Three-phase three-wire series active power filter,which compensates for harmonics and reactivepower

Y.S. Kim, J.S. Kim and S.H. Ko

Abstract: A new control algorithm for a three-phase three-wire series active power filter isproposed. With the proposed control algorithm, the series active power filter compensates for theharmonics and reactive power that are generated by non-linear loads, such as diodes or thyristorrectifiers. The proposed control algorithm is based on the generalised p–q theory. It may be appliedto both harmonic voltage sources and harmonic current sources. In this algorithm, thecompensation voltage references are extracted directly. Therefore, the calculation of thecompensation voltage reference will be much simpler than for other control algorithms. Inaddition, the difficulty of finding the voltage reference gain disappears. The compensation principleof the proposed control algorithm is presented in detail. To verify the effectiveness of the proposedalgorithm, a prototype of a three-phase three-wire series active power filter has been manufacturedand some experiments carried out.

1 Introduction

Recently, the use of semiconductor switching equipment,such as diodes and thyristor rectifiers, has sharply increased.Power quality degradation generally results from these andother non-linear loads. The more non-linear loads increase,the more complex steps are required to avoid power qualitydegradation, such as harmonic increase, power factordegradation etc.Passive filters have traditionally been used to eliminate

harmonic currents, which are generated by nonlinear loads.To eliminate the harmonics in broadband, too manypassive filters would be required. In addition, the hazardof resonance with the source impedance would becomequite difficult to avoid [1].Studies on active power filters began in the late 1970s to

overcome the defects of the passive filter. The active powerfilter is more expensive than the passive, but the former hasan advantage in that it can simultaneously eliminate thebroadband harmonic at the source stage. Active powerfilters are categorised as follows: the parallel active powerfilter, which injects compensation currents [2]; the seriesactive power filter, which injects compensation voltagesthrough a transformer [3]; and the combined system ofparallel passive filters and series active power filter [1, 4, 5].Generally, if the DC smoothing inductor is sufficientlylarge, nearly constant DC current flows in the DC link of arectifier. So this type of load can be called a harmoniccurrent source. The parallel active power filter is suitable forcompensating for these harmonic current sources, while theseries active power filter is appropriate for compensating for

the harmonic voltage source, which has sufficient capaci-tance component in the DC link of the rectifier [6]. Inparticular, the solution for a harmonic voltage source iscritical because the loads that act as harmonic voltagesources, such as copiers, fax machines, fluorescent lamps,air conditioners etc., have continued to increase.In this paper, the proposed control algorithm for series

active power filters is applicable to harmonic voltage sourceloads as well as to harmonic current source loads. Thiscontrol algorithm is applied under the basic concept of thegeneralized p–q theory [7]. However, this generalised p–qtheory is valid for compensating for the harmonics andreactive power using the parallel active power filter in thethree-phase power system. To overcome such limits, arevised p–q theory is proposed. This revised algorithm maybe effective not only for the three-phase three-wire seriesactive power filter with harmonic current voltage loads, butalso for the combined system of parallel passive filters andactive filter.Another drawback of the generalised p–q theory is that

the compensation voltage will be determinated by multi-plying the gain, which is dependent on the value of current.To obtain the current, some computational efforts areneeded using the instantaneous real power and imaginarypower [8]. The proposed control algorithm directly extractscompensation voltage references without multiplying thegain. Therefore, the calculation of the compensation voltagereference will turn out to be simpler than for other controlalgorithms. To verify the effectiveness of the proposedcontrol algorithm, a series active power filter has beenmanufactured and experiments carried out.

2 Principle of compensation

This Section introduces the control algorithm of the seriesactive power filter, which compensates for harmoniccurrents and reactive power.The three-phase voltages va, vb and vc and currents ia, ib

and ic for the three-phase three-wire power distribution

The authors are with the Research Center for Next-generation High Voltageand Power Technology, Inha University, 253 Yong Hyun-Dong, Nam-Gu,Inchon, 402-752, Korea

r IEE, 2004

IEE Proceedings online no. 20040208

doi:10.1049/ip-epa:20040208

Paper first received 5th June 2003 and in revised form 17th December 2003.Originally published online: 19th March 2004

