optimal reconfiguration and capacitor placement by robust searching hybrid differential evolution

Optimal reconfiguration and capacitor placement by robust searching hybriddifferential evolution

Chung-Fu Chang*,y

Department of Electrical Engineering, WuFeng Institute of Technology, Chiayi 62153, Taiwan

SUMMARY

This paper aims to study distribution system operations by the robust searching hybrid differential evolution (RSHDE) method. The objective ofthis study is to present new algorithms for solving the optimal feeder reconfiguration problem, the optimal capacitor placement problem, and theproblem of a combination of the two.Mathematically, the problem of this research is a nonlinear programming problemwith integer variables. Thispaper presents a new approach which employs the RSHDE algorithm with integer variables to solve the problem. Depending on the initialassignment of the integer variables, the HDE may fail to find the initial search direction for large-scale integer system. This is because theHDE applies a random search at its initial stages. Therefore, two new schemes, the multi-direction search scheme and the search space reductionscheme, are embedded into the HDE. These two schemes are used to enhance the search ability before performing the initialization step of thesolution process. The proposed approach is demonstrated using an IEEE illustrative example and one practical distribution network of TaiwanPower Company (TPC). Moreover, the previous HDE, simulated annealing (SA) and genetic algorithms (GA) methods are also applied to the sameexample systems for the purpose of comparison. Copyright # 2009 John Wiley & Sons, Ltd.

key words: RSHDE; feeder reconfiguration; capacitor placement

1. INTRODUCTION

Generally, distribution systems consist of groups of interconnected radial circuits. The configuration of the system may be varied by

switching operations to transfer loads among the feeders. Two types of switches are applied in primary distribution systems, which

are normally closed switches (sectionalizing switches) and normally open switches (tie switches). Both types of switches are

designed for protection configuration management. Feeder reconfiguration is the process of changing the topology of distribution

systems by altering the open or closed status of switches.

Capacitors have been commonly employed to provide reactive power compensation in distribution systems. They are used to

reduce power losses and to maintain a voltage profile within acceptable limits. The benefits of the compensation depend greatly on

how the capacitors are placed in the system, specifically on the location and size of the added capacitors.

Civanlar et al. [1] conducted the early research on feeder reconfiguration for loss reduction. Baran et al. [2] modelled the problem

of loss reduction and load balancing as an integer programming problem. In References [3,4], the authors used a genetic algorithm to

look for the minimum loss configuration. In Reference [5], the authors presented the use of the power flow method based on a

heuristic algorithm to determine the minimum loss configuration for radial distribution networks. In References [6,7], the authors

proposed a solution procedure which employed simulated annealing (SA) to search for an acceptable non-inferior solution. In

Reference [8], the authors proposed an economic operation model to solve distribution network configuration. In Reference [9], the

authors proposed a tree encoding and two genetic operators to improve the EA performance for network reconfiguration problems.

EUROPEAN TRANSACTIONS ON ELECTRICAL POWEREuro. Trans. Electr. Power 2010; 20:1040–1057Published online 1 September 2009 in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/etep.383

*Correspondence to: Chung-Fu Chang, Department of Electrical Engineering, WuFeng Institute of Technology, Chiayi 62153, Taiwan.yE-mail: [email protected]

Copyright # 2009 John Wiley & Sons, Ltd.

In Reference [10], the authors proposed a fuzzy multi-objective approach to solve the network reconfiguration problem. In

Reference [11], the authors proposed an evolutionary algorithm (EA) based on 1/5 success rule of evolution strategies (ESs) to solve

distribution network configuration.

Optimal capacitor placement is a combinational optimization problem which is commonly solved by employing mathematical

programming techniques. Grainger et al. [12,13] formulated the problem as a nonlinear programming problem by treating the

capacitor sizes and the locations as continuous variables. Duran [14] considered the capacitor sizes as discrete variables and used

dynamic programming to find the optimal solution. A simple heuristic numerical algorithm that is based on the method of local

variation is proposed in Reference [15], and a sensitivity-based method to solve optimal capacitor placement problems is presented

in Reference [16]. Chiang et al. [17] used the optimization techniques, SA to search the global optimum solution to the capacitor

placement problem. In References [18–20], the authors used the genetic algorithm (GA) to select capacitors for radial distribution

systems. In Reference [21], the authors proposed a single dynamic data structure for an evolutionary programming (EP) algorithm to

solve the optimal capacitor allocation. In Reference [22], the authors proposed an EA embedded with the concept of ant colony

search to solve optimal capacitor placement problems.

However, most of the previous studies handled feeder reconfiguration problems without consideration of capacitor placement [1–

11] or handled capacitor placement problems without consideration of feeder reconfiguration [12–22]. They dealt with the feeder

reconfiguration and capacitor placement in a separate manner [1–22], which may result in unnecessary losses and cannot yield the

minimum loss configuration. On the other hand, there are only a few examples in the literature of loss minimization applying

heuristic techniques for feeder reconfiguration and capacitor placement [23–26]. Therefore, we present a new algorithm to solve the

feeder reconfiguration and capacitor placement problems for optimal loss minimization of distribution systems.

Evolutionary algorithms (EAs) are a class of stochastic search and optimization methods that include genetic algorithms (GA),

evolutionary programming (EP), evolution strategies (ES), genetic programming (GP) and their variants. These algorithms, based

on the principles of natural biological evolution, have received considerable and increasing interest over the past decade. EAs

operate on a population of potential solutions, applying the principle of survival of the fittest to produce successively better

approximations to a solution. Recently, some engineering optimization problems have been successfully solved by EAs [27]. EAs

have proved to be one class of the most important methods for solving the severest problems.

