size capacitor and placement

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Optimal Placement and Sizing of Fixed and Switched

Capacitor Banks Under Nonsinusoidal Operating Conditions

M. A. S. Masoum M. LadjevardiDepartment of Electrical Engineering

Iran University of Science & TechnologyTehran, Iran, 168440.

E. F. Fuchs, IEEE FellowDepartment of Electrical and

Computer EngineeringUniversity of Colorado

Boulder, Colorado, 80309-0425.

W. M. Grady, IEEE FellowDepartment of Electrical and

Computer EngineeringThe University of Texas at Austin

Austin, Texas, 78712.

Abstract: An iterative nonlinear algorithm is generated for optimalsizing and placement of fixed and switched capacitor banks on radialdistribution lines in the presence of linear and nonlinear loads. TheHARMFLOW algorithm and the Maximum Sensitivities Selection(MSS) method are used to solve the constrained optimization problemwith discrete variables. To limit the burden of calculations andimprove convergence, problem is decomposed into two subproblems.Objective functions include minimum system losses and capacitorcost while IEEE-519 power quality limits are used as constrains.Results are presented and analyzed for the 18 bus IEEE distortedsystem. The advantage of proposed algorithm as compared to

previous works is consideration of harmonic couplings and reactionsof actual nonlinear loads in the distribution system.

Indexing Terms: Capacitor Banks, Placement, Sizing, MSS,Harmonic Power Flow.

1- IntroductionCapacitor placement has become the most popular solution forreducing system losses, regulating bus voltages and improving power factor at distribution levels. The general capacitor placement problem consists of determining the optimallocations, types and sizes of compensation capacitors such thatmaximum yearly benefit due to loss reduction againstinstallation cost of capacitors is achieved.

Most of the reported techniques for capacitor placementassume sinusoidal conditions and ignore the presence ofnonlinear loads [1-7]. Some of the recent researches haveconsidered the presence of distorted substation voltages for

capacitor placement problem [8-11]. Unfortunately, mostpresented techniques ignore some of the following problems:

discrete sizes of commercially available capacitors with adifferent cost per kVar,

presence of current and voltage harmonics due to thewide-spread use of energy-efficient appliances and power-electronic devices,

interactions and couplings between harmonic voltages andcurrents caused by actual nonlinear loads,

amplification of harmonic currents due to possibleharmonic resonance,

additional harmonic copper and core losses.The presented mathematical optimal methods of shuntcapacitor placement problem include analysis methods [1-2]gradient search method [5], dynamic programming (DP)method [3,8-9], and Maximum Sensitivities Selection (MSS)method [11].

Application of analysis method for solving the shuntcapacitor problem with power quality constrains (e.g., voltageTHD and distortion factor) is quite difficult. The gradientsearch method works well with continuos variables. However,the shunt capacitor placement is an optimal problem of discretevariables. Furthermore, the present of harmonic resonance

make the gradient search method very difficult to produceglobal optimal results. Dynamic programming (DP) is asuitable method of shunt capacitor placement, but it requireslong calculations.

In order to limit the heavy burden of calculation associatedwith DP method, some type of sensitivity analysis could beused to sort system buses before applying any capacitor unitTherefore, MSS method works quite well for capacitor placement but it dos not have the precise accuracy of DPtechnique.

This paper reformulates the capacitor placement and sizingproblem, taking into account fixed and switched capacitors aswell as potential harmonic interactions such as harmonic

losses, harmonic resonance and harmonic distortion factorsThe proposed algorithm is implemented using HARMFLOWcodes [14] and results are presented for the distorted 18-busIEEE system. The advantage of this algorithm as comparedwith [11] is consideration of harmonic couplings and reactionsof actual nonlinear loads in the distribution system.

2- System Model at Harmonic FrequenciesFor modeling distribution system at fundamental and harmonicfrequencies the formulation and notations of reference [14] are

0-7803-7519-X/02/$17.00 2002 IEEE807


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used. System solution is achieved by forcing total(fundamental and harmonic) mismatch active and reactive powers as well as mismatch active and reactive fundamentaland harmonic currents to zero using Newton-Raphson method.