276 IEE Proc.-Electr. Power Appl., Vol. 151, No. 3, May 2004

system in Fig. 1 can be expressed as the space vectors v andi. The three-phase load voltages vL(a,b,c) and the three-phasesource currents is(a,b,c) are represented as:

vLða;b;cÞ ¼vLavLbvLc

24

35; iSða;b;cÞ ¼

iSaiSbiSc

24

35 ð1Þ

The load voltage vector vL(a,b,c) and the source currentvector is(a,b,c) of (1) are transformed into ab0 co-ordinates bythe substituting (3) into (2), as shown in Fig. 2:

vLða;b;0Þ ¼ ½T �vLavLbvLc

24

35 ¼

vLavLbvL0

24

35; iSða;b;0Þ ¼ ½T �

iSaiSbiSc

24

35

¼iSaiSbiS0

24

35 ð2Þ

½T � ¼ffiffiffi2

3

r 1 �1=2 �1=20

ffiffiffi3

p=2 �

ffiffiffi3

p=2

1=ffiffiffi2

p1=

ffiffiffi2

p1=

ffiffiffi2

p

24

35 ð3Þ

The active power p can be expressed as (4) by the innerproduct of the load voltage vector vL(a,b,0) and the sourcecurrent vector is(a,b,0) of (2), where the active power p is theinstantaneous active power at the load side of the CT inFig. 1:

p ¼ vLða;b;0ÞiSða;b;0Þ ¼ vLaiSa þ vLbiSb þ vL0iS0 ð4Þ

Also, the reactive power qL(a,b,0) is represented as (5) by thecross product of vLða;b;0Þ and iSða;b;0Þ:

qLða;b;0Þ ¼ vLða;b;0Þ iSða;b;0Þ ¼qLaqLbqL0

24

35

¼

vLb vL0iSb iS0

��������

vL0 vLaiS0 iSa

��������

vLa vLbiSa iSb

��������

26666664

37777775

ð5Þ

q ¼ qLða;b;0Þ

¼ vLða;b;0Þ isða;b;0Þ ð6Þ

where q is the instantaneous reactive power at the load sideof the CT in Fig. 1.For a three-phase system without zero sequence voltage

and current, i.e. va+vb+vc¼ 0 and ia+ib+ic¼ 0(vL0 ¼ 1

3ðva þ vb þ vcÞ ¼ 0 and iS0 ¼ 1

3ðia þ ib þ icÞ ¼ 0),

(4) and (5) can be expressed as follows:

p ¼ vLða;b;0ÞiSða;b;0Þ ¼ vLaiSa þ vLbiSb ð7Þ

qLða;b;0Þ ¼ vLða;b;0Þ iSða;b;0Þ ¼qLaqLbqL0

24

35

¼

0j j0j j

vLa vLbiSa iSb

��������

2664

3775 ð8Þ

From (1)–(5), the active voltage vector vpða;b;0Þ and thereactive voltage vector vqða;b;0Þ are defined as follows:

vpða;b;0Þ ¼p

iða;b;0Þ � iða;b;0Þiða;b;0Þ ð9Þ

vqða;b;0Þ ¼qða;b;0Þ iða;b;0Þ

iða;b;0Þiða;b;0Þð10Þ

The active voltage vector and the reactive voltage vector canbe obtained by the vector norm of the three-phase loadvoltage vector, which is known from (9), (10) and Fig. 2. Inother words, vpða;b;0Þ represents the parallel component ofthe load voltage vector vLða;b;0Þ to the current vector iSða;b;0Þ;vqða;b;0Þ represents the perpendicular component of the loadvoltage vector vLða;b;0Þ to the current vector iSða;b;0Þ. As aresult, the load voltage vector is represented by the sumof the active voltage vector vpða;b;0Þ and the reactive voltagevector vqða;b;0Þ as follows:

vLða;b;0Þ ¼ vpða;b;0Þ þ vqða;b;0Þ ð11ÞThe active voltage vector vpða;b;0Þ is induced as follows,using the projection of the load voltage vector vLða;b;0Þ ontothe current vector iSða;b;0Þ:

vpða;b;0Þ ¼proji vLða;b;0Þ ¼vLða;b;0ÞiSða;b;0Þ

iSða;b;0Þ 2 iSða;b;0Þ

¼ vLaiSa þ vLbiSb þ vL0iS0i2Sa þ i2Sb þ i2S0

iSða;b;0Þ

¼ pi2Sa þ i2Sb þ i2S0

iSða;b;0Þ

ð12Þ

The reactive voltage vector vqða;b;0Þ, which is perpendicularto the active voltage vector vpða;b;0Þ, is also induced