Differential evolution (DE) developed by Storn and Price [28–30] is one of the most superior evolution algorithms (EAs). DE is a

simple method based on stochastic searches, in which function parameters are encoded as floating-point variables. This method has

proved to be candidate to solve real-valued optimization problems. The computational algorithm of DE is very simple to understand

and implement. Only a few parameters in this algorithm are required to be set by the user. The fitness of an offspring is one-to-one

competed with that of the corresponding parent in DE. This one-to-one competition will have a faster convergence speed than other

EAs. Unfortunately this faster convergence yields in a higher probability of searching towards a local optimum or getting premature

convergence. This drawback could be overcome by employing a larger population. However, by doing so, more computation time is

required to estimate the fitness function. In order to avoid employing a large population, Chiou and Wang [31] had developed a

hybrid algorithm of differential evolution (HDE) to overcome such drawbacks. The HDE had been successfully applied to solve

parameter estimation, optimal control and fuzzy decision-making problems of fermentation processes [32,33]. Though the HDE has

been applied to real-valued optimization problems, but it still has some difficulty in the application to large-scale integer

optimization systems. To handle the integer variables, the HDE may cause the mechanism of the mutation strategy to deteriorate.

Although the HDE is capable of finding the initial search-direction for small-sized or medium-sized integer systems, it may fail to

find the initial search-direction for large-scale integer systems. This is because the HDE applies a random search in its initial stages.

In order to avoid such a drawback, a RSHDE method equipped with two additional schemes, the multi-direction search scheme and

the search space reduction scheme, is proposed.

In this study, the RSHDE for solving the feeder reconfiguration and capacitor placement of distribution systems is proposed. Due

to the discrete nature of network reconfiguration and reactive compensation devices in the real-scale system, both feeder

reconfiguration and capacitor placement lead to a nonlinear programming problem with integer (discrete) variables. This paper

presents a new approach which employs robust searching hybrid differential evolution (RSHDE) algorithm with integer variables

to solve the discrete problem of the feeder reconfiguration and capacitor placement. In order to demonstrate the effectiveness,

the proposed approach is applied to two example systems. One is a three-feeder distribution system from the literature and the other

is a practical distribution network of Taiwan Power Company (TPC), both are solved respectively by the proposed method,

HDE, GA and SA. From the computational results, it is observed that the performance of the proposed method is better than the

other methods.

Copyright # 2009 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2010; 20:1040–1057

DOI: 10.1002/etep

DISTRIBUTION SYSTEM OPERATIONS BY RSHDE 1041

2. ROBUST SEARCHING HYBRID DIFFERENTIAL EVOLUTION

In this section, a brief overview to the HDE [31] is provided. Next, based on this HDE, an RSHDE enhanced by two additional

schemes is introduced.

2.1. Hybrid differential evolution

Hybrid differential evolution is a novel parallel direct search method for minimizing nonlinear and non-differential functions. It is a

simple method based on stochastic searches; also it can be efficiently used to solve a non-differentiable optimization problem. The

main procedures of the HDE include

(1) Initialization.

(2) Mutation operation.

(3) Crossover operation.

(4) Estimation and selection.

(5) Migration phase, if necessary.

(6) Repeat steps 2 to 5.

(7) The steps of HDE are further described below.

Step 1. Initialization

The initial populations are chosen randomly and attempt to cover the entire parameter space uniformly. Uniform probability

distribution for all random variables is assumed, that is

X0i ¼ roundðXmin þ riðXmax � XminÞÞ; i ¼ 1; . . . ;Np (1)

where ri 2 [0,1] is a random number, and round (r) represents the nearest integer of the real number r. The initial process produces

NP individuals of X0i randomly.

Step 2. Mutation operation

A mutant vector is generated based on the present individual XGi as follows:

YGþ1i ¼ roundðXG

i þ FððXGr1 � XG

r2Þ þ ðXGr3 � XG

r4ÞÞÞ (2)

where mutation rate F2 [0,1.2], the upper limit of 1.2 for Fwas determined empirically; subscripts i2 {1,2,. . .,NP}; r1, r2, r3 and r4

are randomly selected.

Step 3. Crossover operation

In order to increase the diversity among the individuals of the next generation, a perturbed individual, YGþ1i , and a present

individual, XGi , are chosen by a binominal distribution to progress the crossover operation to generate an offspring. Each

gene of the ith individual is reproduced from the mutant vector YGþ1i ¼ ðYGþ1

1i ; YGþ12i ; ::::; YGþ1

ki Þ and the present individual

XGi ¼ ðXG

1i;XG2i; ::::;X

GkiÞ. That is

YGþ1hi ¼ XG

hi

YGþ1hi ;

�; if a random number>Cr

otherwise(3)

where i¼ 1,2,..,NP; h¼1,. . .,nC; nC is the dimension of decision parameters; k is the number of gene; and the crossover factor Cr 2[0,1] is arranged by the user.

Step 4. Estimation and selection

The parent is replaced by its offspring if the fitness of the offspring is better than that of the parent. The parent is retained in the

next generation if the fitness of the offspring is worse than that of the parent. Two forms are presented as follows:

XGþ1i ¼ arg minfFðXG

i Þ;FðYGþ1i Þg (4)

XGþ1b ¼ arg minfFðYGþ1

i Þg (5)

where arg min means the argument of the minimum, and XGþ1b is the best individual.


DOI: 10.1002/etep

1042 C.-F. CHANG

Step 5. Migration if necessary

In order to effectively enhance the investigation to the search spaces and reduce the choice pressure to a small population, a

migration operation is introduced to regenerate a new diverse population of individuals. The new populations are yielded based on

the best individual XGþ1b . The hth gene of the ith individual is as follows:

XGþ1hi ¼ roundðXGþ1

hi þ r1ðXhmin � XGþ1hb ÞÞ; if r2 <

XGþ1hi

�Xhmin

Xhmax�Xhmin

roundðXGþ1hi þ r1ðXhmax � XGþ1

hb ÞÞ; otherwise

((6)

where r1, r2 are randomly generated numbers uniformly distributed in the range of [0,1]; i¼1,. . .,Np; h¼1,. . .,nC. The migration

in HDE is executed only if a measure fails to match the desired tolerance of population diversity. The measure is defined as

follows:

r ¼XNP

i¼1i6¼b

Xncj¼1

xji=ncðNP � 1Þ < "1 (7)

where

xji ¼ 1; ifXGþ1ji

�XGþ1jb

XGþ1jb

�� > "2

0; otherwise

8<: (8)

Parameters "1 2 ½0; 1� and "2 2 ½0; 1� respectively express the desired tolerance for the population diversity and the gene diversityof the best individual. Here xji is defined as an index of gene diversity. A value of zero of xji denotes that the jth gene of the

ith individual is very close to the jth gene of the best individual. From Equations (7) and (8), it can be seen that the value of r is in the

range of [0,1]. If r is smaller than "1, then the HDE performs the migration to generate a new population to escape the local point;

otherwise, the HDE breaks off the migration which keeps an ordinary search direction.