Define bus 1 to be the conventional swing bus, buses 2through m-1 to be the conventional linear (PQ and PV) buses,and buses m through n as nonlinear buses (n = total number ofbuses). We assume nonlinear load models are given either in

frequency domain (e.g., )h(V and )h(I characteristics) or in time

domain (e.g., v(t) and i(t) characteristics). These models areavailable for many nonlinear loads and systems such asdischarge lighting [13], power electronic devices [12],nonlinear transformers [14], EHV and HVDC networks [15-16]. The Newton-Raphson method [12] is used to compute the

correction terms, ,U by forcing the appropriate mismatchesto zero:

UJM = (1)for the harmonic power flow analysis, we have (L = maximum

harmonic order considered):

=

)L(

)(

)(

)(

)()L,(),(),L(

)L()L,L(),L(),L(

)()L,(),(),(

)()L,(),(),(

)L()()(

)(

)L(

)(

)(

V

V

V

V

HYGYGYG

HYGYGYG

HYGYGYG

HYGYGYG

JJJ

I

I

I

I

W

M

r

r

L

L

MMOMM

L

L

oL

M

7

5

1

11511

51

775717

555515

51

1

7

5

(2)

where subvectors and submatrices are defined in [12, 14].In the above formulation of harmonic power flow, we

assume capacitor banks are shunt capacitors with variablereactances and capacitor placement is possible for MC number

of buses. The proposed algorithm will determine optimalnumber, types (e.g., fixed or switching), locations and sizes ofcapacitor banks in the presence of nonlinear loads.

3. Problem FormulationIn this paper, the following assumptions are made:

discrete loads at LLM different levels ),LLM....,,,k( 21=

two types of capacitors; fixed capacitors )C( f that are in

service at all load levels and switched capacitors )C( s that

are switched on or off according to load levels,

presence of linear and nonlinear load in a balanced threephase system.

Under these assumptions, the problem can be decomposed intotwo isolated sets of subproblems. This decomposition makessolution much easier to resolve.

Set of Subproblems AThis set includes the capacitor placement problem for load

levels .LLM....,,,k 121 = At load level 1=k , only fixedcapacitors are considered, while the results at load levels

2k consists of fixed and switched capacitors. There are 1LLM subproblems in this set and each could be

optimized independently with its own objective function. Theresult of each subproblem (e.g., optimal sizes and locations of

fC and sC ) are used as the initial condition for the next

subproblem. For the first load level, initial capacitor sizes areassumed zero.

Subproblem BThis subproblem consists of a single problem. That is theoptimal placement and sizing of capacitor banks at the last

load level ( LLMk= ). Results of subproblem A are used asthe initial condition.

Objective FunctionsVoltage constrains will be taken into account by specifiedupper and lower bounds of rms magnitude voltage

max

ih

)h(

k,i

min

i VVV (3)The distortion factor constraints of voltage is considered byspecifying maximum Total Harmonic Distortion (THD) of busvoltages

n...,,j,THDVVTHD maxj)(

k,jh

)h(

k,jj 11001

1== (4)

Bounds of Eqs(3-4) are specified by IEEE-519 standard [17].These bounds constitute a set of functional inequalityconstrains of the from

o)C,V,...,V(H k)h(k)(

kk 1

(5)

Based on Eqs(3-5), the problem of shunt capacitor placementand sizing under nonsinusoidal operating conditions can beexpressed as

++=

oHtosubject

FFFFmin

k

tcoscapacityloss(6)

where lossF is the energy loss cost, capacityF is the costcorresponding to peak active losses (e.g., used capacity of

system), and tcosF is the cost of fixed and switched capacitors.

Therefore, the objective functions corresponding tosubproblems A and B can be defined as Eqs(7A) and (7B),respectively:

=

o)C,V,...,V(Htosubject

)C,V,...,V(PFmin

k)L(

k

)(

kk

k)L(

k

)(

kk,lossA

1

1

(7A)

++=

oHtosubject

FFFFmin

k

tcoscapacitylossB(7B)

=

++

+=

MCj MCj

LLM

sjcsfjcfj

LLM)L(

LLM

)(

LLMLLM,lossA

LLM

k

k)L(

k

)(

kk,losskEB

CKCK)C,V,...,V(Pk

)C,V,...,V(PTKF

1

1

1

where

=k,lossP total system active losses at load level K

808


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=)h(kV bus voltage vector at harmonic h [11,14]=L the highest order of harmonics considered

=kC connected (fixed, switched) capacitors at load level K=AK saving per kW for reduction in peak active losses (e.g.,

MWh/$,KA 000120= )

=EK cost per kWh (e.g., MWh/$KE 50= )

=KT duration of load level K=MC set of possible shunt capacitor buses

=cfjK cost per unit of fixed capacitance of size fjC (Table 1)

=csK cost per unit capacitance of switched capacitor (e.g.,

kVar/$.Kcs 350= ).

