Electrical Power Quality and Utilisation, Journal Vol. XII, No. 2, 2006
Harmonic Compensation Using the Sliding DFT
Algorithm for Three-phase Active Power Filter
K. P. SOZAÑSKI, Member, IEEE
University of Zielona Góra, Poland
Summary: This paper describes the sliding discrete Fourier transform (DFT) algorithm as an
alternative for typical DFT used for spectrum analysis and synthesis. As an example a control
circuit for a three-phase 75 kVA parallel active power filter (APF) is used. Such filters have
been built and tested. Some illustrative, experimental results are also presented in the paper.
The control algorithms of the proposed APF are implemented in the fixed-point digital signal
processor TMS320C50. The presented control APF algorithm is a good alternative to classical
APF control algorithms, because it allows easy selection of control parameters in mains currents:
imbalance, reactive power or harmonics contents. In the proposed circuit transient performance
of APF is improved using non-causal predictive current compensation.
1. INTRODUCTION
Conventionally, passive LC filters and capacitors have been
used to eliminate line current harmonics and to increase the
power factor. However, in some practical applications, in which
the amplitude and the harmonic content of the distortion power
can vary randomly, this conventional solution becomes
ineffective. To suppress such harmonics, an active powerharmonic-compensation filter should be used. The parallel (shunt)
active power filter (APF) permits compensation of the harmonics
and asymmetries of the mains currents caused by nonlinear
loads. Two versions of harmonic compensation circuit with
current-fed active power filter are possible: without feedback
(with unity gain) and with feedback. Because of its better stability
an APF without feedback was chosen. The three-phase active
power filter tests circuit is shown in Fig. 1.
The shunt active power filter injects AC power current iC
(Fig. 1) to cancel the main AC harmonic content. To suppress
power line harmonics, an active power-harmoniccompensation filter can be used. The parallel (shunt) APF
permits compensation of the harmonics and asymmetries of
the mains currents caused by nonlinear loads.
When It is possible to find in the literature many control
algorithms for three-phase active power filters, for example
in: [3, 4, 6, 10]. The control algorithm presented in this paper
offers many advantages to existing ones. In this paper a new
control algorithm with non causal harmonics compensation
circuit is presented too.
Key words:
active filter,
harmonics,
DSP,
Power Quality,
Power factor correction
spectrum analysis is the DFT, using the efficient
implementation fast Fourier transform (FFT). Typically,
N-point DFT is calculated for every sample of input signal,
for example described in [12]. For this kind of task the sliding
DFT algorithm is better than ordinary DFT. Such a solution is
very simple and efficient, especially in coherent sampling.
The z-domain transfer function for the kth bin of the sliding
DFT filter is described by equation
H SDFT ( z ) =
1 − z−N
1 − e j 2 ð k N z −1
(1)
where:
N — length of the signal block, typically equal to signal period,
k — number of frequency bin.
The single-bin sliding DFT filter structure is shown in
Figure 2. The magnitude frequency characteristic of a singlebin sliding filter for N = 10 and k = 1 is shown in Figure 3a.
2. THE CONTROL CIRCUIT
The control algorithm is based on a sliding discrete Fourier
transform (DFT). The sliding DFT is well described by
Jacobsen and Lyons in [1] and its application in a singlephase APF is presented by author in [8, 9].
2.1. A Short Introduction to Sliding DFT
Spectrum analysis of signal in power electronics is an
important measuring technique. The usual method for
Fig. 1. Three-phase active power filter tests circuit
K.P. Sozañski: Harmonic Compensation Using the Sliding DFT Algorithm for Three-phase Active Power Filter
#
Fig. 2. Block diagram of single-bin sliding filter
in Figure 1 and 5a. To model the nonlinear load a thyristor
power controller with resistive loads is used. The power circuit
consists of a three-phase IGBT power transistor bridge IPM
(intelligent power module) connected to the AC mains
through inductors: LC1, LC2, LC3. The APF circuit contains a
DC energy storage component, two capacitors C1 and C2.
The control circuit is realized using the fixed-point digital
signal processor TMS320C50. The digital signal processor is
synchronized with the mains voltage by a synchronization
unit. It consists of: a low-pass filter and phase locked loop
circuit (PLL).
The active power filter injects the harmonic current iC,
into the power network and offers a notable compensation
for harmonics and reactive power. The compensating current
can be determined by
iC1 (t ) = iL1 (t ) − I m1 sin (wt + j1 )
(2)
where:
I m1 — amplitude of first harmonic.
In the case when harmonics compensation is ideal the line
current iS1(t) consists in only the first harmonics of mains
current
iS1 (t ) = I m1 sin (wt + j1 )
Fig. 3. Sliding DFT characteristics for N=10 and k=2: a) magnitude,
b) z-domain pole/zero location
When phase angle between mains voltage u1 and line
current iS is equal to zero the reactive power is compensated
too.
