>>.>>

3F 7 37 >.3>4> >.)>45 7>.>> 3F.>>

37 37 38 >.3953 >.)558 7>.>> 3F.>>

38 38 39 ).>F >..>> 3>.>>

39 39 3< >.9>53 >.8>>7 )3>.>> 8>.>>

3< 3< 4> >.F>8F >.3F9F 3>>.>> 7>>.>>

4> 4> 4) >.. )F>.>> 8>.>>

4) 4) 43 >.4)>F >.47)< 3)>.>> )>>.>>

43 43 44 >.45)> >.F4>3 7>.>> 5>.>>

Tie-s'itc*es data

44 9 3) 3.>>>> 3.>>>> ( (45 < )F 3.>>>> 3.>>>> ( (

4F )3 33 3.>>>> 3.>>>> ( (

47 )9 44 >.F>>> >.F>>> ( (

48 3F 3< >.F>>> >.F>>> ( (

$ase "A, 1?? B 12.

Ta/&e A-0, Line3 &oad and tie s'itc* data o =-node radia& distri/ution

net'or8

7)


62/63

$r.

No.

#endin+

end

node

Receivin+-

end node

Resistance

4Reactance

4

Rea&

)o'er

B!4

Reactive

)o'er

BAr4

#(a@BA4

) ) 3 >.>>>F >.>>)3 >.>> >.>> )87

3 3 4 >.>>>F >.>>)3 >.>> >.>> )874 4 5 >.>>)F >.>>47 >.>> >.>> )>448

5 5 F >.>3F) >.>3.>> >.>> 3F38

F F 7 >.477> >.)975 3.7> 3.3> 773

7 7 8 >.49)) >.).5> 4>.>> 75.>.>58> 8F.>> F5.>> )4).>5.>3F) 4>.>> 33.>> )9>4

< < )> >.9) >.38>8 39.>> )> 553

)> )> )) >.)983 >.>7)< )5F.>> )>5.>> .8))5 >.34F) )5F.>> )>5.>> 58F

)3 )3 )4 ).>4>> >.45>> 9.>> F.F> 455> >.45F> 9.>> F.F> 4F9> >.45.>> >.>> 49.).>7F> 5F.F> 4>.>> 4

)7 )7 )8 >.4855 >.)349 7>.>> 4F.>> 7F5

)8 )8 )9 >.>>58 >.>>)7 7>.>> 4F.>> F95>

)9 )9 )< >.4387 >.)>94 >.>> >.>> 8>>

)< )< 3> >.3)>7 >.>7> >.7> 983

3> 3> 3) >.45)7 >.))3< ))5.>> 9).>> 79F

3) 3) 33 >.>)5> >.>>57 F.4> 4.F> 4495

33 33 34 >.)F.>F37 >.>> >.>> )>>5

34 34 35 >.4574 >.))5F 39.> 3>.>> 79>

35 35 3F >.8599 >.358F >.>> >.>> 5743F 3F 37 >.4>9< >.)>3) )5.>> )>.>> 83>

37 37 38 >.)843 >.>F83 )5.>> )>.>> .>>55 >.>)>9 37.>> )9.7> 7>4F

39 39 3< >.>75> >.)F7F 37.>> )9.7> )F94

3< 3< 4> >.4.)4)F >.>> >.>> 74F

4> 4> 4) >.>8>3 >.>343 >.>> >.>> )F))

4) 4) 43 >.4F)> >.))7> >.>> >.>> 787

43 43 44 >.94 >.39)7 )5.>> )>.>> 548

44 44 45 ).8>9> >.F757 ) )5.>> 4>7

45 45 4F ).585> >.5984 7.>> 5.>> 44>

4F 4 47 >.>>55 >.>)>9 37.>> )9.FF 7>4F

47 47 48 >.>75> >.)F7F 37.>> )9.FF )F94

48 48 49 >.)>F4 >.)34> >.>> >.>> )345

49 49 4< >.>4>5 >.>4FF 35.>> )8.>> 33 >.>>)9 >.>>3) 35.>> )8.>> 5> 5) >.8394 >.9F>< ).3> ).>> 57.4)>> >.4734 >.>> >.>> 8).>5)> >.>589 7.>> 5.4> ).>>.>))7 >.>> >.>> 5)85

55 55 5F >.)>9< >.)484 4 )3)4

5F 5F 57 >.>>>< >.>>)3 4 )445F

57 5 58 >.>>45 >.>>95 >.>> >.>> 7977

58 58 59 >.>9F) >.3>94 8> F7.5> )48359 59 5< >.39.8> 385.F> 855

5< 5< F> >.>933 >.3>)) 495.8> 385.F> )4 9 F) >.>.>584 5>.F> 39.4> )4)5

F) F) F3 >.44)< >.)))5 4.7> 3.8> 7.)85> >.>997 5.4F 4.F>

F4 F4 F5 >.3>4> >.)>45 37.5> )> 99.3953 >.)558 35.5> )8.3> 8F)

FF FF F7 >.39)4 >.)544 >.>> >.>> 8FF

F7 F7 F8 ).F> >.F448 >.>> >.>> 4)9

F8 F8 F9 >.8948 >.374> >.>> >.>> 5F3

F9 F9 F< >.4>53 >.)>>7 )>>.>> 83.>> 837F< F< 7> >.497) >.))83 >.>> >.>> 755

7> 7> 7) >.F>8F >.3F9F )355.>> 999.>> F73

7) 7) 73 >.>.>5> 34.>> )394

73 73 74 >.)5F> >.>849 >.>> >.>> )>F)

74 74 75 >.8)>F >.47)< 338.>> )73.>> 58F

75 75 7F ).>5)> >.F4>3 F> 53.>> 4.3>)3 >.>7)) )9.>> )4.>> 9.>>58 >.>>)5 )9.>> )4.>> F95>

78 )3 79 >.84.3555 39.>> 3>.>> 577

79 79 7< >.>>58 >.>>)7 39.>> 3>.>> F95>

Tie s'itc* data

7< )) 54 >.F >.F ( ( F77

8> )4 3) >.F >.F ( ( F77

8) )F 57 ).> >.F ( ( 5>>

83 F> F< 3.> ).> ( ( 394

84 38 7F ).> >.F ( ( 5>>

$ase "A, 1?? $ase B, 12.

ant colony search algorithm (acsa)

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