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EYEWIRE

1077-2618/07/$25.002007 IEEE

N THE ATTEMPT TO MINIMIZE THE

harmonic distortion from nonlinear loads,

the choice of active power filters (APFs)

emerges to improve the filtering efficiency

and solve the issues of classical passive filters. One of the

key points for proper implementation of an APF is to use a

reliable method for current/voltage reference generation.

Currently, there is a large variety of practical implementa-

tions supported by different theories (either in the time or

frequency domain), the performances of which are contin-

uously debated while ever-better

solutions are proposed. This article

gives a survey of the commonly

used methods for harmonic detec-

tion in APFs. The work proposes a

simulation setup that decouples the

harmonic detection method from the active filter model

and its controllers. In this way, the selected methods canbe equally analyzed and compared with respect to their

performance, which helps in anticipating possible imple-

mentation issues. A comparison is given that may be used

to decide the future hardware setup implementation. The

comparison shows that the choice of numerical filtering is

a key factor for obtaining a good accuracy and dynamic

performance of an active power filter.

Mitigation of Harmonic Currents by Shunt APFsIt is a fact that the continuous proliferation of electronic

equipment either for home appliances or for industrial

use has the drawback of increasing the nonsinusoidal

currents into the power network. Different mitigation

solutions are currently proposed and used in practice,

involving passive, active, and hybrid filtering tech-

niques, magnetic wave shaping, or reconfiguration of

the power system. In the last decades, the use of active

filtering techniques has become more attractive due to

the technological progress in

power switching devices, increased

performance of the DSPs,

enhanced numerical methods, and

new control algorithms. As a

result, if initially the APFs were

tested in laboratory conditions, nowadays they are

implemented more and more in real-life applications.Therefore, to have good harmonic mitigation perfor-

mance, there is an increased interest in developing the

best implementation of both hardware and software

components of an APF.

Regarding the software components, two main parts

are important for a proficient APF: the harmonic detection

method and the inner (current/voltage) controller.

I

DETECTIONIS KEYHarmonic detection

methods for active power filter applications

B Y L U C I A N A S I M I N O A E I ,F R E D E B L A A B J E R G , &

S T E F FA N H A N S E N

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Figure 1 shows a typical diagram of a shunt APF for anadjustable speed drive (ASD) application. The harmonic

detection method has the task of detecting the harmoniccurrents/voltages that have to be compensated by the APF;hence, a reliable method must be used. Once the harmonicsare detected, they are given as reference signals to the innercurrent/voltage controller, which has the task of producingan output signal identical to its reference.

There are numerous published references that describedifferent topologies and different algorithms used for activefiltering. Most of them present a single type of harmonicdetection method implemented in an APF which mitigatesthe existing harmonics according to the initial expectations.Therefore, it is difficult to assess if the overall performanceof the APF is due to the selected harmonic detectionmethod or due to another design choice. There are also pub-lications [1][8] that explain and compare different har-monic detection methods, describing their advantages anddrawbacks by giving as final indices the dynamics, the totalharmonic distortion (THD) reduction, the inverter efficien-cy, or the cost of the entire active filter.

Usually, the comparisons are made between different APFtopologies, each being considered as a whole unit, whereonly the implemented harmonic detection method is stressedout and the rest of the control is left behind [1][5].

However, the inner and outer loop controllers may havedifferent performances depending on tuning, which influ-

ence also the harmonic compensation. Consequently, thecomparison between different APFs that use different har-

monic detection methods is not a direct effect of only theperformance of the used harmonic detection block, but alsoof the quality of the existing controllers.

This article analyzes the harmonic detection methodsdecoupled from the entire APF structure in order to revealits contribution. At first, the work summarizes the theo-retical background of several commonly used harmonicdetection methods for APFs.

The actual study contributes to the existing compar-isons in [1][8] by adding a new trend in harmonicdetection by using resonant controllers. Then, as an inves-tigation this article extends the existing comparisons in[2] with a proposed simulation study. The study separatesthe harmonic detection method from the APF control andstudies its behavior with respect to the detection accuracyof a specific harmonic component.

The input signal from sensors is replaced with a priorknown signal, artificially constructed, by summing differ-ent known characteristic harmonics to the fundamentalfrequency. Thus, the output obtained from each analyzedharmonic detection method is recorded and compared tothe input harmonic signal.

The results indicate that the choice of numerical filtersis a key factor for obtaining good accuracies and dynam-ics. Finally, the conclusions are compared with respect to

One-line diagram of a shunt active filter in a feed-forward configuration with an adjustable speed drive as load. The diagram

shows the placement of the harmonic detection block in the APF control.

