Power electronic handbook

Подождите немного. Документ загружается.

1068 L. Morán and J. Dixon

39.2 Types of Active Power Filters

The technology of active power ﬁlter has been developed

during the past two decades reaching maturity for harmon-

ics compensation, reactive power, and voltage balance in ac

power networks. All active power ﬁlters are developed with

pulse width modulated (PWM) converters (current-source

or voltage-source inverters). The current-fed PWM inverter

bridge structure behaves as a non-sinusoidal current source to

meet the harmonic current requirement of the non-linear load.

It has a self-supported dc reactor that ensures the continuous

circulation of the dc current. They present good reliability,

but have important losses and require higher values of parallel

capacitor ﬁlters at the ac terminals to remove unwanted cur-

rent harmonics. Moreover, they cannot be used in multilevel

or multistep modes conﬁgurations to allow compensation in

higher power ratings.

The other converter used in active power ﬁlter topologies is

the PWM voltage-source inverter (PWM-VSI). This converter

Non-Linear

Load

AC Mains

Active Filter

Non-Linear

Load

AC Mains

Active Filter

(a) (b)

Non-Linear

Load

AC Mains

Series Active

Filter

Shunt Passive

Filter

(c)

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

active power ﬁlter.

is more convenient for active power ﬁltering applications since

it is lighter, cheaper, and expandable to multilevel and multi-

step versions, to improve its performance for high power rating

compensation with lower switching frequencies. The PWM-

VSI has to be connected to the ac mains through coupling

reactors. An electrolytic capacitor keeps a dc voltage constant

and ripple free.

Active power ﬁlters can be classiﬁed based on the type

of converter, topology, control scheme, and compensation

characteristics. The most popular classiﬁcation is based on

the topology such as shunt, series, or hybrid. The hybrid

conﬁguration is a combination of passive and active compen-

sation. The different active power ﬁlter topologies are shown

in Fig. 39.1.

Shunt active power ﬁlters (Fig. 39.1a) are widely used to

compensate current harmonics, reactive power, and load cur-

rent unbalanced. It can also be used as a static var generator

in 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 coupling

transformer to eliminate voltage harmonics and to balance and

regulate the terminal voltage of the load or line. The hybrid

conﬁguration is a combination of series active ﬁlter and pas-

sive shunt ﬁlter (Fig. 39.1c). This topology is very convenient

for the compensation of high power systems, because the rated

power of the active ﬁlter is signiﬁcantly reduced (about 10% of

the load size), since the major part of the hybrid ﬁlter consists

of the passive shunt LC ﬁlter used to compensate lower-

order current harmonics and reactive power at fundamental

frequency.

Due to the operation constraint, shunt or series active power

ﬁlters can compensate only speciﬁc power quality problems.

Therefore, the selection of the type of active power ﬁlter to

improve power quality depends on the source of the problem

as can be seen in Table 39.1.

The principles of operation of shunt, series, and hybrid

active power ﬁlters are described in the following sections.

TABLE 39.1 Active ﬁlter solutions to power quality problems

Active ﬁlter

connection

Load on ac supply AC supply on load

Shunt Current harmonic ﬁltering

Reactive current compensation

Current unbalance

Voltage ﬂicker

Series Current harmonic ﬁltering Voltage sag/swell

Reactive current compensation Voltage unbalance

Current unbalance Voltage distortion

Voltage ﬂicker Voltage interruption

Voltage unbalance Voltage ﬂicker

Voltage notching

39.3 Shunt Active Power Filters

Shunt active power ﬁlters compensate current harmonics by

injecting equal but opposite harmonic compensating current.

