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Abstract--This paper gives an overview of techniques to
determine the harmonic content of the currents of three-phase
power systems. The different methods, such as the Fourier
transformation, the FBD-method, the instantaneous reactive
power theory, the dq-transformation and the multiple rotating
transformation method are described in detail. Advantages and
disadvantages are given and the most favorable methods are
compared.
As an example, these methods are used for the control of a
Shunt Active Power Filter to investigate them in detail. For this
purpose, the filter is simulated in the PSpice environment. Two
different methods to calculate the harmonic content in three-
phase power systems are implemented in a special DSP-model to
simulate the time discrete behavior of processors. The simulation
results are presented in this paper.
Index Terms - Harmonic Measurement, Harmonics, Active
Filters.
I. INTRODUCTION
With the increasing use of non-linear loads such as diode-
bridge rectifiers, adjustable speed-drives and cycloconverters
the generation of harmonic currents has steadily increased. Thedisadvantages [1] of these harmonic currents are well known.
Among others, additional heat in power cables, transformers,
electrical machines and capacitors is generated due to the
increased RMS current. In three-phase four-wire systems, the
triplen harmonics all add up in the neutral conductor which
can lead to an unacceptable high current in this conductor. The
power factor is typically reduced, and harmonics accelerate the
system’s aging process causing extra costs.
In the past, different passive, active or a combination of
both, the so-called hybrid filters have been investigated [2, 3].
Especially the Shunt Active Power Filter (SAPF) has proven to
be very effective and is therefore available on the commercialmarket and has been installed in industrial power systems.
The control of these filters requires the calculation of the
harmonic content of the load currents. In this paper different
methods such as the Fourier analysis, FBD-Method, multiple
rotating transformation method, the instantaneous reactive
power theory, and the dq-transformation are described in detail
and are discussed. Based on this discussion two methods are
selected for comparison with the help of the simulation tool
PSpice.
II. SHUNT ACTIVE POWER FILTER
The single phase diagram of a Shunt Active Power Filter is
depicted in Fig. 1. The source supplies a harmonic generating
load. A SAPF is installed in parallel to the load to compensate
the harmonics making the source current is sinusoidal. In this
case, the active filter current iAF has to be equal to the
harmonic content of the load current iL. Therefore, depending
on the control, the load current iL or the source current iS can
be measured and its harmonic content can be calculated. In all
examples in this paper the load current is measured. The
calculated harmonic content is used as the current command
signal for the SAPF.
In this paper, the authors mention that the goal of an activefilter is to generate sinusoidal source currents. The other
option is to control the SAPF in a way that the source currents
become proportional to the source voltage, i.e. the filter tries
to model a resistive load. Opinions vary about which control is
the preferred one. It is the authors' opinion that from the utility
customers view-point the sinusoidal source current generation
is the most favorable because in this case active power is
drawn only from the fundamental voltage.
iS i
L
iAF
Fig. 1: Single-phase diagram of the Shunt Active Power Filter installed in a
power system.
For the investigations in this paper we assume a three-
phase 50 Hz / 400 V voltage source with a source inductance
of 50 µH. This voltage source feeds a non-linear load whichconsists of a B6 diode bridge rectifier with a series connected
resistor (R = 1 Ω) and inductor (L = 8.3 mH). Capacitive loads
are not considered in this paper because it is known that series
active filters are better suited for compensating harmonics
generated by voltage inducing loads.
The SAPF consists of a current controlled three-phase
IGBT voltage source inverter, filter inductors ( Lf = 500 µH) to
smooth the inverter's output current and a 800 V dc-link. In the
simulations, the dc-link is assumed to be ideal. The filter
current is controlled with a hysteresis controller.
The control (including the calculation of the harmonic
content of the load current) of the SAPF is implemented in aDSP-model to simulate the time discrete behavior of
microprocessors or DSPs [11]. This model allows to write the
An Overview of Methods to Determine the
Harmonics in Three-Phase SystemsJoep Jacobs, Dirk O. Detjen, Rik W. De Doncker
Institute for Power Electronics and Electrical Drives
Aachen University of Technology, Germany
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complete control algorithm in C -code. The sample frequency
of the controller is f S = 12.8 kHz which results in a number of
256 cycles per fundamental period.
