Active Filters - chiataimakro.vicp.cc:8880

39Active Filters

Luis MoránElectrical Engineering Dept.,

Universidad de ConcepciónConcepción, Chile

Juan DixonElectrical Engineering Dept.,

Universidad Católica de ChileSantiago, Chile

39.1 Introduction ........................................................................................ 106739.2 Types of Active Power Filters .................................................................. 106839.3 Shunt Active Power Filters...................................................................... 1069

39.3.1 Power Circuit Topologies • 39.3.2 Control Scheme • 39.3.3 Power Circuit Design• 39.3.4 Technical Specifications

39.4 Series Active Power Filters ...................................................................... 108539.4.1 Power Circuit Structure • 39.4.2 Principles of Operation • 39.4.3 Power Circuit Design• 39.4.4 Control Issues • 39.4.5 Control Circuit Implementation • 39.4.6 ExperimentalResults

39.5 Hybrid Active Power Filters .................................................................... 109439.5.1 Principles of Operation • 39.5.2 The Hybrid Filter Compensation Performance• 39.5.3 Experimental Results

Further Reading ................................................................................... 1101

39.1 Introduction

The growing number of power electronics-based equipmenthas produced an important impact on the quality of electricpower supply. Both high power industrial loads and domesticloads cause harmonics in the network voltages. At the sametime, much of the equipment causing the disturbances is quitesensitive to deviations from the ideal sinusoidal line voltage.Therefore, power quality problems may originate in the systemor may be caused by the consumer itself. Moreover, in thelast years the growing concern related to power quality comesfrom:

• Consumers that are becoming increasingly aware of thepower quality issues and being more informed about theconsequences of harmonics, interruptions, sags, switch-ing transients, etc. Motivated by deregulation, they arechallenging the energy suppliers to improve the qualityof the power delivered.

• The proliferation of load equipment with microprocessor-based controllers and power electronic devices which aresensitive to many types of power quality disturbances.

• Emphasis on increasing overall process productivity,which has led to the installation of high-efficiency equip-ment, such as adjustable speed drives and power factorcorrection equipment. This in turn has resulted in anincrease in harmonics injected into the power system,

causing concern about their impact on the systembehavior.

For an increasing number of applications, conventionalequipment is proving insufficient for mitigation of power qual-ity problems. Harmonic distortion has traditionally been dealtwith the use of passive LC filters. However, the application ofpassive filters for harmonic reduction may result in parallelresonances with the network impedance, over compensationof reactive power at fundamental frequency, and poor flexibil-ity for dynamic compensation of different frequency harmoniccomponents.

The increased severity of power quality in power networkshas attracted the attention of power engineers to developdynamic and adjustable solutions to the power quality prob-lems. Such equipment, generally known as active filters, arealso called active power line conditioners, and are able tocompensate current and voltage harmonics, reactive power,regulate terminal voltage, suppress flicker, and to improvevoltage balance in three-phase systems. The advantage of activefiltering is that it automatically adapts to changes in the net-work and load fluctuations. They can compensate for severalharmonic orders, and are not affected by major changes in net-work characteristics, eliminating the risk of resonance betweenthe filter and network impedance. Another plus is that theytake up very little space compared with traditional passivecompensators.

Copyright © 2007, 2001, Elsevier Inc.

All rights reserved.

1067

1068 L. Morán and J. Dixon

39.2 Types of Active Power Filters

The technology of active power filter has been developedduring the past two decades reaching maturity for harmon-ics compensation, reactive power, and voltage balance in acpower networks. All active power filters are developed withpulse width modulated (PWM) converters (current-sourceor voltage-source inverters). The current-fed PWM inverterbridge structure behaves as a non-sinusoidal current source tomeet the harmonic current requirement of the non-linear load.It has a self-supported dc reactor that ensures the continuouscirculation of the dc current. They present good reliability,but have important losses and require higher values of parallelcapacitor filters at the ac terminals to remove unwanted cur-rent harmonics. Moreover, they cannot be used in multilevelor multistep modes configurations to allow compensation inhigher power ratings.

The other converter used in active power filter topologies isthe PWM voltage-source inverter (PWM-VSI). This converter

Non-LinearLoad

AC Mains

IS IL

IC

Active Filter

Vdc

Non-LinearLoad

AC Mains

IS IL

Active Filter

Vdc

VAF

(a) (b)

Non-LinearLoad

AC Mains

IS IL

Series ActiveFilter

Vdc

VAF

IC

Shunt PassiveFilter

(c)

FIGURE 39.1 Active power filter topologies implemented with PWM-VSI: (a) shunt active power filter; (b) series active power filter; and (c) hybridactive power filter.

is more convenient for active power filtering applications sinceit is lighter, cheaper, and expandable to multilevel and multi-step versions, to improve its performance for high power ratingcompensation with lower switching frequencies. The PWM-VSI has to be connected to the ac mains through couplingreactors. An electrolytic capacitor keeps a dc voltage constantand ripple free.

Active power filters can be classified based on the typeof converter, topology, control scheme, and compensationcharacteristics. The most popular classification is based onthe topology such as shunt, series, or hybrid. The hybridconfiguration is a combination of passive and active compen-sation. The different active power filter topologies are shownin Fig. 39.1.

Shunt active power filters (Fig. 39.1a) are widely used tocompensate current harmonics, reactive power, and load cur-rent unbalanced. It can also be used as a static var generatorin power system networks for stabilizing and improving volt-age profile. Series active power filters (Fig. 39.1b) is connected

39 Active Filters 1069

before the load in series with the ac mains, through a couplingtransformer to eliminate voltage harmonics and to balance andregulate the terminal voltage of the load or line. The hybridconfiguration is a combination of series active filter and pas-sive shunt filter (Fig. 39.1c). This topology is very convenientfor the compensation of high power systems, because the ratedpower of the active filter is significantly reduced (about 10% ofthe load size), since the major part of the hybrid filter consistsof the passive shunt LC filter used to compensate lower-order current harmonics and reactive power at fundamentalfrequency.

Due to the operation constraint, shunt or series active powerfilters can compensate only specific power quality problems.Therefore, the selection of the type of active power filter toimprove power quality depends on the source of the problemas can be seen in Table 39.1.

The principles of operation of shunt, series, and hybridactive power filters are described in the following sections.

TABLE 39.1 Active filter solutions to power quality problems

Active filterconnection

Load on ac supply AC supply on load

Shunt Current harmonic filteringReactive current compensationCurrent unbalanceVoltage flicker

Series Current harmonic filtering Voltage sag/swellReactive current compensation Voltage unbalanceCurrent unbalance Voltage distortionVoltage flicker Voltage interruptionVoltage unbalance Voltage flicker

Voltage notching

39.3 Shunt Active Power Filters

Shunt active power filters compensate current harmonics byinjecting equal but opposite harmonic compensating current.In this case, the shunt active power filter operates as a currentsource injecting the harmonic components generated by theload but phase shifted by 180. As a result, components ofharmonic currents contained in the load current are cancelledby the effect of the active filter, and the source current remainssinusoidal and in phase with the respective phase-to-neutralvoltage. This principle is applicable to any type of load con-sidered as an harmonic source. Moreover, with an appropriatecontrol scheme, the active power filter can also compensate theload power factor. In this way, the power distribution systemsees the non-linear load and the active power filter as an idealresistor. The compensation characteristics of the shunt activepower filter is shown in Fig. 39.2.

Non-LinearLoad

AC Mains

Active Filter

Vdc

IS ILLoad CurrentSource Current

Lsystem

ICCompensation

Current

LLoadLcoupling

FIGURE 39.2 Compensation characteristics of a shunt active powerfilter.

39.3.1 Power Circuit Topologies

Shunt active power filters are normally implemented withPWM-VSIs. In this type of application, the PWM-VSI oper-ates as a current-controlled voltage source. Traditionally, twolevel PWM-VSI have been used to implement such systemconnected to the ac bus through a transformer. This type ofconfiguration is aimed to compensate non-linear load ratedin the medium power range (hundreds of kVA) due to semi-conductors rated values limitations. However, in the last yearsmultilevel PWM-VSIs have been proposed to develop activepower filters for medium voltage and higher rated power appli-cations. Also, active power filters implemented with multipleVSIs connected in parallel to a dc bus but in series through atransformer or in cascade has been proposed in the technicalliterature. The different power circuit topologies are shown inFig. 39.3.

The use of VSI connected in cascade is an interesting alter-native to compensate high power non-linear loads. The useof two PWM-VSI with different rated power allows the useof different switching frequencies, reducing switching stresses,and commutation losses in the overall compensation system.The power circuit configuration of such a system is shownin Fig. 39.4.

The VSI connected closer to the load compensates for thedisplacement power factor and lower frequency current har-monic components (Fig. 39.5b), while the second compensatesonly high-frequency current harmonic components. The firstconverter requires higher rated power than the second and


Non- LinearLoad

(a)

Non-LinearLoad

(b)

FIGURE 39.3 Shunt active power filter topologies implemented withPWM-VSIs: (a) a three-phase PWM unit and (b) three single-phase unitsin parallel to a common dc bus.

can operate at lower switching frequency. The compensationcharacteristics of the cascade shunt active power filter is shownin Fig. 39.5.

In recent years, there has been an increasing interest in usingmultilevel inverters for high power energy conversion, espe-cially for drives and reactive power compensation. The useof neutral-point-clamped (NPC) inverters (Fig. 39.6) allowsequal voltage shearing of the series-connected semiconductorsin each phase. Basically, multilevel inverters have been devel-oped for applications in medium voltage ac motor drives andstatic var compensation. For these types of applications, theoutput voltage of the multilevel inverter must be able to gen-erate an almost sinusoidal output current. In order to generatea near sinusoidal output current, the output voltage should notcontain low-frequency harmonic components.

However, for active power filter applications, the three-level NPC inverter output voltage must be able to generatean output current that follows the respective reference currentcontaining the harmonic and reactive component required by

Gating SignalsGenerator

Current ControlSystem

AC Source

CurrentReferenceGenerator

VoltageControl DC Voltage Reference

LinkInductor PWM - VSI #2


Current ControlSystem

CurrentReferenceGenerator

VoltageControl DC Voltage Reference

LinkInductor PWM - VSI #1

Non-LinearLoad

FIGURE 39.4 A shunt active power filter implemented with two PWM-VSI connected in cascade.

the load. Current and voltage waveforms obtained for a shuntactive power filter implemented with a three-level NPC-VSIare shown in Fig. 39.7.

39.3.2 Control Scheme

The control scheme of a shunt active power filter must cal-culate the current reference waveform for each phase of theinverter, maintain the dc voltage constant, and generate theinverter gating signals. The block diagram of the controlscheme of a shunt active power filter is shown in Fig. 39.8.

The current reference circuit generates the reference cur-rents required to compensate the load current harmonics andreactive power, and also try to maintain constant the dc voltageacross the electrolytic capacitors. There are many possibilitiesto implement this type of control, and the most popular ofthem will be explained in this chapter. Also, the compensation


−1,5−1

−0,50

0,51

1,5

0 0,02 0,04 0,06 0,08 0,1

p.u

.

(a)

0 0,02 0,04 0,06 0,08 0,1

(b)

0 0,02 0,04 0,06 0,08 0,1

(c)

−0,8

−0,4

0

0,4

0,8

p.u

.

−0,8

−0,4

0

0,4

0,8

p.u

.

−1,5−1

−0,50

0,51

1,5

0 0,02 0,04 0,06 0,08 0,1

p.u

.

(d)

0 0,02 0,04 0,06 0,08 0,1

(e)

−1,5−1

−0,50

0,51

1,5

p.u

.

FIGURE 39.5 Current waveforms of active power filter implementedwith two PWM-VSI in cascade: (a) load current waveform; (b) currentwaveform generated by PWM-VSI no. 1; (c) current waveform generatedby PWM-VSI no. 2; (d) power system current waveform between thetwo inverters (THDi = 13.7%); and (e) power system current waveform(THDi = 4.5%).

effectiveness of an active power filter depends on its ability tofollow with a minimum error and time delay, the referencesignal calculated to compensate the distorted load current.Finally, the dc voltage control unit must keep the total dc

bus voltage constant and equal to a given reference value. Thedc voltage control is achieved by adjusting the small amountof real power absorbed by the inverter. This small amountof real power is adjusted by changing the amplitude of thefundamental component of the reference current.

39.3.2.1 Current Reference GenerationThere are many possibilities to determine the reference currentrequired to compensate the non-linear load. Normally, shuntactive power filters are used to compensate the displacementpower factor and low-frequency current harmonics generatedby non-linear loads. One alternative to determine the currentreference required by the VSI is the use of the instantaneousreactive power theory, proposed by Akagi [1], the other one isto obtain current components in d–q or synchronous referenceframe [2], and the third one to force the system line currentto follow a perfectly sinusoidal template in phase with therespective phase-to-neutral voltage.