LoadPowerSupply Z S

Generated Harmonic Currents

Adjustable Speed Drive

VoltageSensors

[I h ]

[I h ]

ProtectionCircuit

Load CurrentSensors I L

Filter Current

Sensor I F

EnergyStorageElement

C dc

V dc

Boost InductorLF

I F I L

PowerConverter

Inner CurrentController

H a r d w a r e

Shunt ActivePower Filter

D S P S o

f t w a r e

Harmonic CurrentReference Generator

dc-VoltageController

A/D A/D A/DPWM Modulation

HarmonicDetectionMethod+

+

1

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different criteria, which gives an overview and helps indeciding on the final experimental setup for an active filter.

Harmonic Detection MethodsThe harmonic detection method is the part of the APFscontrol that has the capability of determining specificattributes of the harmonics (frequency, amplitude, phase,time of occurrence, duration, energy) from an input signal(which can be voltage or current) by using a special mathe-matical algorithm. Then, with the achieved information,the inner controller (current controller in Figures 1 and 2) isimposed to compensate for the existing harmonic distortion.

Figure 2 presents a classical control diagram of a shuntAPF. The outer dc voltage control loop maintains the dc

capacitor charged at the voltage reference ( V DC ) in orderfor the inverter to be able to operate against the line volt-age. Thus, the dc voltage controller produces a current ref-erence ( I

DC ) that keeps the capacitor charged and coversthe losses in the passive and active elements. The harmon-ic detection method receives at the input the current fromthe nonlinear load and extracts the harmonic content( I Harm ) according to the selected algorithm. Then, bothcurrents are summed into a new reference ( I F ), which isprovided to the inner current controller. Errors given bythe harmonic detection method degrade the overall perfor-mance of the APF.

Different harmonic detection algorithms led to a largescientific debate concerning which part the focus should

Principle algorithm of the synchronous fundamental dq

frame for harmonic detection.

HPFFiltering

abc

dq

dq

abc

i a i b i c

i d = i d i a i b i c

PossibleLocations

of the InnerCurrent

Controller

Phase LockedLoop (PLL)

u a u b u c

I n n e r

C o n

t r o

l l e r

I n n e r

C o n

t r o

l l e r

1 1

1

Harmonic Detection Method

i d = i d + i d

i q = i q + i q

i q = i q

6

Bode plot of RDFT calculated as in (3) for the 5th harmonic

(h = 5) and a sampling frequency of 3.2 kHz ( N = 64).

Phase0

Magnitude0 dB

Frequency250 Hz

500

100

18090

200

0 90

180 P h a s e

( )

M a g n

i t u

d e

( d B )

1 10 100 1,000Frequency (Hz)

5

The principle of the recursive DTF applied for a rectangular

moving window.

RDFT

NewSample

RectangularMoving

WindowFundamental Period

SamplingFrequency

LastSample

SampledInput Signal

4

Example of the decimation in time algorithm for the FFT.

N/ 2PointDFT

N/ 2

PointDFT

N/ 4PointDFT

N/ 4PointDFT

N/ 4PointDFT

N/ 4PointDFT

N PointRecombine

Algebra

HarmonicSpectrum

X (0)X (1)

X (N /2 1)X (N /2)

X (N 1)x (N 1)

x (N 2)

x (1)

x (0)x (2)

x (3)

x (n )n = (1,N )

SampledInput

Signal

3

Control diagram of a shunt active filter in a feed-forward configuration.

HarmonicDetection Method

dc-VoltageController

Inner CurrentController

PWMInverter

BoostInductor

dc-LinkCapacitor

Feedback Paths

I L

V *dc

I *dc

I *F +

++

V dcI F

I *Harm

I F V

dc

1sLF + R F

1sC dc

2

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be placed upon: the detection accuracy, the speed, the fil-ter stability, easy and inexpensive implementation, etc.

The classification of the harmonic detection methodscan be done depending on the mathematical algorithmsinvolved [7]. Thus, two directions are described here: thetime-domain and the frequency-domain harmonic detec-tion methods, as shown in Table 1.

Frequency-Domain MethodsThe frequency-domain methods are mainly identifiedwith Fourier analysis, rearranged to provide the result asfast as possible with a reduced number of calculations, toallow a real-time implementation in DSPs.

The discrete Fourier transform (DFT) is a mathematicaltransformation for discrete signals (1), which gives both theamplitude and phase informations of the selected harmonic

Xh = N 1

n= 0

x(n) cos2 h n

N

j

N 1

n= 0 x(n) sin

2 h n N

Xh = Xh( real ) + j Xh( imag)

| Xh | = X 2h( real ) + X2h( imag) ;

h = arctanXh( imag)Xh( real )

. (1)

where N is the number of samples per fundamental peri-od; x(n) is the input signal (voltage or current) at thesampled point n; Xh is the complex Fourier vector of thehth harmonic; Xh(real) and Xh(imag ) are the real part

respective the imaginary part of Xh; | Xh | and h are theamplitude respective the phase of Xh .

