distribution systems reconfiguration using ant colony optimization and harmony search algorithms

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Distribution Systems ReconfigurationUsing Ant Colony Optimization andHarmony Search AlgorithmsAlmoataz Y. Abdelaziz a , Reham A. Osama a & Salem M. Elkhodary aa Electrical Power and Machines Department, Faculty of Engineering,Ain Shams University, Cairo, Egypt

To cite this article: Almoataz Y. Abdelaziz , Reham A. Osama & Salem M. Elkhodary (2013):Distribution Systems Reconfiguration Using Ant Colony Optimization and Harmony Search Algorithms,Electric Power Components and Systems, 41:5, 537-554

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Electric Power Components and Systems, 41:537–554, 2013

Copyright © Taylor & Francis Group, LLC

ISSN: 1532-5008 print/1532-5016 online

DOI: 10.1080/15325008.2012.755232

Distribution Systems Reconfiguration Using Ant

Colony Optimization and Harmony

Search Algorithms

ALMOATAZ Y. ABDELAZIZ,1 REHAM A. OSAMA,1 and

SALEM M. ELKHODARY 1

1Electrical Power and Machines Department, Faculty of Engineering,

Ain Shams University, Cairo, Egypt

Abstract One objective of the feeder reconfiguration problem in distribution systems

is to minimize the distribution network total power loss for a specific load. Forthis problem, mathematical modeling is a non-linear mixed integer problem that

is generally hard to solve. This article proposes two heuristic algorithms inspiredfrom natural phenomena to solve the network reconfiguration problem: (1) “real ant-

behavior-inspired” ant colony optimization implemented in the hyper cube frameworkand (2) the “musician behavior-inspired” harmony search algorithm. The optimization

problem is formulated taking into account the operational constraints of distributionsystems. A 32-bus system and a 118-bus distribution were selected for optimizing

the configuration to minimize the losses. The results of reconfiguration using theproposed algorithms show that both of them yield the optimum configuration with

minimum power loss for each case study; however, the harmony search requiredshorter simulation time but more practice of the iterative process than the hyper cube–

ant colony optimization. Implementing the ant colony optimization in the hyper cubeframework resulted in a more robust and easier handling of pheromone trails than

the standard ant colony optimization.

Keywords ant colony optimization, distribution networks, harmony search, recon-figuration, power loss

1. Introduction

Distribution network reconfiguration is the process of changing the topology of distri-

bution systems by altering the open/closed status of switches to transfer loads among

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

are normally closed switches (sectionalizing switches) and normally open switches (tie

switches). Those two types of switches are designed for both protection and configuration

management. In 1975, the network reconfiguration for loss reduction concept was first

introduced by Merlin and Back [1] by applying the branch-and-bound heuristic technique.

Since then, several reconfiguration techniques have been proposed, which can be

grouped into three main categories: (1) techniques based upon a blend of heuristics

Received 20 June 2012; accepted 29 November 2012.Address correspondence to Prof. Almoataz Y. Abdelaziz, Electrical Power and Machines

Department, Faculty of Engineering, Ain Shams University, Abdu Pasha Square, Abassia, Cairo,11517, Egypt. E-mail: [email protected]

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538 A. Y. Abdelaziz et al.

and optimization techniques [2], which proved to be very time consuming for large

distribution systems therefore not practical for real-time implementation; (2) techniques

based upon purely heuristic techniques [3, 4], where the optimality is not guaranteed

and the algorithms are more likely to fall into local optimum; and (3) techniques based

on artificial intelligence (AI) and modern heuristics, such as genetic algorithms [5],

particle swarm optimization [6], tabu search (TS) [7], simulated annealing (SA) [8],

hybrid algorithms [9], and so on. These AI-based algorithms overcome the shortcoming of

the conventional methods in saving computation time, ensuring accuracy and optimality,

and thus are suitable for real-time applications.