39.3.2.1.1 Instantaneous Reactive Power Theory This con-cept is very popular and useful for this type of application,and basically consists of a variable transformation from the a,b, c, reference frame of the instantaneous power, voltage, andcurrent signals to the α, β reference frame. The transforma-tion equations from the a, b, c, reference frame to the α, β

coordinates can be derived from the phasor diagram shown inFig. 39.9.

The instantaneous values of voltages and currents in the α,β coordinates can be obtained from the following equations:

[vα

vβ

]= [A] ·

va

vb

vc

[iα

iβ

]= [A] ·

ia

ibic

(39.1)

where A is the transformation matrix, derived from Fig. 39.9and is equal to

[A] =√

2

3

[1 −1/2 −1/20

√3/2 −√

3/2

](39.2)

This transformation is valid if and only if va(t) + vb(t) +vc(t) is equal to zero, and also if the voltages are balancedand sinusoidal. The instantaneous active and reactive powerin the α, β coordinates are calculated with the followingexpressions:

p(t ) = vα(t ) · iα(t ) + vβ(t ) · iβ(t ) (39.3)

q(t ) = −vα(t ) · iβ(t ) + vβ(t ) · iα(t ) (39.4)

It is evident that p(t) becomes equal to the conventionalinstantaneous real power defined in the a, b, c reference frame.


Non-LinearLoad

AC Source

LICIS

IL

C1

C2

2VDC

su1

su2

su3

sv1

sv2

sv3

su4 sv4

sw1

sw2

sw3

sw4

FIGURE 39.6 A shunt active power filter implemented with a three-level neutral point-clamped VSI.

(a)

i_La

linv vinv

200.00

100.00

0.00

−100.00

−200.00

200.00

100.00

0.00

−100.00

−200.00

i_La200.00

100.00

0.00

−100.00

−200.000.00 10.00 20.00 30.00

Time (ms)

40.00 50.00

(b)

0.00 10.00 20.00 30.00

Time (ms)

40.00 50.00

(c)

0.00 10.00 20.00 30.00

Time (ms)

40.00 50.00

(d)

0.00−1.50K

−1.00K

−0.50K

0.00K

0.50K

1.00K

1.50K

10.00 20.00 30.00

Time (ms)

40.00 50.00

FIGURE 39.7 Current and voltage waveforms for a shunt active power filter implemented with a three-level NPC-VSI: (a) load current;(b) compensated system current (THD = 3.5%); (c) current generated by the shunt active power filter; and (d) inverter output voltage.

However, in order to define the instantaneous reactive power,Akagi introduces a new instantaneous space vector defined byexpression (39.4) or by the vector equation:

q = vαxiβ + vβxiα (39.5)

The vector q is perpendicular to the plane of α, β coordi-nates, to be faced in compliance with a right-hand rule, vα isperpendicular to iβ , and vβ is perpendicular to iα . The physical

meaning of the vector q is not “instantaneous power” becauseof the product of the voltage in one phase and the current inthe other phase. On the contrary, vαiα and vβ iβ in Eq. (39.3)obviously mean “instantaneous power” because of the productof the voltage in one phase and the current in the same phase.Akagi named the new electrical quantity defined in Eq. (39.5)“instantaneous imaginary power,” which is represented by theproduct of the instantaneous voltage and current in differentaxes, but cannot be treated as a conventional quantity.


Reference CurrentGenerator

Active, Reactiveand Harmonic

decomposition ofthe Load Current

iar(t) ibr(t) icr(t)

12

Driver

Voltage ControlUnit

P*(t)

Pavg

Current Reference Generator

DC Voltage Control

Current Control and Gating Signals Generator

Vdc gen.

Vdc ref.

Gating SignalsGen.

CurrentControlUnit

va(t) vb(t) vc(t)

va(t)

vb(t)

vc(t)

IaLoad(t) IbLoad(t) IcLoad(t)

CompensatedCurrent Component

dc Current ControlComponent

IaGen(t)

IbGen(t)

IcGen(t)

FIGURE 39.8 The block diagram of a shunt active power filter control scheme.

c

a

b

α

β

Va, ia

Vα, iα

Vβ, iβVb, ib

30°

120°

Vc, ic

ω t

FIGURE 39.9 Transformation diagram from the a, b, c reference frameto the α, β coordinates.

The expression of the currents in the α–β plane, as a func-tion of the instantaneous power is given by the followingequation:

[iαiβ

]= 1

v2α + v2

β

·[

vα vβ

vβ −vα

]·[

p0

]+[

vα vβ

vβ −vα

]·[

0q

]

≡[

iαp

iβp

]+[

iαq

iβq

](39.6)

and the different components of the currents in the α–β planeare shown in the following expressions:

iαp = vαp

v2α + v2

β

(39.7)

iαq = vβq

v2α + v2

β

(39.8)

iβp = vβp

v2α + v2

β

(39.9)

iβq = −vαq

v2α + v2

β

(39.10)

From Eqs. (39.3) and (39.4), the values of p and q canbe expressed in terms of the dc components plus the accomponents, that is:

p = p + p (39.11)

q = q + q (39.12)

where,

p dc component of the instantaneous power p, and is relatedto the conventional fundamental active current.

p is the ac component of the instantaneous power p, it doesnot have average value, and is related to the harmonic


currents caused by the ac component of the instantaneousreal power.

q is the dc component of the imaginary instantaneous powerq, and is related to the reactive power generated by thefundamental components of voltages and currents.

q is the ac component of the instantaneous imaginary powerq, and it is related to the harmonic currents caused by theac component of instantaneous reactive power.

In order to compensate reactive power (displacement powerfactor) and current harmonics generated by non-linear loads,the reference signal of the shunt active power filter mustinclude the values of p, q, and q. In this case the reference cur-rents required by the shunt active power filters are calculatedwith the following expression:

[i∗c ,α

i∗c ,β

]= 1

v2α + v2

β

·[

vα vβ

vβ −vα

]·[

pL

qL + qL

](39.13)

The final compensating currents including the zerosequence components in a, b, c reference frame are thefollowing:

i∗c ,a

i∗c ,b

i∗c ,c

=

√2

3·

1√2

1 01√2

−12

√3

21√2

−12

−√3

2

·

−i0i∗c ,α

i∗c ,β

(39.14)

where the zero sequence current component i0 is equal to 1/√

3(ia + ib + ic). The block diagram of the circuit required togenerate the reference currents defined in Eq. (39.14) is shownin Fig. 39.10.

The advantage of instantaneous reactive power theory is thatreal and reactive power associated with fundamental compo-nents are dc quantities. These quantities can be extracted with alow-pass filter. Since the signal to be extracted is dc, filtering ofthe signal in the α–β reference frame is insensitive to any phase

a-b-cto

α - β

α - βto

a-b-c

a-b-cto

α - β

laload

Van

Vbn

Vcn

laload

laload

Vα

Iα

Iβ

Vβ

PandQ

Va2 and Vb

2

P

QLPF Qref

Pref

∆Vdc

lαref

and

lβref

laref

lbref

lcref

Va2

Vb2

lαref

lβref

FIGURE 39.10 The block diagram of the current reference generator using p–q theory.

shift errors introduced by the low-pass filter, improving com-pensation characteristics of the active power filter. The sameadvantage can be obtained by using the synchronous referenceframe method, proposed in [2]. In this case, transformationfrom a, b, c axes to d–q synchronous reference frame is done.

Effects of the Low-pass Filter Design Characteristics inCompensation PerformanceA Butterworth filter is normally used due to the adequatefrequency response. A second-order filter offers an appropri-ate relation between the transient response and the requiredattenuation characteristic. Higher-order filters achieve bet-ter filtering characteristic, but the settling time is increased.Since the low-pass filter cannot eliminate completely the low-frequency harmonic contained in the p and q signals, the shuntactive power filter cannot compensate the entire low-frequencyharmonic contained in the load current. Normally the cut-off frequency is equal to 127 Hz with an attenuation factor of15 dB for the first ac component to be eliminated, which meansan 82.2% attenuation of the fifth and seventh harmonic com-ponents (Fig. 39.11). The expression that relates the system linecurrent harmonic distortion with the LPF cut-off frequencyand load power factor is shown in Eq. (39.15).

THDisys

=

√∞∑

h=6k

1

(fn /fc)4+1

[((h2 +1)/(h2 −1))−(1/(h2 −1))cos(2ϕ)

]

cos(ϕ)(39.15)

with k = 1, 2, 3, . . . and fh is the frequency of the harmoniccomponent of order h, and fc is the LPF cut-off frequency.

Figure 39.12 shows the total harmonic distortion (THD) inthe line currents introduced by the second-order Butterworthfiltering characteristics, as a function of the cut-off frequency,and considering a 50 Hz ac mains frequency. The harmonicdistortion of the compensated line current depends on the


Low-passfilter

+

+

pAC

−p

FIGURE 39.11 The low-pass filter block diagram used to extract the accomponent of p.

40 60 80 100 120 140 160 180

0

5

10

15

20

Cut-off frequency

TH

Di [

%]

PF = 1PF = 0.8PF = 0.5PF = 0.2

FIGURE 39.12 Harmonic distortion of the compensated line current asa function of the low-pass filter cut-off frequency. PF = cos(ϕ).

load displacement power factor, as shown in Eq. (39.15). If thecut-off frequency of the low-pass filter is changed, the activepower filter compensation performance is affected as well asthe transient response of the control scheme.

Effects of the Supply Voltage Distortion in CompensationPerformanceOne of the most important characteristics of the instantaneousimaginary power concept is that in order to obtain the currentreference signal required to compensate reactive and harmoniccurrent components, the system phase-to-neutral voltages areused. In general, purely sinusoidal voltages are considered inpreviously reported analysis. In case voltage is purely sinu-soidal, the dc component of p and q in the α–β plane arerelated with the fundamental components in the real a, b, creference system. This is not the case if the system voltages aredistorted or unbalanced, as it is demonstrated below.

It is assumed that the supply voltages have harmonicdistortion, and these are represented by:

va (t ) = V1cos (ωt ) + Vhcos [h (ωt − δh)]

vb (t ) = V1cos

(ωt − 2π

3

)+ Vhcos

[h

(ωt − δh − 2π

3

)]

0

2

4

6

8

10

THDv [%]

TH

Di [

%]

PF = 1PF = 0.8PF = 0.5PF = 0.2

51 2 3 4 6 7 8 90 10

FIGURE 39.13 Harmonic distortion in the compensated line current asa function of the harmonic distortion in the system voltages. PF = cos(ϕ).

vc (t ) = V1cos

(ωt + 2π

3

)+ Vhcos

[h

(ωt − δh + 2π

3

)]

(39.16)

Since the harmonic voltage component introduces a dccomponent in p and q, compensation performance of theshunt active power filter is reduced, as shown in Fig. 39.13.The larger the harmonic distortion in the system voltage is,the active power filter performance is more affected.

Effects of the Supply Voltage Unbalance in CompensationPerformanceVoltage unbalance also affects the active power filter com-pensation performance. In this analysis, the phase-to-neutralvoltages of the ac supply are equal to:

va (t ) = V1cos (ωt )

vb (t ) = V1 (1 + m) cos

(ωt − 2π

3

)

vc (t ) = V1 (1 − m) cos

(ωt + 2π

3

)(39.17)

with 0 < m < 1.If the low-pass filter is considered as ideal, and the active

filter follows exactly the current references, the compensatedsupply current (phase a) is:

iSa = I1cos (ϕ) cos (ωt ) +√

3

3m [I1sin (ϕ) cos (3ωt )

+I1cos (3ωt − ϕ)] (39.18)


In this case, the active power filter is not able to fullycompensate the system line current, since compensationperformance depends on the voltage unbalance magnitude.

If the unbalance is defined as a function of the positive andnegative sequence component as:

Va1 = 1

3

[Va + a2Vb + aVc

]

Va2 = 1

3

[Va + aVb + a2Vc

], a = 1∠120

Unbalance = |Va2||Va1| =

√3

3m

(39.19)

The harmonic distortion is equal to:

THDi =√

3

3m (39.20)

The relation between line current harmonic distortion andvoltage unbalance is shown in Fig. 39.14.

6

5

4

TH

Di [

%]

3

2

1

00 1 2 3

Desb. [%]

4 5 6

FIGURE 39.14 Harmonic distortion in the compensated line current asa function of the system voltages unbalance.

