Harmonic Detection Methods for Active PowerFilters Based on Discrete Cosine Transform and

DitheringAlejandro G. Yepes, Francisco D. Freijedo, Jesus Doval-Gandoy,

Oscar Lopez, Pablo Fernandez-Comesana and Jano MalvarDepartment of Electronic Technology, University of Vigo,

ETSEI, Campus Universitario de Vigo, 36200, Spain.Email:agyepes,fdfrei,jdoval,olopez,pablofercom,janomalvar@uvigo.es

Abstract—Harmonic detection is a very important factor incontrol of active power filters, since it generates the referencesignal for the current controller. The performance of the wholesystem will be affected by the suitability of the implementedharmonic identification method. In this work, the equivalencesand advantages of the discrete cosine transform with respect toother Fourier based algorithms are demonstrated. It is provedin this paper that both the transient length and the phaseerror in presence of grid frequency deviations are halved incomparison to DFT. Therefore, it is convenient to provide afrequency adaptation feature to the discrete cosine transform. Inthis work it is proposed to adjust the window width to the gridfundamental frequency, and to enhance the frequency resolutionby means of dithering. The theoretical approach is verified byexperimental results in an APF lab prototype.

I. INTRODUCTION

The ac current distribution network pollution and the reac-tive power demand have been considerably increased, due tothe proliferation of equipment with switching non-linear loads[1]. Different mitigation solutions have been proposed andused, involving passive filters, active power filters (APFs) andhybrid active-passive filters. Currently, due to the technologicaladvance in digital control techniques, APFs have become anefficient alternative to passive filters. Therefore, there is anincreasing interest to develop and use better active filteringsolutions [2]–[10].

In this work, it is studied the use of the discrete cosinetransform (DCT) for reference generation algorithms for APFs.The DCT permits to use a window length of a fundamentalsemi-period, which implies that the transient provided bya DCT filter lasts just half a cycle. This is an importantimprovement over the most widely used algorithms, such asDFT, FFT and RDFT [10]–[12], which need a full fundamentalcycle [10], and reference generation techniques based on p-qtheory, which even for low order filters may require more thantwo cycles [12]. Moreover, the recursive formulation of theRDFT may yield larger numerical errors [13], so its advantageon implementation simplicity is not always decisive.

In this paper, it is analytically demonstrated that the methodbased on Synchronous Reference Frame (SRF) and MovingAverage Filters (MAFs) of [14], [15], for which the transient

lasts half a cycle, actually implements the DCT in a recursivemanner. However, when considering fixed point implementa-tions, FIR digital filters based on DCT have some advantageswith respect to SRF-IIR based structures: recursive schemesfeature a larger quantization error due to the finite word length[13], [16].

SRF schemes provide a good adaptation in presence of gridfrequency deviations when they are employed in combinationwith PLLs. This approach can be extended to DCT by somealternatives. In [16] it is proposed to employ variable samplerate for repetitive current control based on DCT, but itsdiscrete-time implementation is not a trivial task. In this work,it is proposed another alternative. The limitation of beingonly possible to set integer values for the window width isovercome by the application of dither signals. The ditheringtechnique consists in the application of noise in order toincrease the accuracy by reducing quantization patterns. It hasbeen widely used in fields such as digital audio and imageprocessing. Dithering is not so extended in power electronicscontrol, although several approaches have proposed to employit in power converters. Some significant applications were theuse of dither signals in combination with hysteresis bands andSigma-Delta modulation [17]–[19], and the enhacement of thefrequency resolution of resonant converters [20].

The theoretical approach is supported by experimental re-sults. It is demonstrated the advantages of the DCT over theDFT for APFs. The effectiveness of the proposed adaptivemethod is also proved, even when considerable frequencydeviations occur.

II. STUDY OF DCT EQUIVALENCES AND ADVANTAGES

A. Reference Generation by DCT Filter

The reference generation algorithm should detect the har-monic components (iLh) of a distorted current (iL) demandedby non-linear loads. It can be achieved by the subtraction ofthe fundamental component (iL1), so iLh = iL − iL1. The role ofthe algorithm is to extract iL1. If ω1n defines the nominal valueof the grid fundamental frequency1, a DCT filter for harmonic

1ω1n is 2π50 rad/s in Europe.

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identification can be expressed as:

F(z) =IL1(z)IL(z)

=2N

N−1

∑n=0

cos(ω1n nTs)z−n, (1)

N being the window width, which coincides with the numberof samples that represents half a fundamental cycle.

Equation (1) matches a DCT-I type expression, and itcorresponds to a N − 1 order digital filter. Its N coefficientsare constants stored in a table in memory.

