passive harmonic filter planning in a power system

8/18/2019 Passive Harmonic Filter Planning in a Power System

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208 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 1, JANUARY 2009

Passive Harmonic Filter Planning in a Power SystemWith Considering Probabilistic Constraints

Gary W. Chang , Senior Member, IEEE , Hung-Lu Wang , Student Member, IEEE , Gen-Sheng Chuang, andShou-Yung Chu , Student Member, IEEE

Abstract—This paper presents a new method for planning

single-tuned passive harmonic filters to control harmonic voltage

distortion throughout a power system. In the problem, the prob-

abilistic characteristics of the harmonic source currents andnetwork harmonic impedances in the filter planning are taken into

account. The objective is to minimize the total filter installation

cost, while the harmonic voltage limits and filter component

constraints are satisfied with predetermined confidence levels. Toobtain the optimal size of each filter component of the planning

problem, the proposed procedure is first to find the candidate filter

buses based on the sensitivity analysis. Next, the formulated prob-

ability-constrained problem is transformed into a deterministicnonlinear programming problem and is solved by a genetic-algo-

rithm-based optimizer. The proposed solution procedure is tested

with an actual distribution network and is verified by the conven-

tional deterministic approach and by the Monte Carlo simulation.

Numerical experiences show that the proposed method yields

favorable results compared with the other two approaches.

Index Terms—Chance-constrained programming model, geneticalgorithm (GA), passive harmonic filter, sensitivity analysis.

I. INTRODUCTION

IN the power system, nonlinear loads may be categorizedas the harmonic current or voltage source types. Therefore,

the effectiveness of the shunt passive harmonic filter for har-

monic and reactive power compensation of a nonlinear load

depends highly on the harmonic source type [1]. In addition,

the interactions between the grid voltages and the nonlinear

loads may slightly change the harmonic current injections [2].

However, as suggested by IEEE Std. 519 [3], electric utilities

are responsible for controlling individual and total harmonic

voltage distortions (i.e., IHDv and THDv) at their network

buses, while the customers are responsible for controlling their

harmonic current injections into the utility network. Therefore,

it is practical to control harmonic voltage distortion throughoutthe utility power network by the placements of passive filters at

selected candidate buses instead of installing filters at specified

nonlinear load buses. Assume that the harmonics-producing

loads are under balanced operations. The distributed nonlinear

loads can be modeled as aggregated harmonic current injections

Manuscript receivedJanuary 25, 2008; revised June01, 2008. Current versionpublished December 24, 2008. Paper no. TPWRD-00028-2008.

The authors are with the Department of Electrical Engineering, Na-tional Chung Cheng University, Min-Hsiung, Chia-Yi 621 Taiwan (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.2008.2005371

at certain buses and the direct current injection method can be

applied to perform the linear harmonic analysis.

When considering passive harmonic filter planning in a

power system, most approaches adopt the deterministic model

for network harmonic impedance and harmonics-producing

loads [4]–[8]. However, the deterministic approach for har-

monic studies may fail to model the actual behaviors of

nonlinear loads and network harmonic impedances. In many

cases, harmonic currents produced by nonlinear loads may have

probabilistic characteristics due to the variation of load levels

which change over time [9]–[11]. For the passive harmonic

filter planning, if harmonic currents and the system impedance

are regarded as deterministic, the capacities of installed passive

harmonic filters may not be properly sized, which also leads

to excessive cost. Therefore, the probabilistic characteristics

of harmonic currents and impedances of the system should be

considered when performing filter planning.

In recent years, major standards for controlling power system

harmonics have set up limits for harmonic current injections

based on probabilistic characteristics. For instance, IEC Std.

61000-3-6 considers the extensive influences of harmonic cur-

rents generated by electrical equipment and requires that the

electrical equipment work normally within 95% probabilityvalue of maximum daily and weekly allowed harmonic currents

[12]. That is, the harmonic currents cannot exceed the allowed

limit of 95% of the time during measurement. IEEE Std. 519 also

suggests considering the probabilistic constraints for harmonic

currents generated by nonlinear loads and the standard interprets

that the harmonic level can exceed the recommended limits in a

short time without causing damage to the equipment [3].

