International School on Nonsinusoidal Currents and Compensation LagoH', Poland, 2008

Complete Harmonic-Domain Modeling and PerformanceEvaluation of an Optimal-PWM-Modulated STATCOM in a

Realistic Distribution Network

H. Valizadeh Haghi l and M. Tavakoli Bina l ,*, Senior Member, IEEE

1 Faculty of Electrical Engineering, K. N. Toosi University of Technology* Dr M. Tavakoli Bina, Seid Khandan, P. O. Box 16315-1355, Tehran 16314-Iran

Tel: 98(21 )88462174-7, Fax: 98(21 )88462066, E-mail: tavakoli@kntu.ac.ir

Abstract-Power systems use STATCOM for compensatingpurposes that is subjected to the high switching frequencies.Various PWM techniques make selective harmonicelimination possible, which effectively control the harmoniccontent of voltage source inverters. On the other hand,distribution systems have to supply unbalanced nonlinearloads, transferring oscillations to the DC-side of theconverter in a realistic operating condition. Thus, additionaluncharacteristic harmonics are modulated through theSTATCOM at the point of common coupling (PCC). Thisrequires more attention when switching angles arecalculated offline using the optimal-PWM technique. Thispaper suggests a harmonic-domain model in order torealistically evaluate the injected harmonics at the PCC.This model properly takes into account the DC capacitoreffect, effects of other possible varying parameters such asvoltage unbalance as well as network harmonics, and effectsof operating conditions on the STATCOM harmonicperformance. The model is programmed, and can be easilyrun with other algorithms such as Monte Carlo sim ulation.Further, a semi-stochastic method is proposed to predictand simulate the three-phase voltage unbalance, leading toan analytical tool for prediction of harmonic performance ofSTATCOM. The predictive method is developed based onthe measured data obtained from a low-voltage distributionnetwork. Finally, the modeled STATCOM is linked with thedistribution substation, applying the proposed voltageunbalance modeling to evaluate aggregate harmonics of theload.

I. INTRODUCTION

A STATCOM consists of a voltage source converter(VSC), a dc capacitor and a coupling transformer. It maybe seen as a basic building block with which severalpower system objectives can be achieved. A variety ofapplications include improving power quality problems inlow-voltage distribution networks, voltage regulation andreactive power control. Reference signals, containingthese objectives, are then modulated by PWM switchingfrequencies that are much higher than the synchronousfrequency.

In practice, as with any VSC-based applications,STATCOM will act as a source of producing harmonicsfor power systems. It could also interact with possibleharmonic distortions and unbalances of the power network(e.g. distributed systems are coined with many powerelectronic loads). These interactions would be complex,making the analysis of steady-state harmonic levels andfull assessment of their dynamic behavior challengingtasks. This is necessary since STATCOM can be betterdesigned, and power quality problems are more efficientlytreated. Thus, it is required to pursue evaluation of these

978-1-4244-2130-5/08/$25.00 ©2008 IEEE

harmonic interactions through a suitable combination ofpractical measurements and analytical studies [1], [2].

Various modulation techniques and topologies aresuggested for STATCOM to effectively remove thegenerated low-order characteristic harmonics, includingmulti-module PWM techniques, selective harmonicelimination and multilevel topologies [3], [4]. An optimalpulse-width modulation (OPWM) uses pre-calculatedswitching angles based on assuming an ideal fixed DC busvoltage. This method presents several advantages incomparison to the conventional carrier-based sinusoidalPWM schemes [5].

On the other hand, load-terminal harmonics andunbalance of distribution systems impose distortion onboth DC and AC sides [2, 12], introducing additionaluncharacteristic harmonics generated by STATCOM.Considering the OPWM, the pre-calculated choppingangles will not then be optimal under these conditions.Hence, the amount of uncharacteristic harmonics that isinjected to a distribution system depends on severalfactors such as ratings and operating conditions of bothSTATCOM and distribution network. Therefore, thefollowing steps should be performed to obtain a morereliable harmonic performance evaluation of STATCOM:

• Developing a comprehensive model which takesinto account relevant factors and interactions in harmonicgeneration such as DC capacitor oscillations, capacitancelimitation, and unbalance of the grid system. This modelwill provide accurate and efficient analytical capability.