vsa vLa

vLb

vLc

vsb

isa

isb CT

CT

CT

vsc isc

load

active

power

filter

5th 7th

Fig. 1 Main circuit configuration

a

b

c

iS�vL�

vL�VL

Vq

Vp

is

iS�

�

�

�

�

0

Fig. 2 Vector diagrams of three-phase currents and voltages

IEE Proc.-Electr. Power Appl., Vol. 151, No. 3, May 2004 277

through (13)–(16):

qLða;b;0Þ ¼ vLða;b;0Þ iSða;b;0Þ ð13Þ

iSða;b;0ÞqLða;b;0Þ ¼ iSða;b;0Þ vLða;b;0Þ iSða;b;0Þ� �

¼ iSða;b;0ÞiSða;b;0Þ� �

vLða;b;0Þ � iSða;b;0ÞvLða;b;0Þ� �

iSða;b;0Þ

¼ iSða;b;0Þ 2vLða;b;0Þ � piSða;b;0Þ

ð14Þ

vLða;b;0Þ ¼isða;b;0Þ qLða;b;0Þ

iSða;b;0Þ 2 þ p

isða;b;0Þ 2 iSða;b;0Þ ð15Þ

After taking a cross product on both sides of (13), (14) isobtained when the right side of (13) is unfolded by means ofthe relations of inner and cross product. After transposingthe current vector component of the right-hand side to theleft side in (14), (15) can be obtained. The second term ofthe right-hand side of (15) is the active voltage vectorvpða;b;0Þ and the first term of the right-hand side of (15)becomes the reactive voltage vector vqða;b;0Þ:

vqða;b;0Þ ¼iSða;b;0Þ qLða;b;0Þ

iSða;b;0Þ 2 ¼

iSða;b;0Þ qLða;b;0ÞiSða;b;0ÞiSða;b;0Þ

ð16Þ

where qLða;b;0Þ is equal to the reactive power, which is

defined in the instantaneous reactive power theory. Thevoltage compensation reference of the series active powerfilter can be represented as (17), using vpða;b;0Þ and vqða;b;0Þ in(9) and (10):

v�Cða;b;0Þ ¼~rr

iSða;b;0ÞiSða;b;0ÞiSða;b;0Þ þ

iSða;b;0Þ qLða;b;0ÞiSða;b;0ÞiSða;b;0Þ

ð17Þ

As shown in Table 1, the active power and the reactivepower can be divided into DC components �pp and �qq, whichare generated from the fundamental components of theload voltages and the source currents, and AC components~pp and ~qq, which are generated from the negative sequencecomponents and the harmonic components of the loadvoltages and the source currents. If the reactive power q isreplaced by the AC component of reactive power ~qq, a newvoltage compensation reference compensates for the ACcomponent of the active power ~pp and the reactive power ~qq.In this case, only the harmonic components can becompensated for. As above, the voltage compensationreference, which meets the compensation target, can beobtained from Table 1 and (17). In this paper, experimentsare carried out so that the harmonics and reactive power arecompensated for simultaneously by means of compensation~pp and q.

The compensation voltage reference in ab0 co-ordinatesis obtained from (17) and the final compensation voltagereference by transforming this compensation voltagereference in ab0 co-ordinates into the compensation voltagereference of three-phase co-ordinates. Equation (19) is the

a, b, 0/three-phase transformation matrix:

v�Cða;b;cÞ ¼ T½ ��1v�Cav�Cbv�C0

24

35 ¼

v�Cav�Cbv�Cc

24

35 ð18Þ

T½ ��1¼ 23

1 0 1=2�1=2

ffiffiffi3

p=2 1=2

�1=2 �ffiffiffi3

p=2 1=2

24

35 ð19Þ

The block diagram of the entire control algorithm is shownin Fig. 3. First, three-phase load voltages and sourcecurrents are transformed into ab0 co-ordinates. Then, theactive power and the reactive power can be calculated. TheAC component of the active power ~pp is extracted by simplefiltering. The compensation voltage reference in ab0 co-ordinates is calculated by substituting the obtained ACcomponent of the active power, the reactive power and thethree-phase currents into (17). The final voltage compensa-tion reference for the harmonics and the power factorcompensation are obtained by transforming the voltagecompensation reference in ab0 co-ordinates into the voltagecompensation reference in three-phase co-ordinates. Thedifference between the reference value of the DC-linkvoltage and the monitored DC-link voltage serves as aninput signal to the inverter DC-link voltage controller. Theinverter DC-link voltage can be controlled to follow theinverter DC-link voltage reference V �

dc�inv through a PIcontroller. In Fig. 3, Isaf ; Isbf and Iscf are the fundamentalcomponents of three-phase source currents, which areobtained by lowpass filtering.