In [13], the authors have developed a probabilistic approach

for planning passive harmonic filters connected to an indus-

trial power system. To tackle the more complex system-wide

filter planning problem, which requires efficiently controlling

the harmonic voltage and voltage distortion at each network bus,the authors extend a previously developed method for planning

the single-tuned passive filters in a power system, where the

probabilistic characteristics of the system parameters are taken

into account. The planning problem is formulated as a proba-

bilistic-constrained optimization problem and is then solved to

find the optimal sizes of filter components. The objective func-

tion to be minimized is the total filter component cost. The con-

straints are the individual and total harmonic voltage distortion

limits imposed on the nonlinear load, tuned frequency varia-

tion limits of the filter, and the root mean square (rms) voltage

and current limits imposed on the filter capacitor and inductor,

respectively.

0885-8977/$25.00 © 2008 IEEE


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CHANG et al.: PASSIVE HARMONIC FILTER PLANNING IN A POWER SYSTEM 209

Fig. 1. Schematic diagram for the th order of harmonic current source at bus injected into the power system.

Fig. 2. Thevenin equivalent circuit for a passive filter connected to bus

To solve the probabilistic optimization problem, the authors

propose a practical two-phase solution procedure. The sensi-

tivity analysis-based placement procedure is first used to quickly

locate the best candidate filter buses. Next, the linear approx-

imation method is employed to obtain the approximate mean

value and variance of each probabilistic constraint function, and

the chance-constrained programming-based model is proposed

to transform the probabilistic problem into a deterministic one.

The deterministic programming problem is then solved by a

GA-based optimizer that is built upon the Matlab environment.

Finally, the proposed method is tested for an actual distribution

system to show its usefulness.

II. HARMONIC ANALYSIS

Figs. 1 and 2 show a typical -bus network with an th order

of a harmonic current source at bus and the equivalent network

after the passive filter installed at bus , respectively. Before the

passive filter is installed, the th order of the harmonic voltage

at each bus is given in (1)

(1)

where is the th order of the harmonic

transfer impedance between buses and .

If there are more than one harmonic current sources of the

th order existing in the network, the corresponding harmonic

voltage at any bus becomes

(2)

where are the bus numbers of the harmonic

sources.

As shown in Fig. 2, after the single-tuned passive filter is con-nected to bus for controlling the th order harmonic current

in the system, the harmonic current drawn by the passive filter

is given in (3)

(3)

where

(4)

and where is the filter tuned harmonic

order, is the base apparent power; and is the reactive

power capacity of the filter at the fundamental frequency. Also,

is the Thevenin equivalent voltage at bus

before siting the passive filter and is

the driving point harmonic impendence at bus before placing

the passive filter.

The new harmonic voltage at any bus for the th harmonic

after the filter installed at bus is then obtained by

(5)

where is the harmonic transfer

impedance between buses and before siting the passive

filter and is the harmonic voltage at bus

. If more filters are installed at different buses, the harmonic

voltage at bus is given

(6)

where are bus numbers of passive filters for

the th harmonic.When considering the probabilistic characteristics of the pas-

sive harmonic filter planning problem, the real and imaginary

components of the harmonic current and the system harmonic

impedance are treated as random variables. Then, (5) and (6)

can be rewritten as

(7)

where represents probabilistic

parameters and is the solution variable (i.e., filter

harmonic impedance) to be determined for the passive filter.

III. PROBABILISTIC APPROACH OF THE PASSIVEHARMONIC FILTER PLANNING

A. Linear Approximation for a Function of Random Variables

Let be a function of random vari-

ables. After expanding the in a Taylor’s series about the mean

value, the following is obtained:

(8)


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where is the mean value of the corresponding random vari-

able . From the aspect of engineering applications, it is prac-

tical to approximate the mean value and the variance of by

the transmission of variance formulae, as shown in (9) and (10),

to simplify the calculation of and [14], [15]

(9)

(10)

where is the covariance between and . If

and are uncorrelated for all and , the second term at the

right-hand side of (10) can be ignored.