Step 6. Repeat steps 2–5 until the maximum iteration quantity or the desired fitness is reached

2.2. Robust searching hybrid differential evolution

For a large-scale integer-programming problem, the huge number of combinations in the solution space makes the solution

searching process a heavy burden. Also, the HDE when handling non-integer variables, can cause deterioration in the mechanism of

the mutation strategy itself and consequently weaken its solution-searching ability. In order to avoid such a drawback, the HDE is

modified by incorporating two schemes—the multi-direction search scheme and the search space reduction scheme. These two

schemes help find the search direction for solutions before performing the initialization step. In doing so, the performance of the

RSHDE is significantly improved. These two new schemes embedded in the RSHDE are briefly introduced below.

2.2.1. Multi-direction search scheme. The HDE finds the initial search direction determined by the initial populations which are

chosen randomly. Actually, this is not an efficient way. In order to effectively enhance the initial search and accommodate itself to a

small population, the RSHDE uses a new scheme called multi-direction search scheme. The idea of this scheme is to create a group

of new vectors to perform a multi-direction search at the beginning, and to start the migration immediately. Figure 1 shows a two-

dimensional solution space which helps to illustrate the concept of this scheme. In Figure 1, XGP represents the best individual of the

randomly generated initial population. XGr1 and X

Gr2 are two randomly selected vectors of the population. The difference D between

XGr1 and XG

r2 is then obtained from Equation (9).

D ¼ ðXGr2 � XG

r1Þ (9)

This difference vector D is first multiplied by a scaling factor F and then added to the original vector XGP to get a new vector YGþ1

i ,

as was shown in Equation (10).

YGþ1i ¼ roundðXG

P þ F � DÞ (10)


DOI: 10.1002/etep


Repeating the similar procedures for Np times, we have NP newly created vectors, ðYGþ11 ;YGþ1

2 ; :::; YGþ1NP

Þ. In other words, there

are NP new search directions created emanating from the original individual XGP , to provide search direction for next generation.

Besides, due to less diversity among the individuals at the beginning stages, migration must be started. Accordingly, the multi-

direction search and the migration operation will provide an effective and fast convergent solution search.

2.2.2. Search space reduction scheme. Another scheme called search space reduction scheme which is proposed to help solution

search before performing the initialization step of the solving process. This scheme confines its search to a limited range called the

reduced space rather than the overall solution space. This may lead to a premature convergence and fall into a local solution, if the

reduced space is fixed during the process. Therefore, the use of C1 and C2 cannot guarantee that the global optimization point is still

within the reduced space. However, when the migration is executed, an overall solution space with diverse individuals is generated

to replace the previous reduced space with closely clustered individuals. Accordingly, the RSHDE still retains the global search

ability but searches more efficiently.

To further illustrate this scheme, assume that the solution space is bounded as follows:

Xmin � X � Xmax (11)

where Xmin and Xmax are the original lower and upper bounds of the decision parameters, respectively. A reduced space with new

bounds can be made by applying Equations (12) and (13).

X0max ¼ roundðXmaxð1� C1Þ þ C1 � XminÞ (12)

X0min ¼ roundðXminð1� C2Þ þ C2 � XmaxÞ (13)

where X0min and X

0max are the new lower and upper bounds of the reduced space for the decision parameters, respectively. Parameters

C1 2 [0,1] and C2 2 [0,1] are determined empirically. In the solution process, different reduced spaces are created by using

Equations (12) and (13). Figure 2 shows a two-dimensional solution space which helps to illustrate this scheme. Assume that

X1;min ¼ 1, X1;max ¼ 28 and X2;min ¼ 1, X2;max ¼ 28. Setting C1 ¼ 0:5 and C2 ¼ 0:1, and applying Equations (12) and (13), the newbounds for the reduced space are obtained as X0

1;min ¼ 4, X01;max ¼ 15 and X0

2;min ¼ 4, X02;max ¼ 15. In Figure 2, area ABCD is the

overall solution space, and area A’ B’ C’ D’ represents the reduced space.

Figure 1. A schematic diagram showing the concept of the multi-direction search scheme.


DOI: 10.1002/etep

1044 C.-F. CHANG

3. PROBLEM FORMULATION

This paper aims to minimize the system power loss, subject to operating constraints under a certain load pattern. The mathematical

model of the problem can be expressed below:

MinF ¼ min PT ;Loss þ lV � SCV þ lI � SCI� �

(14)

where PT ;Loss is the total real power loss of the system. Parameters lV and lI are the penalty constants, SCV is squared sum of the

violated voltage constraints and SCI is squared sum of the violated current constraints. Moreover, the penalty constants are given

below:

(1) Constant lV (lI) is given a value of 0, if the associated voltage (current) constraint is not violated.

(2) A significant value is given to lV (lI) if the associated voltage (current) constraint is violated, this makes the objective

function move away from the undesirable solution.

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 expressed as follows:

Vmin � Vij j � Vmax (15)

Iij j � Ii;max (16)

where Vij j is voltage magnitude of bus i, Vmin and Vmax are minimum and maximum bus voltage limits, respectively. Iij j is currentmagnitude and Ii;max is maximum current limit of branch i.