3 System configuration

Figure 4 shows the three-phase three-wire series activepower filter and the combined system of the parallel passivefilters and the series active power filter, which were usedin the experiments. In the three-phase three-wire seriesactive power filter, the harmonic voltage source consistsof the parallel connection of a resistor and a capacitor inthe DC-link of the three-phase diode rectifier, whereasthe harmonic current source consists of the series connec-tion of a resistor and an inductor in the DC-link of thethree-phase diode rectifier. The input voltage of the loadis 110V, 60Hz, and the source inductance is 0.1mH.The converter with RC load is used as a harmonicvoltage source, while the converter with the RL load isused as a harmonic current source. The value of the inverterDC-link capacitance is 2350mF. The LC filter to reduce theswitching ripple of the inverter switching frequency iscomposed of inductance Lr¼ 4mH and capacitanceCr¼ 0.5mF. Table 2 shows the system parameters, whichare used to form the active power filter system. The

Table 1: Separation of active and reactive power

�pp and �qq The DC components of the active power and thereactive power, which are generated by thefundamental component of the load voltages andthe source currents

~pp and ~qq The AC components of the active power and thereactive power, which are generated by thenegative sequence and the harmonic componentof the load voltages and the source currents

vL(a,b,c)

is(a,b,c)

vL(�,�,0)

is(�,�,0)

vC(�,�,0)

p

q

� − � − 0

� − � − 0

∗

∗∗

∗

three−phase

three−phase

p and qcalculation

LPFp

q

~

p~p_

_+

Vdc_inv

Vdc_inv

VCa

VCb

VCc

iSaf

iSbf

iScf

∆Vdc_

+∗ PI

__

_

×

×

×

voltagereferencecalculation

Fig. 3 Block diagram of control algorithm

278 IEE Proc.-Electr. Power Appl., Vol. 151, No. 3, May 2004

parameter values for the parallel passive filters arerepresented in Table 3. The 5th passive filter is made upwith inductance Lf¼ 2mH and capacitance Cf¼ 140mF,and the 7th passive filter is made up with the inductanceLf¼ 2mH and capacitance Cf¼ 70mF.

Ls vSa

vSb

vSc

iSa

iSb

iSc

vLaCL

CrLr

RL

LL

LrCinv

Cinv

Cr Lf

Cf

RL

vLb

vLc

Ls vSa

vSb

vSc

iSa

iSb

iSc

vLa

vLb

vLc

+

_

+

_

a

b

Fig. 4 Three-phase three-wire series active power filter circuitsa RC load circuitb RL load circuit

Table 2: System parameters of series active power filterand combined system of the passive filters and seriesactive power filter

Type parameters Three-phasethree-wireseries APF

Combined sys-tem of passiveand active filter

Supply voltage, frequency 110V, 60Hz 110V, 60Hz

Source inductor (Ls) 0.1mH 0.1mH

Transformer turn ratio 1 :2 1 :2

Load capacitor (Cload) 2400mF F

Load inductor (Lload) F 35mH

Load resistance (Rload) 30O 30O

Inverter DC-link capacitor 2350mF 2350mF

LC-filter inductor (Lr) 4mH 4mH

LC-filter capacitor (Cr) 0.5mF 0.5mF

Table 3: System parameters of the parallel passive filters

5th passive filter inductor 2mH

capacitor 140mF

7th passive filter inductor 2mH

capacitor 70mF

Fig. 5 Experimental waveforms for harmonic current sourcea Source current and voltage waveforms without compensationb Source current and voltage waveforms when compensated for withparallel passive filtersc Source current and voltage waveforms when compensated for withcombined system of parallel passive filters and series active power filterd Load voltage and compensation voltage waveforms when compen-sated for with combined system of parallel passive filters and seriesactive power filter