III. HARMONIC CURRENT CALCULATION
In literature, different methods to calculate the harmonic
content of voltages and currents have been presented during
the past 25 years [4 - 11]. In this paper the authors will
describe the following methods:
• Fourier Transformation (FFT[5], DFT [6])
• Fryze-Bucholz-Depenbrock (FBD)-Method [7]
• Instantaneous Reactive Power Theory [4]
• dq-Transformation [9, 10]
• Multiple dq-Transformations [8]
A. Fourier Transformation
The result of a Fourier Transformation is a continuous
frequency spectrum of a function in the time domain. The
discrete equivalent is called the Discrete FourierTransformation (DFT). For active filter applications the source
or load currents are measured. The DFT of a time discrete
current i[n] sampled with frequency f s is determined as
∑−
−=
−⋅=
1 2 j
][][a
N an
N
kn
enik I (1)
k = 0, 1, 2, ..., N -1 (2)
The spectrum ][k I represents all harmonic values of the
fundamental frequency f 0 up to ( N -1)th. The number of samples
taken during one fundamental period is
0
S
f
f N = (3)
With the Euler-equation we obtain for the spectrum
∑∑−
−=
−
−=
⋅⋅−
⋅=
11 2sin][ j
2cos][][
a
N an
a
N an N
knni
N
knnik I (4)
With the help of this equation the harmonics that need to
be compensated can be determined. The inverse DFT is used
to generate the inverter’s reference signals in the time domain.
An advantage of the DFT is that all harmonics can be
identified individually. Hence, harmonics that need to be
compensated can be selected and compensated. Furthermore,the DFT can be implemented in single-phase systems as well
as multi-phase-systems.
Mayor disadvantage, however, is that a large amount of
computing power is required. Especially, in multi-phase
systems the frequency spectrum has to be calculated for every
phase. On the other hand, if all harmonics need to be
compensated only the amplitude of the fundamental has to be
determined, so saving calculation time. Although reducing the
required computing power drastically the DFT method still
requires powerfull processors. Another drawback is that in
transient conditions the harmonics can not be accurately
detected rapidly, because the spectrum is calculated from thesamples of the previous fundamental cycle. In commercial
harmonic filter products the DFT is not applied due to these
disadvantages.
B. Fryze-Bucholz-Depenbrock (FBD)-Method
The FBD-Method [7] tries to shape the source currents iSa,
iSb and iSc in such a way that they become proportional to the
source voltages eSa, eSb and eSc. Therefore, the source voltagesand the load currents are measured. With the help of these
quantities the equivalent conductance can be calculated:
SiSi
LiSi
i
ee
ieG
⋅⋅
= , i = a, b or c (5)
The index i indicates the three phases a, b or c. The terms
in the formula can be calculated with
Pt t it eT
ie
t
T t
=⋅⋅=⋅ ∫ −
d)()(1
LiSiLiSi(6)
2d)()(
1 2
SiSiSiSi
E t t et e
T ee
t
T t
=⋅⋅=⋅ ∫ −
(7)
This means that the minimum current required by the load
can be determined:
)(Sii
*
imin,,Lt eGi ⋅= (8)
To achieve unity power factor the current difference has to
be compensated. Therefore, the active filter has to inject a
current:
)()(SiiiL,
*
iAF,t eGt ii ⋅−= (9)
The advantages of this method are that it is very easy to
implement and that it requires little computing power. This
method can be used in single- as well as in multi-phase
systems. Especially, in stationary systems the damping
performance is quite good.
On the other hand this method can not distinguish between
reactive and distortion power, i.e. it is not possible to
compensate the harmonic load currents only. Single harmonics
such as the 5th and the 7th can not be detected and
compensated, too. A second disadvantage is that source
voltage harmonics result in proportional source current
harmonics, causing additional power flow. Lastly, the
equivalent conductance G is gained with voltage and currentsamples of the last fundamental period, which means that this
method requires one fundamental period to respond to
transient conditions making the active filter rather slow.
C. The instantaneous reactive power theory
The instantaneous reactive power theory [2 - 4] or pq-
theory (Fig. 2) is already used in a lot of filter applications.
The phase voltages ea, eb, and ec and the load currents iLa,
iLb and iLc are measured and transformed into α-β orthogonal
coordinates.