B. Siting Index for Determining Passive Filter Locations

The first step of the solution procedure is to find the best can-

didate bus one at a time among existing capacitor busses in the

network for siting passive filters while minimizing the system

voltage distortion of a specific harmonic order under considera-

tions [16]. Based on sensitivity analysis, the best filter bus can

be identified by the siting index

(11)

where and where is the

highest order of harmonic under considerations and is the

th harmonic voltage change at bus after the filter is installed

at bus . Also

(12)

and

(13)

where and

are the mean value and the variance of the corre-

sponding probabilistic parameter. The detailed derivation of

the siting index is given in Appendix A. Equation (11) implies

the effectiveness for controlling voltage distortion across the

network corresponding to the filter placement at bus . The top

priority bus is the filter bus that yields the least value among allsiting indices.

Fig. 3. Applying the limit to the probabilistic harmonic voltage.

Fig. 4. Harmonic voltage estimation in the confidence interval range.

C. Chance Constraints for Harmonic Voltage at Each Bus

Figure 3 illustrates the immunity of harmonic pollutions at

each network bus in the system, where is the recommendedharmonic voltage limit and represents the actual har-

monic voltage. In is the vector of known parame-

ters with probabilistic characteristics for the harmonic passive

filter and is the vector of unknown variables.

Let denote the probability of event . By observing

Fig. 3, the probability constraint of the harmonic voltage ex-

pressed by the chance-constrained programming model of (14)

is to guarantee that the bus voltage is satisfied within the re-

quired probability level given by

(14)

where is the limit level of the probability.

For convenience, Fig. 3 can be expressed as a confidence in-

terval shown in Fig. 4, which includes the predetermined con-

fidence level. The permissive value of the harmonic voltage at

the point of common coupling is then estimated. Therefore, the

probabilistic constraint of the harmonic voltage expressed by

(14) can be replaced by the deterministic constraint given in

(15). In this situation, the probabilistic optimization problem is

treated as a deterministic nonlinear programming problem

(15)

In (15), and represent the mean value andthe variance of the probabilistic harmonic voltage, respectively.


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is the tabulated value of the inverse cumulative distri-

bution function of the standard normal distribution evaluated at

. By substituting the mean value and the variance of (9) and

(10) into (15), the deterministic constraint corresponding to the

probability confidence interval of the harmonic voltage shown

in Fig. 4 is determined. The proof of (15) can be found in [13,

Appendix].

IV. PROBLEM FORMULATION

In the probabilistic passive harmonic filter planning problem,

the objective function is to minimize the filter installation cost.

The constraints of the problem include IEEE-519 individual

harmonic voltage and total harmonic voltage limits at each

bus, tuned frequency variation limits of the passive filter due

to manufacturing errors and environmental changes, as well as

the IEEE-1531 capacitor rms voltage and inductor rms current

limit of the filter [17]. The details of the problem formulation

will be listed.

A. Constraints

1) Voltage Harmonics and Voltage Distortion Constraints:

The IEEE-519 individual harmonic voltage distortion (IHDv)

and total harmonic voltage distortion (THDv) constraints im-

posed on any network bus after the filter placement are

(16)

for , and

(17)

where is the fundamental voltage at bus and are

usually 3% and 5%, respectively.

For convenience, substituting (7) into (16) to obtain the

squared function, the individual harmonic voltage limit can be

expressed by (18)

(18)

Similarly, the constraint of the total harmonic voltage distortion

at any network bus of (18) can be written as

(19)

The specified confidence levels of individual harmonic voltage

and total harmonic voltage distortion can be defined by the

chance-constraints of (20) and (21), respectively

(20)

(21)

where and represent the specified confidence levels of

the individual harmonic current and the total harmonic current

injections, respectively.

