EMP 31(8) #5895
Electric Power Components and Systems, 31:777–789, 2003
c Taylor & Francis Inc.
Copyright ISSN: 1532-5008 print/1532-5016 online
DOI: 10.1080/15325000390219767
Park’s Transformation Application for Power
System Harmonics Identification and
Measurements
S. A. SOLIMAN
R. A. ALAMMARI
University of Qatar
Electrical Engineering Department
P. O. Box 2713
Doha, Qatar
M. E. EL-HAWARY
Dalhousie University
Electrical and Computer Engineering Department
Halifax, Nova Scotia, Canada
A. H. MANTAWAY
King Fahad University
Electrical Engineering Department
Dhahran, Saudi Arabia
This paper presents the application of Park’s transformation for identifying
and measuring power system harmonics. One of the advantages of the proposed
technique is that it does not need a model for the harmonics components or a
prior knowledge of the number of harmonics expected to be in the voltage or
current signal. The proposed algorithm uses the digitized samples of the three
phases of voltage or current to identify and measure the harmonics content in
these signals. Sampling frequency is the only parameter tied to the harmonic
order in question to verify the sampling theorem. The identification process is
simple and applicable. Results for simulated data generated from EMTP for an
actual system are reported in the text.
Keywords Park’s transformation, harmonics identification and measurements, unbalanced three-phase system
Manuscript received in final form on 10 June 2002.
Address correspondence to S. A. Soliman. E-mail: solimans@qu.edu.qa
777
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S. A. Soliman et al.
1. Introduction
The application of electric devices and electronic equipment in power system control
as well as nonlinear loads produces nonsinusoidal waveforms of voltage and current.
To assess the quality of delivered power, especially in open markets, it is necessary
to estimate the harmonic components in a power system. The quality of power
delivered necessitates knowledge regarding the magnitude of harmonic components
and phase angle of these components. The reduction of harmonics in a system leads
to good system quality. Installing filters at feeding points along the network can
do this if the harmonic components’ magnitudes as well as their phase angles are
known in advance.
Reference [1] presents an optimal measurement scheme for tracking the harmonics in power system voltage and current waveforms. The proposed scheme was
based on Kalman filtering. Reference [2] implements the well-known LES algorithm
for identification and measurement of power system harmonics. The mathematical
model for the identification and measurement process is presented in this reference.
The samples used in this reference are for one of the three-phase voltage or current
signals.
Reference [3] presents a comparative study for power system harmonic estimation. It compares the results obtained using a discrete Fourier transform (DFT),
the well-known least errors square (LES) parameter estimation algorithm, and the
least absolute value (LAV) parameter estimation algorithm. It has been concluded
that the three algorithms produce the same estimate if the signal under study is
free of noise. However, if some data samples are missed, the least absolute value
produces better estimates than the DFT and LES algorithms. The algorithms in
this reference use the samples of a one-phase voltage or current signal.
An approach based on singular value decomposition (SVD) for estimating harmonic components in a power system is presented in [4]. Three different techniques
are investigated: the standard averaged SVD, total LS, and double SVD. Reference
[5] implements the neural network in its analogue form for estimation of harmonics.
It has been shown that such problem formulation leads to a quadratic objective of
the global minimum, which can be found by using simple electronic circuitry in real
time.
Reference [6] presents an approach for the estimation of harmonic components
of a power system using a linear adaptive neuron called Adaline. The proposed estimator tracks Fourier coefficients of signal data corrupted with noise and decaying
DC components. Reference [7] develops a fast Newton-type solution of the six-pulse
rectifier and dc system in the harmonic domain. The nonlinear equations are solved
using Newton’s method, which employs a Jacobian matrix of partial derivatives.
A 12-state Kalman filtering algorithm is applied in [8], using an 8 bit microprocessor, for continuous real-time tracking of the harmonics in the voltage or current
waveforms of a power system to obtain in real time the instantaneous values for a
maximum of six harmonics as well as the existing harmonic distortion.
Reference [9] reviews the problems associated with direct application of the
Fast Fourier Transform to compute harmonic levels of nonsteady-state distorted