Total active losses at load level Kcan be computed usingharmonic power flow outputs:

==

= ==

=

)cos(YVV

PP

)h(

ij

)h(

k,j

)h(

k,i

)h(

ij

)h(

k,j

n

i

n

j

)h(

k,i

L

h

L

h

)h(

k,lossk,loss

1 11

1

(8)

where )h(k,iV and

)h(

k,i are magnitude and phase ofthh

harmonic voltage at bus i for load level K, and)h(

ijY and)h(

ij

are magnitude and phase ofthh harmonic line admittance

between buses i and j, respectively.

Table 1. The yearly cost of fixed capacitors [11]

]kVar[Qj 300 600 900 1200 1500

]kVar/[$Kcfj 0.350 0.220 0.183 0.204 0.302

Sensitivity FunctionsSensitivities of objective function (Eq(7)) to capacitancevalues at compensation buses can be computed using partialderivatives:

=

=

L

h)h(

)h(

k,loss

h

k,loss

Q

PW

dQ

dP

1

(9A)

j)j(

)j(

tcos

)j(

tcos

j

tcos

QQ

FF

dQ

dF

=

+

+

1

1

(9B)

where h1Wh = is the weight function of harmonic h,)h(

kP is

the total system active losses at harmonic h for load level

Kand)j(

tcosF1+

is the cost corresponding to capacitor of size

)j(Q 1+ . Note the application of linear interpolation in Eq(9B)

due to discrete nature of capacitor cost (Table 1). Partial

derivatives of)h(

k,lossP are computed as follows:

=

)h(

)h(

k,loss

)h(

)h(

k,loss

)h(

)h(

)h(

)h(

)h(

)h(

)h(

)h(

)h(

)h(

k,loss

)h(

)h(

k,loss

V

P

P

V

Q

V

P

QP

Q

PP

P1

(10)

where all right hand side entries are computed using outputs ofharmonic power flow.

4. Solution MethodologyThe shunt capacitor placement and sizing problem in the presence of linear and nonlinear loads is an optimizatio problem with discrete variables (e.g., discrete values ocapacitors). This problem is solved using harmonic power flowalgorithm and the maximum sensitivities selection (MSS)method as follows:Step 1: Input system parameters (e.g., line and loadspecifications, system topology, and number of load levels)

Select the first load level (e.g., 1=k ) and set initial capacitorvalues to all possible compensation nodes. Use zero capacitor

values for load level 1=k and previous values for 2k .Step 2: Calculate harmonic power flow (Eqs(1-2)) at load

level kand compute the appropriate objective function (e.g.,Eq(7A) for subproblems A and Eq(7B) for subproblem B).Step3: Calculate sensitivities of objective function tocapacitances at compensation buses (Eqs(9-10)).Step 4: Select several candidate buses with maximumsensitivities.Step 5: Add one unit of capacitor to one of the candidate buses. Calculate harmonic power flow and the relateobjective function. Remove the newly added capacitor unitfrom the corresponding candidate bus.Step 6: Repeat Step 5 for all candidate buses.Step 7: Among all candidate buses, select the one withminimum objective function and place one unit of capacitor atthis bus. Record the corresponding system topology, F, THDand Vrms.Step 8: If the total capacitance of shunt capacitors does notexceed the sum of reactive loads, go to Step 3.Step 9: According to the recorded information (objectivefunctions, THD and Vrms values), select the topology withminimum objective function that satisfies problem constrains

(Eq(5)). This is the optimal solution at load level k. If suchtopology does not exist (no solution at load level

k), go to

Step 11.Step 10: Select the next load level. If the final load level is notreached, go to Step 2.Step 11: Print the solution and stop.

Figure 1 shows the iterative algorithm for capacitor placementand sizing which consists of solving "LLM" 1 subproblem oftype A and one subproblem of type B.

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Figure 1. The proposed iterative algorithm for optimal placement and sizing of capacitor banks under nonsinusoidal operating conditions

using harmonic power flow and MSS method.