• Simulating a realistic distribution system underunbalance and distorted situation. The environment shouldbe also capable of integrating deterministic and stochasticapproaches, if necessary.

• Making the simulation components efficient andfast to avoid unnecessary iterative procedures that slowdown complementary simulations such as Monte Carlomethod.

The first step is a challenging topic that has beendevoted a noticeable amount of research work over thelast few years [6-14]. A detailed overview of harmonicmodeling methods can be found in [15]-[17]. Generallyspeaking, there are three philosophies in modeling ofdevices, network and their interactions. The methods areembedded in time-domain, frequency-domain, andharmonic-domain. The time-domain formulation consistsof relevant differential equations representing the dynamicbehavior of the interconnected power system components.The resultant set of equations is normally solved usingnumerical methods as done in typical simulating software

International School on Nonsinusoidal Currents and Compensation Lagow, Poland, 2008

+

C

Coupling TransKmrer

[>A ... a5 7t-a5 ... 7t-a l

0.2>--

Figure 1: Three-phase voltage source converter connected to the powersystem through a transformer.

such as PSCAD/EMTDC, PSpice, and SABER. Harmonicinformation is then obtained using the fast Fouriertransform (FFT) in steady state [2]. This requiresconsiderable computation steps even for relatively smallsystems [18]. Other problems attached to the time-domainalgorithms for harmonic. studies are the difficulty ofdescribing distributed or frequency-dependent parameters[15], their inflexibility related to the advanced stochasticalgorithms, and the needed delay to converge steady state.

Meanwhile, a model is suggested in [10, 11] based onthe switching function, introducing a direct solution of thesteady state in a closed form. The presented algorithmrequires a periodic steady state to be quantified in thetime-domain. Thus, this constraint implies similarsituation when FFT analysis is performed on the resultingwaveforms from the time-domain simulations. Also, non­linearity cannot be well-performed by the closed-formsolution of the steady state.

Another direct solution for the effect of an individualharmonic (or frequency) is discussed in [15] in thefrequency-domain, excluding the harmonic interactionbetween the network and the nonlinear compensator.When the harmonic injection from each source dependson those of other sources along with the state variables ofthe system, accurate results can only be obtained inharmonic-domain. The harmonic-domain is a restrictedfrequency-domain, while all non-linear interactions aremodeled [19]. While some research works take the effectsof control system into account [7, 14] in the harmonic­domain, another method works on harmonic power flow[6]. These are not relevant to evaluation of steady stateharmonic performance of STATCOM. In [8], modeling ofSTATCOM is carried out by representing switchingfunctions by their Fourier spectra. This approachpotentially offers an accurate solution, considering allharmonics and their interactions. This model is extendedand improved in this paper using harmonic-domaintechniques.

This paper introduces a more realistic evaluation ofharmonic penetration of STATCOM connected across adistribution substation. The structure and modeling ofSTATCOM for harmonic analysis is presented, whereharmonic performance of STATCOM using the OPWM isinvestigated by means of a harmonic-domain model. Adetailed semi-stochastic modeling is then proposed at thePCC for the voltage unbalance. The suggested techniquesare extensions of the proposed methods in [20, 21]. Then,this realistic representation of the PCC is applied to themodel of STATCOM. Resultant outcomes show themaximum uncharacteristic harmonic injection by theOPWM-STATCOM. This can be easily performed withother modulation techniques under a realistic operating

978-1-4244-2130-5/08/$25.00 ©2008 IEEE 2

B(rad)

-

o1_- lJ- - ---.L.lI_--I-'-.J...I • ~ -----.J

5 10 15 20 25 30 35 40 45 50Harmonic Order

Fi~ure 2: A typical selective harmonic elimination (5th, 7th

, 11 th, and13 1) based on quarter-cycle symmetry: the waveform and its harmonicspectra.

condition. Further, a real case study is arranged in whichthe voltage unbalance and background harmonics aresimulated at a distribution substation. This substation ischosen from part of Tehran north-east distribution system.Also, a STATCOM will be connected across this realsubstation. It is eventually examined uncharacteristicharmonics produced by STATCOM that is subjected tothe interaction with the realistic model of the PCC. Theobtained conclusions are finally discussed and evaluated.