In this case, the shunt active power ﬁlter operates as a current

source injecting the harmonic components generated by the

load but phase shifted by 180

◦

. As a result, components of

harmonic currents contained in the load current are cancelled

by the effect of the active ﬁlter, and the source current remains

sinusoidal and in phase with the respective phase-to-neutral

voltage. This principle is applicable to any type of load con-

sidered as an harmonic source. Moreover, with an appropriate

control scheme, the active power ﬁlter can also compensate the

load power factor. In this way, the power distribution system

sees the non-linear load and the active power ﬁlter as an ideal

resistor. The compensation characteristics of the shunt active

power ﬁlter is shown in Fig. 39.2.

Non-Linear

Load

AC Mains

Active Filter

Load CurrentSource Current

system

Compensation

Current

Load

coupling

FIGURE 39.2 Compensation characteristics of a shunt active power

ﬁlter.

39.3.1 Power Circuit Topologies

Shunt active power ﬁlters are normally implemented with

PWM-VSIs. In this type of application, the PWM-VSI oper-

ates as a current-controlled voltage source. Traditionally, two

level PWM-VSI have been used to implement such system

connected to the ac bus through a transformer. This type of

conﬁguration is aimed to compensate non-linear load rated

in the medium power range (hundreds of kVA) due to semi-

conductors rated values limitations. However, in the last years

multilevel PWM-VSIs have been proposed to develop active

power ﬁlters for medium voltage and higher rated power appli-

cations. Also, active power ﬁlters implemented with multiple

VSIs connected in parallel to a dc bus but in series through a

transformer or in cascade has been proposed in the technical

literature. The different power circuit topologies are shown in

Fig. 39.3.

The use of VSI connected in cascade is an interesting alter-

native to compensate high power non-linear loads. The use

of two PWM-VSI with different rated power allows the use

of different switching frequencies, reducing switching stresses,

and commutation losses in the overall compensation system.

The power circuit conﬁguration of such a system is shown

in Fig. 39.4.

The VSI connected closer to the load compensates for the

displacement power factor and lower frequency current har-

monic components (Fig. 39.5b), while the second compensates

only high-frequency current harmonic components. The ﬁrst

converter requires higher rated power than the second and

1070 L. Morán and J. Dixon

Non-

Linear

Load

(a)

Non-

Linear

Load

(b)

FIGURE 39.3 Shunt active power ﬁlter topologies implemented with

PWM-VSIs: (a) a three-phase PWM unit and (b) three single-phase units

in parallel to a common dc bus.

can operate at lower switching frequency. The compensation

characteristics of the cascade shunt active power ﬁlter is shown

in Fig. 39.5.

In recent years, there has been an increasing interest in using

multilevel inverters for high power energy conversion, espe-

cially for drives and reactive power compensation. The use

of neutral-point-clamped (NPC) inverters (Fig. 39.6) allows

equal voltage shearing of the series-connected semiconductors

in each phase. Basically, multilevel inverters have been devel-

oped for applications in medium voltage ac motor drives and

static var compensation. For these types of applications, the

output voltage of the multilevel inverter must be able to gen-

erate an almost sinusoidal output current. In order to generate

a near sinusoidal output current, the output voltage should not

contain low-frequency harmonic components.

However, for active power ﬁlter applications, the three-

level NPC inverter output voltage must be able to generate

an output current that follows the respective reference current

containing the harmonic and reactive component required by

Gating Signals

Generator

Current Control

System

AC Source

Current

Reference

Generator

Voltage

Control

DC Voltage Reference

Link

Inductor

PWM - VSI #2

Gating Signals

Generator

Current Control

System

Current

Reference

Generator

Voltage

Control

DC Voltage Reference

Link

Inductor

PWM - VSI #1

Non-Linear

Load

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 shunt

active power ﬁlter implemented with a three-level NPC-VSI

are 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 the

inverter, maintain the dc voltage constant, and generate the

inverter gating signals. The block diagram of the control

scheme 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 and

reactive power, and also try to maintain constant the dc voltage

across the electrolytic capacitors. There are many possibilities

to implement this type of control, and the most popular of

them will be explained in this chapter. Also, the compensation

39 Active Filters 1071

−1,5

−1

−0,5

0,5

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

0,4

0,8

p.u.