In this article, two meta-heuristic algorithms are employed to solve the network

reconfiguration problem. The first algorithm is the gradient search ant colony optimiza-

tion (ACO) algorithm proposed by Dorigo [10], implemented in the hyper cube (HC)

framework. It is inspired by the foraging behavior of real ant colonies finding the shortest

path between their nest and the food source. The second algorithm is a random search

musician-behavior-inspired evolutionary algorithm, harmony search (HS). It is inspired

by the observation that the aim of music is to search for a perfect state of harmony

that is analogous to finding the optimality in an optimization process. In this article,

the two proposed algorithms are implemented to solve the minimum loss distribution

network reconfiguration problem and are applied to 32-bus and 118-bus systems from

the literature. The results obtained by both algorithms are then compared to previously

applied algorithms to prove their effectiveness.

This article is organized as follows. Section 2 explains the distribution network

minimum loss reconfiguration problem, its objective function, and constraints. Section 3

illustrates the ACO paradigm and how it is developed in the HC framework to solve the

network reconfiguration problem. Section 4 explains the main steps of the HS algorithm,

its operators, and parameters. Section 5 shows the numerical results of applying both

algorithms to the two test systems. Finally, the conclusion is given in Section 6.

2. Formulation of the Network Reconfiguration Problem forLoss Reduction

The reconfiguration problem can be formulated as follows:

Min f D

NRX

iD1

Ri jIi j2 (1)

subjected to the following constraints:

� the voltage magnitude:

V min � jVi j � Vmax;8i 2 NbI (2)

� the current limit of branches:

jIi j � Ij max;8j 2 NRI (3)

� radial topology:

det.A/ D 1 or �1 ! radial,

det.A/ D 0 ! not radial,(4)

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Reconfiguration of Electrical Distribution Networks 539

where

f is the fitness function to be minimized, corresponding to the total power loss

in the system;

Ri is the resistance of branch i ;

Ii is the magnitude of the current flowing through branch i ;

Vi is the voltage on bus i ;

Vmin and Vmax are minimum and maximum bus voltage limits, respectively;

Ij and Ij max are current magnitude and maximum current limit of branch i ,

respectively; and

Nb and NR are the total number of buses and branches in the system, respectively.

The objective function is calculated starting from the solution of the power flow

equations that can be solved using the forward/backward sweep method, which is very

robust and proven to be efficient for solving radial distribution networks [11]. To check

the radiality constraints for a given configuration, a method based on bus incidence matrixOA is used [12] in which a graph may be described in terms of a connection or incidence

matrix. Of particular interest is the branch to node incidence matrix OA, which has one

row for each branch and one column for each node with a coefficient aij in row i and

column j . The value of aij D 0 if branch i is not connected to node j , aij D 1 if branch i

is directed away from node j , and aij D �1 if branch i is directed toward node j . For

a network calculation, a reference node must be chosen. The column corresponding to

the reference node is omitted from OA, and the resultant matrix is denoted by A. If the

number of branches is equal to the number of nodes, then a square branch-to-node matrix

is obtained. The determinant of A is then calculated. If det.A/ is equal to 1 or �1, then

the system is radial; else, if det.A/ is equal to zero, this means that either the system is

not radial or a group of loads are disconnected from service.

3. ACO

3.1. Behavior of Real Ants

As is well known, real ants are capable of finding the shortest path between their nest

and food sources by the indirect communication between them via pheromone trails, and

this behavior forms the fundamental paradigm of the ACO algorithm.

3.2. ACO Paradigm

In the ACO method, a set of artificial ants cooperate in finding optimal solutions to

difficult discrete optimization problems. These problems are represented as a set of points

called “states,” and the ants move through adjacent states. Exact definitions of state and

adjacency are problem specific. The ACO adopts three main rules [13].

1. The state transition rule (“the pseudo random proportional rule”). At first, each

ant is placed on a starting state. Each will build a full path from the beginning to

the end state through the repetitive application of the state transition rule given

in Eq. (5):

Pk.i; j / DŒ�.i; j /�˛Œ�.i; j /�ˇ

X

m2Jk.i/

Œ�.i; m/�˛Œ�.i; m/�ˇ;8j 2 Jk.i/; (5)

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where Pk.i; j / is the probability with which ant k in node i chooses to move to

node j ; �.i; j / is the pheromone deposited on the edge between nodes i and j ;

�.i; j / is the visibility of the edge connecting nodes i and j , which is problem

specific (e.g., inverse of the edge distance); and Jk.i/ is the set of nodes that

remain to be visited by ant k positioned on node i . ˛ and ˇ are parameters

that determine the relative importance of pheromone versus the path’s visibility.