B. Equivalence to SRF Schemes Based on MAFs

In [14], [15] it is proposed a method for fundamentalcomponent identification based on MAFs, which is shown inFig. 1. The transfer function of an MAF can be expressed as:

H(z) =1N

N−1

∑n=0

z−n =1N

1− z−N

1− z−1 , (2)

where N is chosen to match the number of samples of half afundamental cycle, in order to cancel all odd harmonics.

The algorithm of Fig. 1 can be simplified in the form of aFIR digital filter in stationary frame, so the scheme is replacedby just one block.

wx(k) = cos(ω1nkTs +θR)vx(k) =

= cos(ω1nkTs +θR)2N

N−1

∑n=0

ux(k−n) =

=2N

N−1

∑n=0

cos(ω1nkTs +θR)cos(ω1n(k−n)Ts +θR)iL(k−n).

(3)

The same reasoning is applied to the orthogonal signal:

wy(k)=2N

N−1

∑n=0

sin(ω1nkTs +θR)sin(ω1n(k−n)Ts +θR)iL(k−n).

(4)Adding both expressions and using trigonometric identitiesresults in:

iL1(k) = wx(k)+wy(k) =

=2N

N−1

∑n=0

cos(ω1n(k−n)Ts +θR −ω1nkTs −θR)iL(k−n) =

=2N

N−1

∑n=0

cos(ω1n nTs)iL(k−n),

(5)

which leads to (1), so the equivalence between the algorithmsis demonstrated.

Following an analogous mathematical reasoning, it is easyto prove that to employ a DCT of 2N samples is equivalentto the RDFT based scheme shown in Fig. 2 of [11].

If selective harmonic identification is desired, the stationaryframe implementation is more simple than using SRFs, sinceit can be performed for an arbitrary number of harmonicswithout increasing the computational burden. On the contrary,in the case of SRF methods, the complexity increases with

xuX X2 H(z)

xv xw

Li

+

+X X2 H(z)

1Li

ywyvyu

cos

sin

Figure 1. SRF scheme based on MAFs for fundamental componentidentification.

the number of harmonics to compensate, since a scheme asthe one shown in Fig. 1 should be used for each component.

On the other hand, the frequency adaptation of the DCTfilter is not a trivial task. It can be achieved by modifyingthe sample rate in real time [16], but the adaptation of SRFalgorithms is easier to obtain.

C. Equivalence Between DFT and DCT

The DFT can be defined as:

G(z) =1N

2N−1

∑n=0

(cos(ω1n nTs)− j sin(ω1n nTs))z−n. (6)

For real and even-symmetric signals, (6) performs exactlythe same as (1) in steady state and ideal conditions (when thegrid frequency equals ω1n). This is because of the fact thathalf of the data becomes redundant, so half of the coefficientsare no longer necessary, and the imaginary part of (6) becomeszero.

However, if (6) is implemented instead of (1), 2N coeffi-cients are employed. Therefore, the behavior differs duringtransients and/or in presence of grid frequency deviations.Those differences are analyzed in the following section, study-ing the effect of the window length.

In fact, a 2N samples DCT is often implemented as anequivalent to a DFT [16] or, in a recursive manner, to anRDFT [11].

D. Advantages of DCT for APFs Reference Generation

Fig. 2 shows the frequency response of DCT filters withN = 100 and N = 200 (corresponding to half a cycle andto a full cycle, respectively, at 10 kHz commutation). Thedouble window length provides a better filtering at the cost ofdoubling the transient time.

It is a common practice to neglect even harmonics insystems voltages and currents, which is accurate for most ofpractical electric circuits [21].

Therefore, it could be said that DCT with N = 100 is a moreefficient alternative. Another benefit appears when frequencydeviations are considered: as it can be observed in Fig. 2, whenthe input wave frequency is not nominal, the phase error ishalved.

III. FREQUENCY ADAPTIVE DCT

If the grid frequency deviates from nominal, the transferfunction of (1) should be modified in order to maintain zerosteady state error. A direct method to adapt the filter would be

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-50

-40

-30

-20

-10

0

10

Ma

gn

itu

de

(d

B)

0 50 100 150 200 250 300 350 400 450 500-180

-90

0

90

180

Ph

as

e (

de

g)

Frequency (Hz)

N=100

N=200

Figure 2. Frequency response of DCT with N = 100 and N = 200 (half andfull fundamental cycle, respectively).

to recalculate each of the coefficients of (1) according to theactual value of the grid fundamental frequency (ω1). However,that would imply a huge computational effort, reducing theperformance. An alternative approach is proposed below.

Applying trigonometric identities:

F(z) =2N

N−1

∑n=0

cos(ω1n nTs)cos(∆ω1 nTs)z−n−

− 2N

N−1

∑n=0

sin(ω1n nTs)sin(∆ω1 nTs)z−n,

(7)

where ∆ω1 is the deviation from nominal frequency in rad/s(∆ω1 = ω1 −ω1n).