waveforms, and various ways to describe recorded data in statistical terms. Reference [10] presents an approach based on fuzzy linear regression for the measurement
of power system harmonic components. The nonsinusoidal voltage or current waveform is written as a linear function. The parameters of this function are assumed
Power System Harmonics Identification
779
to be fuzzy numbers having certain middle and spread value. The problem in this
reference is formulated as a linear optimization problem, where the objective is to
minimize the spread of voltage or current samples.
The online digital measurement on power systems for the power quality analysis
under nonsinusoidal conditions is considered in [11]. The proposed instrument in
this reference adopts a floating point digital signal processing (DSP) hosted on an
IBM PC and interfaced with a special high-speed data acquisition system (DAS).
Voltage and current waveforms are acquired and processed by using a fast recursive
least-square (FRLS) measurement algorithm. Using such an algorithm, different
quantities are obtained, such as the current and voltage rms values, their harmonic
content, the active power, the harmonic active power, the power factor, etc.
Reference [12] applies a technique to the computation of individual harmonics
in digital protections, where only certain isolated harmonics, rather than the full
spectrum, are needed. This leads to O(log2 N ) computations per harmonics. A
technique based on modeling and identification method is proposed in [13] using a
mathematical model describing the signal in question. The recursive least square
error identification algorithm is used to identify the harmonic parameters. These
include the frequency, amplitude, and phase angle.
Because of the limitations associated with conventional algorithms, particularly under supply-frequency drift and transient situations, an approach based on
nonlinear least squares parameter estimation has been proposed in [14]. To reduce
the computational time the Hopfield-type feedback neural networks for real-time
harmonic evaluation are adopted. The neural network implementation determines
simultaneously the supply frequency variation, the fundamental-amplitude/phase
variation, as well as the harmonics-amplitude variation. Reference [15] considers
state estimation of harmonic signals with time-varying magnitudes. Harmonic signals are modeled using elliptical set-theoretic methods, and an optimal reducedorder estimator, which has one-half the dimension of the state vector, is developed
for predicting the unknown time-varying harmonic magnitudes.
Reference [16] proposes the optimization of spectrum analysis to reduce the
restriction on the Fast Fourier Transform (FFT) [17], resulting in a mismatch of
the frequency scale with signal characteristics. By using this method, both the
picket-fence effect and the leakage effect are reduced, and it makes the harmonic
parameters show on spectrum more accurately. Reference [18] proposes a harmonic
model based on the Wavelet Transform (WT) for online tracking of a power system
using Kalman filtering. The close relation between the Wavelet and Multiresolution
analysis is utilized to express the harmonic magnitudes and phase angle as a sum
of Wavelet and scaling function.
Park’s transformation is a well-known technique in the analysis of electric
machines, where the three rotating phases abc are transferred to three equivalent
stationary dq0 phases (d-q reference frame). This paper presents the application of
Park’s transformation for identifying and measuring power system harmonics. The
proposed technique does not need a harmonics model or prior knowledge of the
number of harmonics expected to be in the voltage or current signal. The proposed
algorithm uses the digitized samples of the three phases of voltage or current to
identify and measure the harmonics content in these signals. Sampling frequency is
the only parameter tied to the harmonic order in question to verify the sampling
theorem. The identification process is very simple and easy to apply. Results for
simulated and data generated from EMTP for an actual system are presented.
780
S. A. Soliman et al.
2. Proposed Identification Processes
In the following steps we assume that m samples of the three-phase currents
or voltage are available at a preselected sampling frequency satisfying the sampling theorem. Sampling frequency changes according to the order of harmonic
to be identified, for example, to identify the 9th harmonics in the signal; in this
case, the sampling frequency must be greater than 2 × 50 × 9 = 900 Hz and
so on.
The forward transformation matrix at harmonic order n; n = 1, 2, N , where N
is the total harmonics order to be expected in the signal, resulting from the multiplication of the modulating matrix to the αβ signal and the αβ0-transformation
matrix is given as (dq0 transformation or Park’s transformation)