Input System Parameters

Set initial capacitance=0, set initial load level; K=1

Set Objective Function

F=Eq(7A), if K=LLM then F=Eq(7B)

HARMFLOW, Eqs(1-2)

Compute sensitivities, Eqs(9-10)

Temporary Bus Sorting

Select MC candidate buses

Set temporary counter; i=1

next candidate bus

i=i+1i = MC

Change System Topology

Add one capacitor unit at the best candidate busRecord information (topology, F, THD, Vrms)

< LC QQ

Optimal Topology Selection

Choose topology with lowest F and constrains; Eq(6)

topology exist?No solution

for problem

Print solution for load level K

Select Next Load Level

K=K+1

K > LLM

Stop

Select

subproblemA

orB

Bussortingby

sensitivities

Temporary Change of System Topology

Add one capacitor unit to bus i

Run HARMFLOW

Temporary record of i and F

Com

puteFfor

candidatebuses

Selectoptimal

solutionwith

noconstrains

Selectoptimal

solutionwith

constrains

Printdataand

changeloadlevel

MSSmethod

no

yes

yes

no

yes

no

no

yes

810


5/7

Convergence of the Proposed AlgorithmThe burden of calculation in the algorithm of Fig.1 is muchless than the analysis and DP methods of references [1-10].

AssumingcN number of capacitor units and cK number of

compensation buses, the average number of power flow

calculations in DP algorithm is )N)(K(K ccc 11 ++ . Total

number of power flow calculations of Fig.1 isccKN which is

much less than DP method, specially when cN is large.

Under sinusoidal operating conditions, algorithm of Fig.1gives global optimal since power loss is an unique peakfunction of capacitance of the shunt capacitors at various busesof distribution system [11].

When capacitor banks are added under nonsinusoidaloperating conditions, harmonic resonance may occur andincrease transmission losses. With further increase of capacitorbank value, the harmonic resonance condition diminishes. Thiswill not affect the proposed method in locating global optimal,since increasing and decreasing of the objective function is not

used as the criterion for convergence [11].

5. Simulation ResultsThe proposed method of loss reduction by capacitor placementin the present of voltage and current harmonics was tested onthe 23 kV, 18-bus, distorted IEEE distribution system (Fig.1).Specifications of this system are given in reference [18-20].Simulations results are shown in Table 2 and Fig.3 for thefollowing three operating conditions:

Case 1: High Harmonic Distortion The nonlinear loadin Fig.1 is a six pulse rectifier with active and reactive powersof 0.3 pu (3 MW) and 0.226 pu (2.26 MVAR), respectively.Outputs of harmonic power flow show a maximum voltage

THD of 8.486% for this system. The algorithm of Fig.1 wasapplied to this system for optimal placement and sizing ofcapacitor banks. Results show a yearly benefit of 20360 dollarsper year (last row of Table 2) and different locations and sizesof capacitor banks before (column 2 of Table 2) and after(column 3 of Table 2) optimization. In addition, maximumvoltage THD is limited to 6.37% (row 19 of Table 2 andFig.3). For this system, optimal capacitor placement results inconsiderable yearly benefit but it does not limit voltage THDto the desired level of 5%. This is expected for rich harmonicconfigurations where capacitor placement is not the primarysolution for harmonic mitigation. For such systemsapplications of passive filters, active filters or Active PowerLine Conditioners (APLC) before capacitor placement arerecommended [19-20].

Case 2: Normal Harmonic Distortion The size ofnonlinear load is adjusted such that its active and reactive powers are decreased to 0.12 pu and 0.166 pu, respectivelyAfter optimal placement and sizing of capacitor banks, voltage

THD is decreases from 5.18% to 5.0% and a yearly benefit of11650.6 dollars is achieved (columns 4 and 5 of Table 2 andFig.3).

Case 3: Low Harmonic Distortion The size ofnonlinear load is further decrease such that its active andreactive powers are limited to 0.02 pu and 0.046 pu,respectively. After optimal placement and sizing of capacitorbanks, voltage THD increases from 2.1% to 2.95% (which islower that the desired limit of 5%), total capacitance ofcapacitor banks are decreased from 1.005 pu to 0.63 pu and ayearly benefit of 12608 dollars is achieved (columns 6 and 7 ofTable 2 and Fig.3).

Figure 2. Single-line diagram of the 18-bus IEEE distorted system [18-20] used for simulation and analysis.

8

7

6 5

Six-PulseConverter

4 3 2

9

1Substation

5051

20

21

2223

25

26

24

Swing Bus

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(a) (b)

Figure 3. Simulated results of Fig.2; (a) yearly benefit after optimal sizing and placement of capacitor banks, (b) maximum voltage THDbefore and after optimal capacitor placement.

Table 2. Simulation results for the 18-bus, distorted IEEE distribution system (Fig.2) at different distortion levels (per unit VA = 10 MVA,

per unit V = 23 kV, swing bus voltage = 1.05 pu).