Assuming ∆ω1 nTs ≈ 0:

F(z) =2N

N−1

∑n=0

cos(ω1n nTs)︸ ︷︷ ︸

an

z−n−

− 2N

N−1

∑n=0

sin(ω1n nTs)nTs︸ ︷︷ ︸

bn

∆ω1z−n,

(8)

where an and bn are constants, so they can be calculatedoff-line before the control is implemented. The filter can beseparated in two blocks, one of which is multiplied by a gainof value ∆ω1, obtained from a PLL. Each block is actually adigital filter of N coefficients.

The only remaining parameter that should be also modifiedis N, which is easily obtained in real time by the identityN = 2 fs/ f1 = 4π fs/ω1, where fs is the sampling frequency ofthe controller and f1 = ω1/2π is the actual value of the gridfundamental frequency in Hz. The variation of N is feasiblewithout an important increase of the operations needed, sinceit means in practice just taking into account or not the lastcoefficients of the filter, as well as changing the gain 2/N.Focusing on implementation, the correction of the windowwidth is achieved by simply modifying the maximum value(N) of the pointer that sweeps through both tables in eachcycle of the controller.

As N can only take integer values, this approach to correct

X

1N

0n

nnza

N

2

1N

0n

nnzb

N

2

+

-

1Li

1

Li

Dither0.9

0+

+1

4 sf

1

'N

N

Figure 3. DCT improved with frequency adaptation and dithering.

the window width implies that there is going to be a limitedresolution in the frequency adaptation. If fs is set to 10 kHz,the most reduced feasible variation in N (±1) corresponds to±0.5 Hz, which is a considerable error. This issue can besolved by means of dithering, which permits to improve reso-lution by means of the addition of noise. Several applicationsof this technique to power converters where exposed in [17]–[20].

In this case, a number in the interval [0,0.9] is generatedwith uniform probability, and it is added to the expressionN′ = 4π fs/ω1. Then, the result is truncated.

This process causes an oscillatory behavior, alternatingbetween the two closest integer values of N, and the average ofN matches N′. Given that the error introduced in iL1 is almostlinear, its medium value also fits the actual magnitude of thefundamental component.

It should be remarked that the amplitude of the resultingnoise in iL1 is minimal. Actually, iL1 is always containedbetween the two signals that would be obtained if the closestinteger values of N were chosen. Moreover, these oscillationsin the current reference (i∗f ) are filtered by the bandwidth ofthe current controller and the plant. Finally, the harmoniccomponents of the current demanded by the APF (i f ) areexpected to average the original harmonics of iL.

In most applications the dither signal consists in a randomnumber, so the noise is spread across the frequency spectrumand, therefore, it becomes less noticeable. However, in thiscase, it is preferable to concentrate the noise in higher fre-quencies, in order to achieve a better filtering by means of theinductance. The added dither is chosen to follow the pattern:0.9, 0.4, 0.8, 0.3, 0.7, 0.2, 0.6, 0.1, 0.5, 0. In this manner, thelowest harmonic appears at 1000 Hz, and most of the noisepower is concentrated at 5 kHz, which is attenuated 24 dB inclosed loop.

Fig. 3 depicts the resulting scheme for the fundamentalcomponent identification.

It should be remarked that the proposed technique forenhancing the resolution by dithering can be also appliedto other harmonic identification methods that include integer

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sin( 1 )

if 1

*

Vs

PCCis

ifVSC

Lf

CVdc

PWM

-

PLL

PI

+

+

Vdc*

-

+

+if*

if

iLh

HARMONIC

DETECTION

iL

iL

DIGITAL CONTROLLER

Vs

Vdc

1 1

Ls

X

Rf

PR CURRENT

CONTROLLER

AC REGULATOR

UNCONTROLLED

RECTIFIER

R1

R2

L1

L2

|if 1

*|

Vf*

Vf

Figure 4. Prototype diagram of the APF power circuit and control.

delays, such as DFT, MAFs and the delayed signal cancellation(DSC) [22] algorithm.

IV. APF PROTOTYPE AND CONTROLLER

Fig. 4 shows the tested single-phase APF prototype. It iscomposed of an IGBT based voltage source converter (VSC)connected to the point of common coupling (PCC) through theinterfacing inductor L f . The equivalent series resistance (R f )of this inductor has been measured and taken into accountfor the modelling. Table I shows the values of the powercircuit components. fsw is the IGBTs switching frequency. Aprogrammable ac source (Chroma 61501) is employed, andthe control has been implemented in a prototyping platform(dSpace DS1103). Two non-linear loads are connected to thePCC: an AC regulator and a diode-bridge rectifier.

