1.a New Approach for Allocation and Sizing Of

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1026 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 25, NO. 2, APRIL 2010

A New Approach for Allocation and Sizing ofMultiple Active Power-Line Conditioners

Iman Ziari, Student Member, IEEE, and Alireza Jalilian

AbstractIn this paper, a particle swarm optimization (PSO) al-gorithm is developed for allocation and sizing of multiple activepower-line conditioners (APLCs) in power systems. The utilizedobjective function comprises four factors as total harmonic dis-tortion, total size of active power-line conditioners (APLCs), har-monic transmission-line loss, and motor load loss. To evaluate thecapability of the proposed method, the IEEE 18-bus test system isemployed and investigated in three different cases. These cases arebased on the assumption of continuous/discrete and limited/unlim-ited size for APLCs, requiring the optimization method to solvediscrete/nondiscrete nonlinear problems. Therefore, in addition toPSO, an integer nonlinear optimizer (discrete PSO called DPSO)algorithm is also developed. Simulation results are compared withresults obtained by genetic algorithm as well as with nonlinearpro-gramming (discrete nonlinear programming). It is demonstratedthat analytical methods enjoy higher accuracy in smooth objectivefunctions while they are not accurate enough in practicalsituationswhere nonsmooth objective functions are involved. If improper ini-tial solutions are provided, achieving optimized results cannot beguaranteed by using analytical approaches. It is also shown thatheuristic algorithms due to their randomness characteristics aremore useful in practical cases where a number of local minima arepresent. Simulation results confirmed the capability and effective-ness of the proposed PSO-based algorithm in the allocation andsizing of multiple APLCs in a test power system compared withanalytical methods and other heuristic algorithms.

Index TermsActive power-line conditioner (APLC), genetic

algorithm (GA), harmonics, nonlinear programming, particleswarm optimization (PSO).

I. INTRODUCTION

IN modern electrical distribution systems, there has beena remarkable growth in the use of nonlinear loads, such

as rectifiers, converters, adjustable speed drives, arc furnaces,

computer power supplies, etc. Nonlinear loads act as currentsources injecting harmonic currents into the power systems.

These power-electronic-based loads have caused severelydistorted voltage waveforms at the point of common coupling

(PCC). Other linear loads connected at the same PCC willreceive a distorted supply voltage, which may lead to various

Manuscript received September 30, 2008; revised May 08, 2009. First pub-lished December 28, 2009; current version published March 24, 2010. Paper no.TPWRD-00750-2008.

I. Ziari is with the Department of Electrical Engineering, Iran University ofScience and Technology (IUST), Narmak, Tehran 1684613114, Iran (e-mail:[email protected]).

A. Jalilian is with the Department of Electrical Engineering, Centre of Excel-lence for Power System Automation and Operation, Iran University of Scienceand Technology (IUST), Tehran 1684613114, Iran (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPWRD.2009.2036180

unwanted effects. The overheating of motors, transformers,

cables, maloperation of some protection devices, and resonance

with capacitors are some of these effects [1].

Conventionally, passive L-C filters are used to reduce

harmonics; nevertheless, they have demerits of fixed compen-

sation, large size, and resonance. In order to overcome these

problems, active power filters (APFs), which are also called

APLCs, have been developed [2]. The basic compensation prin-

ciples of APLCs were proposed in the 1970s. APLCs provide

injected equal-but-opposite currents to the PCC that completely

eliminate the nonsinusoidal requirements of the nonlinear

loads. However, in practice, a more reasonable requirement is

to reduce harmonic distortion to a minimum acceptable level

for a given condition [3][5].

The first APLC with power rating of 800 kVA was put into

practical use for harmonic compensation in 1982. Later, in 1986,

a combined system of an APLC of 900 kVA and a passive filter

of 6600 kVA was practically installed with steel mill drives

[6]. Currently, APLCs have attracted a lot of attention owing to

their capabilities, particularly in the elimination of harmonics

[7][9].