Case 1

(High Distortion)

Case 2

(Normal Distortion)

Case 3

(Low Distortion)Case

Number BeforeOptimizing

AfterOptimizing

BeforeOptimizing

AfterOptimizing

BeforeOptimizing

AfterOptimizing

Q2 = 0.105 Q2 = 0.030 Q2 = 0.105 Q2 = 0.120 Q2 = 0.105 Q2 = 0.090

Q3 = 0.060 Q3 = 0.090 Q3 = 0.060 Q3 = 0.060 Q3 = 0.060

Q4 = 0.060 Q4 = 0.180 Q4 = 0.060 Q4 = 0.060 Q4 = 0.060 Q4 = 0.060Q5 = 0.180 Q5 = 0.240 Q5 = 0.180 Q5 = 0.180 Q5 = 0.180 Q5 = 0.180

Q7 = 0.060 Q7 = 0.120 Q7 = 0.060 Q7 = 0.060 Q7 = 0.060 Q7 = 0.060

Q20= 0.060 Q20= 0.090 Q20= 0.060 Q20= 0.060 Q20= 0.060 Q20= 0.030

Q21= 0.120 Q21= 0.120 Q21= 0.120 Q21= 0.120 Q21= 0.120

Q24= 0.150 Q24= 0.150 Q24= 0.150 Q24= 0.150 Q24= 0.180

Q25= 0.090 Q25= 0.030 Q25= 0.090 Q25= 0.090 Q25= 0.030

Q50= 0.120 Q50= 0.120 Q50= 0.030 Q50= 0.120

Q1 = 0.090

Q8 = 0.060

Capacitor

Bank

Locations

Q9 = 0.030

Total Capacitor [pu] Qt = 1.005 Qt = 0.900 Qt = 1.005 Qt = 1.020 Qt = 1.005 Qt = 0.630

Minim Voltage [pu] 1.029 1.016 1.044 1.048 1.050 1.020

Maxim Voltage [pu] 1.055 1.056 1.062 1.074 1.073 1.050

Maximum THD [%] 8.486 6.370 5.182 5.000 2.100 2.950

Losses [kW] 282.93 246.43 213.69 192.81 189.91 167.32

Capacitor Cost [$] 2432.70 1959.90 2432.70 2963.40 2432.70 1754.70

Total Cost [$ / year] 157872.98 137512.00 119237.90 107587.47 105969.20 93361.15

Benefits [$ / year] 20360 11650.43 12608

20350

11650.412608

0

5000

10000

15000

20000

25000

Case1 Case2 Case3

Benefits[$/year]

8.486%

5.182%

2.1%

5%

2.95%

6.37%

0.000%

2.000%

4.000%

6.000%

8.000%

10.000%

Case1 Case2 Case3

befor optimization

after optimization

THD(%

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6. ConclusionsMaximum Sensitivities Selection (MSS) is used for thediscrete optimization problem of fixed and switched shuntcapacitor placement and sizing under nonsinusoidal operatingconditions with different load levels. In order to reduceproblem complexity and improve convergence, the problem isdecomposed into several isolated subproblems. Saving due toreduction in peak loss, saving of the energy loss cost due toinstalled capacitors and cost of fixed and switched capacitorsare considered in the objective function. The power qualitylimits of IEEE-519 standard are used as constrains. Based onanalysis of this paper, the following conclusions are stated:

Compared with the local variation (DP), MSS methodrequires less computational time specially when manycompensation buses for capacitor placement areconsidered.

The proposed algorithm takes into account the interactionof harmonics and generates optimal locations and sizes ofcapacitor banks under nonsinusoidal (e.g., nonlinearloads) operating conditions.

Different harmonic voltages and various linear andnonlinear load levels correspond to different optimalcapacitor sizes (Table 2). Therefore, harmonic effectsshould be considered in system planning as well as systemoperation stages.

Simulation results for the distorted 18-bus IEEEdistribution system show that proper placement and sizingof capacitor banks result in considerable yearly benefit(Fig.3) and prevent undesired harmonic resonanceconditions.

For rich harmonic systems (Case1 of Table 2), optimalplacement of capacitor banks can not limit system THD tostandard levels of IEEE-519. For these systems,

application of passive filters, active filters or APLCsbefore capacitor placement are recommended.