According to the linear approximation of (9), the mean valueof given in (18) at the th harmonic order can be written as

(22)

where

is the vector of mean values of associated probabilistic

parameters of . With (10), the approximate variance of (18) at

the th harmonic order then becomes

(23)

where

(24)

(25)

The corresponding coefficients used in (23)–(25) are given in

Appendix B.

2) RMS Voltage Limit Across the Capacitor of the Filter:

Under the specified confidence level, the chance-constraint of

the rms voltage across the capacitor can be expressed by

(26)

where

(27)

(28)

where is the capacitor rated rms voltage of the th filter

and is the recommended percentile limit of the rated voltagevalue; represents the specified confidence level of the rms


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voltage across a filter capacitor at the filter bus and is the

tuned harmonic order.

The mean value and variance of the rms voltage across the

capacitor of the th filter then can be approximated by

(29)

and

(30)

where

(31)

(32)

3) RMS Current Limit Flowing Through the Filter: Similarly,

the chance-constraint of the rms current flowing through the in-

ductor can be expressed by (33)

(33)

where

(34)

(35)

where is the rated inductor rms current of the th filter and

is the recommended percentile limit of the rated current value

and represents the specified confidence level of the rms

current flowing through the filter inductor at the filter bus.

The mean value and variance of the rms current flowing

through the th filter then can be approximated by

(36)

and

(37)

where

(38)

(39)

4) Filter-Tuned Frequency Variation Constraint: The manu-facturing and environment caused deviations for filter inductors

and capacitors are in the range of % to 3% and % to 12%

from their nominal values, respectively; the LC variation range

of the filter is usually between 0.92 and 1.06 at the system fun-

damental frequency [8]. Therefore, the deviation limits for any

tuned harmonic order will be controlled in the range; that is,

, where is the

exact tuned harmonic order and is the closest integer tuned

harmonic order for the th filter.

The chance-constraint corresponding to the th filter-tuned

frequency variation can then be expressed by

(40)

where is the specified confidence level of the filter tuned har-

monic order. Assume that the variation range of the tuned har-

monic order is normally distributed and the system frequency

is constant. The approximate probability distribution can be ex-

pressed by .

B. Probabilistic Optimization Problem

According to the described objective function and con-

straints, the problem formulation of the passive filter planning in

terms of each filter-tuned harmonic order becomes “Minimize”

(41)

subject to

(42)

(43)

(44)

(45)

(46)

where

and are unit costs associated with the fundamental re-

active power of the capacitor (i.e., ) and the inductor (i.e.,

) of the th filter, respectively; is the fundamental re-

active power of the th filter; , and repre-

sent the confidence levels, where the first four terms are 95% and

the last is 99.9%, respectively;

and .

V. SOLUTION PROCEDURE

The formulated filter planning problem is categorized as aprobabilistic-constrained optimization problem. After adopting


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the chance-constrained programming model, the probabilistic

problem is transformed into a deterministic nonlinear program-

ming problem where the optimal solution by conventional non-

linear programming approaches is still very difficult to obtain

since the problem is highly nonlinear and nonconvex. Since

the genetic algorithm (GA)-based method can be used in con-

junction with local-improvement heuristics for nonlinear func-tion optimization and is very suitable for solving nontypical

nonlinear programming problems with relatively high solution

speed, the Genetic Algorithm Optimization Toolbox (GAOT)

from Matlab is adopted as the problem solver [18]. The solu-

tion procedure for GAOT typically includes creating an initial

population, evaluating all individuals, selecting a new popula-

tion, creating new solutions by the mutation and crossover, and

termination criteria check [18], [19].

The following list summarizes the major steps of the pro-

posed solution procedure for passive harmonic filter planning

in a power system.

Step 1) Input the probability parameters of harmonic source

data and harmonic transfer impedance of the net-work.