sin nωt

P = cos nωt
0
=

cos nωt
sin nωt
0
sin nωt
2
cos
nωt


3
1
√
2


1

0
2
 0

0 ×

3
 1
1
√
2
sin(nωt + 120n)
cos(nωt + 120n)
1
√
2
−0.5
√
3
2
1
√
2

−0.5
√ 
− 3

2 

1 
√
2
(1)

sin(nωt + 240n)
cos(nωt + 240n)


1
√
2
The matrix of Equation (1) can be computed offline if the frequencies of the
voltage or current signal and the order of harmonic to be identified are known in
advance, as well as the sampling frequency and the number of samples used. If this
matrix is multiplied digitally by the samples of the three-phase signals of the voltage
or current, which sampled at the same sampling frequency of matrix (1), a new set
of three-phase samples are obtained. We call this set a dq0 set (reference frame).
This set of new three-phase samples contains the ac component of the three-phase
voltage or current signals as well as the dc offset. The dc off set components can be
calculated as
m
Vd (dc) =
1 (Vd )i ,
m i=1
Vq (dc) =
1 (Vq )i ,
m i=1
m
(2)
m
1 (V0 )i .
V0 (dc) =
m i=1
If these dc components are eliminated from the new pq0 samples, new ac harmonic samples are produced. We call this set of samples Vd (ac), Vq (ac), and V0 (ac).
If we multiply this set of samples by the inverse of the matrix of Equation (1),
Power System Harmonics Identification
781
which is given as

P −1
sin nωt
cos nωt


2
sin(nωt + 240n) cos(nωt + 240n)
=

3


sin(nωt + 120n) cos(nωt + 120n)