Despite a large number of benefits that APLCs enjoy, their

huge installation and operation costs prevent electrical en-

gineers from employing these profitable instruments greatlywithout any restriction at all buses in power systems. Owing to

this fact, a variety of solution techniques has been utilized to

solve the APLCs allocation and sizing (AAS) problem. These

techniques are mainly categorized into two groups: 1) alloca-

tion of only one APLC and 2) allocation of multiple APLCs.

References [10][12] introduced the initial steps toward solving

the AAS problem; nonetheless, using only one APLC may not

guarantee satisfaction of the harmonic limits at all buses, in

general, and when many nonlinear loads are available in a

power system in particular.

References [13][15] are based on allocating and sizing

of multiple APLCs using an analytical optimization algo-

rithm called nonlinear mixed integer programming. Although

the analytical optimization algorithms do not suffer from a

time-consuming problem, the local minimum is their main

drawback. The need for the initial solution and difficulty in

differentiation from various types of nonlinear objective func-

tions is among other disadvantages of these methods. It should

be noticed that an incorrect selection of initial values leads to

inaccurate results when these methods are applied.

As the essential part of the objective function, THD level and

APLCs size have been considered in [11][14], [16]. Harmonic

transmission-line loss (HTLL) and motor load loss (MLL), dueto harmonics and telephone influence factor (TIF), are included

in the objective function in other studies [10], [15].

0885-8977/$26.00 2010 IEEE


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ZIARI AND JALILIAN: NEW APPROACH FOR ALLOCATION AND SIZING OF MULTIPLE APLCS 1027

In [16], a genetic-based algorithm (GA) is presented to solvethe AAS problem. Compared to the analytical algorithm, GA asa heuristic algorithm enjoys the luxury of simple concept andeasy implementation. It also does not need an initial solutionand differentiation from the nonlinear objective functions. Nev-ertheless, time consuming, local minimum, and the need to ad-just several parameters and lack of memory are still among sub-stantial imperfections.

Particle swarm optimization (PSO), as another heuristic-based approach, enjoys all plus characteristics of heuristic al-gorithms [17]. Compared to the other heuristic algorithms, suchas GA, the PSO method, to some extent, is not time consuming,does not suffer severely from local minimum, needs to adjustonly a few parameters, and has memory. These advantages leadPSO to be applied to different areas of electric power systems,such as economic dispatch, reactive power control, powerlosses reduction, optimal power flow, power system controldesign, and unit commitment [17][20].

In this paper, a PSO-based algorithm is proposed for allo-

cation and sizing of multiple APLCs in a standard IEEE testsystem. The objective function consists of four different fac-tors as: 1) THD, 2) total APLC current, 3) MLL, and 4) HTLL.Furthermore, APLCs with continuous/discrete and limited/un-limited sizes are considered in this investigation. Simulation re-sults are finally compared with the results obtained by anotherheuristic method (GA in this case) and a conventional analyticaloptimization technique to validate the capability of the proposedmethod.

II. PROBLEM FORMULATION

In this section, first of all, the APLC model assumed in this

paper is presented; then, the AAS problem and the relative ob-jective function and constraints are formulated.

A. APLC Model

The common model for APLC employed in almost all refer-

ences in the field of AAS [10][16] is a current source injecting

harmonics to the PCC in a power system. In order to assume

the APLC current with certain amplitude and phase angle, the

phasor model of (1) is used

(1)

where

APLC current at bus for harmonic order ;

real part of APLC current at bus for harmonic

order ;

imaginary part of APLC current at bus for

harmonic order .

The indices and represent the real and imaginary parts of the

APLC current, respectively.

B. Objective Function

The most important objectives of placement and sizing of

APLCs in a power system are to reduce total harmonic distor-tion (THD), harmonic transmission line losses (HTLL), motor

load losses (MLL), and to minimize the cost of APLCs. The con-

straints also maintain individual and total harmonic distortions

within a standard level. The mentioned parts will be lumped into

the objective function which is expressed as follows:

(2)

where

weight coefficient for THD;

weight coefficient for MLL;

weight coefficient for HTLL;

weight coefficient for total APLCs current.