A set of simplified feeder-line flow formulations is employed to avoid complex power flow computation. Referring to Figure 3,

this set of simplified equations can be described as in Reference [2]

Pi ¼ Piþ1 þ PLiþ1 þ Ri;iþ1

P2i þ Q2

i

Vij j2" #

(17)

Figure 2. A schematic diagram showing the concept of the search space reduction scheme.


DOI: 10.1002/etep


Qi ¼ Qiþ1 þ QLiþ1 þ Xi;iþ1

P2i þ Q2

i

Vij j2" #

(18)

Viþ1j j2¼ Vij j2�2ðRi;iþ1Pi þ Xi;iþ1QiÞ þ ðR2i;iþ1 þ X2

i;iþ1ÞðP2

i þ Q2i Þ

Vij j2 (19)

where Pi and Qi are the real and reactive line powers flowing out of bus i, respectively; PLi and QLi are the real and reactive load

powers at bus i. The resistance and reactance of the line section between buses i and iþ 1 are denoted by Ri;iþ1 and Xi;iþ1,

respectively. The power loss of the line section connecting buses i and iþ 1 may be computed as

PLossði; iþ 1Þ ¼ Ri;iþ1

P2i þ Q2

i

Vij j2 (20)

The power loss of the feeder PF;Loss may then be determined by summing the losses of all line sections of the feeder, given by

PF;Loss ¼ Sn�1

i¼0PLossði; iþ 1Þ (21)

The total system power loss PT ;Loss is the sum of power losses of all feeders in the system.

4. COMPUTATIONAL PROCEDURES OF THE PROPOSED METHOD

A technique employing feeder reconfiguration and capacitor placement to reduce power loss for distribution systems is presented.

The RSHDE is employed as the optimization technique. The proposed method mainly involves power loss computation using

Equations (17) and (18), bus voltage determination using Equation (19), and RSHDE application. The computational procedures

find a series of configurations with different status of switches and with the addition of capacitors such that the objective function is

successively reduced.

To solve only the capacitor placement problem in distribution systems by the proposed method, the genotype of the capacitor

placement is built on a single array, and each population individual is composed of n genes. Where n represents the bus number. The

ith gene represents the capacitive compensations for bus i. To solve only feeder reconfiguration problem, the genotype of the

reconfiguration be built on a single array, the genes of each population individual is composed of tie switches’ positions. Moreover,

the solution process begins with encoding parameters. A tie switch (TS) and some sectionalizing switches with the feeders form a

loop. A particular switch of each loop is selected to open to make a loop radial such that the selected switch naturally becomes a tie

switch. The network reconfiguration problem is identical to the problem of selecting an appropriate tie switch for each loop to

minimize the power loss. A coding scheme that recognizes the positions of the tie switches is proposed. The total number of tie

switches is kept constant, regardless of any change in the system’s topology or the tie switches’ positions. Figure 4 shows an

Figure 3. Single-line diagram of a main feeder.

Figure 4. Composition of an individual.


DOI: 10.1002/etep

1046 C.-F. CHANG

individual that is composed of tie switches’ positions. Different switches from a loop are respectively selected for cutting the loop

circuit and trying to become a tie switch. After each loop is made radial, a configuration is proposed.

Furthermore, both feeder reconfiguration and capacitor placement are simultaneously applied to distribution systems. The

genotype of the composite is similarly built on a single array; the genotype of each individual includes the genes of feeder

reconfiguration and capacitor placement. In other words, the genes of each individual that decide the minimum loss configuration

are the status of the switches and setting of the switched capacitors. Figure 5 shows a flowchart of the main computational

procedures. The main computational processes are briefly stated below:

(1) At first, details of the data of system are read and run for feeder-line flow for system, then its fitness to determine the initial

power loss and bus voltage is computed. Also, relevant parameters are set up. Furthermore, the initial populations are chosen

randomly to cover the entire parameter space uniformly. Again, execute the multi-direction scheme and the search space

reduction scheme.

(2) Run feeder-line flow programme for each new chromosome and calculate its fitness value.

(3) Execute the mutation operation and crossover operation using Equations (2) and (3).

Figure 5. Flowchart of the proposed method.


DOI: 10.1002/etep


(4) Estimation and selection using Equations (4) and (5). In this step, the parent is replaced by its offspring if the fitness of the

offspring is better than that of the parent. In contrast, the parent is retained in the next generation if the fitness of the offspring

is worse than that of the parent.

(5) The migration in RSHDE is executed only if a measure fails to match the desired tolerance of population diversity. The

measure is defined by Equations (7) and (8). When r is smaller than "1, then the RSHDE performs the migration to generate a

new population to escape the local point; otherwise, the RSHDE breaks off the migration which keeps an ordinary search

direction.

(6) End the process if the maximum iteration number or the desired fitness is reached; otherwise repeat the outer loop. In

addition, the population size and the number of iterations were experimentally determined.

5. APPLICATION EXAMPLES

Two application systems, one from the literature and one from TPC, are investigated using the proposed RSHDE, HDE, GA and SA

methods, and the results are compared. These methods have been programmed using MATLAB and run on a Pentium IV-1.3GHz

personal computer.

5.1. Example 1

The first example is a three-feeder distribution system [1] shown in Figure 6. The system consists of 3 feeders, 13 normally closed

sectionalizing switches, and 3 normally open tie switches. The system load is assumed to be constant and Sbase ¼ 100MVA. Details

of the data of this example system are shown in Table I. Moreover, practical sizes available for the switched capacitor banks are 300,

600, 900, 1200, 1500 and 1800 kVAr.