7. References[1]. J.J. Grainger and S.H. Lee, Optimal Size and Location ofShunt Capacitors for Reduction of Losses in DistributorFeeders, IEEE Trans. on PAS, Vol. PAS-100, No.3, PP.1105-1118, 1981.[2]. J.J. Grainger and S.H. Lee, Capacity Release by ShuntCapacitor Placement on Distribution Feeders: a new VoltageDependent Model, IEEE Trans. on PAS, Vol. PAS-101, No.5,PP. 1236-1244, 1982.[3] P.K.S.P. Ponnavaikko, Optimal Choice of Fixed and

Switched Capacitors on Radial Distributors by the Method ofLocal Variations, IEEE Trans. on PAS, Vol. PAS-102, No.6,PP.1607-1615, 1983.[4]. S. Civanlar and J.J. Grainger, Volt/Var Control onDistribution Systems with Lateral Branches using ShuntCapacitors and Voltage Regulators: Parts I, II, III, IEEETrans.on PAS, Vol. PAS-104, No.11, PP. 3278-97, 1985.[5]. M.F. Baran and F.F. Wu, Optimal Capacitor Placementon Radial Distribution Systems, IEEE Trans. on PowerDelivery, Vol. PD-4, No.1, PP.725-743, 1989.

[6]. Y. Baghzouz and S. Ertem, Shunt Capacitor Sizing forRadial Distribution Feeders with Distorted SubstationVoltage IEEE Trans. on Power Delivery, Vol. PD-5, No.2,PP. 650-657, 1990.[7]. M.H. Hague, Capacitor Placement in Radial DistributionSystems for Loss Reduction, IEE Proc. Gener. Transm. Dist.,Vol. 146, No.5, PP. 501-505, 1999.[8]. H.D. Chiang, J.C. Wang, O. Cocking and H.D. Shin,Optimal Capacitor Placements in Distribution Systems: Parts1 and 2, TEEE Trans. on Power Delivery, Vol. PD-5, No.2,PP. 634-649, 1990.[9]. Y. Baghzouz, Effects of Nonlinear Loads on OptimalCapacitor Placement in Radial Feeders, IEEE Trans. onPower Delivery, Vol. PD-6, No.1, PP.245-251, 1991.[10]. D.T. Rizy, E.W. Gunther and M.F. Mc Granaghan,Transient and Harmonic Voltages Associated with AutomatedCapacitor Switching on Distribution Systems, IEEE Trans. onPower Delivery, Vol. PAS-2, No.3, PP. 713-723, 1987.[11]. Z.Q. Wu and K.L. Lo, Optimal Choice of Fixed and

Switched Capacitors in Radial Distributions with DistortedSubstation Voltage, IEE proc. Gener. Trans. Distrib, Vol.142No.1, PP. 24-28, 1995.[12]. D. Xia and G.T. Heydt, Harmonic Power Flow StudiesParts I and II, IEEE Trans. on PAS, Vol. PAS-100, No.6, PP1257-1270, 1982.[13]. W.M. Grady and G.T. Heydt, Prediction of PowerSystem Harmonics due to Gaseous Discharge Lighting, IEEETrans. on PAS, Vol. PAS-104, No.1, PP. 558-561, 1985.[14]. M.A.S. Masoum and E.F. Fuchs, TransformerMagnetizing Current and Iron Core Losses in Harmonic PowerFlow, IEEE Trans. on Power Delivery, Vol. 9, No.1, PP. 10-20, 1994.

[15]. A.H. El-Abiad and D.C. Tarsi, Load Flow Solution ofUntransposed EHV Networks, Proceeding of 5 th PowerIndustry and Computer Applications Conference, Pittsburgh,pp. 377-384, 1967.[16]. W. Song, G.T. Heydt and W.M. Grady, The Integrationof HVDC Subsystems into Harmonic Power Flow Algorithm,IEEE Trans. on PAS, Vol. PAS-103, No.8, PP. 1953-1961,1984.[17]. IEEE Recommended Practices and Requirements forHarmonic Control in Electric Power Systems, IEEE std 519-1992, New York, NY, 1993.[18]. W.M. Grady, M.J. Samotyj and A.H. Noyola, TheApplication of Network Objective Functions for Actively

Minimizing the Impact of Voltage Harmonics in PowerSystems, IEEE Trans. On Power Delivery, Vol.7, No.3, 1992.[19]. M.A.S. Masoum and A. Asadi Zareh, Decreasing PowerSystem Harmonics by APLC, 13th International PowerSystem Conference, pp. 489-500, PSC 1998.[20]. M.A.S. Masoum and H. Sajadian, Optimal Placement ofAPLCs with Limited Current Rating for Decreasing PowerSystem Harmonics, the 8th Iranian Conference on ElectricalEngineering, pp. 625-634, ICEE 2000.

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