II. STATCOM STRUCTURE AND MODELING

Here it is presented briefly the OPWM, followed by aproposal on modeling of STATCOM for harmonicanalysis and performance evaluation.

A. The OPWM modulation technique

A three-phase VSC, using IGBT switches, is shown inFig. 1. Like other PWM schemes, the DC-link voltage ismodulated by the converter using the programmedharmonic elimination technique, Le. the OPWM. Bothswitching instants and durations are pre-calculated off-linesuch that certain chosen harmonics are eliminated; also, adesired value is assigned to the fundamental component ofthe output. A typical waveform is shown in Fig. 2 inwhich it is programmed to eliminate non-triple oddharmonics up to 17. The quarter-wave symmetry iscommonly used to reduce the complexity of the resultingequations. A general technique is also proposed withoutthis assumption in [22]. Normally, a set of solutions for acertain range of fundamental voltage magnitudes is pre­calculated and stored in a look-up table for on-lineimplementations. Detailed discussions on implementingthe OPWM can be found in [23].

B. STATCOM Modeling for Harmonic Analysis

Frequency-dependent approximate representation ofAC/DC converters already used for small levels ofdistortion using transfer function approach [15].Approximations can be avoided by modeling the converterin the time-domain in the expense of the solution speed. Itis noticeable, however, that advanced probabilistic

International School on Nonsinusoidal Currents and Compensation Lagow, Poland, 2008

Figure 4: STATCOM equivalent impedance matrix structure implyingthe phases a, band e components cross coupling for harmonics biggerthan 0.02 P.D. up to 50th

.

and Zeap is the DC-link impedance. Then, the followingrelationship can be established based on (4):

(5)

ba

a

PCC

analysis and Monte Carlo simulation needs fast andflexible procedures.

Hence, building on the work of [8], this paper proposesa Thevenin equivalent model for STATCOM in theharmonic-domain. This model, shown in Fig. 3, provides afast procedure with sufficient accuracy in addition to theflexibility in programming capabilities. Considering apulse train for the converter's switches, assume Sa, Sb andSe are the three-phase switching functions. Threecombinations Sab= Sa-Sb, Sbe= Sb-Se and Sea= Se-Sa aredefined based on the three switching functions of thePWM converter (e.g. see Fig. 2). Then, the followingequations can be obtained in harmonic-domain:

Figure 3: Suggested three-phase modeling for a STATCOM in theharmonic-domain that is connected to adistribution substation.

Where the converter outputs Yam Vbm V en are the phasevoltages, all represented as complex harmonic vectors,and Sab, Sbe, Sea are convolution matrices constituted ofthe harmonic coefficients of the Sab, Sbe and Sea at a certainharmonic. Also, Vde is the harmonic-domain vector of theDC-side voltage. Considering Fig. 1, the DC current canbe introduced as:

[Van] [ 1 0VdeV = Vbn = -3- -1 1

Ven 0 -1

-1][Sab]o Sbe1 Sea

(1)Substituting lile and V de from (3) and (5) in (1), both the

Thevenin equivalent voltage (Et,,) and impedance (Zt") ofthe converter can be represented as below (shown in Fig. 3in which v,,, is the resultant equivalent converter outputvoltage):

V=Zthl+Eth (6)

Zth = st Zeap S , E th = [Etha Ethb Ethe ] t = E de S

S = .!.[Sab -Sea Sbe -Sab Sea -Sber (6a)3

An integral formula relates both DC quantities in the time­domain (see Fig. 1) as follows:

Assuming the DC-side harmonics are originated fromharmonic content of the ide(t), Ede is the pure DC voltage

III. HARMONIC BEHAVIOR OF STATCOM

The harmonic performance of STATCOM can now beestablished using the developed model in (6) under idealoperating conditions. Also, the OPWM scheme assumesideal switching transitions (having no tum-on or tum-offdelays). Further, assume the balanced system parametersare Iv~1 = 1 P.U., Zs= 0.0043 + jO.1561 P.U., Zt= 0.012 +jO.5612 P.U. at 50 Hz. Simulating the presented model of(6) results in the voltage and current waveforms shown inFig. 5 (Harmonics up to the 100th are considered).