Once the harmonics are detected and isolated in the fre-quency domain, it is just a matter of reconstruction back in thetime domain of the reference signal for the inner controller [9].

The fast Fourier transform (FFT) follows the same math-ematical representation as in (1) but in a different form[10] to reduce the number of operations and, hence, therequired DSP calculation time. The FFT algorithm uses anoperation called decimation (in the time or frequencydomain) that relies on the recursive decomposition of an N -point DFT into two DFT transforms of N / 2 points (Figure3). This process can be applied to any N -sampled signal if N is a regular power of two, so the decomposition can berepeatedly applied until the trivial 1-point transform isreached and calculated. Thus, the total number of calcula-tions is reduced from N 2 to N log2( N ) . Once the har-monic spectrum is determined, the harmonic referencecurrent is the summation of all sinus functions with theknown amplitude, frequency, and phase [11].

The recursive discrete Fourier transform (RDFT) usesthe same principle as the DFT (1) but calculated on a slid-ing window [12] (Figure 4). The window shifts every sam-pling time with a fixed number of samples, usually justone for simplicity. Thus, the DFT analysis can actually be

performed on the newly achieved samples. The only differ-

ences between the actual and the previous windows are thefirst and last samples. All the other samples are the same;therefore, it is not necessary to sample again.

Since the result of the DFT was calculated before forthe old window, a recursive expression as in (2) can befound to avoid the same calculation for the new window.It is also demonstrated that (3) can be rearranged as atransfer function (3) in the form of a finite impulseresponse (FIR) filter. Such a selective FIR filter with zeroattenuation and zero phase-shift at the selected frequency(see Figure 5) is very useful to isolate a specific harmonicfrom the input signal. Different types of windows (e.g.,triangular, exponential, Hamming, Hanning) can beused, but their implementation becomes more complex

Xh =1 N

N 1

n= 0

x(n) W hi ; W = exp j 2 N

Xh(k) =1 N

( x( k) x( k N )) + W h Xh( k 1) , (2)

Hh(z) =Xh(z)Xh(z)

=1 N

1 z N

1 W hz 1(3)

The common drawbacks of the Fourier-based harmonicdetection methods include: proper design of the antialias-ing filter, careful synchronization between the samplingand fundamental frequency, careful application of the win-dowing function, proper usage of the zero-padding toachieve a power of two number of samples, large memoryrequirements to store the achieved samples, large compu-tation power required for the DSP, and imprecise results intransient conditions [3].

Time-Domain MethodsThe time-domain methods offer increased speed and fewercalculations compared to the frequency-domain methods.

TABLE 1. CLASSIFICATION OF THE MOST-USEDHARMONIC DETECTION METHODS IN APF S .

Domain Harmonic Detection Method

Frequency Domain Discrete Fourier transform(DFT)

Fast Fourier transform (FFT)

Recursive discrete Fourier transform (RDFT)

Time domain Synchronous fundamentaldq frame

Synchronous individual har-monic dq frame

Instantaneous power pqtheory and variants

Generalized integratorsand variants

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A synchronous fundamental d q frame is derived fromthe space vector transformation of the input signals. The

load currents are achieved in the ab c coordinates (sta-tionary reference frame) from the current sensors andthen transformed into the d q coordinates (rotating refer-ence frame with fundamental frequency) by means of thePark transformation (4). The dq frame rotates with theangular speed of the fundamental frequency which makesthe fundamental currents a dc component (i d , i

q ) and

the harmonics ac components (i d , i

q ). Thus, the detec-tion of the harmonics is done with a high-pass filter(HPF), as seen in Figure 6 with a cutting frequency of 25120 Hz [14]

id

iq=

2

3

cos(1) cos(1 2/ 3) cos(1 + 2/ 3) sin(1) sin(1 2/ 3) sin(1 + 2/ 3)

i aibic

. (4)

where: id , iq and i a , ib , ic are the currents in the dqrespective ab c frame and 1 is the reference angle of thefundamental.

The synchronous harmonicd q frame is similar to the fun-damental dq-frame method,but the dq frame rotates nowwith an angular speed of theselected harmonic frequency.Thus, in the harmonic dqframe, only the respective har-

monic is a dc signal ( i

dn , i

qn )and all other frequencies,including the fundamental, areac components ( i dn , i

qn ). Thedetection of the selected har-monic is done with low-passfilters (LPFs) as seen in Figure7 [15].

Regarding the d q-frame-related harmonic detectionmethods, one drawback is thenecessity of the reference angle,possibly from a phase lockedloop (PLL), which requires a

careful implementation if theline voltages are not balancedand sinusoidal. Another issue isthe implementation of thenumerical filters (HPF respec-tive LPF for the fundamentalrespective harmonic d q frame)which influence the APF accu-racy and dynamics.