To solve the capacitor placement problem, for RSHDE application, parameters were selected as: population size NP ¼ 5;

maximum generation Gmax ¼ 450; scaling factor F ¼ 0:3; crossover factor Cr ¼ 0:5; two tolerances, e1 and e2, both are set to 0.1;

C1 is set to 0.4, and C2 is set to 0.1, according to the authors’ experience, the selection of C1 and C2 is problem-dependence,

generally, both are set to [0.1–0.9]; for HDE application, parameters were selected as: population sizeNP ¼ 5; maximum generation

Gmax ¼ 450; scaling factor F ¼ 0:3; crossover factor Cr ¼ 0:5; two tolerances, e1 and e2, both are set to 0.1; for SA application,

parameters selected were the initial temperature; 5000, the temperature reduction ratio; 0.98 and the maximum iteration was 4000;

for GA application, parameters were selected such as population size; 10, the crossover ratio; 0.5, the mutation ratio; 0.03, and the

maximum generation; 450. To solve the feeder reconfiguration problem, for RSHDE application, parameters were selected as:

population size NP ¼ 5; maximum generation Gmax ¼ 50; scaling factor F ¼ 0:3; crossover factor Cr ¼ 0:5; two tolerances, e1 ande2, both are set to 0.1; reduce space parameters, C1 and C2, both are set to 0.2; for HDE application, parameters were selected as:

population size NP ¼ 5; maximum generation Gmax ¼ 50; scaling factor F ¼ 0:3; crossover factor Cr ¼ 0:5; two tolerances, e1 ande2, both are set to 0.1; for SA application, parameters were selected as the initial temperature; 500, the temperature reduction ratio;

0.95, and the maximum iteration; 200, for GA application, parameters were selected as population size; 5, the crossover ratio; 0.5,

the mutation ratio; 0.03 and the maximum generation; 50.

Figure 6. A three-feeder distribution system for example 1.


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1048 C.-F. CHANG

In addition, to solve both the feeder reconfiguration and capacitor placement problems simultaneously, for RSHDE application,

parameters were selected as: population size NP ¼ 5; maximum generation Gmax ¼ 500; scaling factor F ¼ 0:3; crossover factorCr ¼ 0:5; two tolerances, e1 and e2, both are set to 0.1, reduce space parameters, C1 is set to 0.5, and C2 is set to 0.2; for HDE

application, parameters were selected as: population size NP ¼ 5; maximum generation Gmax ¼ 500; scaling factor F ¼ 0:3;crossover factor Cr ¼ 0:5; two tolerances, e1 and e2 both are set to 0.1; for SA application, parameters were selected as the initial

temperature; 5000, the temperature reduction ratio; 0.99, and the maximum iteration; 5000; for GA application, parameters were

selected as population size; 10, the crossover ratio; 0.5, the mutation ratio; 0.03 and the maximum generation; 500. Further

information concerning parameter settings of different algorithms are shown in Table II.

Table III shows the computational results of simultaneously considering the feeder reconfiguration and capacitor placement in

RSHDE, which including the tie switch, power loss, voltage profile, required capacitor addition and CPU time before and after

operation. It can be observed from Table III, that the voltage profile and power loss reduction of the system are improved. Moreover,

it can be seen from Table I, the reactive power is 5.9MVAr mainly includes the systemic reactive load and fixed capacitor. And, it

can be observed from Table III, after handled feeder reconfiguration and capacitor placement, the total reactive power of capacitor

setting is 6MVAr. The difference from above both data is 0.1MVAr. Namely the error rate is 1.69%. This is very reasonable and

acceptable from the viewpoint of compensation. For comparison, we also considered two different operation situations; one

Table I. Input data for example 1.

Section busto bus (pu)

Sectionresistance (pu)

Sectionreactance (pu)

End bus realload (MW)

End bus reactiveload (MVAr)

End bus fixedcapacitor (MVAr)

1–4 0.075 0.1 2.0 1.6 —4–5 0.08 0.11 3.0 1.5 1.14–6 0.09 0.18 2.0 0.8 1.26–7 0.04 0.04 1.5 1.2 —2–8 0.11 0.11 4.0 2.7 —8–9 0.08 0.11 5.0 3.0 1.28–10 0.11 0.11 1.0 0.9 —9–11 0.11 0.11 0.6 0.1 0.69–12 0.08 0.11 4.5 2.0 3.73–13 0.11 0.11 1.0 0.9 —13–14 0.09 0.12 1.0 0.7 1.813–15 0.08 0.11 1.0 0.9 —15–16 0.04 0.04 2.1 1.0 1.85–11 0.04 0.04 — — —10–14 0.04 0.04 — — —7–16 0.12 0.12 — — —

Table II. Parameter setting of different algorithms in example 1.

Items Different operation situations

Only capacitor placement Only feeder reconfiguration Reconfiguration and capacitor placement

SA initial temperature ¼ 5000,temperature reduction ratio ¼ 0:98,

maximum iteration ¼ 4000

initial temperature ¼ 500,temperature reduction ratio ¼ 0:95,



maximum iteration ¼ 5000GA NP ¼ 10, Gmax ¼ 450,

crossover ratio ¼ 0:5,mutation ratio ¼ 0:03

NP ¼ 5,Gmax ¼ 50,crossover ratio ¼ 0:5,mutation ratio ¼ 0:03

NP ¼ 10, Gmax ¼ 500,crossover ratio ¼ 0:5,mutation ratio ¼ 0:03

HDE NP ¼ 5, Gmax ¼ 450,F ¼ 0:3, Cr ¼ 0:5, "1 ¼ "2 ¼ 0:1

NP ¼ 5, Gmax ¼ 50,F ¼ 0:3, Cr ¼ 0:5,

"1 ¼ "2 ¼ 0:1

NP ¼ 5, Gmax ¼ 500, F ¼ 0:3,Cr ¼ 0:5, "1 ¼ "2 ¼ 0:1

RSHDE NP ¼ 5, Gmax ¼ 450,F ¼ 0:3, Cr ¼ 0:5, "1 ¼ "2 ¼ 0:1,

C1 ¼ 0:4; C2 ¼ 0:1

NP ¼ 5, Gmax ¼ 50,F ¼ 0:3, Cr ¼ 0:5,

"1 ¼ "2 ¼ 0:1, C1 ¼ C2 ¼ 0:2

NP ¼ 5, Gmax ¼ 500,F ¼ 0:3, Cr ¼ 0:5,

"1 ¼ "2 ¼ 0:1, C1 ¼ 0:5; C2 ¼ 0:2


DOI: 10.1002/etep


considering only capacitor placement and one considering only feeder reconfiguration. Table IV shows the computational results of

three different operation situations in RSHDE. Computational results show that simultaneously taking into account both feeder

reconfiguration and capacitor placement is more effective than considering them separately.