The state transition rule favors transitions toward nodes connected by shorter

edges with greater amount of pheromone.

2. Local updating rule. While constructing the solution, each ant modifies the phero-

mone on the visited path. It is an optional step intended to shuffle the search

process. It increases the exploration of other solutions by making the visited lines

less attractive:

�.i; j / D .1 � �/�.i; j /C ��0; (6)

where �.i; j / is the amount of pheromone deposited on the path connecting

nodes i and j , �0 is the initial pheromone value, and � is a heuristically defined

parameter.

3. Global updating rule. When all solutions are completed, the global updating rule

is applied to increase the pheromone concentration of the edges belonging to the

best ant solution:

�.i; j / D .1 � �/�.i; j /C �ı�1; (7)

where ı is a parameter belonging to the globally best solution, and � is the

pheromone evaporation factor element in the interval [0–1]. This rule is intended

to make the search more directed, enhancing the capability of finding the optimal

solution.

3.3. Formulation of ACO in the HC Framework for Solving

Minimum Loss Reconfiguration Problem

The HC framework is a recently developed framework for the standard ACO [14]. It

is based on changing the pheromone update rules used in ACO algorithms so that the

range of pheromone variation is limited to the interval [0–1], thus providing automatic

scaling of the objective function used in the search process and resulting in a more

robust and easier to implement version of the ACO procedure. The distribution system

is represented as an undirected graph G.B; L/ composed of set B of nodes and set L of

arcs indicating the buses and their connecting branches (switches). Artificial ants move

through adjacent buses, selecting switches that remain closed (sectionalizing switches)

to minimize the system power losses. The solution iterates over three steps.

Step 1. Initialization. The solution starts with encoding parameters by defining

� system parameters—set of supply substations S ; set of buses NB ; set of

branches NR; (where each branch has two possible states, either “0” for

an opened tie switch or “1” for a closed sectionalizing switch); load data

Pload, Qload; branch data Rb , Xb ; base configuration of the system C .0/

defined by the system‘s tie switches; initial power losses of the system

f .C .0// by solving the power flow for C .0/ and evaluating the fitness

function f and

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� algorithm parameters—number of artificial ants in each iteration N ; initial

pheromone quantity �0 assigned to each switch, which is arbitrarily chosen

in the range [0–1]; evaporation factor of pheromone trails �; parameters

˛ and ˇ that determine the relative importance of the line’s pheromone

versus its visibility; a counter h for the number of iterations; a counter

x that is updated at the end of the iteration with no improvement in the

objective function; maximum number of iterations Hmax; and maximum

number of iterations Xmax with no improvement in the objective function,

respectively; the base configuration is then set as an initial reference

configuration and as the best configuration found so far such that Cbest D

C.0/best D C .0/.

Step 2. Ants’ reconfiguration and pheromone updating. In each iteration h, a refer-

ence configuration is set as the best configuration of the previous iteration

such that C.h�1/best D C

.h/ref . N ants are initially located on N randomly

chosen open switches and are sent in parallel in such a way that each ant

n in the hth iteration introduces successive configuration changes to the

reference configuration by applying the state transition rule giving a new

radial configuration C.h/n . Once all ants finish their tour, the configuration

corresponding to each ant is evaluated in three steps:

1. check the radiality constraints; if radial, go to Step 2; otherwise, this trial

configuration is discarded;

2. running the load flow and check for voltage and loading limits; if either

limit is violated, the configuration is discarded; else, go to Step 3;

3. computing the objective function f .C.h/n /.

The best configuration of the hth iteration C.h/best is identified, which is the

configuration corresponding to the minimum evaluated objective function of

all ants (minimum power loss). The best configuration of the hth iteration

C.h/best is compared to the best Cbest configuration so far such that if f .C

.h/best/ <

f .Cbest/, the overall best configuration is updated such that Cbest D C.h/best.