In Figure 5 three versions of the control circuit are shown:
for imbalance, reactive power and harmonics compensations
(Fig. 5a), for reactive power and harmonics compensations
(Fig. 5b), for harmonics compensation (Fig. 5c).
A detailed block diagram of the control algorithm for
compensating: imbalance, reactive power and harmonics
content is depicted in Figure 6. In this diagram the circuit
controlling voltage of DC energy storage (C1 and C1) is
omitted for simplicity.
The sampling periods can be calculated with the formula
Tp =
Fig. 4. The block diagram of first harmonic detector
The passband and stopband of the filter are very poor but
they are adequate for coherent sampled signals. Using the
single-bin sliding filter (from Fig. 1) it is possible to build a
first harmonic detector. The block diagram of such a circuit is
depicted in Figure 4.
2.2. Realization of Control Circuit
In this research a three-phase parallel active power filter is
used. For the laboratory experiments the parallel APF without
feedback was used because of its greater stability. A simplified
block diagram of the proposed active power compensation
circuit with the parallel APF for power of 75 kVA is depicted
$
(3)
T1
N
(4)
where:
T1
— period of the mains voltage,
f1 = T1–1 — frequency of the mains voltage,
N
— total number of samples per mains period.
The algorithm is performed N times per mains period. For
the mains voltage frequency of f1 = 50 Hz and the chosen
number of samples N = 256, the sampling periods is equal
Tp = 78.125ms and the sampling rate is equal to fp = 12800
samples/s.
The signals representing load currents iL1(t), iL2(t), iL3(t)
are converted to digital domain by a 14-bit analog-to-digital
converter with sampling ratio fp. For the control algorithm
a first harmonic detector using sliding DFT is employed. In
this solution only one single-bin sliding DFT filter structure
for detecting first harmonic of load current is used. The first
harmonic spectral component signal of the load current
is calculated by the equation:
Power Quality and Utilization, Journal • Vol. XII, No 2, 2006
m
Fig. 5. Three-phase active power filter with control circuits for: a) unbalance, reactive power and harmonics compensations, b) reactive power
and harmonics compensations, c) harmonics compensation
(
)
(
)
s1 (nTp ) = s1 (n − 1)T p ⋅ e j 2 ð/ N − iL1 (n − N )Tp + iL1 (nTp ) (5)
where:
iL1(nTp)
s1(nTp)
s1((n-1)Tp)
— discrete signal representing first phase
load current,
— discrete signal representing first harmonic
complex spectral component of first phase
load current,
— discrete signal representing previous first
harmonic complex spectral component of
first phase load current.
The discrete signal representing the first harmonic signal
of a load current with zero phase angle between mains voltage
u1(t) and line current iS1(t) can be described by the equation:
( ) (
ih1 (nTp ) = 2 N s1 nTp sin 2 p50nTp + j1
)
(6)
Compensating current signal is the result of a difference
between the load current signal and the first harmonic
reference sinusoidal signal:
( ) (
)
iC1 (nTp ) = iL1 (nTp ) − 2 N s1 nTp sin 2 p50nTp + j2 (7)
K.P. Sozañski: Harmonic Compensation Using the Sliding DFT Algorithm for Three-phase Active Power Filter
%
Fig. 6. Simplified block diagram of active power filter control circuit using sliding DFT for: current unbalance, reactive power and harmonics
compensation
In the case when current imbalance must be compensated
current ic1(nTp) has to be calculated by the formula:
iC1 (nTp ) = iL1 (nTp ) − 2 N
(
( )
( )
( )
s1 nT p + s2 nT p + s3 nT p
⋅ sin 2 p50nTp + j1
3
)
⋅
(8)
where:
s2(nTp), s3(nTp) — discrete signal representing first
harmonic complex spectral component
signal of second and third phase load
current.
&
In the summing block the resultant magnitude of three
phase current is calculated. The magnitude is used to
modulate the amplitude of each phase first harmonic reference
signal. By subtracting these signals with appropriate input
signals the output compensating signals iC1(nTp), iC2(nTp),
iC3(nTp) are calculated.
3. RESULTS ACHIEVED
A prototype of the three-phase active power filter was
built and tested in the laboratory. The simplified diagram of
the test circuit is depicted in Figures 1 and 5. As the nonlinear
load a thyristor power controller RI31 (from METROL) with
the resistive load was used.
Power Quality and Utilization, Journal • Vol. XII, No 2, 2006
Fig. 7. Experimental waveforms of active power filter in steady-state
with the resistive load: load current iL1 (blue), compensating current
iC1 (green), line current iS1 (red)
Fig. 9. Simplified block diagram of harmonics compensation with
non-causal control circuit
Fig. 8. Harmonic spectrum of line current iS1 of active power filter in
steady-state with the resistive load
Figure 7 shows the steady-state performance of the active
power filter. Depicted are: load current IL1, compensating
current IC1, line current IS1. The harmonic spectrum of the
line current IS1 is depicted in Figure 8.