1
√
2

1 
√ 
,
2

1 
√
2
(3)
then the resulting samples represent the samples of the harmonic components in
each phase of the original three phases. The following are the identification steps:
Step 1. Decide the order of harmonic you would like to identify, and then
adjust the sampling frequency to satisfy the sampling theory. Obtain
m digital samples of harmonics-polluted three-phase voltage or current
samples, sampled at the specified sampling frequency Fs . Or you can
obtain these m samples at one sampling frequency that satisfies the
sampling theorem and cover the entire range of harmonic frequency
you expect to be in the voltage or current signals. Simply choose the
sampling frequency to be greater than double the highest frequency
you expect in the signal.
Step 2. Calculate the matrices, given by Equations (1) and (3) at the m
available samples as well as the order of harmonics you would like
to identify. To this stage, we assume that the signal frequency ω is
constant and equals the nominal frequency 50 or 60 Hz.
Step 3. Multiplying the samples of the three-phase signal (abc phases), by the
transformation matrix given by Equation (1).
Step 4. Remove the dc offset from the original samples simply by subtracting
the average of the new samples generated in step 3 and calculated by
using Equation (2) from the original samples. The generated samples,
in this step, are the samples of the ac dq0 signal.
Step 5. Multiply the resulting samples of step 4 by the inverse matrix given in
Equation (3). The resulting samples are the samples of harmonics that
contaminate the three-phase abc signals except for the fundamental
components.
Step 6. Subtract these samples from the original samples; we obtain m samples
for the harmonic component in question.
Step 7. Use the least error squares algorithm, explained in the preceding section, to estimate the amplitude and phase angle of the components.
If the harmonics are balanced in the three phases, the identified component will be the positive sequence for harmonics of order 1st, 4th,
7th, etc., and no negative or zero sequence components. Furthermore,
it will be the negative sequence for the 2nd, 5th, 8th, etc. component
and will be the zero sequence for the 3rd, 6th, 9th, etc. components.
But if the expected harmonics in the three phases are not balanced,
go to step 8.
Step 8. Replace ω by −ω in the transformation matrix of Equation (1) and
the inverse transformation matrix of Equation (3) and repeat steps
1–7 to obtain the negative sequence components.
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S. A. Soliman et al.
3. Magnitudes and Phase Angle Measurement of a
Harmonic Component
Assume that the harmonic component of the phase a of the voltage signal is
presented as
va (t) = Vam cos(nωt + ϕa ),
(4)
where Vam is the amplitude of harmonic component n in phase a, ω is the fundamental frequency, and ϕa its phase angle measured with respect to a certain
reference. Using the trigonometric identity, Equation (4) can be written as
va (t) = Xa cos nωt + Ya sin nωt,
(5)
Xa = Vam cos ϕa ,
(6)
Ya = Vam sin ϕa .
(7)
where we define
As we stated earlier, in step 5, if m samples are available for a certain harmonic
component of phase a, sampled at a preselected rate, then Equation (5) can be
written in vector form as
Z = Aθ + ζ,
(8)
where Z is an m × 1 vector of samples of the voltage of any of the three phases, and
A is an m × 2 matrix of measurement that can be calculated offline providing that
the sampling frequency as well as the signal frequency is known in advance. The
elements of this matrix are a1 (t) = cos nωt, a2 (t) = sin nωt; θ is a 2 × 1 parameters
vector to be estimated; and ζ is an m × 1 errors vector due to the filtering process
to be minimized. The solution to Equation (8) based on least error square (LES) is
θ∗ = [AT A]−1 AT Z.
(9)
Having identified the parameters vector θ∗ , the magnitude and phase angle of
the voltage of phase a can be calculated as follows:
1
Vam = [Xa2 + Ya2 ] 2 ,
ϕa = tan−1
Xa
.
Ya
(10)
(11)
4. Testing the Proposed Algorithm for Simulated Data
The proposed algorithm is tested using simulated data for a highly harmonic
contaminated voltage signal of phase a as
va (t) = sin(ωt − 30◦ ) + 0.25 sin(3ωt) + 0.10 sin(5ωt) + 0.07 sin(7ωt).
The harmonics in the other two phases are displaced backward and forward from
phase a by 120◦ and equal in magnitudes and in a balanced harmonics contamination. The sampling frequency is chosen to be Fs = 4 × fo × n, where fo = 50 Hz,
where n is the order of harmonic to be identified, n = 1, . . . , N , N is the largest
Power System Harmonics Identification
783
order of harmonics to be expected in the waveform. In this example N = 7. A number of sample equaling 50 are chosen to estimate the parameters of each harmonic
component. Table 1 gives the results obtained when n takes the values of 1, 3, 5, 7
for the three phases.
Examining Table 1 reveals that the proposed algorithm is succeeded in estimating the harmonics contents of a balanced three-phase system. Furthermore, there
is no need to model each harmonic component, as was the case in the literature.
In this section another test is conducted, in which harmonics in the three phases
are assumed to be unbalanced. The algorithm produces, in this case, the positive