Finally, the pheromone updating rules are applied. For all switches that

belong to the best configuration, the pheromone values are updated using

Eq. (8); otherwise, the pheromone is updated using Eq. (9):

� .h/ D .1 � �/� .h�1/ C ��; (8)

� .h/ D .1 � �/� .h�1/; (9)

where � .h/ is the new pheromone value after the hth iteration, � .h�1/ is the

old value of pheromone after the .h � 1/th iteration, � is arbitrarily chosen

from the interval [0–1], and � is a heuristically defined parameter that was

chosen to equal

� Df .Cbest/

f .C.h/best/

: (10)

Since f .C.h/

best/ cannot be lower than f .Cbest/, the pheromone assigned

to any switch cannot fall outside the range [0–1] so that the pheromone

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update mechanism is fully consistent with the requirements of the HC

framework [15].

Step 3. Termination of the algorithm. The solution process continues until maximum

number of iterations reaches h D Hmax or until no improvement of the

objective function has been detected after specified number of iterations

x D Xmax.

4. HS Algorithm

The HS algorithm is a meta-heuristic population search algorithm proposed by Geem

et al. in 2001 [16]. HS was derived from the natural phenomena of musicians’ behavior

when they (population members) collectively play their musical instruments (decision

variables) to come up with a pleasing harmony (global optimal solution). This state is

determined by an aesthetic standard (fitness function). When a musician is improvising,

he has three possible choices: playing any famous tune exactly from his memory (memory

Consideration), playing something similar to the aforementioned tune (pitch adjustment),

or composing new or random notes from the pitch range (random selection). The main

steps of HS are as follows [17].

Step 1. Initialize the problem, algorithm parameters and harmony memory (HM).

The optimization problem is specified as follows:

minimize f .x/

subjected to xi 2 Xi ; i D 1; 2; 3; : : : ; N;(11)

where f .x/ is an objective function, x is the set of each decision variable

xi , N is the number of decision variables, and Xi is the set of the possible

range of values for each decision variable. The HS algorithm parameters

are also specified in this step. These are the HM size (HMS), HM consid-

ering rate (HMCR), pitch adjusting rate (PAR), and maximum number of

improvisations (NI). The HM is a memory location where all the solution

vectors (sets of decision variables) are stored. Here, the HMCR and PAR are

parameters used to improve the solution vector, defined in Step 2. The initial

HM consists of a certain number of randomly generated solutions for the

optimization problem under consideration. For a problem of N variables, an

HM with the size of HMS can be represented as in Eq. (12):

HM D

2

6

6

6

6

6

6

6

6

6

4

X11 x1

2 � � � x1N�1 x1

N

::: f1

x21 x2

2 � � � x2N�1 x2

N

::: f2

� � � � � � � � � � � � � � �::: � � �

xHMS1 xHMS

2 xHMSN�1 xHMS

N

::: fHMS

3

7

7

7

7

7

7

7

7

7

5

; (12)

where .xi1; xi

2; : : : ; xiN / represents a candidate solution for the optimization

problem, and f1 is the value of the fitness function corresponding to the

first solution vector. For the network reconfiguration problem, the solution

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vector is represented by the set of tie switches in the radial configuration of

the network [18].

The configuration of the 32-bus system, shown in Figure 1, can be rep-

resented as in Eq. (13):

configurationD Œ13; 37; 22; 26; 29�; (13)

where 13 is the tie switch from loop 1, 37 is the tie switch from loop 2, etc.

The HM is sorted in ascending order with respect to the fitness function such

that configuration with the least power loss (best configuration) is at the top

and the one with the highest power loss (worst configuration) is at the bottom.

The initial randomly generated HM of HMS D 5 is shown in Eq. (14). As

shown, each row in the initial HM represents a radial configuration with the

corresponding power losses. The final HM at the end of the iterative process

is as shown in Eq. (15); the first row of the final HM represents the optimum

solution with the least power losses:

HM D

2

6

6

6

6

6

6

6

6

6

4

10 35 6 32 27::: 172:25

10 34 6 33 28::: 188:53

9 36 21 25 29::: 195:8

12 35 20 26 29::: 296:61

3 34 21 27 28::: 405:02

3

7

7

7

7

7

7

7

7

7

5

; (14)

HM D

2

6

6

6

6

6

6

6

6

6

4

13 18 23 33 25::: 139:50

13 18 23 33 30::: 141:55

13 37 23 33 25::: 142:42

13 18 7 33 25::: 143:11

10 18 23 33 25::: 143:56

3

7

7

7

7

7

7

7

7

7

5

: (15)