In fact, the fundamental and harmonic contents areobtained directly from the equivalent circuit of Fig. 1.Then, the time-domain waveforms in Fig. 5 are analyzedthrough an inverse Fourier transform. This is different

This is the converter model of STATCOM that isconnected through an equivalent inductance to the PCC.The phase components cross coupling structure of the Zthhas been plotted in Fig. 4 for harmonic interactions up tothe 50th harmonic that are bigger than 0.02 P.U. It isnoticeable that the matrix excludes control systemtransfer elements because of steady state analysis (allsub-matrices are diagonal).

(4)

(2)

t

V de (I ) = ~ Jide (I )dl +v de (0+) .

o

Where converter currents la' Ib and Ie are the complexharmonic vectors of the line currents, and Ide is theharmonic-domain vector of the DC-side voltage. Equation(2) could also be presented in matrix form of

978-1-4244-2130-5/08/$25.00 (g2008 IEEE 3

International School on Nonsinusoidal Currents and Compensation Lagow, Poland, 2008

120

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100

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TypeC

604020

In\erter Current Harmonic Spectra

In\erter Phase Voltage Harmonic spectra

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Figure 6: Harmonic spectra of Vtl, and I inv.

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Figure 7: Single-line diagram of the distribution network under study.

6000 8000 10000 12000 14000

One week time period

substation transformer (at 400 V-end), which are obtainedusing a three-phase load flow program in MATLAB.

B. Voltage Unbalance Simulation

The following procedure is suggested to be used herefor prediction and simulating voltage unbalance:

1) Active and reactive powers are split into two parts:probabilistic and deterministic components. This isperformed using the wavelet transform. Also, the analysisis managed separately for two categories of working daysand week-end like those of the load forecastingalgorithms.

2) The approximation parts (obtained from thewavelet transform) are then directly predicted as theaverage daily power curve.

3) The detail parts of the wavelet transform aremodeled using a Gaussian linking function which

Figure 8: Recorded three-phase active and reactive powers by datalogger that is installed at 400 V substation of Mavad.

0.040.015 0.02 0.025 0.03 0.0350.005 0.01

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

-:-~

86

~- -10 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Time (sec)

~>

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Figure 5: Simulations of the model (6) under balanced condition whenSTATCOM supplies 1 P.U. reactive power to the PCC; pictures fromthe top one are the Thevenin voltages (EIII and VII,), the PCC voltage atphase a (Vpcc), and the converter current at phase a (Iinv) all in P.U.

from a normal procedure when a time-domain simulationis taken place. The OPWM is optimized such thatselective harmonics (5th

, 7th, 11 th

, 13th and 17th) are

eliminated. Figure 6 shows the harmonic spectra of theconverter phase voltage and current. Note that theSTATCOM current spectra exclude the third harmonics.

Recently, a detailed study on the harmonic performanceof STATCOM is introduced in [2] using an OPWMswitching pattern under realistic operating conditions.While that method can be examined with the proposedtheoretical modeling in harmonic-domain (see (6)), but amore realistic case is investigated here in which anadditional system examination is performed with regard tothe degrees of unbalance and distortion. This could beuseful when commissioning study of a distribution systemfocuses on the levels of uncharacteristic harmonicproduced by a STATCOM at a distribution substation.

IV. MODELING AND SIMULATION OF SYSTEM

UNBALANCE

Operation of STATCOM under three-phase unbalancedvoltages affects noticeably the penetration ofuncharacteristic harmonics which are not usually targetedby the engaged modulation technique [2]. The properprediction of the level of voltage unbalance could beuseful in applying further measures to the OPWM,influencing the harmonic performance of STATCOM.Here an improved and enhanced version of simulationapproach in [21] is utilized for statistical prediction ofvoltage unbalance.