The results achieved are comparable to results achieved
using classical control algorithms. For example, in the same
active power filter an instantaneous reactive power control
algorithm was implemented, which results are described in:
[10, 11].
4. NON-CAUSAL SOLUTION
The APF control current dynamics is dependent on the
inverter output time constant, resulting from APF output
inductance and resultant impedance of load and mains. When
the value of load current changes rapidly, as in current iL in
Figure 7, the APF transient response is too slow [13, 14] the
line current iS suffers from dynamic distortion. This distortion
causes an increase of harmonic content in the line current,
which is dependent on a time constant. In the APF shown in
Figure 1 the THD ratio is increased by about 10%.
The mains loads can be divided into two main categories:
predictable loads and noise-like loads. Most loads belong to
the first category. For this reason it is possible to predict
current values in subsequent periods, after a few periods
of observation.
In this solution, for predictable loads, it is possible to use
a circuit with non-causal current compensation as shown in
Figure 9. This compensation is dependent on the inverter
output time constant. Current samples iC(nTp) are stored in
DSP memory (sample register) and in subsequent periods of
mains current are compared with present samples, then if
respective sample differences are less than assumed values
the non-causal current compensation algorithm is switched
on (switch S1 in position 1), sending to the output non-causal
samples iC(nTp).
Previous current compensation signal samples iC(nTp) are
stored in memory, and are sent to present output in advance.
In the experimental circuit the advance time TA is about several
hundred microseconds. Because the time constant is
dependent on the load parameter an adaptive algorithm to
calculate advance time is employed. If load current is changed
during a mains period the non-causal current predictive
algorithm is switched off (switch S1 in position 2), and the
algorithm waits for a steady-state and when detected it once
again switches on (switch S1 in position 1).
Figure 10 shows the experimental waveforms in the same
conditions as the classical control algorithm (Fig. 7). All
waveforms (Fig. 10) of the active power filter with non-causal
predictive current compensation are in steady-state with the
resistive load. Depicted are the following waveforms: load
currents iL1, compensating currents iC1, line currents iS1.
The harmonic spectrum of the line current iS1 for the circuit
without compensation is depicted in Figure 11. Using the
new control algorithm with predictive harmonic compensation
it is possible to decrease the harmonic contents in power line
currents from THD=12.5% to THD=4.3%.
K.P. Sozañski: Harmonic Compensation Using the Sliding DFT Algorithm for Three-phase Active Power Filter
'
Fig. 10. Experimental waveforms of active power filter in steadystate with the resistive load for non-causal control circuit: load current
iL1 (blue), compensating current iC1 (green), line current iS1 (red)
5. CONCLUSION
The sliding discrete Fourier transform (DFT) algorithm
proves to be a good alternative for typical DFT used for
signal spectrum analysis and synthesis. The passband and
stopband of filter sliding DFT are very poor and are adequate
only for coherent sampled signals
In this research control circuits for three-phase 75 kVA
parallel active power filter (APF) were implemented. The filter
has been built and tested in our laboratory. Some illustrative,
experimental results have also been presented in the paper.
The new control algorithms of the proposed APF are
implemented in the floating-point digital signal processor
TMS320C50. The control algorithm is easy to implement,
especially using floating point digital signal processors. For
fixed point digital processors this algorithm is more difficult,
particularly for more precise calculation of a complex signal
module.
Using non-causal predictive current compensation it is
possible to decrees harmonics contents for predictable mains
loads.
The presented control APF algorithm is a good alternative
to classical APF control algorithms because it allows easy
selection of mains current parameters: imbalance, reactive
power or harmonics contents.
The sliding discrete Fourier transform (DFT) algorithm is
well suited for other applications in power electronics where
spectrum analysis is necessary, for example: power spectrum
analyzers, energy meters etc.
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Krzysztof Sozañski (M’1997)
was born in Czerwieñsk in Poland, on July 25, 1957. In
1981 he was awarded an MSc degree in electrical
engineering, specializing in the field of automation and
electric metrology, from the Electrical Engineering
Department at the Zielona Góra University of
Technology, in Zielona Góra, Poland. In 1999 he received
his PhD degree in telecommunications from the Institute
of Electronics and Telecommunications at Poznañ
University of Technology, in Poznañ, Poland. He is an Associate Professor
at the Institute of Electrical Engineering, University of Zielona Góra. His
current research interests are in digital signal processing, implementation
of digital signal processing methods in digital signal processors, power
electronics, and active power filters. He is author or coauthor of more
then 70 periodical and conference papers and 1 patent.
Address:
University of Zielona Góra; ul. Podgórna 50, 65-246 Zielona Góra
phone: +4868-3282567; fax: +4868-3254615;
e-mail: K.Sozanski@iee.uz.zgora.pl;web page: www.uz.zgora.pl\~ksozansk
Electrical Power Quality and Utilization, Journal • Vol. XII, No 2, 2006