Step 2. Improvise a new harmony. A new harmony vector (x0

1; x0

2; : : : ; x0

N ) is gen-

erated based on three main rules: (1) memory consideration, (2) pitch ad-

justment, and (3) random selection. Generating a new harmony is called

“improvisation.” Each component of the solution is chosen either from the

HM or by randomness, depending on the value of the HMCR, which varies

between 0 and 1, and defined as the rate of choosing one value from the

historical values stored in the HM, while 1-HMCR is the rate of randomly

selecting one value from the possible range of values:

If (rand. /hHMCR),

x0

i x0

i 2 fx1i ; x2

i ; : : : ; xHMSi gI

else,

x0

i x0

i 2 Xi I

end,

(16)

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Figure 1. Initial configuration of the 32-bus system. Tie switches and sectionalizing switches are

represented by dotted and solid lines, respectively.

where rand. / is a uniformly distributed random number between 0 and 1,

and Xi is the set of the possible range of values for each decision variable.

Every variable x0

i obtained by the memory consideration is examined to

determine whether it should be pitch-adjusted. This operation uses the PAR

parameter, which is the rate of pitch adjustment, and the value 1-PAR is the

rate of doing nothing as follows:

If .rand. /hPAR/,

x0

i x0

i ˙ rand. / � BWI

else,

x0

i x0

i I

end,

(17)

where BW is an arbitrary distance bandwidth for the continuous design

variable, and rand. / is uniform distribution between �1 and 1. If the

problem is discrete in nature, BW is taken as 1. In Step 2, HM consideration,

pitch adjustment, or random selection is applied to each variable of the new

harmony vector in turn.

Step 3. Update HM. If the new harmony vector (x0

1; x0

2; : : : ; x0

N ) is better than the

worst harmony in the HM, judged in terms of the objective function value

(yields to a better fitness than that of the worst member in the HM), the new

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harmony is included in the HM and the existing worst harmony is excluded

from the HM. The HM is rearranged in ascending order according to the

fitness function. Otherwise, the new harmony is discarded.

Step 4. Check stopping criteria. If the stopping criterion (maximum NI) is satisfied,

computation is terminated, and the best one among the solution vectors

stored in the HM is finally selected, which is the optimum solution of the

problem. Otherwise, Steps 3 and 4 are repeated.

5. Worked Examples

Two test systems from literature are investigated using the HC-ACO and the HS al-

gorithms. The results are presented and discussed in detail. The solution algorithms

were implemented using MATLAB V7 (The MathWorks, Natick, Massachusetts, USA).

Several simulations were done to come up with typical values and determine the best

setting for the HC-ACO and HS parameters. For the HC-ACO, it was concluded that

the greater the amount of ants, the stronger the global searching ability will be. But the

computing time of this algorithm is proportional to N ; the greater the amount of ants,

the longer the computing time will be. Thus, is it preferable for N to fall in the range

of the number of variables to twice the number of variables. The smaller the value of

˛ and ˇ, the parameters that determine the relative attaching degree of the information

traces and the heuristic search, the more possible the ants may choose the paths that have

been chosen. If ˛ and ˇ are too small (less than 0.5), the search may be go to the local

minimum prematurely. But if ˛ and ˇ are too large (greater than 1), the affection of

stimulating function will be enhanced but the convergent speed of algorithm decreases;

it is also very difficult to obtain a good solution. And since the optimization process is

the result of the mutual affection of the information traces and the inspiring function, so

˛ and ˇ should typically be in the range 0.5–1. The evaporation factor � represents the

volatile extent of the information in the pheromone updating rules. Experiments showed

that as � increases, the pheromone volatilizes faster and the global searching ability

improves, but the convergent speed of the algorithm decreases, and when � decreases,

the pheromone volatilizes slower and the solutions that have been searched before may be

chosen as more possible, which could affect the global searching ability of the algorithm.

So, based on the simulations, the result is best if 0:01 � � � 0:1.