Step 2) Calculate the mean value and the variance of the har-

monic voltage, and determine the 95% confidence

interval value of individual and total harmonic volt-

ages at each bus.

Step 3) Check whether the 95% confidence value of each in-

dividual harmonic voltage violates the limit. If yes,

go to step 5. Otherwise, proceed to the next step.

Step 4) Check whether the 95% confidence interval of the

total harmonic voltage at each bus violates the limit.

If yes, proceed to the next step. Otherwise, stop.

Step 5) Find the filter siting index by (11) for each harmonicorder under consideration and for all network buses

one at a time and prioritize the indices. The bus with

the least value of (11) is the best candidate bus for

siting the filter.

Step 6) Convert the capacitor bus with the top priority of the

siting index to a filter bus for each harmonic order

under consideration.

Step 7) Formulate the probabilistic-constrained pro-

gramming problem of (41)–(46) and apply the

chance-constrained programming model to trans-

form the problem into a deterministic-constrained

problem.

Step 8) Solve the deterministic optimization problem corre-

sponding to (41)–(46) by the GAOT solver.

Step 9) Check whether all constraints are satisfied. If yes,

stop and output. Otherwise, proceed to the next step.

Step 10) Use the equivalent resistance approach of [7] to de-

termine the filter siting index for each harmonic

order under consideration and for all network buses

one at a time and prioritize the indices. The bus with

the least value of (11) is the best candidate filter bus

for that tuned harmonic order. Proceed to the next

step.

Step 11) For the candidate filter bus with an existing capacitor

bank, repeat step 6. Otherwise, for all commerciallyavailable capacitor sizes, sequentially start with the

Fig. 5. Eighteen-bus test system.

TABLE IPROBABILISTIC CHARACTERISTICS OF HARMONIC CURRENT SOURCES (%)

smallest unit and convert it into a passive filter at the

candidate bus.

Step 12) Repeat Steps 7) and 8) for each converted filter.

Step 13) Stop and output after the filter component sizes with

the least objective function value in Step 12) are

found.

VI. CASE STUDY

The proposed solution algorithm for passive filter planning istested by using an actual 18-bus distribution network, as shown

in Fig. 5 [20]. The base power of the sample system is 10 MVA

and all buses except 17 and 18 are 12.5-kV buses. In the system,

the harmonic current sources, the system impedances, and the

load at each bus have probabilistic characteristics.

Assume that the real and imaginary components for each

order of harmonic current of the nonlinear load are normally

distributed; the corresponding expected values and standard de-

viations are shown in Table I, where the correlation coefficients

related to real and imaginary components for each harmonic are

practically assumed to be 0.5 and the standard deviations of real

and reactive load power, P (%) and Q (%), are assumed to be

30% [10], [21]. Before the placement of passive filters, the IHDvand THDv at each bus are determined by PCFLO [20].


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Fig. 6. 95% confidence intervals for THDv at each network bus before placingthe filters.

Fig. 7. Largest individual harmonic voltages at the upper bound of the 95%confidence interval before placing the filters.

A. Results

The 95% confidence interval of the total harmonic voltage

and the upper/lower bounds of the 95% confidence interval of

individual harmonic voltage at 12.5-kV buses before siting fil-

ters are shown in Figs. 6 and 7, respectively. It is observed that

the upper limits of the largest values of the 95% confidence in-

terval of individual harmonic voltages for the 5th, 7th, 11th,

13th, 17th, and 19th harmonics are 7.5%, 4.9%, 3.11%, 2.0%,

2.7%, and 0.3%, respectively. The highest 95% confidence in-

terval of THDv is 9.92%, occurring at bus 14. Therefore, IHDv

and THDv at several network buses violate the IEEE-519 limits.