Effects of the Time Delay Introduced by the DSP inCompensation PerformanceIf a DSP is used to derive the reference generation signals,a time delay, T, associated with the processing time is intro-duced in the calculation of the reference signals. Consideringa simultaneous sampling, the ac mains compensated linecurrent with an ideal low-pass filter in the control scheme(phase a) is:

iSa = I1cos (ωt ) [cos (ϕ) + sin (ωT )] + VDC cos (ωt )

+∞∑

n=2k−1

In [1 − cos (nωT )] cos [n (ωt − ϕ)]

+∞∑

n=2k−1

Insin (nωT ) sin [n (ωt − ϕ)] (39.21)

The time delay introduced by the DSP affects the compen-sation performance, especially when fast changes are presentin the load current. Nevertheless, when T is small (T > 50 µs),the effect in compensation performance is negligible.

In four-wire systems, unbalances in the load current alsoaffect the generation of the adequate current reference signalthat will assure high performance active power filter com-pensation. This assumption can be proved by considering thefollowing load currents:

iLa = I1cos (ωt − ϕ) +∞∑

n=2k−1

Incos [n (ωt − ϕ)]

iLb = I1 (1 + l) cos

(ωt − ϕ − 2π

3

)

+∞∑

n=2k−1

In (1 + l) cos

[n

(ωt − ϕ − 2π

3

)]

iLc = I1 (1 − l) cos

(ωt − ϕ + 2π

3

)

+∞∑

n=2k−1

In (1 − l) cos

[n

(ωt − ϕ + 2π

3

)]

iLN = iLa + iLb + iLc

(39.22)

with 0 < l < 1, k = 1, 2, 3, . . .Equation (39.23) shows the presence of low-frequency even

harmonics in the active power expression introduced by theload current unbalanced. These low frequency current com-ponents forces to reduce the cut-off frequency of the low-passfilter, close to 100 Hz, which affects the transient response ofthe active power filter.

p = 3

2V1

I1cos (ϕ) +

∞∑n=2k−1

Incos [(n + 1) ωt − nϕ]

+∞∑

m=2k−1

Imcos [(m − 1) ωt − mϕ]

with k = 1, 2, 3, . . .

(39.23)

39.3.2.1.2 Synchronous Reference Frame Algorithm Theblock diagram of a current reference generator that uses thesynchronous reference frame algorithm is shown in Fig. 39.15.

In this case, the real currents are transformed into asynchronous reference frame [2]. The reference frame is syn-chronized with the ac mains voltage, and is rotating at the


PLL

cos(θ)

sin(θ)

Iaload

Ibload

Icload

Van

Vbn

Vcn

a-b-cto

α - β

α - βto

d - q

Id

IqLPF

Idref

∆Vdc

+

+d - q

toα - β

Iqref Iβref

Iαrefα - β

toa-b-c

Iaref

Ibref

Icref

Iα

Iβ

FIGURE 39.15 The circuit block diagram required to obtain current reference waveforms using the synchronous reference frame theory.

same frequency. The transformation is defined by:

[idiq

]=[

cos (ωt ) sin (ωt )sin (ωt ) cos (ωt )

] [iαiβ

](39.24)

As for the instantaneous reactive power theory, d and qterms are composed by a dc and multiple ac components,such as id = iddc + idac and iq = iqdc + iqac . The compensationreference signals are obtained from the following expressions:idref = −idac and iqref = −iqdc − iqac . The compensated cur-rents generated by the shunt active power filter are obtainedfrom Eq. (39.25).

iaref

ibref

icref

=

√2

3

1√2

1 01√2

− 12

√3

21√2

− 12 −

√3

2

[−1 0 0

0 cos(ωt ) −sin(ωt )0 sin(ωt ) cos(ωt )

][i0

idrefiqref

]

(39.25)

One of the most important characteristics of this algorithmis that the reference currents are obtained directly from theloads currents without considering the source voltages. This isan important advantage since the generation of the referencesignals is not affected by voltage unbalance or voltage distor-tion, therefore increasing the compensation robustness andperformance. However, in order to transform from the α–β

plane to the d–q synchronous reference frame, sine and cosinesignals synchronized with the respective phase-to-neutral volt-ages are required. A phase-locked loop (PLL) per each phase,as the one shown in Fig. 39.16 must be used.

Since the algorithm used to obtain the reference currentpresents the same mathematical procedure and operations thatthe ones required in the instantaneous reactive power concept,the effects introduced by the filter and DSP time delay are sim-ilar. Unbalanced load currents generate a different harmonicspectrum in the synchronous reference frame, and low-orderharmonic components appear in the reference signal. In order

PhaseDetector

Low-PassFilter

VoltageControlledOscillator

Input SignalOutput Signal

FIGURE 39.16 The PLL circuit block diagram.

to separate these uncharacteristic low-frequency current com-ponents, the cut-off frequency of the low-pass filter must bereduced.

39.3.2.1.3 Peak Detection Method There are other possi-bilities to generate the current reference signal required tocompensate reactive power and current harmonics. Basically,all the different schemes try to obtain the current refer-ence signals that include the reactive components requiredto compensate the displacement power factor and the cur-rent harmonics generated by the non-linear load. Figure 39.17shows another scheme used to generate the current referencesignals required by a shunt active power filter. In this case,the ac current generated by the inverter is forced to followthe reference signal obtained from the current reference gen-erator. In this circuit, the distorted load current is filtered,extracting the fundamental component, il1. The band-pass fil-ter is tuned at the fundamental frequency (50 or 60 Hz), sothat the gain attenuation introduced in the filter output sig-nal is zero and the phase-shift angle is 180. Thus, the filteroutput current is exactly equal to the fundamental componentof the load current but phase shifted by 180. If the load cur-rent is added to the fundamental current component obtainedfrom the second-order band-pass filter, the reference currentwaveform required to compensate only harmonic distortion isobtained. In order to provide the reactive power required bythe load, the current signal obtained from the second-orderband-pass filter Il1 is synchronized with the respective phase-to-neutral source voltage (see Fig. 39.17) so that the inverter acoutput current is forced to lead the respective inverter output


SecondOrder Filter(50 or 60

Hz)

PeakValue

+

LoadCurrent Dc Reference

Voltage

−

VoltageError

VoltageController

−

+

Dc GeneratedVoltage

xSinusoidalWaveform(EPROM)

SynchronizationSignal with

System Voltage

LoadCurrent

+

−

il1

il1s

Current Reference Generator Dc Control Unit

CurrentController

ReferenceCurrent

GeneratedCurrent

+−

Gating SignalsGeneratorCurrent Error

Current Control Unit

FIGURE 39.17 The block diagram of an active power filter control scheme that does not use the instantaneous reactive power concept.

voltage, thereby generating the required reactive power andabsorbing the real power necessary to supply the switchinglosses and also to maintain the dc voltage constant.

The real power absorbed by the inverter is controlled byadjusting the amplitude of the fundamental current referencewaveform, Il1, obtained from the reference current genera-tor. The amplitude of this sinusoidal waveform is equal to theamplitude of the fundamental component of the load cur-rent plus or minus the error signal obtained from the dcvoltage control unit. In this way, the current signal allowsthe inverter to supply the current harmonic components, thereactive power required by the load, and to absorb the smallamount of active power necessary to cover the switching lossesand to keep the dc voltage constant (Fig. 39.18).

The main characteristic of this method is the direct deriva-tion of the compensating component from the load current,without the use of any reference frame transformation [1, 2].Nevertheless, this technique presents a low frequency oscil-lation problem in the active power filter dc bus voltage. Toimprove this technique, a modification of the previous scheme(Fig. 39.17) is shown in Fig. 39.19. The scheme is necessary foreach phase. The expression for iMa is:

iMa = I1 cos (ϕ)

2+ I1 cos (2ωt − ϕ)

2

+∞∑

n=2k−1

In

2

cos [(n − 1) ωt − nϕ]

+ cos [(n + 1) ωt − nϕ]

(39.26)

with k = 1, 2, 3, . . .

Figure 39.20 shows the current harmonic distortion intro-duced by the low-pass filter used in the control scheme andthe associated cut-off frequency. The current distortion ofthe compensated current depends on the phase angle of thefundamental load current component.

The supply voltage has no effect on the reference currentgeneration. Synchronization with the ac mains voltage is theimportant issue in this scheme as well as in the synchronousreference frame theory. Unbalanced loads do not affect thereference generation. Nevertheless, the method cannot achieveactive power balance in four-wire systems. The control circuitimplementation of the peak detection method is simple anddoes not require complex calculation, so the processing timeon a DSP is lower than the required in the two previous imple-mentations, (T < 10 µs). The use of this method minimizesthe distortion introduced on current harmonics.

Technical Comparison of the Three Different TechniquesIn order to validate the effectiveness of the proposed analy-sis, a common industrial system is considered. The results areobtained using the previous equations and Matlab simulationsresults. The DSP delay introduced is using the processing timeaccording the DSP ADSP2187. The parameters considered are:

THDiL = 29%

THDV = 5%

cos (ϕ) = 0.87

Unbalance = 3%


(a) (b)

2.0V

i1siload

i11

i1s - iload

0V

−2.0V0s 10ms 20ms 30ms 40ms 0s 10ms 20ms 30ms 40ms

2.0V

0V

Fundamentalcomponent

−2.0V

FIGURE 39.18 The procedure for the generation of the current reference waveform: (a) the load current iload, its fundamental component, il1,and the fundamental current component synchronized with the respective phase-to-neutral source voltage, ils and (b) the synchronized fundamentalcurrent signal minus the load current, ils− iload, and its fundamental component.

Low-PassFilter

iL

g = 2

PLL

cos (ωt )

x

x+ +

vf

i*C

++ +

−

iM

FIGURE 39.19 Modified version of the original peak detection method.

4 8 12 16 20 240

2

4

Cut-off frequency

TH

Di [

%]

6

8 PF = 1PF = 0.8PF = 0.5PF = 0.2

FIGURE 39.20 Harmonic distortion of the system line current as afunction of the cut-off frequency of the low-pass filter and the loaddisplacement power factor.

Table 39.2 shows the numeric results:

TABLE 39.2 Results of the considered case

Technique THDiS [%] FP Transient delay td [ms]

PQ 8.2 0.99 8.1DQ 3.1 0.99 8.1DPVM 2.3 0.99 56.8

Table 39.3 shows a comparison of the three different tech-niques analyzed. The effects of the non-ideal conditions aredescribed for each technique.

In conclusion, it can be mentioned that the compensa-tion performance of the different techniques is similar under

TABLE 39.3 Comparison of the techniques

Parameter PQ SRF PDM

Load PF required toachieve fullcompensation

PF > 0.3 PF > 0.3 PF > 0.2

Harmonic distortionvoltage effect on thecompensated current.

THDi ≈ THDv 0 0

Unbalanced voltage effect THDi ≈Unbalance

0 0

Dynamic response underload changes (infunction of fc ,td = 5τ )

Fast (loadbalanced)

Fast(loadbalanced)

Slow

Capability of load balance Yes (It isnecessary toreduce cut-offfrequency fc )

Yes (It isnecessary toreduce cut-offfrequency fc )

No

DSP delay timeintroduced

Minimum Minimum ≈ 0


ideal conditions, but under the presence of unbalanced andvoltage distortion, the synchronous reference frame algorithmpresents the best performance, since it is insensitive to volt-age perturbations. It is fundamental to consider adequately thecut-off frequency in the filter used to extract the ac compo-nent in the different techniques. If the frequency is changed,the compensation performance is affected as well as the tran-sient response of the control scheme. In four-wire systems,unbalance in the load current also affects the generations ofthe adequate current reference. Unbalanced load currents gen-erate a different harmonic spectrum in the dc reference frame,and low-order harmonic components appear in the referencesignal; then the cut-off frequency of the filter must be reduced.

39.3.2.2 Current ModulatorThe effectiveness of an active power filter depends basi-cally on the design characteristics of the current controller,the method implemented to generate the reference templateand the modulation technique used. Most of the modulationtechniques used in active power filters are based on PWMstrategies. In this chapter, four of these methods, whose char-acteristics are their simplicity and effectiveness, are analyzed:periodical sampling control, hysteresis band control, triangu-lar carrier control, and vector control. The first three methodshave been tested with different waveform templates sinu-soidal, quasisquare, and rectifier compensation current andwere compared in terms of the harmonic content and distor-tion at the same switching frequency [3]. The analysis showsthat for sinusoidal current generation the best method is tri-angular carrier, followed by hysteresis band and periodicalsampling. For other types of references, however, one strat-egy may be better than the others. Also it was shown that eachcontrol method is affected in a different way by the switchingtime delays present in the driving circuitry and in the powersemiconductors.

39.3.2.2.1 Periodical Sampling The periodical samplingmethod switches the power transistors of the converter duringthe transitions of a square wave clock of fixed frequency (thesampling frequency). As shown in Fig. 39.21, this type of con-trol is very simple to implement since it requires a comparatorand a D-type flip-flop per phase. The main advantage of this

QD

flip-flopCLK

+I_line

I_refPWM

sampling clock

−

FIGURE 39.21 Control modulator block for periodical samplingmethod.

method is that the minimum time between switching transi-tions is limited to the period of the sampling clock. However,the actual switching frequency is not clearly defined.