Due to the nonideal filteringrejection and the phase shiftintroduced by the numerical fil-

ters, the reference signals ( i ab c ) are not in opposite phase orwith the same amplitude as the acquired harmonic distur-

bance. This limitation adds to the existing delays from theanalog-to-digital converters, the inner current loop, andpulse-width modulated (PWM) inverter, requiring separatecompensation. This can especially be difficult in practicefor the harmonic d q frame methods, where the compensa-tion and the controllers must be individually tuned [15].Another issue is encountered for unbalanced load currents;therefore, the control and the harmonic detection mustinclude positive-, zero-, and negative-sequence compo-nents, which again increases the number of calculations andmakes the tuning of each controller more difficult. Howev-er, the dq theory is extensively used in active filters becauseof the simplicity of the controller design.

The instantaneous power theory determines the har-monic distortion by calculating the instantaneous power ina three-phase system (Figure 8), which is the multiplica-tion of the instantaneous values of the currents and voltages[17]. The calculations are done in coordinates, as in (5):

pq

=v v

v v

ii

(5)

i i

= 1

v2 + v2

v vv v

p

q. (6)

Diagram of multiple synchronous harmonic dq transformations.


abc

dq5th

abc

dq7th

abc

dqn th

Phase Locked Loop (PLL)

1

n n

7

5 5

5=5 1t

7=7 1t

n =n 1t

1 Possible Locations of theInner Current Controller

I n n e r

C o n

t r o

l l e r

I n n e r

C o n

t r o

l l e r

I n n e r

C o n

t r o

l l e r

I n n e r

C o n

t r o

l l e r

LPFFiltering

LPFFiltering

LPFFiltering

---

---

---

i a

i b

i c

u a u b u c

i q 5=i q 5 + i q 5

i d 5=i d 5 + i d 5

i q 7=i q 7 + i q 7

i d 7=i d 7 + i d 7

i dn =i dn + i dn

i qn =i qn + i qn

i a

i b

i c

i a 7

i b 7

i c 7

i an i dn =i dn i bn

i cn

abc

dq7th

abc

dqn th

abc

dq5th

i a 5

i b 5

i c 5

7

i d 7=i d 7

i q 7=i q 7

i qn =i qn

i q 5=i q 5

i d 5=i d 5

7

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For a three-phase system,the values of the instantaneouspowers p and q (real respectiveimaginary powers) contain dcand ac components dependingon the existing active, reactive,and distorted powers in the sys-tem. The dc components of p

and q represent the active andreactive powers in the load cur-rent and can be removed byHPFs (Figure 8) with a cuttingfrequency between 5 and 35Hz. The ac components of the pand q power are afterwards cal-culated back to the ab c frameto obtain the harmonic current distortion ( i , i

) as in (6),given as references for the inner current controller. Again,the presence of the numerical filters influences the dynam-ic and the accuracy of the entire APF response.

The calculation in (5) is affected if the system has a zero-sequence component due to an existing unbalance. Thus,

also a p0 -power (also referred to as 0-power) componentmust be added to the control to provide a complete analysis[18]. Furthermore, if the line voltages are distorted, thenthe calculation of the instantaneous powers and the refer-ence currents is influenced and the mitigation of the har-monics is not suitable, which calls for filtering techniquesfor line voltages.

Other techniques based on the same principle of instan-taneous power improve other features, as like the cancella-tion of the neutral currents [19], the minimization of theenergy storage element [20], and the pre-processing of theinput voltages for extracting the positive sequence [18].

Generalized integrators are types of integratorsdesigned for a given frequency, and they can be used to

create equivalent proportional-integral (PI) controllers forac signals. A limitation of a classical PI controller in d qframe designed for dc signals is that it has a limited track-ing capability for ac signals and therefore creates steady-state errors. A generalized integrator derives theintegration in the time domain, obtaining a second-ordertransfer function in Laplace domain (Figure 9), whichgives an infinite gain and zero phase-shift at the selectedresonant frequency. Thus, two functionalitiesharmonicfiltering and current controllercan be implemented inthe stationary or rotating frame. Since there is no need tocalculate the reference angle, the method does not requirea PLL and therefore reduces the number of calculations.Such an approach leads to a number of different imple-mentations since one can use either individual harmoniccompensation (notch filter) [21], [22] or a broadbandapproach (band-pass filters, HPFs, or LPFs) [23], [24].