−0,8

−0,4

0,4

0,8

p.u.

−1,5

−1

−0,5

0,5

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

−0,5

0,5

1,5

p.u.

FIGURE 39.5 Current waveforms of active power ﬁlter implemented

with two PWM-VSI in cascade: (a) load current waveform; (b) current

waveform generated by PWM-VSI no. 1; (c) current waveform generated

by PWM-VSI no. 2; (d) power system current waveform between the

two inverters (THD

= 13.7%); and (e) power system current waveform

(THD

= 4.5%).

effectiveness of an active power ﬁlter depends on its ability to

follow with a minimum error and time delay, the reference

signal 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. The

dc voltage control is achieved by adjusting the small amount

of real power absorbed by the inverter. This small amount

of real power is adjusted by changing the amplitude of the

fundamental component of the reference current.

39.3.2.1 Current Reference Generation

There are many possibilities to determine the reference current

required to compensate the non-linear load. Normally, shunt

active power ﬁlters are used to compensate the displacement

power factor and low-frequency current harmonics generated

by non-linear loads. One alternative to determine the current

reference required by the VSI is the use of the instantaneous

reactive power theory, proposed by Akagi [1], the other one is

to obtain current components in d–q or synchronous reference

frame [2], and the third one to force the system line current

to follow a perfectly sinusoidal template in phase with the

respective 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, and

current 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 in

Fig. 39.9.

The instantaneous values of voltages and currents in the α,

β coordinates can be obtained from the following equations:





[

]













[

]









(39.1)

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

and is equal to

[

]





1 −1/2 −1/2

√

3/2 −

√

3/2



(39.2)

This transformation is valid if and only if v

(t) + v

(t) +

(t) is equal to zero, and also if the voltages are balanced

and sinusoidal. The instantaneous active and reactive power

in the α, β coordinates are calculated with the following

expressions:

p(t) = v

(t) · i

(t) + v

(t) · i

(t) (39.3)

q(t ) =−v

(t) · i

(t) + v

(t) · i

(t) (39.4)

It is evident that p(t) becomes equal to the conventional

instantaneous real power deﬁned in the a, b, c reference frame.

1072 L. Morán and J. Dixon

Non-Linear

Load

AC Source

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_La

200.00

100.00

0.00

−100.00

−200.00

0.00 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 by

expression (39.4) or by the vector equation:

q = v

(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, v

perpendicular to i

, and v

is perpendicular to i

. The physical

meaning of the vector q is not “instantaneous power” because

of the product of the voltage in one phase and the current in

the other phase. On the contrary, v

and v

in Eq. (39.3)

obviously mean “instantaneous power” because of the product

of 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 the

product of the instantaneous voltage and current in different

axes, but cannot be treated as a conventional quantity.

39 Active Filters 1073

Reference Current

Generator

Active, Reactive

and Harmonic

decomposition of

the Load Current

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

Driver

Voltage Control

Unit

P*(t)

avg

Current Reference Generator

DC Voltage Control

Current Control and Gating Signals Generator

dc gen.

dc ref.

Gating Signals

Gen.

Current

Control

Unit

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

va(t)

vb(t)

vc(t)

Load

(t) Ib

Load

(t) Ic

Load

(t)

Compensated

Current Component

dc Current Control

Component

Gen

(t)

Gen

(t)

Gen

(t)

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

, i

30°

120°

, i

ω t

FIGURE 39.9 Transformation diagram from the a, b, c reference frame

to 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 following

equation:







−v









−v







≡



αp

βp





αq

βq



(39.6)

and the different components of the currents in the α–β plane

are shown in the following expressions:

αp

(39.7)

αq

(39.8)

βp

(39.9)

βq

−v

(39.10)

From Eqs. (39.3) and (39.4), the values of p and q can

be expressed in terms of the dc components plus the ac

components, that is:

p =

p +

p (39.11)

q =¯q +˜q (39.12)

where,

p dc component of the instantaneous power p, and is related

to the conventional fundamental active current.

p is the ac component of the instantaneous power p, it does

not have average value, and is related to the harmonic

1074 L. Morán and J. Dixon

currents caused by the ac component of the instantaneous

real power.