Table IPOWER CIRCUIT VALUES.

Parameter Value Parameter ValueV ∗

dc 220 V C 1 mFVsrms 110 V L f 5 mHR1 149 Ω R f 0.5 ΩL1 8 mH Ls 50 µHR2 153 Ω fsw 10 kHzL2 12.5 mH fs 10 kHz

The proposed algorithm has been tested in a single-phaseAPF, although its application to three phase systems is alsofeasible [14], [23].

As the objective of the experiments is to prove the effec-tiveness of the harmonic identification methods, it has beenemployed Proportional+Resonant (PR) current control, whichassures zero steady state error [24]. The frequency adaptationof the controller is achieved as it was proposed in [25]–[27].The frequency is measured by a phase locked loop (PLL) withfast transient, as those exposed in [28], [29].

The PR controller has been tuned with a proportional gainof 10 and an integral gain of 1000 for each of the resonant

Figure 5. iL1 response to a iL change, obtained by DCT filters of N = 100(Ch1) and N = 200 (Ch3). Ch2 is iL.

regulators. Three of them are implemented, in order to com-pensate 3rd , 5th and 7th harmonics. With these parameters, it isobtained a phase margin PM = 50deg, which assures stability.

V. EXPERIMENTAL RESULTS

A. Advantages of Half-Cycle DCT for Harmonic Identification

Fig. 5 compares the transient response to an abrupt changein the load current (when the ac source is connected) providedby DCT filters of half and full period window lengths. Asexpected, the former is much faster; it achieves steady state inhalf a cycle. Only a resistive load (R2) has been considered,in order to avoid undesired transients in iL.

Figs. 6 shows the fundamental component extracted withboth DCT filters. It is proved that the phase error is doubledin the case of the larger window, as it was stated in sectionII-D.

B. Test of Proposed Frequency Adaptive Method

The resulting load current iL features the highly distortedspectrum shown in Fig. 7, with THD≈ 29%.

The whole system is tested in a very unfavourable condition( f1 = 51.25 Hz) with the adaptive and non adaptive schemes,giving rise to Figs. 8 in steady state.

Figs. 8(a) and 8(b) depict the results with the proposedadaptive method. iL is satisfactorily filtered thanks to the fre-quency adaptation effectiveness of the harmonic identificationalgorithm, and the infinite gain of the implemented resonantcontrollers (for harmonics 3rd , 5th and 7th). Only the highorder components of iL remain present in the source currentis.

Figs. 8(c) and 8(d) show the performance of the APFusing DCT filter (1) with N = 100. The obtained THD ishigher than the one provided by the adaptive method (about11% and 9%, respectively), so the superior behavior of theproposed algorithm is demonstrated. However, the error causedby the original non adaptive half-cycle DCT filter is not verysignificant, and it can be neglected for many real applications.

VI. CONCLUSIONS

A comparative analysis between several implementationsbased on Fourier transforms has been provided. The equiv-alence between the DCT and previous schemes with SRF has

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(a) Comparison of the four waves.

(b) Detail of the extracted iL1 signals.

Figure 6. Comparison of iL1 obtained by DCT filters of N = 100 (Ch1) andN = 200 (Ch3) in steady state, at f1 = 48 Hz and f1n = 50 Hz. Ch2 is iL andCh4 is iL1 obtained with a DCT tuned at f1n = 48 Hz. The phase error isdoubled when a window of a full period is employed.

Figure 7. Fourier spectrum of iL.

been analitically proved. It has been also concluded that theDCT permits to reduce the DFT phase error by half in presenceof grid frequency deviations, as well as the transient length.

Taking advantage of these good features, an algorithm hasbeen developed in order to provide frequency adaptation tothe DCT. The window width is adjusted in real time to thevalue of the grid frequency, and the dithering technique hasbeen employed to increase the frequency resolution.

A shunt APF lab prototype has been built, and the ex-perimental results validate the theoretical approach and theeffectiveness of the novel frequency adaptive algorithm.

REFERENCES

[1] H. Akagi, “Active harmonic filters,” Proceedings of the IEEE, vol. 93,no. 12, pp. 2128–2141, Dec. 2005.

(a) Steady state currents with the pro-posed adaptive algorithm.

(b) Fourier spectrum of is inFig. 8(c).

(c) Steady state currents with DCT filter(1).

(d) Fourier spectrum of is inFig. 8(a).

Figure 8. Steady state currents and FFT with DCT filter (1) and the proposedadaptive algorithm, at f1 = 51.25 Hz and f1n = 50 Hz. Ch2 is iL, Ch3 is isand Ch4 is i f .

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