The transfer function of the system in Figure 9 isgiven in Figure 10 (zoomed around the selected 5th and7th harmonics) for different values of the integration con-stant K I. One design issue is to determine the optimumintegration constant K I, since a small value gives goodselectivity but determines a slow dynamic response, whilea bigger value can lead to instability. Furthermore, thecontroller must be tuned depending on the existing plant

(not shown in Figures 9 and 10), which makes the methoddependent on the transfer function of the existing process.

Other Methods

Other types of harmonic detection methods can be foundin the literature, but not all are widely used in APFs

because some of them either have different issues in non-ideal conditions or they are difficult to implement. Themethods can also be classified as time domain, frequencydomain, or combinations of both, as follows:

Sinusoidal subtraction: where the fundamental isdetected either by a DFT algorithm or by a selectivefilter, then subtracted from the load current to createthe harmonic reference for the inner current con-troller [25]

Filtering: where the fundamental component is fil-tered out by advanced adaptive or vector filters[26], [27]

Nonactive-power-related theories [8]: Many of theseare the precursors of the pq theory, trying to express

the power flow in electrical systems, which definethe real, reactive, and harmonic powers in differentways. Some of these theories are tested in active fil-ters; others are only proven by simulations. Thestate-of-the-art work is orthogonal currents theory[28], inductive and capacitive power [29], polyphase

Example of the generalized integrators used in APFs. Here

the generalized integrators also have the function of current

controller which selectively regulates the filter current of 5th

and 7th harmonics.

Harmonic Detection Methodand Current Controller

BoostInductorInput

SignalI L

I F

+

I F

Feedback Path

PWM

Inverter

1

sLF +R F K I7 *s s 2 + (7* 1)2

K I5 *s s 2 + (5* 1)2

9

Principle diagram of the instantaneous power theory.


abc

abc

Calculationof

InstantaneousPowers p, q

HPFFiltering

Calculationof

ReferenceCurrents

i

, i

i

i

abc I n n

e r

C o n

t r o

l l e r

i a i b i c

i a

i b

i cV a V bV c

i

i

v

v

q ~

p ~ p = p + p ~

q = q + q ~

8

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orthogonal currents (FBD) [30], nonperiodic wave-forms [31], instantaneous orthogonal currents [32],park-power [33], multiphase pq theory [34], polarcoordinates [35], generalized instantaneous reactive

power [36], and generalized nonactive power [37] Online estimation algorithms used for time-varying

harmonics applications: where the harmonic detec-tion must use special prediction algorithms as Pronyanalysis [38] or Kalman filters [39]

Neural networks used also for special applicationslike self-learning and adaptive APF control [40]

Wavelet filtering used for compensation of rapidlychanging harmonics [41].

Simulations and Practical ValidationsA comparison is done next through simulations to revealthe performance of some of the presented methods withrespect to the settling time and the accuracy of theresults. An input signal (the load current from an ASDthat has to be compensated by a shunt APF) is artificial-ly constructed from the 50-Hz fundamental componentand the sum of several harmonics. The 5th harmonic hasan amplitude of about 30% of fundamental and is addedalone for a specific time interval at the beginning. Fig-ure 11 shows only one phase of the test input signal. Itcan be noted that the resultant signal has a similar shapeas the line currents obtained from a three-phase dioderectifier. The parameters used to obtain the resultantsignal in Figure 11 are given in Table 2.

As the 5th harmonic is separately added between the

time interval of 0.050.15 s, the output obtained fromeach detection method should be predictable and easilystudied during this interval. Therefore, the goal is torecord the output of each harmonic detection method withrespect to the detection of the 5th harmonic, as illustratedin Figure 12.

In order to have the same base of comparison for allmethods, the output results will be displayed in the same ab c frame as the original input signal. Thus, for example,for the dq-frame harmonic detection method the output d and q components are transformed back into the ab c framefor comparison.

In the case of a real active filter implementation, theoutput of each method represents the reference imposed tothe inner loop controller (i.e., current controller for ashunt APF). This controller, which has a certain responsetime, will introduce even more delays in the loop due tothe limited tracking capability, but this limitation doesnot appear here. The algebraic reconstruction of the out-put signal into the ab c frame emulates an ideal inner loopcontroller, with a unitary gain and an infinite bandwidth.

The methods considered for this comparison are listedin Table 3 together with their characteristics. The settlingtime, the overshoot, and the phase error are measured foreach detection method and given in Table 4.

TABLE 2. CHARACTERISTICS OF THE SIMULATED INPUT TEST SIGNAL.

Indices Amplitude [A] Phase Angles [ ] Starting Time [s]

Fundamental 30 0, 120, 240 05th harmonic 10 5 (0, 120, 240) 0.05Higher harmonics (7th, 11th, 13th, 17th) 1.5 each h (0, 120, 240) 0.15 All at the same time

TABLE 3. SETTINGS USED FOR CONFIGURING THESELECTED HARMONIC DETECTION METHODS.