Moreover, in order to investigate performance of the proposed algorithm, the HDE, SA and GA are also applied to solve this

problem. This example was repeatedly solved for 100 runs. The best and worst computation results among the 100 runs are listed in

Table V. The average value for the best solutions of those 100 runs and the average CPU time are also shown in this table. From the

computational results of Table V, it is observed that the average loss reduction ratio and average CPU times obtained by the RSHDE

are less than those of the HDE, SA and GA. Therefore, from the above discussions, it could be concluded that the performance of the

proposed RSHDE method is better than the HDE, SA and GA methods.

5.2. Example 2

The second example is a practical distribution network of TPC. Its conductors mainly employ both overhead lines ACSR 477MCM

and underground lines; copper conductors 500MCM. The system is shown in Figure 7 and the relating data are shown in Table VI. It

Table III. The results after reconfiguration and capacitor.

Items Originalconfiguration

Reconfiguration andcapacitor placement

Tie switches 15,21,26 19,17,26Power loss (kW) 511.4 448.1Maximum voltage (pu) Vmax ¼ 1:0000 (Buses 1,2,3) Vmax ¼ 1:0000 (Buses 1,2,3)Minimum voltage (pu) Vmin ¼ 0:9693 (Bus 12) Vmin ¼ 0:9757 (Bus 12)Loss reduction (%) — 12.38CPU times (s) — 16.93Bus Bus voltage (pu) Placed Qc(kVAr) Bus voltage1 1.0000 — 1.00002 1.0000 — 1.00003 1.0000 — 1.00004 0.9907 1500 0.99315 0.9878 1800 0.99036 0.9860 0 0.99017 0.9849 0 0.98948 0.9791 0 0.98449 0.9711 0 0.977510 0.9769 0 0.991011 0.9710 900 0.990312 0.9693 0 0.975713 0.9944 900 0.993314 0.9948 0 0.991715 0.9918 0 0.990716 0.9913 900 0.9901

Table IV. The results of three different situations in RSHDE for example 1.

Items Original configuration Different operation situations

Only capacitorplacement

Only feederreconfiguration


Tie switches 15,21,26 15,21,26 19,17,26 19,17,26Maximum voltage (pu) 1.0000 (buses 1,2,3) 1.0000 (buses 1,2,3) 1.0000 (buses 1,2,3) 1.0000 (buses 1,2,3)Minimum voltage (pu) 0.9693 (bus 12) 0.9734 (bus 12) 0.9716 (bus 12) 0.9757 (bus 12)Power loss (kW) 511.4 487.1 466.1 448.1Loss reduction (%) — 4.75 8.86 12.38CPU times (s) — 15.52 1.12 16.93


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1050 C.-F. CHANG

Table V. Numerical results of example 1.





Original configuration (kW) 511.4 511.4 511.4SA Best (kW) 489.7 466.1 448.3

Worst (kW) 496.6 501.3 451.6Average (kW) 493.3 482.3 449.7Average loss reduction (%) 3.54 5.69 12.06Average CPU times (second) 33.79 2.07 42.15

GA Best (kW) 488.2 466.1 448.2Worst (kW) 492.1 493.2 449.8Average (kW) 489.6 474.8 448.6Average loss reduction (%) 4.26 7.16 12.28Average CPU times (second) 32.39 2.32 38.46

HDE Best (kW) 487.1 466.1 448.1Worst (kW) 488.0 479.3 449.4Average (kW) 487.6 467.1 448.5Average loss reduction (%) 4.65 8.66 12.30Average CPU times (second) 21.21 2.05 23.89

RSHDE Best (kW) 487.1 466.1 448.1Worst (kW) 487.3 473.0 448.2Average (kW) 487.2 466.2 448.1Average loss reduction (%) 4.73 8.84 12.38Average CPU times (second) 15.63 1.15 17.14

Figure 7. A distribution system of Taiwan Power Company for example 2.


DOI: 10.1002/etep


Table VI. The three-phase load and line data of example 2.

Section busto bus

Sectionresistance (V)

Sectionreactance (V)

End bus realload (kW)

End bus reactiveload (kVAr)

A-1 0.1944 0.6624 0 01–2 0.2096 0.4304 100 502–3 0.2358 0.4842 300 2003–4 0.0917 0.1883 350 2504–5 0.2096 0.4304 220 1005–6 0.0393 0.0807 1100 8006–7 0.0405 0.1380 400 3207–8 0.1048 0.2152 300 2007–9 0.2358 0.4842 300 2307–10 0.1048 0.2152 300 260B-11 0.0786 0.1614 0 011–12 0.3406 0.6944 1200 80012–13 0.0262 0.0538 800 60012–14 0.0786 0.1614 700 500C-15 0.1134 0.3864 0 015–16 0.0524 0.1076 300 15016–17 0.0524 0.1076 500 35017–18 0.1572 0.3228 700 40018–19 0.0393 0.0807 1200 100019–20 0.1703 0.3497 300 30020–21 0.2358 0.4842 400 35021–22 0.1572 0.3228 50 2021–23 0.1965 0.4035 50 2023–24 0.1310 0.2690 50 10D-25 0.0567 0.1932 50 3025–26 0.1048 0.2152 100 6026–27 0.2489 0.5111 100 7027–28 0.0486 0.1656 1800 130028–29 0.1310 0.2690 200 120E-30 0.1965 0.3960 0 030–31 0.1310 0.2690 1800 160031–32 0.1310 0.2690 200 15032–33 0.0262 0.0538 200 10033–34 0.1703 0.3497 800 60034–35 0.0524 0.1076 100 6035–36 0.4978 1.0222 100 6036–37 0.0393 0.0807 20 1037–38 0.0393 0.0807 20 1038–39 0.0786 0.1614 20 1039–40 0.2096 0.4304 20 1038–41 0.1965 0.4035 200 16041–42 0.2096 0.4304 50 30F-43 0.0486 0.1656 0 043–44 0.0393 0.0807 30 2044–45 0.1310 0.2690 800 70045–46 0.2358 0.4842 200 150G-47 0.2430 0.8280 0 047–48 0.0655 0.1345 0 048–49 0.0655 0.1345 0 049–50 0.0393 0.0807 200 16050–51 0.0786 0.1614 800 60051–52 0.0393 0.0807 500 30052–53 0.0786 0.1614 500 35053–54 0.0524 0.1076 500 300