As seen, the given objective function is composed of four parts:

1) ; 2) ; 3) ; and 4) .

, , and are the summation of THDs,

MLLs and APLCs current magnitude at all buses, respectively.

is the summation of transmission line losses at all har-

monic orders. These factors are formulated as

(3)

(4)

where

number of buses;

maximum considered harmonic order;

voltage at bus for harmonic order ;

THD at bus .

can also be formulated as follows [10]:

(5)

(6)

where is the MLL at bus . The formulation indicates

that lower order harmonics affect electrical motors more thanhigher order harmonics.

On the other hand, can be obtained as [15], as

shown in (7) at the bottom of the next page, where and

are line resistance and impedance between buses

and for harmonic order h, respectively.

finally can be determined by using (8) as

(8)

where is the amplitude of APLC current at bus for

harmonic order . The rest of the mathematical formulation willbe described in Section III.


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III. IMPLEMENTATION OF PSO FOR AAS PROBLEM

A. Overview of PSO

PSO is a population-based and self-adaptive technique intro-

duced originally by Kennedy and Eberhart in 1995 [19]. This

stochastic-based algorithm handles a population of individuals

in parallel to probe capable areas of a multidimensional spacewhere the optimal solution is searched. The individuals are

called particles and the population is called a swarm. Each

particle in the swarm moves toward the optimal point with

adaptive velocity [20], [21].

Mathematically, the position of particle in an -dimensional

vector is represented as . The ve-

locity of this particle is also an -dimensional vector as

. Alternatively, the best position related

to the lowest value of the objective function for each particle is

represented as and

the global best position among all particles or best pbest is de-

noted as . During

the iteration procedure, the velocity and position of particles are

updated [22].

B. Implementation of the PSO Algorithm

It should be noticed that the APLC current at each bus for

each harmonic is regarded as the position of a particle during the

optimization process. The proposed PSO-based algorithm as an

APLC location and size optimizer is illustrated in Fig. 1. The

description along with the required comments on the proposed

algorithm steps will be given. The coding is written in Matlab

7.4 programming language and executed by using a Pentium IV,

1.8 GHz, with 2-GB random-access memory (RAM) processor.

1) Step 1: (Input System Data and Initialize): In this step,the power system configuration and data, nonlinear loads loca-

tion, current , and constraints, such as maximum-allowed

individual harmonic distortion and THD, and APLCs size are

specified.

The number of population members and iterations are set.

The population of particles (consisting of real and imaginary

parts of APLC current at all buses for all harmonic orders) as

well as their velocity in the search space are initialized in

this step.

Vectors X and V are described as shown in the equation at the

bottom of the next page.

The PSO weight factors are also set in this step.2) Step 2: (Calculate the Objective Function): For this pur-

pose, APLCs current and bus voltages should be available. As

mentioned, APLCs current at each bus for each harmonic is con-

sidered as the position of a particle which is initialized in Step 1.

Fig. 1. Algorithm of proposed PSO-based approach for the AAS problem.

The bus voltages can be attained as

(9)

where and are the bus voltages and injecting current

vectors for harmonic order , respectively. The corresponding

bus admittance matrix for harmonic order is also presented

by . Considering an APLC located at each bus, the bus in-

jecting current is implied as

(10)

where and are the APLC and nonlinear load current

vectors for harmonic order h, respectively.

After calculation of the objective function by using (2), the

best position related to each member of particles can be evalu-ated as described in the next step.