For the HS parameters, simulations showed that if the HMCR, which gives the

probability of memory consideration, is too low (near zero), only a few elite harmonies

are selected and the optimization process may converge too slowly; if this rate is extremely

high (near 1), the pitches in the HM are mostly used, and others are not explored well,

which will not lead to good solutions. Therefore, typically, HMCR D 0:7–0.95. Also a

low PAR, which controls the degree of adjustment, can slow down the convergence of HS

because of the limitation in the exploration of only a small subspace of the whole search

space. On the other hand, a very high PAR may cause the solution to scatter around

some potential optima in a random search. Therefore, typically, PAR D 0:1–0.5. The

HMS is typically in the range of N � 2N for any N variables problem. For the 32-bus

system, the HC-ACO parameters used are N D 10, ˛ D 0:7, ˇ D 0:9, � D 0:04, �0 D 1,

Hmax D 100, and Wmax D 10; the HS parameters are NI D 200, HMS D 5, HMCR D 0:8,

and PAR D 0:2. For the 118-bus system, the setting for the HC-ACO parameters was

N D 20, ˛ D 0:7, ˇ D 0:9, � D 0:01, �0 D 1, Hmax D 100, and Wmax D 10; the HS

parameters were NI D 800, HMS D 15, HMCR D 0:9, and PAR D 0:1.

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Test System 1

The first system is a 12.66-kV radial distribution system whose data is given in [4]. A

schematic diagram of the initial configuration [22, 26, 29, 37, 13] is shown in Figure 1.

The system has 1 supply point, 32 buses, 3 laterals, and 5 tie switches. The total substation

loads of the base configuration are 3715 kW and 2300 kVAR. Table 1 shows the initial

configuration data of the system, the results obtained by the two proposed algorithms

after reconfiguration, and the results obtained by previously applied purely heuristic

algorithms from literature, such as branch exchange and branch-and-bound algorithms,

and AI-based algorithms, such as standard ACO, modified TS (MTS), modified particle

swarm (MPS), SA, and TS. As shown in Table 1, the proposed methods yield to the same

optimum configuration [23, 33, 25, 18, 13] with final loss of 139.5 kW. This amounts

to a reduction of 31.13% in total power loss. The results obtained by the HC-ACO

are identical to those obtained by AI-based algorithms, and this proves the validity of

the proposed approach. In comparison with purely heuristic algorithms, the HC-ACO

provided a better loss reduction than the branch exchange mechanism of Baran and

Wu [4] and the branch-and-bound technique of Shirmohammadi and Hong [2].

Figure 2 shows the voltage profile of the initial and final configurations of the

system. As shown, the system’s voltage profile is improved after reconfiguration such

that before reconfiguration (case 1), the lowest bus voltage was 0.9129 p.u., while after

reconfiguration (case 2), the lowest bus voltage is 0.9378 p.u. with 2.6% improvement.

Table 2 shows the evolution of objective function and maximum and minimum

pheromone during the iterative process of the best solution of the HC-ACO algorithm.

The optimum solution (indicated in bold) was reached at the fourth iteration compared

to 153 iterations for the HS algorithm. The initial pheromone level is fixed �0 D 1, so

that for any switch belonging to all the best configurations found in any iteration, its

pheromone will always remain at unity in all iterations. Conversely, if a switch does not

belong to the best configuration for at least one iteration, its pheromone level at the end

of the iterative process will be lower than the initial value. The maximum pheromone

value remains at unity due to the fact that the best configuration has been at least reached

Table 1

Results of reconfiguration of the 32-bus system

Configuration Total power loss (kW) Tie switches

Initial 203 22, 26, 29, 37, 13

Final using HC-ACO Best: 139.5 23, 33, 25, 18, 13

Worst: 154.7

Average: 150.25

Final using HS Best: 139.5 23, 33, 25, 18, 13

Worst: 141.5

Average: 140.2

Final using MPS [6],

SA C TS [9], MTS [7]

139.5 23, 33, 25, 18, 13

Final using standard ACO [19] 139.5 23, 33, 25, 18, 13

Final using branch exchange [4] 147.89 22, 26, 30, 17, 14

Final using branch and bound [2] 140.28 23, 33, 27, 18, 13

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Figure 2. Voltage profile of the 32-bus system before and after reconfiguration. (color figure avail-

able online)

or improved at each iteration, which is why implementing the ACO in the HC framework

made the pheromone trails easy to handle. Table 3 shows the evolution of the standard

ACO algorithm for the 32-bus system as given in [19]. In comparison with Table 2, it is

clear that the optimum solution of standard ACO was found after 27 iterations compared

to 4 iterations for the HC-ACO, and thus, implementing ACO in the HC framework

comes with the benefit of scaling objective function value, allowing rapid discovery of

good solutions and fast optimum convergence. Figure 3 further explains the convergence

of the HC-ACO and the standard ACO.