sequence component for each harmonic. In this test we assume a three-phase voltage
signal given as
va (t) = sin(ωt − 30◦ ) + 0.25 sin(3ωt) + 0.10 sin(5ωt) + 0.02 sin(7ωt),
vb (t) = 0.9 sin(ωt − 150◦ ) + 0.20 sin(3ωt) + 0.15 sin(5ωt + 120) + 0.03 sin(7ωt − 120),
vc (t) = 0.8 sin(ωt + 90◦ ) + 0.15 sin(3ωt) + 0.12 sin(5ωt − 120) + 0.04 sin(7ωt + 120).
The sampling frequency should be greater than or equal 2×50×7 = 700 Hz. We use,
in this case, a sampling frequency of 1000 Hz, 50-sample data window size. Table 2
gives the results obtained for the positive sequence of each harmonics component
including the fundamental component.
Examining this table reveals that the proposed technique produces good estimates for the positive sequence of each harmonic, such unbalanced harmonics, and
magnitude and phase angle of each harmonics component.
5. Simulated Data Generated from EMTP
In this section the proposed algorithm is tested using simulated data generated
from EMTP for an actual system. The waveform of the voltage signals is given in
Figure 1. The sampling frequency in this case is 1,500 Hz, and a 90-sample data
window size is used.
Table 3 gives the estimated positive sequence for each harmonic component.
Since the sampling frequency is 1,500 Hz, the maximum number of harmonics
to be estimated is approximately 15 to satisfy the sampling theorem. Hence, we
identify up to 10 harmonics, as shown in the table. Furthermore the 3, 6, 9,12, etc.,
harmonics should be calculated using samples of the zero sequence components,
not the samples of individual phases. Indeed, we checked these results using the
conventional symmetrical components.
Example: A Highly Contaminated Three-Phase Voltage Signal
The proposed transformation is used in this section to identify and measure the
harmonics content of the three-phase voltage signal that is highly contaminated
with only odd harmonics in each phase, but they are phase balanced in the phases.
Figure 2 depicts the waveforms of the three phases with respect to time. Table 4
gives the results obtained. Again, harmonics of order 3 and their multiple must be
identified using the zero sequence components, since these components are actually
zero sequence components.
784
V
1.0
1.0
1.0
Phase
A
B
C
−30
−150
89.9
φ
1st harmonic
0.2497
0.2496
0.2496
V
179.95
179.95
179.95
φ
3rd harmonic
0.1
0.1
0.0997
V
0.0
119.83
−119.95
φ
5th harmonic
0.0501
0.04876
0.0501
V
0.200
−120.01
119.8
φ
7th harmonic
Table 1
Estimated harmonic in each phase, sampling frequency = 1,000 Hz and number of samples = 50
785
V
0.9012
0.8986
0.900
Phase
A
B
C
−29.9
−149.97
89.91
φ
1st harmonic
0.2495
0.2495
0.2495
V
179.91
179.93
179.9
φ
3rd harmonic
0.124
0.123
0.123
V
0.110
119.85
−119.96
φ
5th harmonic
Table 2
Estimated positive sequence for each harmonic
0.0301
0.0298
0.0301
V
0.441
−120.0
119.58
φ
7th harmonic
786
S. A. Soliman et al.
Figure 1. Three-phase short-circuited voltage waveforms (p.u.).
Table 3
Estimated positive sequence for each harmonic
Phase A
Phase B
Phase C
Harmonic
V
φ
V
φ
V
φ
1
2
3
4
5
6
7
8
9
10
11
0.88280
0.024120
0.00894
0.01270
0.00732
0.00581
0.00706
0.00982
0.00445
0.0120
0.0187
100.3
−144.6
−12.68
−147.3
154.2
100.47
177.48
47.12
−86.14
10.78
172.24
0.8828
0.0241
0.00894
0.0127
0.00732
0.00581
0.00703
0.00982
0.00445
0.01198
0.01868
−19.7
−24.6
−12.68
92.8
−85.8
100.47
57.5
161.12
−86.14
−109.23
−67.758
0.883
0.02412
0.00894
0.01268
0.00732
0.00581
0.00703
0.00982
0.00445
0.012
0.01866
−139.7
95.5
−12.68
−27.32
−34.2
100.47
−62.53
−72.89
−86.14
130.78
52.24
6. Conclusions
We present in this paper a new algorithm based on Park’s transformation to identify
and measure harmonic components in a power system for quality analysis. The main
features of the proposed algorithm are the following:
• It needs no model for the harmonic components in question.
• It filters out the dc components of the voltage or current signal under
consideration.
Power System Harmonics Identification
787
Figure 2. Three-phase voltage contaminated with harmonics.
Table 4
Estimated positive sequence for each harmonic
Phase A
Phase B
Phase C
Harmonic
V
φ
V
φ
V
φ
1
2
3
4
5
6
7
8
9
10
11
13
19
1.001
0.0
0.1
0.0
0.05
0.0
0.0
0.0
0.08
0.0
0.06
0.05
0.03
9.995
0.0
20
0.0
30
0.0
0.0
0.0
40
0.0
50.0
60.0
70.0
1.00
0.0
0.1
0.0
0.05
0.0
0.0
0.0
0.08
0.0
0.06
0.05
0.03
−110.0
0.0
20
0.0
150
0.0
0.0
0.0
40.0
0.0
170.0
−60.0
−50.0
1.000
0.0
0.1
0.0
0.05
0.0
0.0
0.0
0.08
0.0
0.06
0.05
0.03
130.0
0.0
20
0.0
90
0.0
0.0
0.0
40.0
0.0
−70.0
180.0
190.0
• The proposed algorithm avoids the drawbacks of the algorithms published
earlier in the literature such as FFT, DFT, etc.
• It uses samples of the three-phase signals that give a better view of the system
status, especially in the fault conditions.
• It has the ability to identify a large number of harmonics, since it does not
need a mathematical model for harmonic components.
788
S. A. Soliman et al.
The only drawback, like other algorithms, is that if there is a frequency drift, it
produces an accurate estimate for the components under study. Thus a frequency
estimation algorithm is needed in this case. Also, we assume that the amplitude
and phase angles of each harmonic component are time independent, steady-state
harmonics identification.
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