3) Step 3: (Calculate Pbest): The objective function related

to each particle in the population in the current iteration is com-

pared with it in the previous iteration and the position of the par-

(7)


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ticle enjoying a lower objective function as Pbest for the current

iteration is recorded

if

if(11)

where is the number of iterations, and is the objective

function evaluated for particle .4) Step 4: (Calculate Gbest): In this step, the best objective

function associated with the Pbests among all particles in the

current iteration is compared with it in the previous iteration

and the lower value is chosen as the current overall Gbest

if

if(12)

5) Step 5: (Update Velocity): After calculation of Pbest and

Gbest, the velocity of particles for the next iteration should be

modified by using

(13)

where

velocity of particle j at iteration k;

inertia weight factor;

acceleration coefficients;

position of particle j at iteration k;

best position of particle j at iteration k;

best position among all particles at iteration k.

In the velocity updating process, as inertia weight, and

and as acceleration coefficients should be determined in ad-

vance. The acceleration coefficients are two different random

values in the interval and are defined as follows:

(14)

where

initial inertia weight factor;

final inertia weight factor;

current iteration number;

maximum iteration number.

Fig. 2. Single-line diagram of the 18-bus IEEE distorted system [6], [10][16].

6) Step 6: (Update Position): The position of each particle

at the next iteration is modified as

(15)

7) Step 7: (Check Convergence Criterion): If

or , the program is terminated and the results

are printed; otherwise, the programs goes to Step 2.

IV. SIMULATION RESULTS

To assess the applicability of the proposed approach, the18-bus IEEE power system is considered. The system contains

several linear loads and three nonlinear loads located at buses 7,

24, and 25, where the 5th, 7th, 11th, 13th, 17th, 19th, 23rd, and

25th harmonics are produced. The single-line diagram for the

testing system is depicted in Fig. 2. The detailed specification

of this system is given in [10].

In this power system, three identical harmonic current sources

are employed as nonlinear loads. The harmonic contents of the

employed harmonic current sources (the nonlinear loads) are

shown in Fig. 3. Voltage distortions for all harmonic orders and

at all buses as well as THD and MLL at all buses can be calcu-

lated by using the admittance matrix for all harmonic orders and


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Fig. 3. Harmonic contents of the nonlinear loads in the test system.

TABLE IVOLTAGE CONDITIONS AT DIFFERENT BUSES IN THE ABSENCE OF APLCS

the harmonic contents of nonlinear loads shown in Fig. 3 [using

(9) and (10)]. It should be noted that because no APLC is in-

stalled, APLCs current in (10) is considered a zero matrix.

Table I illustrates the obtained simulation results in no APLC

state. It can be found from Table I that the system is badly dis-

torted by nonlinear loads.

As shown in Table I, the resulting THD at all buses is, on av-

erage, 10.25%, which represents a rather unallowable harmonic

distortion level in reference to the IEEE standard limit of 5%.

Using (5) and (7), the average MLL and HTLL in no APLCs

state were calculated to be 0.0035 and 0.0394 p.u., respectively.

The proposed AAS problem is evaluated in three differentcases. In case 1, no constraints are applied to the APLCs size.

The rms value of the APLCs current is limited to 0.05 p.u. in

the second case. At last, the employed APLCs size is assumed

to be integer multiples of 0.01 p.u.; plus, the size of APLCs is

limited to 0.07 p.u. in case 3. Furthermore, the investigations are

performed in six states as represented in Table II.

In Table II, the signs and indicate the availability and un-

availability of APLC in the relative bus, respectively. Therefore,

state 1 considers only buses 7, 24, and 25 as the only candidate

buses for installing APLCs. Buses 7, 23, 24, 25, and 26 are also

the only candidate buses in state 2. The rest of the states can be

simply recognized by using the signs marked by in Table II.