39.3.2.2.2 Hysteresis band The hysteresis band methodswitches the transistors when the current error exceeds a fixedmagnitude: the hysteresis band. As can be seen in Fig. 39.22,this type of control needs a single comparator with hystere-sis per phase. In this case the switching frequency is notdetermined, but it can be estimated.

hysteresis band adjust

PWMI_line

I_ref+

−

FIGURE 39.22 Control modulator block for hysteresis band.

39.3.2.2.3 Triangular Carrier The triangular carrier method,shown in Fig. 39.23, compares the current error with fixedamplitude and fixed frequency triangular wave (the triangularcarrier). The error is processed through a proportional-integral (PI) gain stage before the comparison with the tri-angular carrier takes place. As can be seen, this control schemeis more complex than the periodical sampling and hysteresisband. The values for the PI control gain kp and ki deter-mine the transient response and steady-state error of thetriangular carrier method. It was found empirically that thevalues for kp and ki shown in Eqs. (39.27) and (39.28) give agood dynamic performance under transient and steady-stateoperating conditions.

k∗p = (L + Lo) · ωc

2Vdc(39.27)

k∗i = ωc k∗

p (39.28)

where L+Lo is the total series inductance seen by the converter,ωc is the triangular carrier frequency, whose amplitude is onevolt peak-peak, and Vdc is the dc supply voltage of the inverter.

kp + ki/s+− PWM

V_tri

I_line

I_ref

I_err

+

−

FIGURE 39.23 Control modulator block for triangular carrier method.


39.3.2.2.4 Vector Control Technique This current controltechnique proposed in [4] divides the α–β reference frameof currents and voltages in six regions, phase shifted by 30(Fig. 39.24), identifies the region where the current vector errori is located, and selects the inverter output voltage vector Vinv

that will force i to change in the opposite direction, keepingthe inverter output current close to the reference signal.

Figure 39.25 shows the single-phase equivalent circuit of theshunt active power filter connected to a non-linear load andto the power supply.

The equation that relates the active power filter currents andvoltages is obtained by applying Kirchhoff law to the equivalentcircuit shown in Fig. 39.25:

Vinv = Ldigen

dt+ E0 (39.29)

The current error vector i is defined by the followingexpression:

i = iref − igen (39.30)

α

β

1

2

3

4

5

6

V1

V2

V3

V4

V5

V6

α

β

I

II

IIIIV

V

VI

(a) (b)

FIGURE 39.24 The hexagons defined in the α–β reference frame by thecurrent control scheme: (a) the hexagon defined by the inverter outputcurrent vector and (b) the hexagon defined by the inverter output voltagevector.

Active PowerFilter

Non-linearLoad

PowerSupply

Ls

LL

Vinv

E0

Igen

FIGURE 39.25 The single-phase equivalent circuit of a shunt activepower filter connected to the power system.

where iref represents the inverter reference current vectordefined by the instantaneous reactive power concept. Byreplacing Eq. (39.30) in Eq. (39.29):

Vinv = Ld

dt(iref − i) + E0 ⇒ L

di

dt= L

diref

dt+ E0 − Vinv

(39.31)

If E = L(diref /dt ) + E0 then Eq. (39.31) becomes

Ldi

dt= E − Vinv (39.32)

Equation (39.32) represents the active power filter stateequation and shows that the current error vector variationdi/dt is defined by the difference between the fictitious volt-age vector E and the inverter output voltage vector Vinv. Inorder to keep di/dt close to zero, Vinv must be selectednear E0.

The selection of the inverters’ gating signals is defined bythe region in which i is located and by its amplitude. Inorder to improve the current control accuracy and associatetime response, depending on the amplitude of i the followingactions are defined:

– if i ≤ δ the gating signals of the inverter are not changed,– if h ≤ i ≤ δ, the inverter gating signals are defined

following Mode a,– if i > h, the inverter gating signals are defined following

Mode b;

where δ and h are reference values that define the accuracy andthe hysteresis window of the current control scheme.

39.3.2.2.5 Mode a: Small Changes in i (h ≤ i ≤ δ) Theselection of the inverter switching Mode a can be explainedwith the following example. Assuming that the voltage vector Eis located in region I (Fig. 39.26a) and the current error vectori is in region 6 (Fig. 39.26b), the inverter voltage vectors,Vinv, located closest to E are V1 and V2. The vectors E−V2 andE−V1 define two vectors Ldi/dt, located in region III and Vrespectively, as shown in Fig. 39.26a, so in order to reduce thecurrent vector error i, Ldi/dt must be located in regionIII. Thus the inverter output voltage has to be equal to V1. Inthis way i will be forced to change in the opposite directionreducing its amplitude faster. By doing the same analysis forall the possible combinations, the inverter switching modes foreach location of i and E can be defined (Table 39.4).

Vk represents the inverter switching functions defined inTable 39.5.

39.3.2.2.6 Mode b: Large Changes in i (i > h) If ibecomes larger than h in a transient state, it is necessary tochoose the switching mode in which the di/dt has the largest


V1

V2

V3

V4

V5

V6

I

II

IIIIV

V

VI

E

E-V1

E-V2

1

2

3

4

5

6

∆i

(a) (b)

FIGURE 39.26 Selection of the inverter switching pattern according tothe region where i and E are located: (a) the regions where Ldi/dt arelocated (b) the region where i is located.

TABLE 39.4 Inverter switching modes

E region i region

1 2 3 4 5 6

I V1 V2 V2 V0 − V7 V0 − V7 V1II V2 V2 V3 V3 V0 − V7 V0 − V7III V0 − V7 V3 V3 V4 V4 V0 − V7IV V0 − V7 V0-V7 V4 V4 V5 V5V V6 V0 − V7 V0 − V7 V5 V5 V6VI V1 V1 V0 − V7 V0 − V7 V6 V6

TABLE 39.5 Relationship between switching function and inverteroutput voltage

k Switch onphase a

Switch onphase b

Switch onphase c

Inverter output voltage Vk

0 4 6 2 01 1 6 2 2

3 Vdc

2 1 3 2 23 Vdc e jπ /3

3 4 3 2 23 Vdc e j2π /3

4 4 3 5 23 Vdc e jπ

5 4 6 5 23 Vdc e j4π /3

6 1 6 5 23 Vdc e j5π /3

7 1 3 5 0

opposite direction to i. In this case the best inverter out-put voltage Vinv corresponds to the value located in the sameregion of i.

The switching frequency may be fixed by controlling thetime between commutations and not applying a new switchingpattern if the time between two successive commutations islower than a selected value (t = 1/2fc ).

Figure 39.27 shows the block diagram of the inverter vec-tor current control scheme implemented in a microcontroller.In Fig. 39.27 ∗E represents the region where the vector E islocated, ∗i, the region of i, k1 keeps the same value of k(no commutation in the inverter), k2 selects the new inverteroutput voltage from Table 39.4, and k3 selects Vinv in the sameregion of i.

39.3.2.3 Control Loop DesignActive power filters based on self-controlled dc bus voltagerequires two control loops, one to control the inverter out-put current and the other to regulate the inverter dc voltage.Different design criteria have been presented in the techni-cal literature; however, a classic design procedure using a PIcontroller will be presented in this chapter. In general, thedesign procedure for the current and voltage loops is based onthe respective time response requirements. Since the transientresponse of the active power is determined by the current con-trol loop, its time response has to be fast enough to follow thecurrent reference waveform closely. On the other hand, thetime response of the dc voltage does not need to be fast and isselected to be at least 10 times slower than the current controlloop time response. Thus, these two control systems can bedecoupled and designed as two independent systems.

A PI controller is normally used for the current and thevoltage control loops since it contributes to zero steady-stateerror in tracking the reference current and voltage signals,respectively.

39.3.2.3.1 Design of the Current Control Loop The designof the current control loop gains depends on the selected cur-rent modulator. In the case of selecting the triangular carriertechnique, to generate the gating signals, the error betweenthe generated current and the reference current is processedthrough a PI controller, then the output current error is com-pared with a fixed amplitude and fixed frequency triangularwave (Fig. 39.23) The advantage of this current modulatortechnique is that the output current of the converter has well-defined spectral line frequencies for the switching frequencycomponents.

Since the active power filter is implemented with a voltage-source inverter, the ac output current is defined by the inverterac output voltage. The block diagram of the current controlloop for each phase is shown in Fig. 39.28 whereE phase-to-neutral source voltage,Z(s) impedance of the link reactor,Ks gain of the converter, andGc(s) gain of the controller.

The values of Ks and Gc(s) are given in Eqs. (39.33) and(39.34).

Ks = Vdc

2ξ(39.33)


∆i

h

k2

k3

fsw

iref

ic

*E

*∆i

k1

+

+

+

−

fref+

−

Determinesthe Region

of "E"

Determinesthe Region

of ∆i

Amplitudecomparator

δ < ∆i < hVinv selectedfrom Table I

Selects k1, k2 or k3

Defines Switching Mode

>SourceVoltageEo

>

∆i > hVinv same

Region than ∆i

E=L(diref /dt)+EoPrevious k2

α

δ

FIGURE 39.27 The current control block diagram.

Gc(s) Ks1

Z(s)

+

−

Iref

Igen

+

−

E

Igen

FIGURE 39.28 The block diagram of the current control loop.

Gc (s) = Kp + Ki

s(39.34)

From Fig. 39.27 and Eq. (39.34), the following expression isobtained:

Igen =[Ks(Kp + (Ki/s)

)]/(Rr + sLr )

1 + [Ks(Kp + (Ki/s)

)/(Rr + sLr )

] Iref

− 1/(Rr + sLr )

1 + (Ks((Kp + Ki/s)

)/(Rr + sLr )

)E (39.35)

The characteristic equation of the current loop is given by

1 +(Kps + Ki/s

)s(Rr + sLr )

(39.36)

The analysis of the characteristic equation proves that thecurrent control loop is stable for all values of Kp and Ki . Also,this analysis shows that Kp determines the speed response andKi defines the damping factor of the control loop. If Kp is toobig, the error signal can exceed the amplitude of the triangular

waveform, affecting the inverter switching frequency, and ifKi is too small, the gain of the PI controller decreases, whichmeans that the generated current will not be able to follow thereference current closely. The active filter transient responsecan be improved by adjusting the gain of the proportionalpart (Kp) to equal one and the gain of the integrator (Ki) toequal the frequency of the triangular waveform.

39.3.2.3.2 DC Voltage Control Loop Voltage control of thedc bus is performed by adjusting the small amount of realpower flowing into the dc capacitor, thus compensating for theconduction and switching losses. The voltage loop is designedto be at least 10 times slower than the current loop, hencethe two loops can be considered decoupled. The dc voltagecontrol loop need not to be fast, since it only responds forsteady-state operating conditions. Transient changes in the dcvoltage are not permitted and are taken into consideration withthe selection criteria of the appropriate electrolytic capacitorvalue.

39.3.3 Power Circuit Design

The selection of the ac link reactor and the dc capacitor valuesaffects directly the performance of the active power filter. Staticvar compensators implemented with voltage-source inverterspresent the same power circuit topology, but for this type ofapplication, the criteria used to select the values of Lr and Care different. For reactive power compensation, the design ofthe synchronous link inductor, Lr , and the dc capacitor, C,is performed based on harmonic distortion constrain. Thatis, Lr must reduce the amplitude of the current harmonics


generated by the inverter, while C must keep the dc voltageripple factor below a given value. This design criteria cannotbe applied in the active power filter since it must be able togenerate distorted current waveforms. However, Lr must bespecified so that it keeps the high-frequency switching rippleof the inverter ac output current smaller than a defined value.

39.3.3.1 Design of the Synchronous Link ReactorThe design of the synchronous link reactor depends on thecurrent modulator used. The design criteria presented in thissection is based considering that the triangular carrier modula-tor is used. The design of the synchronous reactor is performedwith the constraint that for a given switching frequency theminimum slope of the inductor current is smaller than theslope of the triangular waveform that defines the switchingfrequency. In this way, the intersection between the currenterror signal and the triangular waveform will always exist. Inthe case of using another current modulator, the design criteriamust allow an adequate value of Lr in order to ensure that thedi/dt generated by the active power filter will be able to followthe inverter current reference closely. In the case of the trian-gular carrier technique, the slope of the triangular waveform,λ, is defined by

λ = 4ξ ft (39.37)

where ξ is the amplitude of the triangular waveform, whichhas to be equal to the maximum permitted amount of ripplecurrent, and ft is the frequency of the triangular waveform (i.e.the inverter switching frequency). The maximum slope of theinductor current is equal to

diLdt

= Van + 0.5Vdc

Lr(39.38)

Since the slope of the inductor current (diL/dt) has to besmaller than the slope of the triangular waveform (λ), and theripple current is defined, from Eqs. (39.37) and (39.38), as

Lr = Van + 0.5Vdc

4ξ ft(39.39)

39.3.3.2 Design of the DC CapacitorTransient changes in the instantaneous power absorbed bythe load, generate voltage fluctuations across the dc capacitor.The amplitude of these voltage fluctuations can be controlledeffectively with an appropriate dc capacitor value. It must benoticed that the dc voltage control loop stabilizes the capacitorvoltage after a few cycles, but is not fast enough to limit the firstvoltage variations. The capacitor value obtained with this cri-teria is bigger than the value obtained based on the maximumdc voltage ripple constraint. For this reason, the voltage acrossthe dc capacitor presents a smaller harmonic distortion factor.