¯q is the dc component of the imaginary instantaneous power

q, and is related to the reactive power generated by the

fundamental components of voltages and currents.

˜q is the ac component of the instantaneous imaginary power

q, and it is related to the harmonic currents caused by the

ac component of instantaneous reactive power.

In order to compensate reactive power (displacement power

factor) and current harmonics generated by non-linear loads,

the reference signal of the shunt active power ﬁlter must

include the values of

p, ¯q, and ˜q. In this case the reference cur-

rents required by the shunt active power ﬁlters are calculated

with the following expression:



∗

c,α

∗

c,β





−v





¯q

+˜q



(39.13)

The ﬁnal compensating currents including the zero

sequence components in a, b, c reference frame are the

following:







∗

c,a

∗

c,b

∗

c,c















√

−1

√

−1

−

√













−i

∗

c,α

∗

c,β







(39.14)

where the zero sequence current component i

is equal to 1/

√

+ i

). The block diagram of the circuit required to

generate the reference currents deﬁned in Eq. (39.14) is shown

in Fig. 39.10.

The advantage of instantaneous reactive power theory is that

real and reactive power associated with fundamental compo-

nents are dc quantities. These quantities can be extracted with a

low-pass ﬁlter. Since the signal to be extracted is dc, ﬁltering of

the signal in the α–β reference frame is insensitive to any phase

a-b-c

α - β

a-b-c

α - β

aload

Van

Vbn

Vcn

laload

Vα

Iα

Iβ

Vβ

and

and V

LPF

ref

Pref

∆Vdc

lαref

and

βref

laref

lbref

lcref

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 same

advantage can be obtained by using the synchronous reference

frame method, proposed in [2]. In this case, transformation

from a, b, c axes to d–q synchronous reference frame is done.

Effects of the Low-pass Filter Design Characteristics in

Compensation Performance

A Butterworth ﬁlter is normally used due to the adequate

frequency response. A second-order ﬁlter offers an appropri-

ate relation between the transient response and the required

attenuation 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 shunt

active power ﬁlter cannot compensate the entire low-frequency

harmonic contained in the load current. Normally the cut-

off frequency is equal to 127 Hz with an attenuation factor of

15 dB for the ﬁrst ac component to be eliminated, which means

an 82.2% attenuation of the ﬁfth and seventh harmonic com-

ponents (Fig. 39.11). The expression that relates the system line

current harmonic distortion with the LPF cut-off frequency

and load power factor is shown in Eq. (39.15).

THD

isys



∞



h=6k

(

)



((h

+1)/(h

−1))−(1/(h

−1))cos(2ϕ)



cos(ϕ)

(39.15)

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

is the frequency of the harmonic

component of order h, and f

is the LPF cut-off frequency.

Figure 39.12 shows the total harmonic distortion (THD) in

the line currents introduced by the second-order Butterworth

ﬁltering characteristics, as a function of the cut-off frequency,

and considering a 50 Hz ac mains frequency. The harmonic

distortion of the compensated line current depends on the

39 Active Filters 1075

Low-pass

filter

−

FIGURE 39.11 The low-pass ﬁlter block diagram used to extract the ac

component of p.

40 60 80 100 120 140 160 180

Cut-off frequency

THDi [%]

PF = 1

PF = 0.8

PF = 0.5

PF = 0.2

FIGURE 39.12 Harmonic distortion of the compensated line current as

a function of the low-pass ﬁlter cut-off frequency. PF = cos(ϕ).

load displacement power factor, as shown in Eq. (39.15). If the

cut-off frequency of the low-pass ﬁlter is changed, the active

power ﬁlter compensation performance is affected as well as

the transient response of the control scheme.