Harmonic DetectionMethod Characteristics

DFT of the 5th Harmonic N = 256 samples/fundamental

RDFT of the 5th Harmonic N = 256 samples/fundamental

Fundamental dq Frame HPF 120 Hz, 2ndorder Butterworth,

10 kHz samplingfrequency

5th Harmonic dq Frame LPF 20 Hz, 2ndorder Butterworth,


Instantaneous pq Theory HPF 10 Hz, 3rd order Butterworth,


5th Harmonic Generalized Integration timeIntegrator constant T I = 300

Bode characteristic of the system in Figure 9 for different

values of the K i constant. The integrators are tuned for the

5th and 7th harmonics of a 50-Hz fundamental frequency.

5th Harmonic,250 Hz

7th Harmonic,350 Hz

K l = 10K l = 100K l = 500

Attenuation of 0 dB

20

0

20

40

60

80

M a g n

i t u

d e

( d B )

Phase Shift of 0 90

45

45

90

0

100 1,000

P h a s e

( )

Frequency (Hz)10

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As expected, the performance of the harmonic detection methodsdepends on the numerical filtercharacteristics. This can be seenalso from the Bode plots (Figure14) and step responses (Figure 15)obtained from three types of numerical filters, respectively, used

in the fundamental dq frame, har-monic dq frame, and pq theory.Other types of filters may also beimplemented according to the userchoice for speed or accuracy.

In the case of the DFT meth-ods, the settling time is limited toat least the window period (onefundamental period in this case),and the requirement is that theharmonic should be constant dur-ing the observation interval. Forthe RDFT method, the outputresponse is faster, but it is still

limited to the duration of the win-dow as is any FIR filter response.

The fundamental dq framemethod has a faster response and asmall overshoot but suffers from alarge phase error due to the phaseshift created by the HPF. Therefore,the compensation currents are not inphase with the existing disturbance,which is an impediment for a preciseharmonic detection. One solutionoften used in practice is that theHPF is implemented by mean of anLPF (HPF = 1 LPF). The dc sig-nal output of an LPF has no phaseshift; therefore, it has no delay.

The harmonic dq frame methoddoes not have the issue with the phase shift because the5th harmonic is a dc component filtered by an LPF. How-ever, here due to a low cutting frequency selected for theLPF, the response time is very slow. Another issue is thelarge existing ripple because the fundamental frequency(much bigger in amplitude compared to the harmonics)appears in the harmonic dq frame as an ac signal, whichmust be removed by the LPF.

Therefore, a good rejection or a lower cutting frequencymust be selected to reduce the fundamental. However,increasing the filter order or decreasing the cutting fre-quency degrades the response time.

For the pq theory, the output is characterized by theHPF used. In this case, the overshoot is larger, while theresponse time is relatively fast. The phase error of the HPFis solved by implementation of HPF = 1 LPF.

Obtaining the input test signal for testing the harmonic detection methods: (a) fun-

damental current, (b) 5th harmonic current, (c) higher harmonics currents, and (d)

resultant input test signal.

Start of the5th Harmonic

(a)

20[A]

20[A]

0

10[A]

10[A]

0

5[A]

5[A]

0

20[A]

20[A]

0

0 0.02 0.04 0.06 0.08 0.12 0.14 0.16 0.18 0.20.1

0 0.02 0.04 0.06 0.08 0.12 0.14 0.16 0.18 0.20.1

0 0.02 0.04 0.06 0.08 0.12 0.14 0.16 0.18 0.20.1

0 0.02 0.04 0.06 0.08 0.12 0.14 0.16 0.18 0.20.1

Higher Harmonics

Start of theHigher Harmonics

Resultant Signal (Phase A)

Time (s)

Fundamental Current

5th Harmonic

(b)

(c)

(d)

11

TABLE 4. RESULTS OBTAINED FROM THE HARMONIC DETECTION METHODS (FIGURE 13).

Harmonic Detection Settling Time 5th Overshoot or Method Harmonic [ms] Phase Error [ ] Ripple [%]

DFT 5th > 30 ms 0 0%Recursive DFT > 20 ms 0 0%Fundamental dq Frame > 10 ms 33.4 1% overshoot5th Harmonic dq Frame < 20 ms 0 10% ripplepq Theory < 10 ms 3.2 30% overshoot5th Generalized Integrator > 30 ms 0 0%

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In the case of the generalized integrator, the main limi-tation is given by the settling time, which depends on theintegration time constant T i used. Such large responsetime (comparable to the DFT case) is not suitable forapplications where the harmonics frequently vary within afew fundamental cycles. However, the method can be suc-cessfully used for harmonic compensation from most of the

typical ASD applications if oneassumes quasistationary condi-tions. Improvements in thedynamic response may beachieved by paralleling or cascad-ing APFs (based on generalizedintegrators) at different switchingfrequencies, as in [42].