A. Case study: System Description and Measuements

A data logger is installed at the distribution substationof Mavad that is located in north-east of Tehran. Themeasured data is gathered during a week in September2002. Figure 7 shows single-line diagram of the localdistribution system in which the transformer T3corresponds to the substation of Mavad (20 kV/400 V, 1MVA). Figure 8 demonstrates the recorded active andreactive powers of the three phases at this substation forone week. Also, Fig. 9 introduces phase voltages of the

978-1-4244-2130-5/08/$25.00 ©2008 IEEE 4

International School on Nonsinusoidal Currents and Compensation Lagow, Poland, 2008

(b)One week time period

. -: ,'-I--L ~_.~, ---.1.__ __~__ ____LJ

2000 4000 6000 8000 10000 12000 14000 16000

One week time period

(a)

2-­o

1000

1200

0'-------------1.5 -1 -0.5 0 0.5 1 1.5Calculated probabilistic component of P a (W) X 105

(c)

Figure 10: Application of the wavelet transform to active power, (a)real (blue trace) and approximated (red trace) logged data for one week,(b) the difference between the approximated and the exact data, and (c)the distribution of active power vs. Gaussian distribution function.

certain voltage unbalance percent (see Fig. II(a)). It cannow be studied the effects of voltage unbalance on theharmonic performance of the OPWM-STATCOM. Theproposed model for STATCOM (see (6) along with Fig.3) is linked with the Monte Carlo simulation of thecalculated unbalance percent (VUF%) in Fig. II(b).

Assume the capacitance of the DC-link capacitor ofSTATCOM is I mF. Then, the resultant THD% iscalculated for all data as shown in Fig. 12. Backgroundharmonics are represented by voltage harmonic sourceobtained from field measurements. It can be seen fromFig. 12 that a realistic voltage unbalance would notdramatically modifies the uncharacteristic THD ofSTATCOM AC current (THOS) except for situations thatSTATCOM supplies or absorbs relatively small amountsof reactive power. Figure 12(a) provides a qualitativeevaluation of the proposed method. Also, to validatequantitatively the suggested method, Figs. 12 (b)-(e)compare the probability distributions of the THOS forrealistic VUF% when four different reactive power aresupplied by STATCOM. It can be seen from Figs. 12 (b)­(e) that reactive powers of -0.8 P.D and 0.9 P.U. alongwith realistic VUF% give a relatively small range of theTHDS variations around I% and 0.96%, respectively.

V. STATCOM HARMONIC PERFORMANCE UNDER

UNBALANCE OPERATION

A realistic voltage unbalance case is studied using theforegoing suggestion at a distribution substation. It is alsoshown that the distribution system may operate under

2000 4000 6000 8000 10000 12000

One week time periodFigure 9: Measured phase voltages of the transformer T3 (Mavad)during a week.

generates six correlated random variables correspondingto active and reactive powers of the three phases.

4) Time-domain active and reactive powers (modeledduring the previous steps) are reconstructed by applyingthe inverse wavelet transform.

5) The reconstructed powers are then used to predictthe three-phase voltages through a three-phase load flowprogram using the Monte Carlo simulation. The IECvoltage unbalance factor (VUF) is calculated from theobtained voltages in step 4.

This procedure provides a more realistic insight into theVUF at the PCC. It is also compatible with the MonteCarlo simulation program and statistical evaluations.Wavelet transform presents suitable filteringcharacteristic. This enables the procedure to introduce thepower as a deterministic component (approximation parts)\vith a quite stable mean and standard variation as shownin Fig. 10 (a). Also, the procedure introduces a nearlyGaussian distributed probabilistic component (detail parts)as shown in Figs. 10 (b)-(c). Assuming the Daubechies(db5) mother-wavelet, up to the fourth filtering level isused to determine approximations and details. It isnoticeable that the probabilistic component in Fig. I0 (b)can be modeled using a Gaussian linking function thatdescribes dependencies among variables, providing a wayto correlate multivariate data [24, 25]. In a practicaldistribution system, active and reactive powers of thethree phases show moderate correlation characteristics,which can be modeled using the Gaussian function inlarge networks.