As a well-established and acceptable optimization method,the genetic algorithm (GA) is used to evaluate and compare the

TABLE IISITUATION OF BUSES REGARDING THE AVAILABILITY OF APLC

TABLE IIITOTAL APLCS CURRENT CALCULATED IN CASE 1

results. Two GA algorithms as GA1 and GA2 are utilized to

solve AAS problem in this application. Population size and gen-

eration number are set in GA1 so that the computation time ofGA-based optimization process would be equal to the PSO one

Population size Generation number . In GA2,

the population size and generation number are selected exactly

the same as those used in the proposed PSO Population size

Generation number . The implementation time of

GA2 to solve the AAS problem of the test power system is about

four times that of PSO and GA1. The GAs output is presented

in two last rows of the relative tables. Furthermore, to illustrate

the priority of PSO over analytical methods in problems with

many local minimums, a comparison with the nonlinear pro-

gramming (NLP and DNLP)-based algorithm as an analytical

method is implemented. This program is written in GAMS soft-

ware [23] and the simulation results are given in the last row ofthe following tables under the NLP heading.

A. Case 1

In this case, no current constraints are applied to the APLCs

size. Tables IIIVII demonstrate the size of APLCs, THD, MLL,

and HTLL indices obtained from the simulation of the system

in six states corresponding to 3, 5, 7, 10, 13, and 16 APLCs,

respectively.

Among the aforementioned states, the first and the sixth states

correspond to the least and the most variables for the PSO algo-

rithm (i.e., and variables), re-

spectively. Numbers 2 and 8 indicate the real and imaginaryparts of the APLCs current and harmonic orders, respectively.


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TABLE IVAVERAGE THD CALCULATED IN CASE 1

TABLE VAVERAGE MLL CALCULATED IN CASE 1

TABLE VI

AVERAGE HTLL CALCULATED IN CASE 1

TABLE VIIOBJECTIVE FUNCTION CALCULATED IN CASE 1

Numbers 16 and 3, on the other hand, are the number of

candidate buses for the installation of APLCs. The population

number is about twice and ten times more than the variables in

state 6 and state 1, respectively. Therefore, the number of vari-

ables exactly impresses the accuracy of the results as shown in

the following tables and Fig. 4. States 5 and 6 suffer from the

highest OF values compared to states 1 and 2 which enjoy the

lowest OF values.

As illustrated in Table IV, the maximum average THD amongthe states, 0.00639 (in state 6), is about 16 times lower than

the average THD before installation of the APLCs, which is

0.1025. MLL and HTLL indices also decrease from 0.0394 to

0.00181 and from 0.0035 to 0.000254, respectively, indicating

the effectiveness of the PSO algorithm.

The best output in all states is that the current of installed

APLCs at candidate buses 7, 24, and 25 would be equal to the

nonlinear loads current located in these buses and zero at the rest

of the candidate buses. The small discrepancy observed among

the total APLCs current in different states given in Table III

and the nonlinear load currents given in Fig. 3 is due to two

points: 1) The nature of PSO which converges to some extent

prematurely and 2) THDwhich is not the only objective functionin this study. This fact is vividly perceived in three APLC states.

Fig. 4. Convergence characteristics for the states in case 1.

0 in this state means that the APLCs current is equal to

nonlinear loads current, 0.29154, which imposes a higher cost

than the APLCs current obtained, which is 0.29066.

In order to improve the accuracy of results, two different

ways can be employed: 1) increasing the number of initial

random inputs and 2) increasing the number of iterations.

However, they aggravate the time-consuming problem as a

remarkable difficulty.

Compared to PSO, the analytical method is expected to rep-

resent more accurate results since no constraint is applied to the

size of APLCs as optimizing variables in case 1. Therefore, the

objective function seems to be smooth. As a result, analytical

methods can be more useful in this case so that the aforemen-

tioned tables clarify the convergence in OFs with smaller values.

The trend of the objective function with respect to the numberof iterations is shown in Fig. 4 during PSO implementation.

This is to test the convergence of the objective function to de-

termine the quickness of PSO in terms of iteration numbers. As

observed, the best and the worst results belong to states 1 and 6,

respectively, which is related to the number of variables.

It is clear that the higher aggregation of APLCs should be

around the nonlinear loads which can be seen in Table V.