Effects of the Supply Voltage Distortion in Compensation

Performance

One of the most important characteristics of the instantaneous

imaginary power concept is that in order to obtain the current

reference signal required to compensate reactive and harmonic

current components, the system phase-to-neutral voltages are

used. In general, purely sinusoidal voltages are considered in

previously reported analysis. In case voltage is purely sinu-

soidal, the dc component of p and q in the α–β plane are

related with the fundamental components in the real a, b, c

reference system. This is not the case if the system voltages are

distorted or unbalanced, as it is demonstrated below.

It is assumed that the supply voltages have harmonic

distortion, and these are represented by:

(

)

= V

cos

(

ωt

)

cos

[

(

ωt −δ

)

]

(

)

= V

cos



ωt −

2π



cos





ωt −δ

−

2π



THDv

[

]

THDi [%]

PF = 1

PF = 0.8

PF = 0.5

PF = 0.2

51234 6789010

FIGURE 39.13 Harmonic distortion in the compensated line current as

a function of the harmonic distortion in the system voltages. PF = cos(ϕ).

(

)

= V

cos



ωt +

2π



cos





ωt −δ

2π



(39.16)

Since the harmonic voltage component introduces a dc

component in p and q, compensation performance of the

shunt 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 Compensation

Performance

Voltage unbalance also affects the active power ﬁlter com-

pensation performance. In this analysis, the phase-to-neutral

voltages of the ac supply are equal to:

(

)

= V

cos

(

ωt

)

(

)

= V

(

1 +m

)

cos



ωt −

2π



(

)

= V

(

1 −m

)

cos



ωt +

2π



(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 compensated

supply current (phase a) is:

= I

cos

(

)

cos

(

ωt

)

√

[

sin

(

)

cos

(

3ωt

)

cos

(

3ωt −ϕ

)

]

(39.18)

1076 L. Morán and J. Dixon

In this case, the active power ﬁlter is not able to fully

compensate the system line current, since compensation

performance depends on the voltage unbalance magnitude.

If the unbalance is deﬁned as a function of the positive and

negative sequence component as:



+aV





+aV



, a = 1∠120

◦

Unbalance =

√

(39.19)

The harmonic distortion is equal to:

THD

√

m (39.20)

The relation between line current harmonic distortion and

voltage unbalance is shown in Fig. 39.14.

THDi [%]

012 3

Desb. [%]

456

FIGURE 39.14 Harmonic distortion in the compensated line current as

a function of the system voltages unbalance.

Effects of the Time Delay Introduced by the DSP in

Compensation Performance

If 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. Considering

a simultaneous sampling, the ac mains compensated line

current with an ideal low-pass ﬁlter in the control scheme

(phase a) is:

= I

cos

(

ωt

)

[

cos

(

)

+sin

(

ωT

)

]

+V

cos

(

ωt

)

∞



n=2k−1

[

1 −cos

(

nωT

)

]

cos

[

(

ωt −ϕ

)

]

∞



n=2k−1

sin

(

nωT

)

sin

[

(

ωt −ϕ

)

]

(39.21)

The time delay introduced by the DSP affects the compen-

sation performance, especially when fast changes are present

in 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 also

affect the generation of the adequate current reference signal

that will assure high performance active power ﬁlter com-

pensation. This assumption can be proved by considering the

following load currents:

= I

cos

(

ωt −ϕ

)

∞



n=2k−1

cos

[

(

ωt −ϕ

)

]