IEE Proc.-Electr. Power Appl., Vol. 151, No. 3, May 2004 279

4 Experimental results

4.1 Experimental results for the harmoniccurrent sourceTypical waveforms with the load considered as a harmoniccurrent source are shown in Fig. 5. Figure 5a shows thesource current and source voltage waveforms at a powerfactor 0.951 lagging. The source current and source voltagetotal harmonic distortion (THD) are 23.02 and 7.47%,respectively. As shown in Fig. 5a, the source voltage andsource current waveforms are distorted by the currentharmonic source.These waveforms, which are shown in Fig. 5b, are the

compensated waveforms using only the parallel passivefilters. Figure 5b shows the source current and sourcevoltage waveforms with a power factor of 0.598 leading.The source current and source voltage THD are 6.26 and1.62%, respectively. The amplitude of the source current isincreased, and the power factor is leading because of theeffectiveness of the parallel passive filters. It is known thatthe 5th and 7th source current harmonics are almostcancelled by the parallel passive filters, but the sourcecurrent THD is over 5%, as usual. Moreover, the powerfactor of the source stage is leading. The leading powerfactor is also a cause of malfunction in the load.Figure 5c shows the source current and source voltage

waveforms when compensated with the combined system ofthe parallel passive filters and the series active power filter.The source current THD is 0.59%, and the source voltageTHD is 2.20%. Power factor is almost unity. Figure 5dshows the load voltage and the compensation voltage as anactive filter output. These experimental results show that theseries active power filter complements the defects of theparallel passive filter and co-operates in harmonic compen-sation. Moreover, the series active power filter cancompensate for the leading power factor as well as thesource current harmonics.Figure 6 shows the transient waveforms of the phase,

which are compensated for by the combined system of theparallel passive filters and the series active power filter. Asshown in Fig. 6a, the phase shift and harmonic compensa-tion is performed within a half-cycle, and Fig. 6b shows thatthe reactive components for improving power factor aremainly compensated for.Figure 7 shows the source current waveform from

three-phase to two-phase transformation. Figure 7a is thewaveform before compensation, while Fig. 7b is the oneafter compensation with only the parallel passive filtersapplied. Finally, Fig. 7c is the waveform after compensationby the combined system of the parallel passive filters andthe series active power filter. From Fig. 7, the harmonic

compensation effect is clearly observed. Comparing thesethree waveforms, the better the harmonics compensate, themore smooth the waveforms become and the closer thecircles get. Because the harmonic components of the sourcecurrent are almost zero, when they are compensated for bythe combined system of the parallel passive filters and theseries active power filter, the waveform of Fig. 7c almostforms a circle.

4.2 Experimental results for harmonicvoltage sourceFigure 8 shows typical waveforms for the case of theharmonic voltage source load. Figure 8a shows the sourcecurrent and the source voltage waveforms at a power factor

Fig. 6 Transient states experimental waveforms for harmoniccurrent source when compensated for with combined system ofparallel passive filters and series active power filtera Source current and voltage waveformsb Load voltage and compensation voltage waveforms

−20 −10 0 10 20

current, A

curr

ent,

A

curr

ent,

A

curr

ent,

A

−30 −15 0 15 30

current, A

−40

−40

−20

−20 0 20 40

current, A0

20

40

−30

−15

0

15

30

−20

−10

0

10

20

a b c

Fig. 7 Three-phase to two-phase vector transformation waveforms of source currentsa Without compensationb Compensation with passive filtersc Compensation with passive filters and active filter

280 IEE Proc.-Electr. Power Appl., Vol. 151, No. 3, May 2004

0.971 lagging. The source current and source voltage THDare 33.44 and 9.35%, respectively. From these waveforms, itis observed that source voltages and currents are distortedby harmonic voltage source as well as the harmonic currentsource. Figure 8b shows the source current and sourcevoltage waveforms when compensated for by the seriesactive power filter. The source current and voltage THD are4.04 and 1.37%, respectively. The power factor is measuredas 0.99 lagging. Figure 8c shows the load voltage andcompensation voltage as an active power filter output.Figure 9 shows the transient waveforms when compen-

sated for by using the series active power filter. Figure 9ashows that harmonic compensation for the source currentresponds slowly, but the voltage compensation performsrapidly. Figure 6b shows that the VA rating of the activepower filter can be reduced, because the passive filters

compensate for the 5th and 7th harmonic components andonly the other harmonics are compensated for by the seriesactive power filter. However, Fig. 9b shows that the activepower filter simultaneously compensates for harmonics andreactive power.Figure 10 shows the source currents in two-phase co-

ordinates for the harmonic voltage source. In Fig. 10, theharmonic compensation effect can be observed as clearly asin Fig. 7. Comparing these two waveforms, the waveformafter compensation constitutes a near circle-like orbit. Butthe source current THD after compensation is about 4%, sothe waveform of Fig. 7b is not a perfect circular orbit.Figure 11 shows the comparison of harmonic compo-

nents, which are generated by the harmonic current sourceand voltage source. When the fundamental component ofthe source current is assumed to be 100, the harmoniccomponents of the source currents are represented as apercentage. The values of THD for both harmonic sourcemeet the limitation of IEEE Standard 519. But it turns outthat the compensation by the combined system of theparallel passive filters and active power filter is moreeffective. Both passive filter and active power filter arecomplementary and co-operate with each other.