By applying the proposed solution procedure for filter plan-

ning, the component values and the tuned harmonic filters are

shown in Table II. Figs. 8 and 9 show the 95% confidence in-

terval of the IHDv and THDv at each 12.5-kV bus after siting

filters. It can be seen that IHDv and THDv at each network bus

are well controlled under IEEE-519 limits. After installing thepassive filters in the system, the average of the THDv expected

values at all 12.5-kV buses is reduced to 2.36%, and the highest

is 4.47%, occurring at bus 8. In addition, as listed in Table III,

the rms capacitor voltage and inductor current for each filter also

satisfy IEEE-1531 limits, where the capacitor rms voltage and

the inductor rms current limits are set to be 110% and 135%,

respectively.

B. Comparisons Between the Proposed And the Deterministic

Approaches

In the conventional deterministic approach, the proposed two-

phase solution procedure is used to solve the same problem [7],where a sensitivity analysis-based placement procedure is first

Fig. 8. Ninety-five percent confidence interval of THDV at each network busafter placing the filters.

Fig. 9. Largest individual harmonic voltages at the upper bound of the 95%confidence interval after placing the filters.

TABLE IIHARMONIC FILTER CANDIDATE BUSES AND COMPONENT SIZES

TABLE IIIFILTER CAPACITOR RMS VOLTAGE (%) AT THE UPPER

LIMIT OF THE 95% CONFIDENCE INTERVAL

used to quickly locate the best candidate filter buses among ex-isting capacitor buses for each harmonic order under consider-

ation. Then, the optimizer GAOT is employed for determining

the near-optimal or global optimal filter sizes while minimizing

filter installation cost and satisfying all associated constraints.

The system parameters are treated as the worst case that har-

monic currents and the system data are their mean plus three

times the standard deviation. Figs. 10 and 11 show the IHDv and

the THDv at each 12.5-kV bus before siting passive harmonic

filters. It is found that before placing the filters, the system is

severely polluted with 5th, 7th, 11th, and 13th harmonics. The

average THDv at all 12.5-kV buses is 6.46 %, and the highest is

10.75%, occurring at bus 14.

The results obtained by the conventional passive filer plan-ning method are shown in Table IV. Figs. 12 and 13 show the


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Fig. 10. IHDv at each 12.5-kV bus before placing passive filters.

Fig. 11. THDv at each 12.5-kV bus before placing passive filters.

Fig. 12. IHDv at each 12.5-kV bus after placing passive filters.

Fig. 13. THDv at each 12.5-kV bus after placing passive filters.

IHDv and THDv, before and after installing passive filters, re-

spectively. It shows that the IHDv and THDv are well controlled

after placement of the filters. By comparing results shown in

Table V, the capacity of the passive filter obtained by the proba-

bilistic model is 2807.33 kVAR and by the deterministic model,

it is 4604.5 kVAR. The cost of the proposed approach is lowerthan that of the deterministic one.

TABLE IVFILTER COMPONENT SIZES OBTAINED BY USING THE DETERMINISTIC METHOD

TABLE VCOMPARISON OF THE PROPOSED AND DETERMINISTIC METHODS

Fig. 14. Largest values of the individual harmonic voltage distortion and totalharmonic voltage distortion (THD) at the upper bound of the 95% confidenceinterval at bus 9.

Fig. 15. Largest values of the individual harmonic voltage distortion and totalharmonic voltage distortion (THD) at the upper bound of the 95% confidenceinterval at bus 12.

C. Comparisons Between the Proposed Approach And Monte

Carlo Simulation

In the same manner, a Monte Carlo simulation program

developed by using Matlab is adopted to simulate the corre-

sponding probabilistic characteristics [22]. Simulations are

performed over 1000 times for the 18-bus test system. Figs. 14

and 15 show the expected values of the individual and total

harmonic voltage distortions (the 5th through the 19th harmonic

order) and the total harmonic voltage distortion (indicated by

THD) at buses 9 and 12, respectively, where results obtained

by the proposed linear approximation method are also shown

for comparisons. As indicated in Figs. 14 and 15, it is observedthat results obtained by the proposed method are very close to


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those obtained by Monte Carlo simulation. The absolute THD

differences between the two approaches for buses 9 and 12 are

0.21% and 0.35%, respectively.