The suggested procedure is applied to the studieddistribution substation (Mavad). Simulations are shown inFig. II over a week. The voltage unbalance percent(VUF%) of the substation of Mavad is calculated at thelow-voltage side (400 V) of the transformer T3 usingMATLAB. Simulations are depicted in Fig. II(a) for aone week period, varying within [0.38%, 1.58%]. Tovalidate the suggested algorithm, the probabilitydistribution functions are obtained from both the proposedunbalance algorithm and field measurements. Both POFsare shown in Fig. II(b). Comparing the exact collecteddata (dotted line) with those of the proposed method (solidline), it can be seen that the suggested algorithm providesan accurate simulation of unbalance variation at the PCC.

978-1-4244-2130-5/08/$25.00 ©2008 IEEE 5

International School on Nonsinusoidal Currents and Compensation Lagow, Poland, 2008

0.60.3

90

80

70

(a)

70

60

...50i409~30o

20

-0.3 0.04Reactive Power (pu)

-0.6

50

250

300 ,..__----r-----r---,---_

[1] 1. H. R. Enslin and P. 1. M. Heskes, "Harmonic interactionbetween a large number of distributed power inverters and thedistribution network," IEEE Trans. Power Electron., vol. 19, pp.1586-1593, Nov. 2004.

[2] S. Filizadeh and A. M. Gole, "Harmonic performance analysis ofan OPWM-controlled STATCOM in network applications," IEEETrans. Power Del., vol. 20, pp. 1001-1008, Apr. 2005.

[3] R. W. Menzies and Y. Zhuang, "Advanced static compensationusing a multi-level GTO thyristor inverter," IEEE Trans. PowerDel., vol. 10, pp. 732-738, Apr. 1995.

[4] L. Ran, L. Holdsworth, and G. A. Purtus, "Dynamic selectiveharmonic elimination of a three-level inverter used for static VArcompensation," Proc. Inst. Elect. Eng., Gen., Transm., Distrib.,vol. 149, pp. 83-89, Jan. 2002.

[5] P. N. Enjeti, P. D. Ziogas, and 1. F. Lindsay, "Programmed PWMtechniques to eliminate harmonics: a critical evaluation," IEEETrans. Ind. Applicat., vol. 26, pp. 302-316, Mar./Apr. 1990.

[6] Y. Sun, G. Zhang, W. Xu, and 1. G. Mayordomo, "A harmonicallycoupled admittance matrix model for ac/dc converters," IEEETrans. PowerSyst., vol. 22, pp. 1574-1582, Nov. 2007.

REFERENCES

90 r--~--.----.------r-.......-- __--.----.------.

8070

60

SO

4030

20

10o094096 098 1

100 ~'

90

80 -

70

60

~0 50:r:E--<

40

30

20

10

0-0.9

~6O

i50~40

°3020

10o

~O 100 150 ~ 10 1~ 20 30(d) TIIDS% (e) TIIDSO,/o

Figure 12: (a) Penetrated THDS due to the 5th, 7th

, 11 th, 13th

, 17th and 19th harmonicsunder various VUF% and reactive power obtained from Monte Carlo Simulation.(b)-(e) distributions of the THDS corresponding to the reactive power loadings of ­0.8 P.U., 0.9 P.U., 0.04 P.U. and -0.1 P.U., respectively.

~200

i~150

°100

.1

I

(a)One week time period

30 ~

I

~ 201

I

VI. CONCLUSION

This paper suggests a comprehensive analysis related tothe penetration level of harmonics by STATCOM intopower systems. Hence, starting with introduction of aharmonic-domain model for STATCOM, an equivalentThevenin circuit is established. This simplifies simulationof the developed model, while it links certain harmonicsto their complex values in a three-dimensionalrepresentation. A case study is arranged in which a datalogger gathers required data from a 20 kV/400 Vdistribution substation located in Tehran for a one weekperiod. Analyzing the exact measured voltage unbalanceof this substation, a probabilistic method is suggested topredict the unbalance percent. Then, the Monte Carlomethod is linked with the STATCOM model. Theresultant model is used to evaluate harmonic performanceof STATCOM under the suggested unbalanced predictionalgorithm. Various simulations are performed to verifY thesuggested model along with the proposed unbalancepredictive algorithm.