The maximum overvoltage generated across the dc capacitoris given by

VC max = 1

C

∫ θ2/ω

θ1/ωiC (t )dt + Vdc (39.40)

where VCmax is the maximum voltage across the dc capacitor,Vdc is the steady-state dc voltage, and iC (t) is the instantaneousdc bus current. From Eq. (39.40)

C = 1

V

∫ θ2/ω

θ1/ωiC (t )dt (39.41)

Eq. (39.41) defines the value of the dc capacitor, C, that willmaintain the dc voltage fluctuation below V p.u. The instan-taneous value of the dc current is defined by the product of theinverter line currents with the respective switching functions.The mean value of the dc current that generates the maximumovervoltage can be estimated by

∫ θ2/ω

θ1/ωiC (t )dt = Iinv

∫ θ2/ω

θ1/ω

[sin(ωt ) + sin(ωt + 120)

]dt

(39.42)

In this expression the inverter ac current is assumed tobe sinusoidal. These operating conditions represent the worstcase.

39.3.4 Technical Specifications

The standard specifications of shunt active power filters arethe following:

• Number of phases: three-phase and three wires or three-phases and four wires (in case neutral currents need tobe compensated).

• Input voltage: 200, 210, 220 ±10%, 400, 420, 440 ±10%,6600 ±10%.

• Frequency: 50/60 Hz ±5%.• Number of restraint harmonic orders: 2–25th.• Harmonic restraint factor: 85% or more at the rated

output.• Type of rating: continuous.• Response: 1 ms or less.

For shunt active power filter the harmonic restraint factor isdefined as

[1 − (IH2 /IH1

)]×100%, where IH1 are the harmoniccurrents flowing on the source side when no measure are takenfor harmonic suppression, and IH2 are the harmonic currentsflowing on the source side when harmonics are suppressedusing an active filter.


39.4 Series Active Power Filters

Series active power filters were introduced by the end ofthe 1980s [5], and operate mainly as a voltage regulatorand harmonic isolator between the non-linear load and theutility system. The series-connected active power filter ismore preferable to protect the consumer from an inadequatesupply voltage quality. This type of approach is specially rec-ommended for compensation of voltage unbalances, voltagedistortion, and voltage sags from the ac supply, and for lowpower applications represents an economically attractive alter-native to UPS, since no energy storage (battery) is necessaryand the overall rating of the components is smaller. The seriesactive power filter injects a voltage component in series withthe supply voltage and therefore can be regarded as a con-trolled voltage source, compensating voltage sags and swellson the load side (Fig. 39.29).

If passive LC filters are connected in parallel to the load,the series active power filter operates as an harmonic isolatorforcing the load current harmonics to circulate mainly throughthe passive filter rather than the power distribution system(hybrid topology) (Fig. 39.30). The main advantage of thisscheme is that the rated power of the series active power filter isa small fraction of the load kVA rating, typically 5%. However,

Load

VLoad

ACSupply

Vsys.

CompensationVoltage

SeriesActive Filter

FIGURE 39.29 The series active power filter operating as a voltagecompensator.

VLoadAC

SupplyVsys.

+−Vak

Isys. Vak

PassiveFilter

IFil.

Non-linearLoad

ILoad

Series ActiveFilter

FIGURE 39.30 Combination of series active power filter and passivefilter for current harmonic compensation.

the rated apparent power of the series active power filter mayincrease, in case voltage compensation is required.

39.4.1 Power Circuit Structure

The topology of the series active power filter is shown inFig. 39.31. In most cases, the power circuit configuration isbased on a three-phase PWM voltage-source inverter con-nected in series with the power lines through three single-phasecoupling transformers. For certain type of applications, thethree-phase PWM voltage-source converter can be replacedby three single-phase PWM inverters. However, this type ofapproach requires more power components, which increasesthe cost.

In order to operate as an harmonic isolator, a parallel LCfilter must be connected between the non-linear loads andthe coupling transformers (Fig. 39.30). Current harmonicand voltage compensation are achieved by generating theappropriate voltage waveforms with the three-phase PWMvoltage-source inverter, which are reflected in the power sys-tem through three coupling transformers. With an adequatecontrol scheme, series active power filters can compensatefor current harmonics generated by non-linear loads, voltageunbalances, voltage distortion, and voltage sags or swells atthe load terminals. However, it is very difficult to compensatethe load power factor with this type of topology. In four-wirepower distribution systems, series active power filters with thepower topology can also compensate the current harmoniccomponents that circulate through the neutral conductor.

39.4.2 Principles of Operation

Series active power filters compensate current system dis-tortion caused by non-linear loads by imposing a highimpedance path to the current harmonics, which forces thehigh-frequency currents to flow through the LC passive fil-ter connected in parallel to the load (Fig. 39.30). The highimpedance imposed by the series active power filter is cre-ated by generating a voltage of the same frequency that thecurrent harmonic component needs to be eliminated. Voltageregulation or voltage unbalance can be corrected by compen-sating the fundamental frequency positive, negative, and zerosequence voltage components of the power distribution system(Fig. 39.29) In this case, the series active power filter injectsa voltage component in series with the supply voltage andtherefore can be regarded as a controlled voltage source, com-pensating voltage regulation on the load side (sags or swells),and voltage unbalance. Voltage injection of arbitrary phasewith respect to the load current implies active power trans-fer capabilities which increases the rating of the series activepower filter, and in most cases requires an energy storage ele-ment connected in the dc bus. Voltage and current waveforms


Van Ls C/T

Vcn Ls C/T

Vbn Ls C/TNonlinear

Single Phaseor

Three PhaseLoads

Three-Phase PWM-VSI(Active Power Filter)

Ripple Filter

PassiveFilter

FIGURE 39.31 The series active power filter topology.

200V

0V

−200V120ms 140ms 160ms 180ms 200ms

FIGURE 39.32 Load voltage waveforms for voltage unbalance compen-sation. Phase-to-neutral voltages at the load terminals before and afterseries compensation. (Compensation starts at 140 ms, current harmoniccompensator not operating.)

shown in Figs. 39.32, 39.33, and 39.34 illustrate the compen-sation characteristics of a series active power filter operatingwith a shunt passive filter.

39.4.3 Power Circuit Design

The power circuit topology of the series active power filter iscomposed by the three-phase PWM voltage-source inverter,the second-order resonant LC filters, the coupling transform-ers, and the secondary ripple frequency filter (Fig. 39.30). Thedesign characteristics for each of the power components aredescribed below.

80ms 100ms 120ms 140ms 160ms 180ms 200ms

Time

200A

0A

−200A

4.0A

0.0A

−4.0A

(a)

(b)

FIGURE 39.33 System current waveforms for current harmonic com-pensation: (a) neutral current flowing to the ac mains before and aftercompensation and (b) line currents flowing to the ac mains before andafter compensation. (Voltage unbalance compensator not operating.)

39.4.3.1 PWM Voltage-source InverterSince series active power filter can compensate voltage unbal-ance and current harmonics simultaneously, the rated powerof the PWM voltage-source inverter increases compared withother approaches that compensate only current harmonics,since voltage injection of arbitrary phase with respect to theload current implies active power transfer from the inverterto the system. Also, the transformer leakage inductance entailsfundamental voltage drop and apparent power, which has tobe supported by the inverter, reducing the series active filter


60ms 80ms 100ms 120ms 140ms 160ms 180ms 200ms

Time

50

−50

200

−200

50

−50

(a)

(b)

(c)

FIGURE 39.34 Load voltages and system currents for voltage unbalanceand current harmonic compensation, before and after compensation:(a) power system neutral current; (b) phase-to-neutral load voltages; and(c) power system line current.

inverter rating available for harmonic and voltage compensa-tion. The rated apparent power required by the inverter can beobtained by calculating the apparent power generated in theprimary of the coupling transformers. The voltage reflectedacross the primary winding of each coupling transformer isdefined in Eq. (39.43).

Vseries =K 2

1

∑k =1

I 2sk

1/2

+ K 22 V2 + V02

1/2

(39.43)

where Vseries is the rms voltage across the primary wind-ing of the coupling transformer. Equation (39.43) shows thatthe voltage across the primary winding of the transformer isdefined by two terms. The first one is inversely proportionalto the quality factor of the passive LC filter, while the sec-ond one depends on the voltage unbalance that needs to becompensated. K1 depends on the LC filter values while K2

is equal to one. The current flowing through the primarywinding of the coupling transformer, due to the harmoniccurrents (Eq. (39.44)), can be obtained from the equivalentcircuit shown in Fig. 39.35.

Isk = Zfk Ilk

Zfk + Zsk + K1(39.44)

where Vseries = −K1Isk . The fundamental component of theprimary current depends on the amplitude of the negative andzero sequence component of the source voltage due to thesystem unbalance.

Zsk Isk

VseriesIfk

L f

Cf

ILoadk

FIGURE 39.35 The equivalent circuit of the series active power filterfor harmonic components.

39.4.3.2 Coupling TransformerThe purpose of the three coupling transformers is not only toisolate the PWM inverters from the source but also to matchthe voltage and current ratings of the PWM inverters withthose of the power distribution system. The total apparentpower required by each coupling transformer is one-third thetotal apparent power of the inverter. The turn ratio of thecurrent transformer is specified according to the inverter dcbus voltage, K1 and Vref . The correct value of the turn ratio “a”must be specified according to the overall series active powerfilter performance. The turn ratio of the coupling transformermust be optimized through the simulation of the overall activepower filter, since it depends on the values of different relatedparameters. In general, the transformer turn ratio must be highin order to reduce the amplitude of the inverter output currentand to reduce the voltage induced across the primary winding.Also, the selection of the transformer turn ratio influences theperformance of the ripple filter connected at the output of thePWM inverter. Taking into consideration all these factors, ingeneral, the transformer turn ratio is selected equal to 1:20.

39.4.3.3 Secondary Ripple FilterThe design of the ripple filter connected in parallel to thesecondary winding of the coupling transformer is performedfollowing the method presented by Akagi in [6]. However, itis important to notice that the design of the secondary ripplefilter depends mainly on the coupling transformer turn ratioand the current modulator used to generate the inverter gat-ing signals. If the triangular carrier is used, the frequency ofthe triangular waveform has to be considered in the design ofthe ripple filter. The ripple filter connected at the output of theinverter avoid the induction of the high-frequency ripple volt-age generated by the PWM inverter switching pattern at theterminals of the primary winding of the coupling transformer.In this way, the voltage applied in series to the power sys-tem corresponds to the components required to compensatevoltage unbalanced and current harmonics. The single-phaseequivalent circuit is shown in Fig. 39.36.


Lfr

CfrZsysVinv

FIGURE 39.36 The single-phase equivalent circuit of the inverteroutput ripple filter.

The voltage reflected in the primary winding of the cou-pling transformer has the same waveform as that of the voltageacross the filter capacitor. For low-frequency components, theinverter output voltage must be almost equal to the voltageacross Cfr . However, for high-frequency components, most ofthe inverter output voltage must drop across Lfr , in which casethe voltage at the capacitor terminals is almost zero. Moreover,Cfr and Lfr must be selected in order not to exceed the burdenof the coupling transformer. The ripple filter must be designedfor the carrier frequency of the PWM voltage-source inverter.To calculate Cfr and Lfr the system equivalent impedance atthe carrier frequency, Zsys , reflected in the secondary must beknown. This impedance is equal to

Zsys(secondary) = a2 · Zsys(primary) (39.45)

For the carrier frequency, the following design criteria mustbe satisfied:

(i) XCfr XLfr – to ensure that at the carrier fre-quency most of the inverter outputvoltage will drop across Lfr .

(ii) XCfr and XLfr Zsys – to ensure that the voltage divideris between Lfr and Cfr .