⋅ −−−⋅=c
b
a
e
e
e
e
e
330
1
21
2
1
21
2
1
32
β
α (10)
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⋅
−−−
⋅=
Lc
Lb
La
L
L
i
i
i
i
i
330
1
21
21
21
21
32
β
α (11)
According to [4], the instantaneous real power pL and the
instantaneous imaginary power qL
on the load side can be
calculated with the help of these equations.
⋅
−
=
L
L
L
L
i
i
ee
ee
q
p(12)
According to [2], the instantaneous real power pL and the
instantaneous imaginary power qL are divided into three
components.
L L L L
L L L L
qqqq
p p p p
~ˆ
~ˆ
++=++=
(13)
In the case the source voltage is undistorted, the dc
component L p corresponds to the fundamental active power
and L
q corresponds to the fundamental reactive power. The
high frequency (≥ 150Hz) components L
p~ and L
q~ correspond
to the harmonics and the low frequency (0.9 Hz < f < 150 Hz)
components L
p and L
q correspond to the subharmonics and
to negative sequence components of the three-phase system.
For example, with a high pass filter with a cut-off frequency of
150 Hz the harmonic components L
p~ and L
q~ can be
determined.
=
∗
∗
L
L
q
p
q
p~
~
(14)
By an inverse transformation, the current command signals
can be calculated.
⋅
−
−
⋅
−−−⋅=
∗
∗
∗
∗
∗
q
p
ee
ee
i
i
i 1
3
3
01
2
1
21
2
1
21
32
AF3
AF2
AF1
(15)
The instantaneous power theory is implemented in the
majority of today’s harmonic filters. Its easy implementation
and relatively small computing power are the driving forces
for this. Furthermore, it can respond rapidly in transient
conditions mainly depending on the high- or band-pass filter.The disadvantage of this method is that the source of the
harmonics, the voltage source or the load current, can not be
determined and that in the case of voltage and current
harmonics not all harmonics are detected. In fact this method
tries to modulate a resistive load, which could be the preferred
control strategy. Minor disadvantages of this method is that it
is only applicable in three-phase power systems and that the
harmonics can not be identified separately.
Fig. 2: Instantaneous Reactive Power Theory
D. dq-Transformation
Another method to generate the harmonic content of the
currents is the dq-transformation [9, 10]. This method is based
on a space vector transformation. The time dependent load
currents iLa, iLb, and iLc are measured in a stationary reference
frame and subsequently transformed into the rotating dq-
reference frame. The dq-system rotates with the fundamental
frequency f 0 of the ac supply. The phase angle θ is provided by
a PLL as mentioned in the next paragraph. The principle of
this control is depicted in Fig. 3
Fig. 3: Calculation of the harmonic content with dq-transformation
⋅
+
−
+
−
=
Lc
Lb
La
Lq
Ld
3
2sin
3
2sinsin
3
2cos
3
2coscos
3
2
i
i
i
i
i (16)
The dq-signals of the load currents can be divided into dc-
and ac-components.
LqLqLq
LdLdLd
~
~
iii
iii
+=
+=(17)
The active dc-component Ldi represents the positivesequence fundamental active power of the load currents, the
reactive dc-componentLq
i represents the positive sequence
fundamental of the reactive power of the load currents, and the
ac-componentsLd
~i and
Lq
~i represent the total harmonic content
of the load current.
If the dq-components are high pass filtered, the dc-
components are eliminated, and one obtains the total harmonic
content.
Lq
*
Lq
Ld
*
Ld
~
~
ii
ii
=
=(18)
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The reference signals *
AFai , *
AFbi , and *
AFci for the current
controller are calculated by multiplying the dq-components *
Ldi
and *
Lqi with the inverse transformation matrix.
⋅
+
+
−
−=
*
Lq
*
Ld
*
AFc
*
AFb
*
AFa
3
2sin
3
2cos
3
2sin
3
2cos
sincos
i
i
i
ii (19)
The phase angle θ of the source voltage eSa has to be
determined to synchronize the reference signals with the
fundamental components of the phase voltages. This phase
angle and the fundamental frequency f 0 are provided by a
Phase Locked Loop (PLL).
The three phase voltages eSa, eSb, and eSc are measured and
the reactive or q-component eq of these voltages is calculated
[9, 10].
⋅
+
−=
Sc
Sb
Sa
q3
2sin
3
2sinsin
e
e
e
e (20)
When the angle θ is equal to the phase angle of the voltage
vector, the q-component of the fundamental is zero. Hence, by
adjusting the angle θ the q-component can be controlled to
zero, realizing a tracking of the phase angle of the phase
voltages.