VII. CONCLUSION

This paper presents a new method of harmonic passivefilter planning for controlling harmonic voltage distortion

by considering the probabilistic characteristics of harmonic

source currents and network harmonic impedances in the

power system. The theory and problem formulation for the

filter-planning problem are described in detail. In the solution

procedure, the linear approximation method, the sensitivity

analysis, the chance-constrained programming, and the genetic

algorithm-based approach are then employed to solve the

probabilistic optimization problem. The proposed solution

procedure has been tested by an actual distribution network. It

is shown that the proposed approach yields favorable results

in comparing those obtained by the conventional deterministic

approach and the Monte Carlo simulation.

APPENDIX

A. Siting Index for Determining Passive Filter Locations

In the unconstrained problem for passive filter planning, the

objective function is the sum of all harmonic voltages across the

power network after the filter placement and can be expressed

as

(A1)

where the th harmonic voltage at any bus before and after

filter placement are given in (A2) and (A3) as follows:

(A2)

(A3)

where is the index for harmonic current source location. Ac-

cording to (2) and (3), the following relationship holds:

(A4)

By substituting (A4) into (A3) and then into (A1), we have

(A5)

Representing (A5) with Taylor’s series around and trun-

cating the series at the linear terms, we have

(A6)

where

(A7)

and

(A8)

where and .

Therefore, the sensitivity factor for placement of the th filterbecomes

(A9)

where

(A10)

(A11)

and where and .

B. Corresponding Coefficients of (23)–(25)

(B1)

(B2)

where

(B3)

(B4)


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(B5)

and

(B6)

(B7)

(B8)

(B9)

(B10)

(B11)

(B12)

(B13)

(B14)

(B15)

(B16)

(B17)

(B18)

(B19)

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[18] C. R. Houck, J. A. Joines, and M. G. Kay, “A genetic algorithm forfunction optimization: A Matlab Implementation,” Raleigh, NC, NorthCarolina State Univ., , 1995.


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[19] Genetic Algorithm Optimization Toolbox User’s Guide for use withMatlab 1999, Version 2, The Math Works Inc.,.

[20] “PCFLO User’s Manual,” 2007, Univ. Texas at Austin.[21] G. Carpinelli, T. Esposito, P. Varilone, and P. Verde, “First-order prob-

abilistic harmonic power flow,” Proc. Inst. Elect. Eng., Gen. Transm. Distrib., vol. 148, no. 6, pp. 541–548, Nov. 2001.

[22] R. Y. Rubinstein , Simulation and the Monte Carlo Method . Reading,MA: Addison-Wesley, 1981.

Gary W. Chang (M’94–SM’01) received the Ph.D. degree from the Universityof Texas at Austin in 1994.

He was with Siemens Power T&D, Brooklyn Park, MN, from 1995 to 1998.He is a Professorin theDepartment of Electrical Engineering at National ChungCheng University, Taiwan. His research interests include power systems opti-mization, harmonics, and power quality.

Dr. Chang is a member of Tau Beta Pi and a registered professional engineerin the state of Minnesota. He chairs the IEEE Task Force on Harmonics Mod-eling and Simulation.

Hung-Lu Wang (S’04) received the M.S.E.E. degree from National ChungCheng University, Chia-Yi, Taiwan, in 2002, where he is currently pursuing thePh.D. degree.

His research interests include power system harmonics and power quality.

Gen-Sheng Chuang received the M.S.E.E. degree from National Chung Cheng

University, Chia-Yi, Taiwan, in 2003, where he is currently pursuing the Ph.D.degree.

His areasof research interestsincludepower system optimizations,short-termresource scheduling, and power quality.

Shou-Yung Chu (S’04) received the M.S.E.E. degree from National ChungCheng University, Chia-Yi, Taiwan, in 2002, where he is currently pursuing thePh.D. degree.

His areas of research interests include power systems optimization and powersystem harmonics.

passive harmonic filter planning in a power system

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