Quick convergence is a remarkable benefit of the PSO-based

AAS compared with other heuristic methods (Fig. 4). The con-

vergence is accomplished in about 100 and 200 iterations in

states 1 and 6, respectively. This fact is due to the number of

particles: 500 in this test, which is about twice the number ofvariables in state 6 and 10 times in state 1.

In comparison with the PSO algorithm, GA outcomes suffer

from lower accuracy. This fact is more clearly recognized in

higher states and more variables. As represented, THD, HTLL,

MLL, and total size of the APLCs in the PSO-based approach

are severely more efficient than GA1; however, the length of

implementation in both methods is the same. For instance, the

total APLCs size obtained by GA1 is almost 38% more than that

obtained by GA1 for state 6.

It is undeniable that the case with no current constraints for

APLCs is impractical because the DC source in the APLC is

implemented by capacitors and inductors which are discrete de-

vices. However, this case is assessed only for showing the capa-bility of the proposed algorithm.


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TABLE VIIITOTAL APLCS CURRENT CALCULATED IN CASE 2

TABLE IXAVERAGE THD CALCULATED IN CASE 2

B. Case 2

The rms value of the APLCs current is restricted to 0.05 p.u.in this case. Tables VIII and IX illustrate the total size of APLCs

and THD for different states, respectively. Despite the previous

case, the obtained outcomes are relatively practically feasible

owing to the assumption of discrete APLCs; however, consid-

ering 16 buses as candidate buses of APLCs is not commercially

feasible.

The proposed heuristic strategy is completely the same as the

algorithm presented in Section III-B, but a change in vector

as shown in (16) and (17) should be done at the beginning of step

2 as shown in (17) at the bottom of the page, where is

the maximum current magnitude of APLCs, which is set to 0.05

p.u. in this test.As mentioned before, there are only three buses as candidates

for installing APLCs in state 1 and the maximum current of

each APLC is also limited to 0.05 p.u. Therefore, the total cur-

rent of compensators is 0.15 p.u., which is much less than the

total current of nonlinear loads, 0.2916. This fact causes voltage

THD at buses 20, 21, 22, 23, 24, 25, and 26 in state 1 to not be

in the standard limit of 5%, as demonstrated in Table VII. Al-

though, states 4, 5, and 6 in this case show acceptable results in

the average THD and total APLCs current, they are not compa-

rable with state 2. State 2 devotes the most reasonable results so

that it gets benefit from almost the lowest APLCs current0.24

p.u.while the voltage THD standard range is also satisfied at

2.23%.

Fig. 5. Convergence characteristics for states in case 2.

As a result, in addition to three APLCs state, the application

of 16 APLCs is also not recommended in this case, because

the APLCs current in this state is the highest one, more than

0.36 p.u. In comparison with the no-current constraint state, theresults are converted to the feasible outcomes

Although the GA2-based average THDs and MLLs as well

as the total APLCs current or the total APLCs size are fairly

close to the PSO-based results in states 1 and 2, they are almost

severely higher in other states.

As illustrated in the aforementioned tables, again in this case,

when the objective function is still smooth and only a constraint

is applied to the variables, nonlinear programming as an analyt-

ical method shows better results compared with PSO and GA;

however, this case also suffers from inapplicability.

Fig. 5 depicts a comparison among the trend of objective

functions yielded from the aforementioned six states in terms of

the iteration number. As seen in Fig. 5, the third and fourth states

present almost similar behavior in convergence and show the su-

periority over other states because they converge in the lowest

objective function values. The quickest convergence character-

istics belong to states 1 and 2, but in the highest objective func-

tion value with respect to other states.