Table 2

Evolution of objective function and maximum and minimum pheromone during the

iterative process for the 32-bus system

Iteration no. 1 2 3 4 5

Power losses (kW) 149.1 142.6 139.8 139.5 139.5

Maximum pheromone 1 1 1 1 1

Minimum pheromone 0.96 0.921 0.8847 0.849 0.8153

Bold indicates the optimum solution.

Table 3

Convergence of the standard ACO for the 32-bus system as in [19]

Iteration no. 5 10 15 20 25 27

Power losses (kW) 145.8 144.3 144.3 142.6 140.1 139.5

Bold indicates best setting.

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(a)

(b)

Figure 3. Convergence curve: (a) HC-ACO and (b) standard ACO [19]. (color figure available

online)

Test System 2

The second system is an 11-kV system with 1 supply point and 118-bus and 15 tie

lines. The system data was given in [20]. The total substation loads for the initial

configuration shown in Figure 4 are 22,709.7 kW and 17,042.2 kVAR, and the total

power loss is 1294.68 kW.

The same optimal configuration shown in Figure 5 was reached after 15 iterations

using the HC-ACO and 549 iterations using the HS algorithm, where the total power

is 865.322 kW with 33.1% reduction than that of the base configuration. The results of

reconfiguration using the two algorithms and previously applied algorithms from literature

are shown in Table 4.

The proposed methods obtained better results than the purely heuristic branch ex-

change mechanism and the AI-based TS and the same results of improved TS (ITS) and

MTS algorithms.

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Figure 4. Initial configuration of the 118-bus system.

Figure 6 shows the voltage profile of the initial and final configurations of the

system. As shown, most of the node voltages have been improved after reconfiguration

such that before reconfiguration, the lowest bus voltage was 0.8685 p.u., while after

reconfiguration, the lowest bus voltage is 0.933 p.u. with 6.9 % improvement. Table 5

shows the different setting for the HS parameters for the 118-bus system. The best

setting is indicated in bold. It can be concluded that if the HMCR is too low, only a

few elite harmonies are selected and it may converge too slowly; if this rate is extremely

high (near 1), the pitches in the HM are mostly used, and others are not explored well,

which will not lead to good solutions. Therefore, typically, HMCR D 0:7–0.95. Also

a low PAR can slow down the convergence of HS because of the limitation in the

exploration of only a small subspace of the whole search space. On the other hand, a very

high pitch adjustment may cause the solution to scatter around some potential optima

in a random search. Therefore, typically, PAR D 0:1–0.5. To verify the performance

of the proposed algorithms, both systems were repeatedly solved for 100 runs using

each algorithm. The best and the worst values among the best solutions, the average

value of these 100 runs, as well as the execution times are listed in Tables 1 and 4,

respectively.

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Figure 5. Final configuration of the 118-bus system.

Tables 6 and 7 show the convergence characteristics of the proposed algorithms for

the systems under study. For the purpose of comparison, it is clear that for the 32-bus

system, the HS algorithms reached the optimum configuration in a shorter time than the

HC-ACO. For the 118-bus systems, the HS algorithm reached the final configuration in

a much shorter time than the HC-ACO, because the HS algorithm is a simple algorithm

with fewer steps than the HC-ACO. The HS also has fewer parameters, since only

four parameters are required to be set for running the HS algorithm compared to seven

parameters for running the HC-ACO algorithm.