The dq-transformation method has the same pros and cons
as the instantaneous reactive power theory. Except for the fact
that if the source voltages are tracked properly the sourcevoltage harmonics do not have an impact on the filtering
performance. On the other hand this phase tracking requires
more computing power.
E. Multiple dq-transformations
Fig. 4: Multiple dq-transformations
The algorithms of the previously described dq-
transformation can also be used to transform the time
dependent currents into a system rotating with the harmonic
frequency f h. Now, the harmonic h is represented as a dc-
quantity. This dc-component can be filtered by a low pass
filter and transformed into the stationary reference frame. If
several harmonics have to be compensated severaltransformations have to be made. An example is shown in
Fig. 4.
The advantage is that the harmonics can be determined
separately and precisely. The computing power demand is, of
course, larger, compared to the standard dq-transformation.
IV. SIMULATION RESULTS
The instantaneous reactive power theory and the dq-
transformation have been implemented in the PSpice
simulation environment. The control algorithms are
implemented in a DSP model created with the help of the
Device Equations-option of PSpice [11]. Herein the algorithms
can be written in C-code. The DSP model simulates the timediscrete behavior of DSPs and microprocessors. All values are
chosen as described in chapter II.
800
-800
-400
400
0
iL
/ A
t / ms
20 40
Fig. 5: Load current
Fig. 6: Filter current iF and source current iS with instantaneous reactive
power theory
The load current iL is depicted in Fig. 5. The filter and sourcecurrents in the case the instantaneous reactive power theory is
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used are depicted in Fig. 6. The impact of the source voltage is
shown very clearly. Because there are no in- and output filters
used in the simulation. In practice the measured source
voltages are filtered at the input. Further, the inverter will have
an output LC- or LCL-filter. Fig. 7 shows the results of the dq-
transformation implementation. Again, no in- and output filters
are implemented, but with this method the source currentbecomes nearly sinusoidal. Only the steep current rise can not
be compensated because of the dynamic limits of the SAPF.
For the dq-transformation no in- and output filters are
necessary to obtain sinusoidal source currents.
Fig. 7: Filter current iF and source current iS with dq-transformation
V. CONCLUSION
In this paper different algorithms to calculate the reference
signals for SAPF are reviewed. Advantages and disadvantages
are given. Based on these advantages and disadvantages two
algorithms, the instantaneous power theory and the dq-
transformation, have been selected for implementation in
three-phase power systems.
Although the implementation of the instantaneous power
theory is more simple because it does not require a PLL the
dq-transformation does not require in- and output filters to
generate sinusoidal source currents. In this way the SAPF canbe made more cost effective.
VI. REFERENCES
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electrical power systems", IEEE 519-1992
[2] Akagi H.: "Trends in active power line conditioners", IEEE
Transactions on Power Electronics, Vol.9, 1994
[3] Akagi H.: " New trends in active filters for power line conditioning",
IEEE transactions on industry applications Vol. 32, No. 6, 1996
[4] Akagi H., Kanazawa Y., Nabae A, "Instantaneous reactive power
compensators comprising switching devices without energy storage
components", IEEE transactions on industry applications Vol. 20, No. 3,
1984
[5] Zhang F., Geng Z., Yuan W., "The algorithm of interpolating windowed
FFT for harmonic analysis of electric power systems", IEEE transactions
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[6] Lev-Ari H., T h x v ü Ã H Ã Lin S., "Application of staggered
undersampling to power quality monitoring", IEEE transactions on
power delivery Vol. 15, No. 3, 2000
[7] Depenbrock M, Skudelny H.-Ch., "Dynamic Compensation of
Nonactive Power Using the FBD-Method. Basic Properties
Demonstration by Benchmark Examples", IEEE Second International
Workshop on Power Definitions and Measurements under Non-
Sinusoidal Conditions, Stresa, Italy, 1993
[8] Bojrup M., Karlsson P., Alaküla M., Gertmar L., "A multiple rotating
integrator controller for active filters", ABB
[9] United States Patent US5,648,894, Jul. 15, 1997, De Doncker R.W.:
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[10] Jacobs J., Detjen D., De Doncker R. W.: A new hybrid filter versus a
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algorithms in PSpice", COMPEL, Blacksburg/Virginia, 2000