The small-rated LC passive filter exhibits a high qualityfactor circuit because of the high impedance on the out-put side. Oscillation between the small-rated inductor andcapacitor may occur, causing undesirable high-frequency volt-age across the ripple filter capacitor, which is reflected inthe primary winding of the coupling transformer generatinghigh-frequency current to flow through the power distributionsystem. It is important to note that this oscillation is very dif-ficult to eliminate through the design and selection of the Lfr

and Cfr values. However, it can be eliminated with the additionof a new control loop. The cause of the output voltage oscil-lation is explained with the help of Fig. 39.37. The transferfunction Gp(s) between the input voltage Vi(s) and the outputvoltage VC (s) is given by the equation:

Gp(s) = ω2n

s2 + 2ξωns + ω2n

(39.46)

where ωn = √1/Lfr Cfr and ξ = rL/2√

Cfr /Lfr .

Vi VC

Lfr Rfr

Cfr Is

ILfr

ICfr

FIGURE 39.37 Single-phase equivalent circuit of series active powerfilter connected to the ripple filter.

Normally, the damping factor ξ is smaller than one, causingthe voltage oscillation across the capacitor ripple filter, Cfr .Generally, relatively low impedance loads are connected to theoutput terminals of voltage-source PWM inverters. In thesecases, the quality factor of the LC filter can be low, and theoscillation between the inductor Lfr and the capacitor Cfr isavoided [7].

39.4.3.4 Passive FiltersPassive filters connected between the non-linear load and theseries active power filter play an important role in the com-pensation of the load current harmonics. With the connectionof the passive filters the series active power filter operates as anharmonic isolator. The harmonic isolation feature reduces theneed for precise tuning of the passive filters and allows theirdesign to be insensitive to the system impedance and elimi-nates the possibility of filter overloading due to supply voltageharmonics. The passive filter can be tuned to the dominantload current harmonic and can be designed to correct the loaddisplacement power factor.

However, for industrial loads connected to stiff supply, it isdifficult to design passive filters that can absorb a significantpart of the load harmonic current and therefore its effective-ness deteriorates. Specially, for compensation of diode rectifiertype of loads, where a small kVA passive filter is required,it is difficult to achieve the required tuning to absorb sig-nificant percentage of the load harmonic currents. For thistype of application, the passive filter cannot be tuned exactlyto the harmonic frequencies because they can be overloadeddue to the system voltage distortion and/or system currentharmonics.

The single-phase equivalent circuit of a passive LC filterconnected in parallel to a non-linear current source and to thepower distribution system is shown in Fig. 39.38. From thisfigure, the design procedure of this filter can be derived. Theharmonic current component flowing through the passive fil-ter ifh and the current component flowing through the source


Ifh

Lf

Cf

Rf

LS

RS

ISh

Ih

FIGURE 39.38 The single-phase equivalent circuit of the passive filterconnected to a non-linear load.

ish are given by the following expressions:

ifh = Zs

Zs + Zfih ; ish = Zf

Zs + Zfih (39.47)

At the resonant frequency the inductive reactance of thepassive filter is equal to the capacitive reactance of the filter,that is:

2π frL = 1

2π frC(39.48)

Therefore, the resonant frequency of the passive filter isequal to:

fr = 1

2π√

LC(39.49)

The passive filter band width is defined by the upper andlower cut-off frequency, at which values the filter current gainis −3 dB, as shown in Fig. 39.39.

The magnitude of the passive filter impedance as a functionof the frequency is shown in Fig. 39.40.

At the resonant frequency the passive filter magnitude isequal to the resistance. If the resistance is zero, the filter is in

1

IfhIh

0.707

fr

BW

f [Hz]

FIGURE 39.39 The passive filter bandwidth.

C Y =

4

32

11

Capacitive Mode

−3 −2 −1 1 2 3 X= Qδ

Inductive Mode

Zω QZ0

L

R

FIGURE 39.40 The frequency response of the passive LC filter.

short circuit. The quality factor of the passive filter is definedby the following expression:

Q = ωnL

R(39.50)

It is important to note that the operation of the series activepower filter with off-tuned passive filter has an adverse impacton the inverter power rating compared to the normal case.The more off-tuned the passive filter, the more rated apparentpower is required by the series active power filter.

39.4.3.5 Protection RequirementsShort circuits in the power distribution system generate largecurrents that flow through the power lines until the circuitbreaker operates clearing the fault. The total clearing time of ashort circuit depends on the time delay imposed by the protec-tion system. The clearing time cannot be instantaneous due tothe operating time imposed by the overcurrent relay and by thetotal interruption time of the power circuit breaker. Althoughpower system equipment, such us power transformers, cables,etc. are designed to withstand short-circuit currents duringat least 30 cycles, the active power filter may suffer severedamage during this short time. The withstand capability of theseries active power filter depends mainly on the inverter powersemiconductor characteristics.

Since the most important feature of series active power fil-ters is the small rated power required to compensate the powersystem, typically 10–15% of the load rated apparent power,the inverter semiconductors are rated for low values of block-ing voltages and continuous currents. This makes series activepower filters more vulnerable to power system faults.

The block diagram of the protection scheme described inthis section is shown in Fig. 39.41. It consists of a varistor con-nected in parallel to the secondary winding of each couplingtransformer, and a couple of antiparallel thyristors [8]. A spe-cial circuit detects the current flowing through the varistorsand generates the gating signals of the antiparallel thyristors.


Control CircuitSw's

Iprim

Isec

Ivar

Iinv

IRsw Rsw

sw1 sw2

FIGURE 39.41 The series active power filter protection scheme.

The protection circuit of the series active power filter mustprotect only the PWM voltage-source inverter connected tothe secondary of the coupling transformers and must notinterfere with the protection scheme of the power distributionsystem. Since the primary of the active power filter couplingtransformers are connected in series to the power distribu-tion system, they operate as current transformers, so thattheir secondary windings cannot operate in open circuit. Forthis reason, if a short circuit is detected in the power distri-bution system, the PWM voltage-source inverter cannot bedisconnected from the secondary of the current transformer.Therefore, the protection scheme must be able to limit theamplitude of the currents and voltages generated in the sec-ondary circuits. This task is performed by the varistors and bythe magnetic saturation characteristic of the transformers.

The main advantages of the series active power filterprotection scheme described in this section are the following:

(i) it is easy to implement and has a reduced cost,(ii) it offers full protection against power distribution

short-circuit currents, and(iii) it does not interfere with the power distribution

system.

When short-circuit currents circulate through the powerdistribution system, the low saturation characteristic of thetransformers increases the current ratio error and reduces theamplitude of the secondary currents. The larger secondaryvoltages induced by the primary short-circuit currents areclamped by the varistors, reducing the amplitude of the PWMvoltage-source inverter currents. After a few cycles of dura-tion of the short circuit, the PWM voltage-source inverteris bypassed through a couple of antiparallel thyristors, andat the same time the gating signals applied to the PWMvoltage-source inverter are removed. In this way, the PWMvoltage-source inverter can be turned off. The principles ofoperation and the effectiveness of the protection scheme areshown in Fig. 39.42.

The secondary short-circuit currents will circulate throughthe antiparallel thyristors and the varistors until the fault is

cleared by the protection equipment of the power distributionsystem.

By using the protection scheme described in this sub-section, the voltage and currents reflected in the secondaryof the coupling transformers are significantly reduced. Whenshort-circuit currents circulate through the power distribu-tion system, the low saturation characteristic of the couplingtransformers increases the current ratio error and reduces theamplitude of the secondary voltages and currents. Moreover,the saturated high secondary voltages induced by the primaryshort-circuit currents are clamped by the varistors, reducingthe amplitude of the PWM voltage-source inverter ac currents.Once the secondary current exceeds a predefined referencevalue, the PWM voltage-source inverter is bypassed througha couple of antiparallel thyristors, and then the gating signalsapplied to the PWM voltage-source inverter are removed. Theeffectiveness of the protection scheme is shown in Fig. 39.43.

By increasing the current ratio error due to the magnetic sat-uration, the energy dissipated in the secondary of the couplingtransformer is significantly reduced. The total energy dissi-pated in the varistor for the different protection conditions isshown in Fig. 39.44.

39.4.4 Control Issues

The block diagram of a series active power filter control schemethat compensates current harmonics and voltage unbalancesimultaneously is shown in Fig. 39.45.

Current and voltage reference waveforms are obtained byusing the Park transformation (instantaneous reactive powertheory) voltage unbalance is compensated by calculating thenegative and zero sequence fundamental components of thesystem voltages. These voltage components are added to thesource voltages through the coupling transformers compen-sating the voltage unbalance at the load terminals. In orderto reduce the amplitude of the current flowing through theneutral conductor, the zero sequence components of the linecurrents are calculated. In this way, it is not necessary to sensethe current flowing through the neutral conductor.

39.4.4.1 Reference Signals GeneratorThe compensation characteristics of series active power fil-ters are defined mainly by the algorithm used to generatethe reference signals required by the control system. Thesereference signals must allow current and voltage compensa-tion with minimum time delay. Also it is important that theaccuracy of the information contained in the reference signalsallows the elimination of the current harmonics and voltageunbalance present in the power system. Since the voltage and


0s 20ms 40ms 60ms 80ms 100ms

Time

200V

0V

−200V

2.0KA

0A

−2.0KA(a)

(b)

0s 20ms 40ms 60ms 80ms 100ms

Time

(c)

(d)

(e)

100A

−100A

80A

−0A

100A

−100A

FIGURE 39.42 Current waveforms of the series active power filter during a three-phase short circuit in the power distribution system set at 50 ms:(a) power distribution system line current; (b) current transformer secondary voltage; (c) inverter ac current; (d) current through an inverter leg; and(e) inverter dc bus current.

0s 20ms 40ms 60ms 80ms 100ms

Time

10A

−10A

20A

−20A

10A

−10A(a)

(b)

(c)

FIGURE 39.43 Current waveforms for a line-to-line short circuit in the power distribution system. The protection scheme is implemented with thevaristor, a couple of antiparallel thyristors, and a coupling transformer with low saturation characteristic: (a) current through the varistor; (b) currentthrough the thyristors; and (c) inverter ac current.


A B C

120

155

0

20

40

60

80

100

120

Type of Protection

Energy [Joule]

FIGURE 39.44 The total energy dissipated in the varistor during thepower system short circuit for different protection schemes implemen-tations: (a) only with the varistor used; (b) the varistor is connectedin parallel to a bidirectional switch; and (c) with the complete pro-tection scheme operating (including the coupling transformer with lowsaturation characteristics).

current control scheme are independent, the equations usedto calculate the voltage reference signals are the following:

va0

va1

va2

= 1√

3

1 1 1

1 a a2

1 a2 a

.

va

vb

vc

(39.51)

The voltages va, vb, and vc correspond to the power sys-tem phase-to-neutral voltages before the current transformer.The reference voltage signals are obtained by making the pos-itive sequence component, va1 zero and then applying the

ParkTransformation

Calculation ofp and qVa, Vb, Vo

Va, Vb, V0

Ia, Ib, I0 p(t)

q(t)

pref

qrefHigh Pass Filter

Ia Ib Ic

Van

Van

Vbn

Vcn

Vbn Vcn InversePark

Transformation

pref

qrefK

Iaref

Ibref

Icref

Gating

Signals

GeneratorInverse

SequenceComponents

Transformation

− V(−)

− V(0)

− 1

FundamentalVoltage

SequenceComponents

V(−)

V(0)

Voltage Reference Generator for UnbalanceCompensation

Voltage Reference Generator for CurrentHarmonics Compensation

FIGURE 39.45 Compensation scheme for voltage unbalance correction.

inverse of the Fortescue transformation. In this way the seriesactive power filter compensates only voltage unbalance and notvoltage regulation. The reference signals for the voltage unbal-ance control scheme are obtained by applying the followingequations:

vrefa

vrefb

vrefc

= 1√

3

1 1 1

1 a2 a1 a a2

.

−vao

0−va2

(39.52)

In order to compensate current harmonics generated by thenon-linear loads, the following equations are used:

iaref

ibref

icref

=

√2

3·

1 0−12

√3

2−12

−√3

2

·[

vα vβ

−vβ vα

]−1 [pref

qref

]+ 1√

3

i0

i0i0

(39.53)

where i0 is the fundamental zero sequence component of theline current and is calculated using the Fortescue transforma-tion Eq. (39.50).

i0 = 1√3

(ia + ib + ic ) (39.54)

In Eq. (39.40) pref , qref , vα , and vβ are defined accord-ing to the instantaneous reactive power theory. In order toavoid the compensation of the zero sequence fundamentalfrequency current the reference signal i0 must be forced to cir-culate through a high-pass filter generating a current i′0 which


is used to create the reference signal required to compensatecurrent harmonics. Finally, the general equation that definesthe references of the PWM voltage-source inverter required tocompensate voltage unbalance and current harmonics is thefollowing:

Vrefa

Vrefb

Vrefc

= K1

√

2

3

1 0−12

√3

2−12

−√3

2

.