(Continues)


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1052 C.-F. CHANG

is a three-phase, 11.4 kV system. The system consists of 11 feeders, 83 normally closed sectionalizing switches, and 13 normally

open tie switches. Three-phase balance and constant load are assumed. Similarly, practical sizes available for the switched capacitor

banks are 300, 600, 900, 1200, 1500 and 1800 kVAr. For comparison, the proposed method, HDE, SA and GA are also applied to

solve this problem, and this example was repeatedly solved for 100 runs.

To solve the capacitor placement problem using RSHDE, parameters were selected as: population size NP ¼ 5; maximum

generation Gmax ¼ 4500; scaling factor F ¼ 0:3; crossover factor Cr ¼ 0:5; two tolerances, e1 and e2, both are set to 0.1; reduce

space parameters, C1 is set to 0.8, and C2 is set to 0.1; for HDE application, parameters were selected as: population size NP ¼ 5;

maximum generation Gmax ¼ 4500; scaling factor F ¼ 0:3; crossover factor Cr ¼ 0:5; two tolerances, e1 and e2, both are set to 0.1;for SA application, parameters were selected as the initial temperature; 10 000, the temperature reduction ratio; 0.99, and the

Table VI. (Continued)

Section busto bus

Sectionresistance (V)

Sectionreactance (V)

End bus realload (kW)

End bus reactiveload (kVAr)

54–55 0.1310 0.2690 200 80H-56 0.2268 0.7728 0 056–57 0.5371 1.1029 30 2057–58 0.0524 0.1076 600 42058–59 0.0405 0.1380 0 059–60 0.0393 0.0807 20 1060–61 0.0262 0.0538 20 1061–62 0.1048 0.2152 200 13062–63 0.2358 0.4842 300 24063–64 0.0243 0.0828 300 200I-65 0.0486 0.1656 0 065–66 0.1703 0.3497 50 3066–67 0.1215 0.4140 0 067–68 0.2187 0.7452 400 36068–69 0.0486 0.1656 0 069–70 0.0729 0.2484 0 070–71 0.0567 0.1932 2000 150071–72 0.0262 0.0528 200 150J-73 0.3240 1.1040 0 073–74 0.0324 0.1104 0 074–75 0.0567 0.1932 1200 95075–76 0.0486 0.1656 300 180K-77 0.2511 0.8556 0 077–78 0.1296 0.4416 400 36078–79 0.0486 0.1656 2000 130079–80 0.1310 0.2640 200 14080–81 0.1310 0.2640 500 36081–82 0.0917 0.1883 100 3082–83 0.3144 0.6456 400 3605–55 0.1310 0.2690 — —7–60 0.1310 0.2690 — —11–43 0.1310 0.2690 — —12–72 0.3406 0.6994 — —13–76 0.4585 0.9415 — —14–18 0.5371 1.0824 — —16–26 0.0917 0.1883 — —20–83 0.0786 0.1614 — —28–32 0.0524 0.1076 — —29–39 0.0786 0.1614 — —34–46 0.0262 0.0538 — —40–42 0.1965 0.4035 — —53–64 0.0393 0.0807 — —


DOI: 10.1002/etep


maximum iteration; 20 000; for GA application, parameters were selected as population size; 10, the crossover ratio; 0.5, the

mutation ratio; 0.02 and the maximum generation; 4500. Table VIII expressed the best and worst computation results among

the 100 runs and the average results for the best solutions of those 100 runs. From the numerical results, it is observed that the

average loss reduction ratio and the CPU times obtained by the RSHDE are respectively less than those of the HDE, SA and GA.

To solve the feeder reconfiguration problem using RSHDE, parameters were selected as: population size NP ¼ 5; maximum

generationGmax ¼ 500; scaling factor F ¼ 0:3; crossover factorCr ¼ 0:5; two tolerances, e1 and e2, both are set to 0.1; reduce spaceparameters, C1 and C2, both are set to 0.2; for HDE application, parameters were selected as: population size NP ¼ 5; maximum

generation Gmax ¼ 500; scaling factor F ¼ 0:3; crossover factor Cr ¼ 0:5; two tolerances, e1 and e2, both are set to 0.1; for SA

application, parameters were selected as the initial temperature; 500, the temperature reduction ratio; 0.99, and the maximum

Table VIII. Numerical results of example 2.





Original configuration (kW) 531.99 531.99 531.99SA Best (kW) 342.14 469.88 309.12

Worst (kW) 362.31 498.22 315.86Average (kW) 351.86 489.82 312.30Average loss reduction (%) 33.86 7.93 41.30Average CPU times (second) 1632 257 1966

GA Best (kW) 330.79 469.88 295.39Worst (kW) 332.25 489.25 299.13Average (kW) 331.84 479.73 297.75Average loss reduction (%) 37.62 9.82 44.03Average CPU times (second) 2332 304 2510

HDE Best (kW) 330.16 469.88 294.81Worst (kW) 332.07 470.95 299.37Average (kW) 331.25 470.01 296.08Average loss reduction (%) 37.73 11.65 44.34Average CPU times (second) 1247 207 1409

RSHDE Best (kW) 329.42 469.88 294.21Worst (kW) 330.54 470.01 298.02Average (kW) 329.71 469.89 294.53Average loss reduction (%) 38.02 11.67 44.64Average CPU times (second) 950 151 1126

Table VII. Parameter setting of different algorithms in example 2.