However, HC-ACO reached the optimum solution in fewer iterations than the HS

algorithm, because the HC-ACO is a constructive and greedy search approach that makes

use of positive feedback, such as the gradient information of the objective function as well

as pheromone trails, and heuristic information that guides the search and leads to rapid

discovery of good solutions, requiring less practice to reach the optimum solution. The

HS, however, is a random search algorithm that does not require any prior information

to generate a solution vector and, thus, needs a lot of practice to identify the solution

space and to reach the optimum solution in a reasonable time. The initial configuration

of the system must be defined for the HC-ACO, but this was not necessary for the HS,

since it HS does not require any initial value setting for the problem variables.

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Table 4

Results of reconfiguration of the 118-bus system

Configuration Total power loss (KW) Tie switches

Initial 1294.68 48-27, 17-27, 8-24, 56-45, 65-51,

38-65, 9-42, 61-100, 76-95, 91-78,

103-80, 113-86, 110-89, 115-123, 25-36

Final using HC-ACO Best: 865.322

Worst: 870.5

Average: 867.32

45-44, 27-26, 22-23, 54-53, 51-50,

64-65, 41-42, 61-100, 76-77, 74-75,

80-79, 85-86, 89-110, 114-115, 33-34

Final using HS Best: 865.3

Worst 868.06

Average: 867.32

45-44, 27-26, 22-23, 54-53, 51-50,

64-65, 41-42, 61-100, 76-77, 74-75,

80-79, 85-86, 89-110, 114-115, 33-34

Final using branch

exchange [13]

885.56 45-44, 17-27, 23-24, 53-52, 51-50,

64-65, 41-42, 61-100, 76-77, 74-75,

79-80, 85-86, 89-110, 114-115, 35-36

Final using TS [21] 884.163 45-44, 27-26, 22-23, 54-53, 51-50,

64-65, 41-42, 61-100, 76-77, 74-75,

80-79, 85-86, 89-110, 114-115, 33-34

Final using ITS [20]

and MTS [7]

865.322 45-44, 27-26, 22-23, 54-53, 51-50,

64-65, 41-42, 61-100, 76-77, 74-75,

80-79, 85-86, 89-110, 114-115, 33-34

Figure 6. Voltage profile of the 118-bus system before and after reconfiguration. (color figure

available online)

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Table 5

Different settings of the HS parameters and the best

corresponding power losses for 118-bus system

HMS HMCR PAR Power losses (KW)

15 0.5 0.2 1030

15 0.6 0.2 951.82

15 0.7 0.2 925.5

15 0.75 0.25 897.7

15 0.75 0.1 907.5

15 0.8 0.2 904.2

15 0.95 0.1 865.3

30 0.8 0.1 889.4

30 0.8 0.2 899.3

30 0.85 0.2 888

30 0.9 0.3 911.7

30 0.9 0.1 876.53

40 0.6 0.1 983.2

40 0.75 0.1 877.7

40 0.85 0.2 893.06

40 0.85 0.3 896.43

Bold indicates best setting.

Table 6

Simulation time for the two case studies

32-bus (time in sec) 118-bus (time in min)

HC-ACO 4.86 7.63

HS 4.65 1.85

Table 7

Number of iterations required for convergence

32-bus (time in sec) 118-bus (time in min)

HC-ACO 4 15

HS 153 549

In comparison with previously applied algorithms, both the HS and the HC-ACO

algorithms reached the optimal configuration in a much shorter time than TS [7], which

converged in 10 min for the 32-bus system and 5 hr for the 118-bus system.

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6. Conclusion

Two heuristic optimization techniques were presented in this article to find the most

appropriate topology of the distribution system that minimizes the total system power loss:

the gradient search ACO implemented in the HC framework (HC-ACO) and the random

search HS algorithms. The objective function is subjected to many constraints, such as

bus voltage limits, branch current limits, and radial configuration. Two test systems with

different numbers of nodes are studied to demonstrate the effectiveness of the proposed

techniques. The convergence characteristics of both algorithms are studied as well as

the benefit of implementing the ACO in the HC framework. Their low computation

effort and short simulation time make them suitable for real-time implementation. For

further research, economic analysis can be made to enhance the significance of the choice

of the adopted algorithms for reconfiguration of electrical distribution networks. These

algorithms can also be applied to various power system non-linear optimization problems,

with a wide range of engineering application prospects.

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distribution systems reconfiguration using ant colony optimization and harmony search algorithms

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