C. Case 3

The APLCs size in this case study is assumed as integer mul-

tiples of 0.01 p.u. and is limited to 0.07 p.u. As a practical and

feasible case, this case is described in detail. All of the afore-

mentioned states are assessed in this case, and the results are

finally compared.

if

if(16)

if

if

(17)


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TABLE XAPLCS CURRENT INJECTING INTO BUSES FOR DIFFERENT STATES IN CASE 3

For including the mentioned limits to the algorithm presented

in this subsection, some changes should be applied to the vector

at the beginning of step 2. The following procedure is ex-

ecuted in order to divide APLCs current into 0.01-p.u. steps:

First, the APLC current is multiplied by 100 to make it a digit

between 0 and 10. Then, the integer part of the obtained digit

is kept, and divided by 100 to return it into the original value,

between 0 and 0.10. Therefore

(18)

(19)

where operator int shows rounding the variable up to the

nearest integer. After that, for limiting the APLCs current to

0.07 p.u., (16) and (17) are used and MaxIF is set to 0.07.

The constraints applied to the APLCs size make case 3 apractical and feasible case and make the optimizing problem

a discrete/nonlinear programming with a nonsmooth objective

function. As mentioned before, the main drawback of analytical

methods is local minimum. This problem clearly occurs in the

nonsmooth objective function in case 3.

Table X demonstrates the simulation results indicating the

size of APLCs installed in the candidate buses in all six states.

To compare with the previous cases, THD and MLL are also

represented in Tables XI and XII, respectively.

As demonstrated in Table XI, the maximum THD in this case

belongs to bus 24, which is 0.0353, 0.0161, 0.0143, 0.0167,

0.0143, and 0.0208 in states 1 to 6, respectively. As a result,

it is expected that the maximum current to be injected by APLCin bus 24 (0.07 p.u. in all states is illustrated in Table X).

TABLE XITHD OF BUSES FOR DIFFERENT STATES IN CASE 3

TABLE XIIMLL OF BUSES FOR DIFFERENT STATES IN CASE 3

As seen, the obtained results in this case are more accept-

able than previous ones owing to the fact that applying restric-

tion and discreteness on APLCs size makes the problem more

feasible and practical. As an illustration, only seven buses are

selected to employ APLCs in states 6, while 16 buses are can-

didate. It is observed that the APLCs with very low current

are eliminated when the discreteness criterion is applied com-

pared with case 1 results. Therefore, the concept of allocation is

vividly demonstrated, because only a few buses are selected forinstalling APLCs, while more choices are available.


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TABLE XIIIAVERAGE HTLL CALCULATED IN CASE 3

TABLE XIVOBJECTIVE FUNCTION CALCULATED IN CASE 3

This restriction and discreteness also cause total APLCs cur-

rent in all states in this case to be much less than these in case

1 and fairly less than these in case 2 except with state 2 (0.27 in

case 3 compared with 0.24 p.u. in case 2).

The reduction of step size of APLCs (0.01 p.u.) leads to

making the problem closer to the continuous size; therefore,

more buses will be selected as candidate buses for them. Since

the total nonlinear loads current is constant, the total APLCs

size is known to some extent. Accordingly, the reduction of

size of APLCs may lead to employ more APLCs (case 1).

It has been simply figured out that the aggregation of APLCs

is around the nonlinear loads such as cases 1 and 2 (Tables V

and VI). For instance, the highest APLCs current are allocated

at buses 7 (0.05), 24 (0.07), and 25 (0.07) in state 6 as shown inTable X.

The results in seven APLCs state are remarkably more

efficient than 10, 13, and 16 APLCs states as represented in

Table XIV. It should be noticed that if THD is considered

only as a constraint instead of a part of the objective function,

3APLCs state is the best because only 0.2100-p.u. APLC cur-

rent is needed while THD is 0.0204 (within the standard limit).

The outcomes vividly show that in integer nonlinear opti-

mization, DPSO leads to more operational results with respect

to GA and discrete nonlinear programming (DNLP). This fact

is illustrated specifically in Table XIV so that a discrepancy

of about 70% and 200% in DPSO-based OF as the main cri-teria for comparison exists with GA and DNLP in state 6 with

a higher number of variables, respectively. This difference is

severely distinguished in APLCs current so that the calculated

total APLCs injected current in DNLP is eight times more than

DPSO.