Only capacitor placement Only feeder reconfiguration Reconfiguration and capacitor placement

SA initial temperature ¼ 10 000,temperature reduction ratio ¼ 0:99,

maximum iteration ¼ 20 000



initial temperature ¼ 10 000,temperature reduction ratio ¼ 0:99,

maximum iteration ¼ 25 000GA NP ¼ 10, Gmax ¼ 4500,

crossover ratio ¼ 0:5,mutation ratio ¼ 0:02



HDE NP ¼ 5, Gmax ¼ 4500,F ¼ 0:3, Cr ¼ 0:5, "1 ¼ "2 ¼ 0:1

NP ¼ 5, Gmax ¼ 500,F ¼ 0:3, Cr ¼ 0:5, "1 ¼ "2 ¼ 0:1

NP ¼ 5, Gmax ¼ 5000,F ¼ 0:3, Cr ¼ 0:5, "1 ¼ "2 ¼ 0:1

RSHDE NP ¼ 5, Gmax ¼ 4500, F ¼ 0:3,Cr ¼ 0:5, "1 ¼ "2 ¼ 0:1,

C1 ¼ 0:8, C2 ¼ 0:1

NP ¼ 5, Gmax ¼ 500, F ¼ 0:3,Cr ¼ 0:5, "1 ¼ "2 ¼ 0:1,

C1 ¼ C2 ¼ 0:2

NP ¼ 5, Gmax ¼ 5000, F ¼ 0:3,Cr ¼ 0:5, "1 ¼ "2 ¼ 0:1,

C1 ¼ 0:5, C2 ¼ 0:1


DOI: 10.1002/etep

1054 C.-F. CHANG

iteration; 4000; for GA application, parameters were selected as population size; 10, the crossover ratio; 0.5, the mutation ratio; 0.03

and the maximum generation; 500. Similarly, from the computational results of Table VIII, it is observed that the average loss

reduction ratio and the CPU times obtained by the RSHDE are respectively less than those of the HDE, SA and GA.

Furthermore, to solve both the capacitor placement and feeder reconfiguration problems simultaneously, for RSHDE application,

parameters were selected as: population size NP ¼ 5; maximum generation Gmax ¼ 5000; scaling factor F ¼ 0:3; crossover factorCr ¼ 0:5; two tolerances, e1 and e2, both are set to 0.1. reduce space parameters, C1 is set to 0.5, and C2 is set to 0.1; for HDE

application, parameters were selected as: population size NP ¼ 5; maximum generation Gmax ¼ 5000; scaling factor F ¼ 0:3;crossover factor Cr ¼ 0:5; two tolerances, e1 and e2, both are set to 0.1; for SA application, parameters were selected as the initial

temperature to be 10 000, the temperature reduction ratio to be 0.99, and the maximum iteration to be 25 000; for GA application,

parameters were selected as population size to be 10, the crossover ratio to be 0.5, the mutation ratio to be 0.02, and the maximum

generation to be 5000. Further information concerning parameter settings of different algorithms are shown in Table VII.

The computational results can be seen from Table VIII. The average loss reduction ratio and the CPU times obtained by the

RSHDE are respectively less than those of the HDE, SA and GA. Similarly, computational results show that simultaneously taking

into account both feeder reconfiguration and capacitor placement is more effective than using only one technique. Moreover, it can

be seen from Table VIII, the loss reduction is higher. This is because the capacitor placement is very effective for reactive power

compensation, and for solving the feeder reconfiguration problem, the power loss is effectively minimized by rerouting the active

power flow through reconfiguring the network. Furthermore, when simultaneously taking into account both feeder reconfiguration

and capacitor placement, the loss reduction is much higher than considering them separately. In addition, the convergence rate of the

proposed RSHDE method compared with that of the HDE, SA and GA methods is depicted in Figure 8. It can be seen that the

proposed method has a relatively fast convergence performance.

6. CONCLUSION

A novel method based on hybrid differential evolution (HDE) to solve the optimal reconfiguration and capacitor placement problem

was developed in this study. The RSHDE is very suitable for solving large-scale integer optimization problems, also it does not need

complex mathematical programming. The RSHDE processes two novel schemes, the multi-direction search scheme and the search

space reduction scheme, are embeded in the HDE. These two schemes are used to enhance the search ability before performing the

initialization step of the solution process. From the application results, it was observed that the feeder reconfiguration and capacitor

placement process not only reduce the power loss but also improve the voltage profile. In addition, computational results show that

simultaneously taking into account both feeder reconfiguration and capacitor placement is more effective than considering them

separately. Obviously, the proposed method is helpful for operating existing systems or planning future systems.

Figure 8. The convergence situation of different algorithms.


DOI: 10.1002/etep


7. LIST OF SYMBOLS AND ABBREVIATIONS

Cr crossover factor

D difference vector

F mutation rate

Gmax maximum generation

Ii; max maximum current limit of branch i

Min F objective function

NP population size

PT ;Loss total power loss

PF;Loss power loss of the feeder

Pi real line powers flowing out of bus i

PLi real load powers at bus i

Qi reactive line powers flowing out of bus i

QLi reactive load powers at bus i

SCV squared sum of the violated voltage constraints

SCI squared sum of the violated current constraints

X0i initial population

XGi present individual

XGþ1i offspring individual

XGþ1b best individual

XGþ1jb j-th gene of the best individual

XGþ1hi h-th gene of the i-th individual

XGþ1ji j-th gene of the i-th individual

Xmin; Xhmin lower bounds

Xmax; Xhmax upper bounds

XGP original vector

YGþ1i mutant vectors

YGþ1hi h-th gene of mutant vectors

Vmin bus minimum voltage limits

Vmax bus maximum voltage limits

r degree of population diversity

ri random number

r1; r2 random number

"1 tolerance of the population diversity

"2 tolerance of the gene diversity

xji index of gene diversity

lV ; lI penalty constant

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optimal reconfiguration and capacitor placement by robust searching hybrid differential evolution

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