[vα vβ

−vβ vα

]−1 [pref

qref

]

+ 1√3

i′0

i′0i′0

+ K2

1√3

1 1 1

1 a2 a1 a a2

.

−va0

0−va2

(39.55)

where K1 is the gain of the coupling transformer which definesthe magnitude of the impedance for high-frequency currentcomponents, and K2 defines the degree of compensation forvoltage unbalance, ideally K2 equals 1.

39.4.4.2 Gating Signals GeneratorThis circuit provides the gating signals of the three-phasePWM voltage-source inverter required to compensate voltageunbalance and current harmonic components. The currentand voltage reference signals are added and then the ampli-tude of the resultant reference waveform is adjusted in orderto increase the voltage utilization factor of the PWM inverterfor steady-state operating conditions. The gating signals of theinverter are generated by comparing the resultant referencesignal with a fixed frequency triangular waveform. The trian-gular waveform helps to keep the inverter switching frequencyconstant (Fig. 39.46).

A higher voltage utilization of the inverter is obtained if theamplitude of the resultant reference signal is adjusted for thesteady-state operating condition of the series active power fil-ter. In this case, the reference current and reference voltagewaveforms are smaller. If the amplitude is adjusted for tran-sient operating conditions, the required reference signals will

++

=

Voltage Reference

Current Reference

+

−

InverterGatingSignals

AmplitudeAdjustment

TriangularWaveform

(4 kHz)

FIGURE 39.46 The block diagram of the gating signals generator.

have a larger value, creating a higher dc voltage in the inverterand defining a lower voltage utilization factor for steady-stateoperating conditions.

The inverter switching frequency must be higher in ordernot to interfere with the current harmonics that need to becompensated.

39.4.5 Control Circuit Implementation

Since the control scheme of the series active power filter musttranslate the current harmonics components that need to becompensated in voltage signals, a proportional controller isused. The use of a PI controller is not recommended sinceit would modify the reference waveform and generate newcurrent harmonic components. The gain for proportionalcontroller depends on the load characteristics and its valuefluctuates between one and two.

Another important element used in the control scheme isthe filter that allows to generate pref and qref (Fig. 39.35). Thefrequency response of this filter is very important and must notintroduce any phase-shift or attenuation to the low-frequencyharmonic components that must be compensated. A high-pass first-order filter tuned at 15 Hz is adequated. This cornerfrequency is required when single-phase non-linear loads arecompensated. In this case, the dominant current harmonic isthe third. An example of the filter that can be used is shownin Fig. 39.47.

The operator “a” required to calculate the sequence com-ponents of the system voltages (Fortescue transformation) canbe obtained with the all-pass filter shown in Fig. 39.48.

The phase-shift between Vo and Vi is given by the followingexpression:

θ = 2 arctan (2π fR2C) (39.56)

Since the phase-shift is negative, the operator “a” is obtainedby using an inverter (180) and then by tuning θ = −60.

R2

Rf

C1 C2

R1

+

−iiref

FIGURE 39.47 The first-order high-pass filter implemented for thecalculation of pref and qref (C1 = C2 = 0.1 µF, R1 = Rf = 150 k,R2 = 50 k).


vi

R

C

vo = vi −θR2

R1

+−

FIGURE 39.48 The all-pass filter used as a phase shifter.

The operator “a2” is obtained by phase-shifting Vi by−120. Vi is synchronized with the system phase-to-neutralvoltage Van .

39.4.6 Experimental Results

The viability and effectiveness of series active power filters usedto compensate current harmonics was proved in an exper-imental setup of 5 kVA. Current waveforms generated by a

(a) (b) (c)

4 4 4

Ch4 20.0mV M4.00ms 6 Aug 199718:59:16

Ch4 20.0mV M4.00ms 6 Aug 199718:59:49 Ch4 20.0mV M4.00ms 6 Aug 1997

19:00:20

FIGURE 39.49 Experimental results without the operation of the series active power filter: (a) load current; (b) current flowing through the passivefilter; and (c) current through the power supply.

(a) (b) (c)

4 4 4

Ch4 20.0mV M4.00ms 6 Aug 199719:07:02 Ch4 50.0mV M4.00ms 5 Aug 1997

17:28:30Ch4 50.0mV M4.00ms 5 Aug 1997

17:26:24

FIGURE 39.50 Experimental results with the operation of the series active power filter: (a) load current; (b) current flowing through the passivefilter; and (c) current through the power supply.

non-linear load and using only passive filters and the combi-nation of passive and the series active power filter are shownin Figs. 39.49 and 39.50.

These figures show the effectiveness of the series activepower filter, which reduces the overall THD of the currentthat flows through the power system from 28.9% (with theoperation of only the passive filters) to 9% with the operationof the passive filters with the active power filter connected inseries. The compensation of the current that flows through theneutral conductor is shown in Fig. 39.51.

39.5 Hybrid Active Power Filters

Hybrid topologies composed of passive LC filters connectedin series to an active power filter have already been proposedby the end of the eighties [2, 5, 6, 9, 10, 11]. Hybrid topologyimproves significantly the compensation characteristics of sim-ple passive filters, making the use of active power filter availablefor high power applications, at a relatively lower cost. More-over, compensation characteristics of already installed passive


(a) (b)

4

Ch4 50.0mV M10.0ms 7 Aug 199716:31:22

Ch4 50.0mV M10.0ms 5 Aug 199717:25:35

4

FIGURE 39.51 Experimental results of neutral current compensation: (a) current flowing through the neutral without the operation of the seriesactive power filter and (b) current flowing through the neutral with the operation of the active power filter.

Non-linearLoad

PassiveFilters

SystemLSVS

Control Circuit 5a 7a

C/T

Lr

Cr

FIGURE 39.52 The hybrid active power filter configuration.

filters can be significantly improved by connecting a seriesactive power filter at its terminals, giving more flexibility tothe compensation scheme. Most of the technical disadvantagesof passive filters described before can be effectively attenuateif an active power filter is connected in series to the passiveapproach as shown in Fig. 39.52.

The hybrid active power filter topology is shown inFig. 39.52. The active power filter is implemented with athree-phase PWM voltage-source inverter operating at fixedswitching frequency, and connected in series to the passive fil-ter through a coupling transformer. Basically, the active powerfilter forces the utility line currents to become sinusoidal and inphase with the respective phase-to-neutral voltage, improvingthe compensation characteristics of the passive filter.

39.5.1 Principles of Operation

Since the active power filter is connected in series to thepassive filter through a coupling transformer, it imposes a volt-age signal at its primary terminals that forces the circulationof current harmonics through the passive filter, improvingits compensation characteristics, independently of the varia-tions in the selected resonant frequency of the passive filter.The block diagram of the proposed control scheme shown inFig. 39.53 consists of three modules: the dc voltage control,the voltage reference generator, and the inverter gating signalgenerator.

The voltage reference waveform required by the invertercontrol scheme is obtained by adjusting the amplitude of a


LS

VS

L5

C5

K

1−1

L7

C7

Base DriveCircuit

Synchro.

Reference VoltageGenerator


DC Voltage Control

VDC reference

Vm

TriangularReference

VSIPWM

PassiveFilter

Non-linearLoad

Source

SinusoidalWaveformGenerator

VoltageController

IS

ISK

SynchronizationCircuit

FIGURE 39.53 The hybrid active power filter topology and associated control scheme.

sinusoidal reference waveform in phase with the respectivephase-to-neutral voltage and then subtracting the respective acline current (Fig. 39.53). The sinusoidal reference signal can beobtained from the voltage system (in case of low voltage dis-tortion) or it can be generated from an EPROM synchronizedwith the respective phase-to-neutral voltage. The amplitude ofthis reference waveform controls the inverter dc voltage andthe ac mains displacement power factor. The inverter dc volt-age varies according with the amount of real power absorbedby the inverter, while the ac mains power factor depends onthe amount of reactive power generated by the hybrid filter,which can be controlled by changing the amplitude of thefundamental component of the inverter output voltage.

Vch = K ISh

ZF

ZS

IL

VT

ISh IFh

Load

Vc = β VT

VS

C

ZS VT

Passivefilter

at 50Hz

(a) (b)

FIGURE 39.54 Single-phase equivalent circuits of the hybrid active power filter scheme: (a) for current harmonic compensation and (b) fordisplacement power factor compensation.

The principles of operation for current harmonic and powerfactor compensation are explained with the help of the single-phase equivalent circuit shown in Fig. 39.54. In the currentharmonic compensation mode, the active filter improves thefiltering characteristic of the passive filter by imposing a voltageharmonic waveform at its terminals with an amplitude valueequals to:

VCh = K · ISh (39.57)

where ISh is the harmonic content of the line current tobe compensated, and K is the active power filter gain. Ifthe ac mains voltage is purely sinusoidal, the ratio between


the harmonic component of the non-linear load current andthe harmonic component of the ac line current (attenuationfactor) is obtained from Fig. 39.54a and is equal to:

ISh

ILh= ZF

K + ZF + ZS(39.58)

Equation (39.58) shows that the filtering characteristics ofthe hybrid topology (ISh/ILh) depends on the value of the pas-sive filter equivalent impedance, ZF . Moreover, since the tunedfactor, δ, and the quality factor, Q, can modify the filter bandwidth and the passive filter harmonic equivalent impedance(Fig. 39.55), their values must be carefully selected in order tomaintain the compensation effectiveness of the hybrid topol-ogy. In particular, a high value of the quality factor, Q, definesa large band width of the passive filter, improving the com-pensation characteristics of the hybrid topology. On the otherhand, a low value in the quality factor and/or a large value inthe tuned factor increases the required voltage generated bythe active power filter necessary to keep the same compensa-tion effectiveness, which increases the active power filter ratedpower.

Figure. 39.55 shows how the active power filter gain, K, inEq. (39.57) affects the harmonic attenuation factor of the linecurrents. The attenuation factor of the line current harmonicsexpressed in percentage is obtained from Eq. (2), and is shownin Fig. 39.55, for a power distribution system with two passivefilters tuned at fifth and seventh harmonics.

Also, the K factor affects the THD of the line current, as itis shown in Eq. (39.59).

THD i =

√∑h=2

[ILh · (ZF /(ZS + ZF + K ))]2

IS1(39.59)

Equation (39.59) indicates that the total harmonic distor-tion of the line current decreases if K increases. In otherwords, a better hybrid filter compensation is obtained for

f/50

K=0

K=5

K=15

0

20

40

60

80

Atte

nuat

ion

fact

or

100

42 6 8 10 12 14 16 18

FIGURE 39.55 The attenuation factor of the line current harmonics fordifferent frequency values.

0 5 10 15 200

2

4

6

8

10

12

14

16

K

%THD

FIGURE 39.56 System line current THD vs K factor.

larger values of voltage harmonic components generated bythe active power filter. Also, it is shown that the compensationcapability of the hybrid filter depends on the compensationcharacteristic of the passive filter, that is the filter impedancevalue and tuned frequency will affect the active filter ratedpower required to satisfy the system line current compensa-tion requirements. Figure 39.56 shows how the line currentTHD is affected for different values of K, for a power distribu-tion system connected to a high power six-pulses rectifier andpassive filters tuned at the fifth and seventh harmonics.

Displacement power factor correction can be achieved bycontrolling the voltage drop across the passive filter capaci-tor. In order to do that a voltage at fundamental frequencyis generated at the inverter ac terminals, with an amplitudeequals to:

VC = β · VT (39.60)

Displacement power factor control can be achieve since atfundamental frequency the passive filter equivalent impedanceis capacitive. The reactive power generated by the passive filteris obtained by changing the voltage imposed by the activepower filter across the passive filter capacitor terminals. Thepassive filter fundamental current component is defined by thefollowing expression:

iF = Cd

dt(vT − βvT ) = (1 − β) C

dvT

dt= Cγ β

dvT

dt(39.61)

Equation (39.61) proves that the equivalent capacitance atfundamental frequency Cγ , can be modified by changing β.The reactive power generated by the active filter is β timesthe reactive power generated by the passive filter and can bedefined by:

Qγ = VC · IF = βVT IF (39.62)


Equation (39.62) shows that if β > 0 the active power fil-ter generates a voltage at fundamental frequency in phase withVT , reducing the reactive power that flows to the load. If β < 0the active power filter generates a voltage at fundamental fre-quency phase shifted by 180 with respect to VT , increasing thereactive power that flows to the load. In other words, by select-ing a β positive or negative the hybrid topology can generateor absorb reactive power at fundamental frequency, compen-sating for leading or lagging displacement power factor of thenon-linear load. This is achieved by changing the voltage acrossthe passive filter capacitor CF .