Regarding the presence of thehigher harmonics, it was observedthat they are well rejected by theharmonic dq frame method, whilethe DFT and RDFT are disturbedat least one fundamental cycle.The generalized integratormethod is also slightly disturbedbecause there was no other con-troller tuned to remove the high-er harmonic components. Theinfluence of the filters may beseen in Figure 13, where three of the presented methods are practi-

cally tested with a dSpace system(TMS320F240) for an active filterimplementation.

The load currents from athree-phase diode rectifier areprocessed by the respective har-monic detection methods andthe signal obtained (i.e., currentreference) is summed with thedistorted current. The summa-tion done here excludes the con-tribution of the currentcontroller and the PWM inverterwhich will actually decrease the

tracking speed even more byintroducing more delays. Theresults are presented in Figure13 where both the THD i and thelevel of the 5th harmonic are cal-culated for each. Based on theconclusions obtained from theabove simulations, the practicalresults in Figure 13 now have aneasier interpretation.

For instance, even if bothmethodsfundamental dq frameand instantaneous pq theoryareimplemented with HPF tech-niques, the second method gives abetter result, explicable by thesmaller phase-shift, as measuredin Table 4.

Table 5 presents other conclusions that are important fora practical implementation of the active filter (where a plussign + indicates an advantage or an increase in perfor-mance). For example, pq theory has good dynamics butneeds sinusoidal input voltages, while the dq frame meth-ods depend on the angular speed, usually obtained from aPLL. The generalized integrator method requires individual

Practical validation of the fundamental dq frame, harmonic dq frame, and instanta-

neous pq theory. The choice of the filter implementation is responsible for a goodcurrent reference framework.

Load Current (Phase-A)

2[A]

2[A]

0

2[A]

2[A]

0

2[A]

2[A]

0

2[A]

2[A]

0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

THDi = 31.0%i 5 = 30%

THDi = 14.6%i 5 = 14%

THDi = 9.2%i 5 = 6.2%

THDi = 6.6%i 5 = 6.4%

Fundamental dq-Frame

5th Harmonic dq-Frame

pq Theory

Time (s)13

Simulation setup realized to study the performance of different harmonic detection

methods separated from the APF model.

50 HzFundamental

3-PhaseSignal Harmonic

Detection MethodTuned for 5th

Harmonic

Recording theResponse of

the 5thHarmonic Compare

to theOriginal

Input

5thHarmonic

HigherHarmonics

12

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31 T A B L E 5 . E V A L U A T I O N O F T H E S T U D I E D H A R M O N I C D E T E C T I O N M

E T H O D S .

F u n

d a m e n t a

l

H a r m o n

i c

G e n e r a

l i z e

d

F F T

D F T

R D F T

d q

F r a m e

d q

F r a m e

p q

T h e o r y

I n t e g r a

t o r s

N u m

b e r o

f S e n s o r s

( F o r

T h r e e c u r r e n

t s


t s


t s


t s ,


t s ,

T h r e e c u r r e n t s ,


t s

a C a s e o

f T h r e e - P

h a s e

t w o

/ t h r e e

t w o

/ t h r e e

t h r e e v o

l t a g e s

A p p

l i c a

t i o n

)

v o l t a g e s

v o l t a g e s

N u m

b e r o

f N u m e r i c a

l

0

0

0

2

H P F s

2

L P F s

N *

2

H P F s

2

N *

F i l t e r s

R e q u i r e

d b y

t h e

H a r m o n i c

D e

t e c

t i o n

A l g o r i t

h m

A d d i t i o n a

l T a s k s

R e q u

i r e d

W i n d o w

i n g ,

W i n d o w

i n g ,

/

P L L

P L L

V o

l t a g e

/

b y

t h e

H a r m o n

i c

s y n c

h r o n i z a

t i o n

s y n c

h r o n i z a t

i o n

p r e p r o c e s s

i n g

D e

t e c

t i o n

A l g o r i t

h m

C a

l c u l a

t i o n

B u r d e n

+

+

+

( E x c

l u d i n g

t h e

N u m e r i c a

l F i l t e r s

)