= I

(

1 +l

)

cos



ωt −ϕ −

2π



∞



n=2k−1

(

1 +l

)

cos





ωt −ϕ −

2π



= I

(

1 −l

)

cos



ωt −ϕ +

2π



∞



n=2k−1

(

1 −l

)

cos





ωt −ϕ +

2π



= i

(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 the

load current unbalanced. These low frequency current com-

ponents forces to reduce the cut-off frequency of the low-pass

ﬁlter, close to 100 Hz, which affects the transient response of

the active power ﬁlter.

p =





cos

(

)

∞



n=2k−1

cos

[

(

n +1

)

ωt −nϕ

]

∞



m=2k−1

cos

[

(

m −1

)

ωt −mϕ

]





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

(39.23)

39.3.2.1.2 Synchronous Reference Frame Algorithm The

block diagram of a current reference generator that uses the

synchronous reference frame algorithm is shown in Fig. 39.15.

In this case, the real currents are transformed into a

synchronous reference frame [2]. The reference frame is syn-

chronized with the ac mains voltage, and is rotating at the

39 Active Filters 1077

PLL

cos(θ)

sin(θ)

aload

Ibload

Icload

Van

Vbn

Vcn

a-b-c

α - β

d - q

LPF

dref

∆V

d - q

α - β

qref I

βref

αref

α - β

a-b-c

aref

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:







cos

(

ωt

)

sin

(

ωt

)

sin

(

ωt

)

cos

(

ωt

)





(39.24)

As for the instantaneous reactive power theory, d and q

terms are composed by a dc and multiple ac components,

such as i

= i

ddc

dac

and i

= i

qdc

qac

. The compensation

reference signals are obtained from the following expressions:

dref

=−i

dac

and i

qref

=−i

qdc

− i

qac

. The compensated cur-

rents generated by the shunt active power ﬁlter are obtained

from Eq. (39.25).





aref

bref

cref













√

−

√

−

√









−10 0

0 cos

(

ωt

)

−sin

(

ωt

)

0 sin

(

ωt

)

cos

(

ωt

)



dref

qref



(39.25)

One of the most important characteristics of this algorithm

is that the reference currents are obtained directly from the

loads currents without considering the source voltages. This is

an important advantage since the generation of the reference

signals is not affected by voltage unbalance or voltage distor-

tion, therefore increasing the compensation robustness and

performance. However, in order to transform from the α–β

plane to the d–q synchronous reference frame, sine and cosine

signals 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 current

presents the same mathematical procedure and operations that

the 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 harmonic

spectrum in the synchronous reference frame, and low-order

harmonic components appear in the reference signal. In order

Phase

Detector

Low-Pass

Filter

Voltage

Controlled

Oscillator

Input Signal

Output 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 be

reduced.

39.3.2.1.3 Peak Detection Method There are other possi-

bilities to generate the current reference signal required to

compensate reactive power and current harmonics. Basically,

all the different schemes try to obtain the current refer-

ence signals that include the reactive components required

to compensate the displacement power factor and the cur-

rent harmonics generated by the non-linear load. Figure 39.17

shows another scheme used to generate the current reference

signals required by a shunt active power ﬁlter. In this case,

the ac current generated by the inverter is forced to follow

the reference signal obtained from the current reference gen-

erator. In this circuit, the distorted load current is ﬁltered,

extracting the fundamental component, i

. The band-pass ﬁl-

ter is tuned at the fundamental frequency (50 or 60 Hz), so

that the gain attenuation introduced in the ﬁlter output sig-

nal is zero and the phase-shift angle is 180

◦

. Thus, the ﬁlter

output current is exactly equal to the fundamental component

of the load current but phase shifted by 180

◦

. If the load cur-

rent is added to the fundamental current component obtained

from the second-order band-pass ﬁlter, the reference current

waveform required to compensate only harmonic distortion is

obtained. In order to provide the reactive power required by

the load, the current signal obtained from the second-order

band-pass ﬁlter I

is synchronized with the respective phase-

to-neutral source voltage (see Fig. 39.17) so that the inverter ac

output current is forced to lead the respective inverter output