5 Conclusion

A new control algorithm, which can compensate both theharmonic current source and the harmonic voltage source,has been introduced. A series active power filter wasmanufactured and was simultaneously applied to the three-phase three-wire series active power filter system and acombined system of the parallel passive filters. Theproposed algorithm extracts the voltage compensation

Fig. 8 Experimental waveforms for harmonic voltage sourcea Source current and voltage waveforms without compensationb Source current and voltage waveforms when compensated for withseries active power filterc Load voltage and compensation voltage waveforms when compen-sated for with series active power filter

Fig. 9 Transient states experimental waveforms for harmonicvoltage sourcea Source current and voltage waveformsb Load voltage and compensation voltage waveforms

IEE Proc.-Electr. Power Appl., Vol. 151, No. 3, May 2004 281

reference directly. Therefore, the calculation methodbecomes simpler than for other control algorithms of seriesactive power filters. Some experiments on the harmoniccurrent source and the harmonic voltage source have beencarried out. It has been observed that the THD of sourcecurrents is lower than 5%, which meets the regulations ofIEEE Standard 519 and the power factor is almost unityafter compensation.These experimental results confirm the enhanced effec-

tiveness of the control algorithm. It is expected that theseries active power filter will contributes to resolving powerquality problems, such as harmonics and power-factordegradation.

6 Acknowledgments

This work was financially supported by MOCIE throughthe EIRC programme.

7 References

1 Peng, F.Z., Akagi, H., and Nabae, A.: ‘A new approach to harmoniccompensation in power systemF a combined system of shunt passiveand series active filter’, IEEE Trans. Ind. Appl., 1990, 26, (6), pp.983–990

2 Peng, F.Z., and Lai, J.-S.: ‘Generalized instantaneous reactive powertheory for three phase power systems’, IEEE Trans. Instrum. Meas.,1996, 45, (1), pp. 293–297

3 Wang, Z., and Wang, Q.: ‘A series active power filter adopting hybridcontrol approach’, IEEE Trans. Power Electron., 2001, 16, (3),pp. 301–310

4 Bhattacharya, S., Divan, D.M., and Banerjee, B.: ‘Synchronous frameharmonic isolator using active series filter’. European Conference onPower Electronics and Application (EPE ’91) Firenze, Italy, 1991,pp. 3-030–3-035

5 Peng, F.Z., Akagi, H., and Nabae, A.: ‘Compensation characteristics ofthe combined system of shunt passive and series active filters’, IEEETrans. Ind. Appl., 1993, 29, (1), pp. 144–152

6 Peng, F.Z.: ‘Harmonic sources and filtering approaches’, IEEE Ind.Appl. Mag., 2001, 7, (4), pp. 18–25

7 Peng, F.Z., Ott, G.W., and Adams, D.J.: ‘Harmonic and reactive powercompensation based on the generalized instantaneous reactive powertheory for three-phase four-wire systems’, IEEE Trans. Power Electron.,1998, 13, (6), pp. 1174–1181

8 Moran, L., Pastorini, I., Dixon, J., and Wallace, R.: ‘Series activepower filter compensates current harmonics and voltage unbalancesimultaneously’, IEE Proc., Gener. Transm. Distrib., 2000, 147, (1),pp. 31–36

−20 −10 0 10 20

current, A

curr

ent,

A

−20

−10

0

10

20

−20 −10 0 10 20

current, A

curr

ent,

A

−20

−10

0

10

20

a

b

Fig. 10 Three-phase to two-phase vector transformation wave-forms of source currentsa Before compensationb After compensation

0

10

20

30

40

50

60

%

70

80

90

100

1st 5th 7th 11th 13th

harmonic voltage source

harmonic current source

0

10

20

30

40

50

60

%

70

80

90

100

1st 5th 7th 11th 13th

harmonic voltage source

harmonic current source

a

b

Fig. 11 Comparison of the harmonic amplitude for harmonicvoltage source and harmonic current sourcea Harmonic orders before compensationb Harmonic orders after compensation

282 IEE Proc.-Electr. Power Appl., Vol. 151, No. 3, May 2004