N u m e r i c a

l

C a

l c u l a

t i o n

C a

l c u l a

t i o n

I n s t a

b i l i t y

f o r

F i l t e r i n g

F i l t e r i n g , T

u n i n g

F i l t e r i n g

T u n i n g c o n

t r o

l

I m p

l e m e n

t a t i o n

I s s u e s

b u r

d e n

b u r

d e n

l o w p r e c

i s i o n

c o n

t r o

l

R e

l a t e d A l g o r i t

h m s o r

S i m

i l a r F

F T

/

R o

t a t i n g

F i l t e r s

t y p e

F i l t e r s

t y p e

F i l t e r s

t y p e ; o

t h e r

R e s o n a n

t

I m p

l e m e n

t a t i o n s

a l g o r i t

h m s

f r a m e

t h e o r i e s p q r , p q

0

f i l t e r s

t y p e

A p p

l i c a

t i o n s

i n S i n g

l e -

B o t h 1 - p

h /

B o t h 1 - p

h /

B o t h 1 - p

h /

I n h e r e n

t l y

I n h e r e n

t l y

I n h e r e n

t l y

B o t h 1 - p

h /

o r T

h r e e - P

h a s e

S y s t e m s

3 - p

h

3 - p

h

3 - p

h

3 - p

h

3 - p

h

3 - p

h

3 - p

h

U s a g e o

f t h e

V o

l t a g e

N o

N o

N o

Y e s

Y e s

Y e s

N o

I n f o r m a

t i o n

i n

t h e

A l g o r i t

h m

M e

t h o

d s P e r f o r m a n c e

+ +

+ +

+ +

+

+

+ +

f o r U n

b a

l a n c e

d a n

d

P r e

d i s t o r t e

d L i n e

V o

l t a g e s

M e

t h o

d s P e r f o r m a n c e

+ +

+ +

+ +

+

+ +

+ +

+

f o r U n

b a

l a n c e

d L o a

d

C u r r e n t s

A p p

l i e d f o r S e

l e c

t i v e

N o

Y e s

Y e s

N o

Y e s

N o

Y e s

H a r m o n i c

C o m p e n s a

t i o n

T r a n s i e n

t R e s p o n s e

T i m e

+

+ +

+

+ +

+

S t e a

d y -

S t a

t e A c c u r a c y

+

+

+

+

+

* N r e p r e s e n t s

t h e n u m

b e r o

f c o m p e n s a

t e d h a r m o n i c s .

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tuning for each frequency, but itneeds no input voltage.

Regarding which of these tech-

niques is better for a given case,some assumptions may be made,such as how a faster DSP reducesthe numerical implementationissues. Also, the acquisition of bothvoltages and currents is not neces-sarily a drawback and can be usefulfor protections or auxiliary featuresprovided by the APF. Therefore,the APF design should mainly con-sider the characteristics of the dis-torted currents, the dominantharmonic currents, their magni-tude, the variation in time, and theexistence of a certain unbalance inthe power system.

ConclusionsThis article evaluated the com-monly used methods for harmonicdetection in APF applications.The simulations show that thechoice of numerical filtering is akey factor for obtaining goodaccuracy and dynamics of an APF.

Bode plots of the numerical filters used for three of the harmonic detection methods: fundamental dq frame, harmonic dq

frame, and pq theory.

5th in Harmonic dq-Frame, (0 Hz, 0 dB)

5th in pq-Theory,(600 Hz, 0 dB)

5th in Fundamental dq-Frame, (600 Hz, 0 dB)

HPF Butterworth 2-Order, 120 HzLPF Butterworth 2-Order, 20 HzHPF Butterworth 3-Order, 10 Hz

0

20

40

60

80

M a g n i t u

d e

( d B )

5th in pq-Theory,(600 Hz, 3.2 )

5th in Fundamental dq-Frame, (600 Hz, 33.4 )

5th in Harmonic dq-Frame, (0 Hz, 0 )

270

180

90

0

90

180

P h a s e

( )

1 10 100 1,000Frequency (Hz)

14

Step responses of the numerical filters used for three of the harmonic detection

methods: fundamental dq frame, harmonic dq frame, and pq theory.

1.2

1

0.8

0.6

0.4

0.2

0

0.2

0.40 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Time (s)

Fast Response forHigher Frequencies

Using HPF's inFundamental dq -

Frame and pq-Theory

Slow Response forLower Frequencies anddc-Signals Using HPFs

in Fundamental dq - Frame and pq-Theory

Step Input Signal Slow Dynamic for dc-Signal,i.e., 5th in Harmonic dq-Frame

HPF Butterworth 2-order, 120 HzLPF Butterworth 2-order, 20 HzHPF Butterworth 3-order, 10 HzStep Input Signal

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Lucian Asiminoaei ([email protected]) is with Danfoss Drives A/S in Graasten, Denmark. Frede Blaabjerg is with AalborgUniversity, Institute of Energy Technology, in Aalborg,

Denmark. Steffan Hansen is with Danfoss Drives A/S inGraasten, Denmark. Asiminoaei and Hansen are IEEE Mem-bers. Blaabjerg is an IEEE Fellow. This article first appeared asHarmonic Detection Methods for Active Power Filter Applica-tions at the 2004 IAS Annual Meeting.

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