Fig. 6 depicts the convergence characteristics of the objective

function evaluated in case 3 for all six states with respect to the

iteration number. Among the mentioned states, the 3 APLCs

state obtains a benefit from quick convergence so that it con-

verges in only 30 iterations, but in the highest objective func-

tion value than others, the second and third states are in the

second and third priorities, respectively. The best results are as-

sociated with the third state which converges in the lowest ob-jective function.

Fig. 6. Convergence characteristics for states in case 3.

V. CONCLUSION

A PSO-based approach has been proposed in this paper for

optimal allocation and sizing of multiple APLCs in power sys-

tems. In this optimization procedure, a comprehensive objec-

tive function is evaluated. The objective function is composed

of four parts: 1) THD; 2) total APLCs current; 3) MLL; and

4) HTLL. The proposed algorithm has been successfully imple-

mented to an 18-bus IEEE power system in which three non-

linear loads are located. The results were achieved in three dif-

ferent cases. Case 1 in which the size of APLCs is continuous

and unlimited, case 2 in which the size of APLCs is continuous

but limited to 0.05 p.u.; and case 3 in which the size is discrete

by 0.01-p.u. steps and limited to 0.07 p.u. These cases were as-

sessed in six different states depending on the number of candi-

date buses to install APLCs.

To determine the capability of the proposed method, twoGA-based algorithms (GA1 with the same computation time

as PSO and GA2 with the same population size and iteration

number as PSO, but with longer computation time) are devel-

oped. An analytical optimization approach based on nonlinear

programming (NLP) is also implemented for comparison pur-

poses.

As expected, when no local minimum exists in the objective

function, analytical methods provide the most accurate results

as observed in cases 1 and 2. It is also demonstrated that when

no constraint is applied to the problem (cases 1 and 2); PSO en-

joys more accurate results compared to GA, but less accurate

results compared with NLP. However, the main shortcoming ofthe analytical methods is the local minimum being highlighted

when the problem is a nonsmooth function. The realistic AAS

problem is a practical case with a nonsmooth function and a

number of local minimums. In this type of function, analyt-

ical methods can optimize properly only when perfect initial

solutions are provided. This, however, is not an easy task, and

hence, to solve this problem, heuristic algorithms are highly rec-

ommended. In this paper, a discrete PSO algorithm (DPSO) is

developed to overcome a discrete nonlinear problem, including

discreteness and limited size for APLCs (case 3). It is demon-

strated that the obtained results using DPSO are more accurate

compared with the results obtained by the discrete nonlinear

programming (DNLP) method and other heuristic approaches,such as GA.


10/10


In terms of computation time, the PSO is more effective than

GA as a well-known heuristic method so that a remarkable dif-

ference is observed between the PSO-based results and GA1

results. This difference is still remarkable when the GA2 algo-

rithm is used for comparison. The results clearly state the men-

tioned plus properties, such as the applicability and capability

of using the PSO for allocation and sizing of multiple APLCsin power systems.

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Iman Ziari (S09) was born in Abadan, Iran, onSeptember 11, 1978. He received the B.Sc. degreefrom Sahand University of Technology (SUT),Tabriz, Iran, and the M.Sc. degree in electricalengineering from Iran University of Science andTechnology (IUST), Tehran, Iran, where he iscurrently pursuing the Ph.D. degree.

Currently, he is a Lecturer in the Electrical En-gineering Department of Islamic Azad University(IAU)-Damavand branch, Tehran. His research

topics include power system harmonics.

Alireza Jalilian was born in Yazd, Iran, in 1961. Hereceived the B.Sc. degree in electrical engineeringfrom Mazandaran University, Babol, Iran, in 1989,and the M.Sc. and Ph.D. degrees from the Universityof Wollongong, Wollongong, New South Wales,Australia, in 1992 and 1997, respectively.

He joined the Power Group of the Department ofElectrical Engineering at the Iran University of Sci-ence and Technology in 1998, where he is AssistantProfessor. His research interests are power-qualitycauses and effects as well as mitigations.

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