39.5.2 The Hybrid Filter CompensationPerformance

The design characteristics of the passive filters have an impor-tant influence on the compensation performance of the hybridtopology. The tuned frequency and quality factor of the passivefilter directly influence the compensation characteristics of thehybrid topology. If these two factors are not properly selected,the rated power of the active filter must be increased in order toget the required compensation performance. The tuned factor,δ, in per unit with respect to the resonant frequency is definedby Eq. (39.63).

δ = ω − ωn

ωn(39.63)

Here, δ, defines the magnitude in which the passive filter res-onant frequency changes due to the variations in the powersystem frequency and modifications in the values of the passivefilter parameters L and C. The values of L and C can changedue to aging conditions, temperature, or design tolerances.The tuning factor, δ, can also be defined as:

δ = f

fn+ 1

2

(L

Ln+ C

Cn

)(39.64)

The quality factor, Q, of the passive filter is defined byEq. (39.65).

Q = X0

R(39.65)

where X0 is equal to:

X0 = ωnL = 1

ωnC=√

L

C(39.66)

The passive filter quality factor, Q, defines the ratio betweenthe passive filter reactance with respect to its resistance. Inorder to have a passive filter low impedance value in a lim-ited frequency band width, defined by the maximum expectedchanges of the rated power system frequency, it is neces-sary to reduce X0 or increase the passive filter resistance R,

reducing the value of Q. However, if R increases, the equiva-lent impedance of the passive filter at the resonant frequencywill increase, with the associated power losses, increasing theoverall passive filter operational cost. On the other hand, thelarger the value of Q, the lower is the passive filter equivalentimpedance at the resonant frequency, increasing the cur-rents harmonic components across the filter, at the resonantoperating point.

39.5.2.1 Effects of the Power System EquivalentImpedance

The influence of the power system equivalent impedance onthe hybrid filter compensation performance is related withits effects on the passive filter, since if the system equivalentimpedance is lower compared to the passive filter equivalentimpedance at the resonant frequency, most of the load currentharmonics will flow mainly to the power distribution system.In order to compensate this negative effect on the hybrid filtercompensation performance, K must be increased, as shown inEq. (39.57), increasing the active power filter rated power.

Figure 39.57 shows how the system equivalent impedanceaffects the relation between the system current THD with theactive filter gain, K, in a power distribution system with passivefilters tuned at the fifth and seventh harmonics. If Zs decreases,the current system THD increases, so in order to keep the samecompensation performance of the hybrid scheme, the activepower filter gain, K, must be increased. On the other hand, ifZs is high, it is not necessary to increase K in order to ensurea low THD value in the system current.

0 5 10 15 200

2

4

6

8

10

12

14

16

K

% T

HD

LS=0.21mH

LS=1.21 mHLS= 4.21 mH

FIGURE 39.57 Relation between the THD of the line current vs theactive power filter gain, K, for different values of the system equivalentimpedance.

39.5.2.2 Effects of the Passive Filter Quality FactorThe quality factor, Q, defines the passive filter bandwidth.A passive filter with a high quality factor, Q, presents


a larger bandwidth and better compensation characteristics.Equation (39.67) defines the quality factor value in terms ofthe passive filter parameters, L, R, and C. This equation showsthat if R or C decreases, Q increases, and the filter bandwithbecomes larger improving the hybrid scheme compensationcharacteristics.

Q = 1

R

√L

C(39.67)

Although by decreasing R or C the passive filter qualityfactor is improved, each element produces a different effectin the hybrid filtering behavior. For example, by increasingR, the filter equivalent impedance at the resonant frequencybecomes bigger, affecting the current harmonic compensationcharacteristics at this specific frequency.

Figure 39.58 illustrates how the system current THDchanges with different values of C, while keeping the filtertuned factor, δ, constant. It is important to note that the largerthe value of C, the better is the compensation characteris-tic of the hybrid scheme. An appropriate value of C must beselected, since C cannot be increased too much. If C is too big,the fundamental component of the filter current will increase,overloading the hybrid filter and generating a large amount ofreactive power to the power distribution system.

0 5 10 15 200

2

4

6

8

10

12

14

16

K

% T

HD

C5=60µF y C7=30µF

C5=80µF y C7=40µF C5=100µF y C7=50µF

FIGURE 39.58 THD of the system line current vs the K factor fordifferent values of filter capacitor.

39.5.2.3 Effects of the Passive Filter Tuned FactorThe tuned factor, δ, is defined in Eq. (39.63). This factor affectsthe hybrid scheme performance especially at the passive filterresonant frequency, since δ defines the changes in the sys-tem frequency, affecting the value of the passive filter resonantfrequency. Figure 39.60 shows how the system current THDchanges with respect to the active power filter gain, K, for dif-ferent values of passive filter tuned factors. The larger the value

0 5 10 15 200

2

4

6

8

10

12

14

16

K

% T

HD

R5=2Ω y R7=2Ω

R5=1Ω y R7=1Ω

R5=0Ω y R7=0Ω

FIGURE 39.59 THD of the system line current vs the K factor fordifferent values of filter resistor.

0 5 10 15 200

2

4

6

8

10

12

14

16

K

% T

HD

δ5=0.1 y δ7=0.07

δ5=0.04 y δ7=0.03

δ5=0 y δ7=0

FIGURE 39.60 THD of the system line current vs the K factor fordifferent values of tuned factor.

of δ, the filtering performance of the hybrid filter becomesworst.

39.5.3 Experimental Results

A laboratory prototype using IGBT switches was implementedand tested in the compensation of a six-pulses controlled rec-tifier. The inverter was operated at 4 kHz switching frequency.Steady-state experimental results are illustrated in Figs. 39.61and 39.62.

In case, resonance is generated between the passive filter andsystem equivalent reactance, the system current THD increasesto 60%. By connecting the active power filter, the line currentTHD is reduced to 4.9%, as shown in Figs. 39.63 and 39.64.


Tek Run: 10.0kS/s Sample Tek Run: 5.00kS/s Hi Res

500mV M5.00ms Ch4 170mV100mV 100 Hz

M2.00ms Ch4 1.00 V

(a) (b)

FIGURE 39.61 Experimental ac line current waveform with passive filtering compensation: (a) line current waveform (THD = 24%) and (b) linecurrent frequency spectrum.

Tek Run: 10.0kS/s Hi Res

500mV M5.00ms Ch4 1.00 V100mV 100 Hz

M2.00ms Ch4 1.00 V

Tek Run: 5.00kS/s Hi Res

(a) (b)

FIGURE 39.62 Experimental ac line current waveform with hybrid filter compensation: (a) line current waveform (THD = 6.3%) and (b) associatedfrequency spectrum.

(a) (b)

Tek Run: 10.0kS/s Hi Res Tek Run: 5.00kS/s Hi Res

1.00 V M5.00ms Ch2 −1.6 V200mV 100 Hz

M2.00ms Ch2 −1.6 V

FIGURE 39.63 Experimental ac line current waveform for resonant compensation: (a) line current waveform (THD = 60%) and (b) associatedfrequency spectrum.


(a)

1.00V

Tek Run: 10.0kS/s Hi Res Tek Run: 5.00kS/s Hi Res

200mV 100 HzM2.00ms Ch2 −1.6 VM5.00ms Ch2 −1.6 V

(b)

FIGURE 39.64 Experimental ac line current waveform with hybrid topology compensation: (a) line current waveform (THD = 4.9%) and(b) associated frequency spectrum.

Acknowledgment

The authors would like to acknowledge the financial sup-port from “FONDECYT” through the 1050067 project.The collaboration of Dr. Víctor Manuel Cárdenas from theUniversity of San Luis Potosí and Pedro Ruminot fromUniversidad de Concepción are also recognized.

Further Reading

1. H. Akagi, Y. Kanzawa, and A. Nabae (1984) Instantaneous reactivepower compensators comprising switching devices without energycomponents. IEEE Trans. Ind. Appl., 20 (3), pp. 625–630.

2. S. Bhattacharaya and D. Divan (1995) Design and implementationof a hybrid series active filter system. IEEE Conference Record PESC1995, pp. 189–195.

3. J. Dixon, S. Tepper, and L. Morán (1996) Practical evaluationof different modulation techniques for current controlled voltage-source inverters, IEE Proc. on Electric Power Applications, 143 (4),pp. 301–306.

4. A. Nabae, S. Ogasawara, and H. Akagi (1986) A novel control schemefor current controlled PWM inverters, IEEE Trans. on Ind. Appl.,22 (4), pp. 312–323.

5. F.Z. Peng, H. Akagi, and A. Nabae (1990) A new approach to har-monic compensation in power system – A combined system of shuntpassive and series active filter, IEEE Trans. on Ind. Appl., 26 (6),pp. 983–989.

6. F.Z. Peng, M. Kohata, and H. Akagi (1993) Compensation charac-teristics of shunt passive and series active filters, IEEE Trans. on Ind.Appl., 29 (1), pp. 144–151.

7. T. Tanaka, K. Wada, and H. Akagi (1995) A new control scheme ofseries active filters, Conference Record IPEC-95, pp. 376–381.

8. L. Morán, I. Pastorini, J. Dixon, and R. Wallace (1999) A fault pro-tection scheme for series active power filters, IEEE Trans. on PowerElectronics, 14 (5), pp. 928–938.

9. S. Fukuda and T. Endoh (1995) Control method for a combinedactive filter system employing a current-source converter and highpass filter, IEEE Trans. Ind. Appl., 31 (3), pp. 590–597.

10. Weimin Wu, Liqing Tong, MingYue Li, Z.M. Qian, ZhengYu Lu,and F.Z. Peng (2004) A novel series hybrid active power filter, IEEE35th Annual Power Electronics Specialists Conference PESC 04, 4,pp. 3045–3049.

11. H. Fujita, T. Yamasaki, and H. Akagi (2000) A hybrid active filter fordamping of harmonic resonance in industrial power systems, IEEETrans. Power Electronics, 15, pp. 215–222.

12. A. Campos, G. Joos, P.D. Ziogas, and J. Lindsay (1994) Analysisand design of a series voltage unbalance compensator based on athree-phase VSI operating with unbalanced switching functions, IEEETrans. on Power Electronics, 9 (3), pp. 269–274.

13. G. Joos and L. Morán (1998) Principles of active power filters, IEEE– IAS 98 Tutorial Course Notes, October.

14. H. Akagi (1997) Control strategy and site selection of a shunt activefilter for damping harmonics propagation in power distributionsystem, IEEE Trans. Power Delivery, 12 (1), pp. 17–28.

15. H. Akagi (1994) Trends in active power line conditioners, IEEE Trans.Power Electronics, 9 (3), pp. 263–268.

16. L. Morán, P. Ziogas, and G. Goos (1993) A solid-state high perfor-mance reactive-power compensator, IEEE Trans. Ind. Appl., 29 (5),pp. 969–978.

17. L. Morán, J. Dixon, and R. Wallace (1995) A three-phase active powerfilter operating with fixed switching frequency for reactive power andcurrent harmonic compensation, IEEE Trans. Industrial Electronics,42 (4), pp. 402–408.

18. H. Fujita, S. Tominaga, and H. Akagi (1996) Analysis and designof a dc voltage-controlled static var compensator using quad-series voltage-source inverters, IEEE Trans. Ind. Appl., 32 (4),pp. 970–978.

19. V. Bhavaraju and P. Enjeti (1996) An active line conditioner to bal-ance voltages in a three-phase system, IEEE Trans. Ind. Appl., 32 (2),pp. 287–292.

20. T. Tanaka and H. Akagi (1995) A new method of harmonic powerdetection based on the instantaneous active power in three-phasecircuit, IEEE Trans. Power Delivery, 10 (4), pp. 1737–1742.


21. E. Watanabe, R. Stephan, and M. Aredes (1993) New concepts ofinstantaneous active and reactive power in electrical system withgeneric loads, IEEE Trans. Power Delivery, 8 (2), pp. 697–703.

22. M. Aredes and E. Watanabe (1995) New control algorithms for seriesand shunt three-phase four-wire active power filter, IEEE Trans.Power Delivery, 10 (3), pp. 1649–1656.

23. T. Tanaka, K. Wada, and H. Akagi (1995) A new control scheme ofseries active filters, Conference Record IPEC-95, pp. 376–381.

24. T. Thomas, K. Haddad, G. Joos, and A. Jaafari (1998) Design and per-formance of active power filters, IEEE Industry Applications Magazine,4 (5), pp. 38–46.

25. S. Bhattacharya, T. M. Frank, D. Divan, and B. Banerjee (1998) Activefilter implementation, IEEE Industry Applications Magazine, 4 (5),pp. 47–63.

26. Sangsun Kim and P.N. Enjeti (2002) A new hybrid active power filter(APF) topology, IEEE Trans. Power Electronics, 17, pp. 48–54.

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