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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 §ionalizing 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
)3
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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
)4
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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.)
)8
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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
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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.
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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
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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
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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.
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remains firmly embedded in biology, and so it is common to discuss &parents,'
&children', &alleles' and so on.
0.
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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 ) ) ) > > > > )
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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.
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!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
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/: 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
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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
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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
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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
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by ants in each iteration should be evaluated. If a better path found in the
process, it will be saved for later reference.
4
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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>
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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 $us #yste(
) , 3 !ode number
), 3,. . . . Branch number
K R K)) F 5)
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transferring loads on eeder() to eeder(3 is e+pected to increase
losses. /onsequently, from rule 3 opening the sectionalizing switch ) or 3 are
regarded as undesirable and need not be considered. Therefore, associated with
closing the tie switch F are three candidate options, viz., opening the sectionalizing
switches )5
53
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inde+ is minimized. In other words, all the branch load balancing indices are set to
be more or less the same value and are also nearly equal to the system load
balancing inde+.
The load(balancing problem is formulated in the form of branch load
balancing and system load balancing indices are
The branch load balancing inde+ ma+-i
-i
-i:
:@B = L5.9
The system load balancing inde+
=
=)nb
)ima+
-i
-i
sys:
:
nb
)@B L5..>F).>F) )5 >.>F).>F).)FF.)FF
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$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>.>>
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34 34 35 >.9 >.8>.>> 3>>.>>
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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>>> ( (
$ase "A, 1?? B 12.
Ta/&e A-0, Line3 &oad and tie s'itc* data o =-node radia& distri/ution
net'or8
7)
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$r.
No.
#endin+
end
node
Receivin+-
end node
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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
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57 5 58 >.>>45 >.>>95 >.>> >.>> 7977
58 58 59 >.>9F) >.3>94 8> F7.5> )48359 59 5< >.39.8> 385.F> 855
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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
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$ase "A, 1?? $ase B, 12.