I.1a I~ -+ - .~",_.-:••;..rl'~· ~.• - ~

I ,.... I .~·ol

I I ....••• .. ,,..,. .. II.' 7·· e.......

a ~~-~-,---~--- L .L __ ,_L,_ "~ ,~_~.~0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

VUF(%)

(b)Figure 11: (a) The VUF% variation over the one week period at thelow-voltage side of T3 obtained using the proposed simulation method,and (b) comparing the PDFs of VUF% obtained from fieldmeasurements (dotted line) and the proposed algorithm (solid line).

However, for reactive powers equal to 0.04 and -0.1 P.U.along with the same realistic VUF% result in aconsiderable range of the THDS variations around 25%and 15%, respectively. It should also be noted that theTHDS is affected by changing the DC-link capacitance.Also, the power system equivalent impedance influencesthe THDS under variation of the voltage unbalancepercent. This analysis can be easily extended to includeother realistic conditions. It is theoretically possible tomodifY the OPWM procedure to remove uncharacteristicharmonics under unbalanced conditions. However, theanalysis become more complex, it is not purelydeterministic and could be difficult to implement online.

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International School on Nonsinllsoidal Currents and Compensation LagoH', Poland, 2008

[7] A. R. Wood and C. M. Osauskas, "A linear frequency-domainmodel of a STATCOM," IEEE Tram.... Power Del., vol. 19, pp.1410-1418, Jul. 2004.

[8] M. Madrigal and E. Acha, "Modeling of custom power equipmentusing harmonic domain techniques," IEEE, pp. 264-269, 2000.

[9] A. Gole, "Steady state frequency response of STATCOM," IE'E'ETrans. Power Del., vol. 16, pp. 18-23, Jan. 2001.

[10] P. W. Lehn, "Direct harmonic analysis of the voltage sourceconverter," IEEE Tram;. Power Del., vol. 18, pp. 1034-1042, Jul.2003.

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[13] L. 1. G. Lima, A. Semlyen, and M. R. Iravani, "Harmonic domainperiodic steady state modeling of power electronics apparatuses:SVC and TCSC," IEEE Tran,'. Power Del., vol. 18, no. 3, pp.960-967, Jul. 2003.

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[16] IEEE Task Force, "Characteristics and modeling of harmonicsources-power electronic devices," IEEE Trans. Power Del., vol.16, pp. 791-800, Oct. 2001.

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[17] K. W. Louie, P. Wilson, R. A. Rivas, A. Wang, and P. Buchanan,"Discussion on power system harmonic analysis in the frequencydomain," in Proc. 2006 IEEE PES Transm., Distrib. ConI, Exp.Latin America, Vene~uela.

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[19] V. Sharma, R. 1. Fleming, and L. Niekamp, "An iterativeapproach for analysis of harmonic penetration in the powertransmission networks," IEEE Trans. Power Del., vol. 6, no. 4, pp.1698-1706, Oct. 1991.

[20] M. 1. Au and 1. V. Milanovic, "Development of stochasticaggregate harmonic load model based on field measurements,"IEEE' Trans. Power Del., vol. 22, no. 1, pp. 323-330, Jan. 2007.

[21] Y.-1. Wang and L. Pierrat, "A method integrating deterministicand stochastic approaches for the simulation of voltage unbalancein electric power distribution networks," IEEE Trans. Power Syst.,vol. 16, no. 2, pp. 241-245, May. 2001.

[22] 1. R. Wells, P. L. Chapman, and P. 1. Krein, "Generalization ofselective harmonic control/elimination," IEEE, pp. 1358-1362,Apr. 2005.

[23] S. R. Bowes and A Midoun, "Microprocessor implementation ofnew optimal PWM switching strategies," Proc. Inst. Elect. Eng.,vol. 135, pp.269-280, Sep. 1988.

[24] R. B. Nelsen, An Introduction to Copulas, Springer, 2nd Edition,1999.

[25] The Math Works, MATLAB. (2008), http://www.mathworks.com.