P P Bedekar

S R Bhide

V S Kale

CiteSeerX

Comparison of Wavelet Transform and             
Fourier Transform based methods of  
Phasor Estimation for Numerical Relaying 
V.S.Kale                   S.R.Bhide                  P.P.Bedekar 
Department of Electrical Engineering, VNIT 
Nagpur, India 
Abstract—Estimation of fundamental frequency component of 
current and / or voltage signals is essential for relaying decision 
in most of the numerical algorithms used for the protection of 
power system components. Modern numerical relays generally 
use discrete Fourier transform for extracting the fundamental 
component from the post fault signals consisting of fundamental 
as well as non-fundamental frequency signals. With the advent of 
wavelet analysis, many researchers used wavelet analysis for the 
phasor estimation with the claims of better performance. The 
paper compares the performance of discrete wavelet transform 
method with discrete Fourier transform method for phasor 
estimation, considering the effect of decaying dc offset. MATLAB 
software was used for programming and simulation.  
Keywords-Decaying dc offset, fundamental frequency 
component, phasor estimation, numerical relaying, wavelet 
transform 
I. INTRODUCTION 
Numerical relays are preferred over their electromechanical 
and static counterparts for economic as well as technical 
reasons. The process of numerical relaying consists of several 
steps. Currents and voltages from instrument transformers are 
band-limited using low pass anti-aliasing analog filters. The 
analog signals are then discritized and quantized to get digital 
signals. The acquired information is then processed by the 
relay algorithm. The algorithm uses signal processing 
techniques to estimate the magnitudes and angles of the 
corresponding current and voltage phasors. These estimates 
are used to calculate other quantities, such as impedances. The 
computed quantities are compared with pre-specified 
thresholds to decide whether the power system is experiencing 
a fault condition or not. If it is, relay sends a command to open 
one or more circuit breakers to isolate the faulted section of 
power system [1]. 
In phasor estimation process, the desired frequency 
component of the signal received is converted to a 
representative phasor. This process is called estimation 
because the true value of the desired component is not known 
upfront. All the methods which attempt to estimate the 
parameters of the signal are based on some assumptions and 
accuracy of phasor estimated depends on method used [2].  
One of the important areas of application of wavelet 
transform is the measurements of various power system 
signals [3]. An algorithm for phasor calculation based on 
wavelet analysis was proposed in [4] and effect of frequency 
deviation on the accuracy of the method was considered. The 
digital distance protection schemes for the transmission line 
distance protection based on wavelet transform were presented 
in [5] and [6]. However, the complete removal of decaying dc 
offset from the signal by some means was required. It may be 
noted that the complete removal of dc offset is still an open 
issue as can be seen from the recent publication [7]. The 
method of calculating effective values of voltage and current 
using wavelet transform was illustrated in [8], but the method 
computed only rms values and not phase angle and therefore, 
has limited application. Hilbert transform and wavelet packet 
transform was used to extract the rms value and phase angle of 
fundamental harmonic in [9]. The equation based and 
experimental data was used for validation but the effect of dc 
offset was not considered. 
This paper attempts to compare the performances of DFT 
and DWT methods of phasor calculations. Fault waveforms 
were generated from the equations as well as simulated using 
MATLAB/ Simulink software package. 
II. REVIEW OF ALGORITHMS
A.  DFT Algorithm 
The most commonly used method of calculating phasors 
from sampled data is that of Discrete Fourier Transform 
(DFT). It is performed by the implementation of following 
equation: 
                              (n)ex
N
2
X
1-N
0n
N
nk   2 j
-
k ∑
=
=
π
                 (1) 
        
In this equation, ‘X’ is the phasor of a signal, ‘k’ is the 
order of harmonic, ‘x’ is the instantaneous value of sinusoidal 
signal such as voltage and current, ‘n’ is the nth sample of data 
window and ‘N’ is the number of samples in data window. 
Fast Fourier Transform (FFT) is a numerical technique to 
make the calculation of DFT faster. It is orders of magnitude 
faster than the calculations implied by (1), if all the Xk are 
desired and if N is large. In relaying applications, however, N 
is generally small (from 4 to 20 for most algorithms) and only a 
few of the Fk are required. Hence, FFT has found little 
application in digital relaying. 
Proc. of the International Conference on Science and Engineering (ICSE 2011)
Copyright © 2011 RG Education Society
ISBN: 978-981-08-7931-0
76
Proc. of the International Conference on Science and Engineering (ICSE 2011)
Recursive implementation of DFT is a more 
computationally efficient method which computes the 
estimated phasor recursively by adding the contribution made 
by the new sample, and subtracting the contribution made by 
the oldest sample. Phasor Measurement Units, which are being 
deployed on power systems as a measurement tool in many 
countries, use recursive DFT algorithm. This paper has used 
this implementation. 
B.  DWT based Algorithm   
The algorithm [4] is briefly described here. The magnitude 
and phase of a signal at fundamental frequency can be 
computed by constructing a unity amplitude sinusoidal 50 Hz 
reference signal (R1). For each window of samples, reference 
signal and measured signal are decomposed into two levels 
using suitable mother wavelet and their approximation 
coefficient vectors of the second level are used for phasor 
estimation. The reason for this can be explained using Fig. 1, 
which shows the filter bank model for a discrete signal sampled 
at 1 kHz. Since the phasor is a component at fundamental 
frequency and fundamental frequency lies in the middle of 
approximation vector at level 2, these coefficients . 
Fig.1 Filter bank model 
Using vector mathematics, angle between reference 
and measured signal is given by: 
                               
)(
cos
212
2121
SR
SR
AA
AA •
=
−
θ                    (2) 
where, A2R1 is the approximation coefficient vector at 
second level of decomposition of reference signal and A2S is  
approximation coefficient vector at second level of 
decomposition of measured signal.  Numerator is the dot 
product of two vectors while ║A2R1║ and ║A2S║ are the 
norms of the corresponding vectors. 
Now a new unity magnitude sinusoidal reference 
signal (R2) with phase shift equal to calculated angle θ is 
constructed. The new signal is decomposed as before. Then 
the magnitude M of measured signal is given by: 
                               
22
2
R
S
A
A
M =                                        (3)  
C.  Selection of mother wavelet 
Performance of signal extraction depends on the mother 
wavelet used. To select the appropriate mother wavelet, a 
typical fault waveform was analyzed with different wavelets. 
The results of phasor estimation using commonly used 
wavelet functions viz. Daubechies 4 (db 4), Daubechies 8 (db 
8), Biorthogonal 1.1 (bior 1.1) and Symlet 5 (sym 5) are 
shown in Fig. 2. As can be seen from the fig. 2, all the 
waveforms reach the final value at the same time but 
waveforms obtained by using db4, db8, sym5 wavelets 
oscillates about the final value for a longer period. Therefore, 
bior 1.1 was selected as mother wavelet. 
0 10 20 30 40 50 60 70 80
0
50
100
Sample number
C
ur
re
nt
 (
A
)
Phasor Estimation using DWT
Selection of mother wavelet
db4
db8
bior1.1
sym5
                                         Fig. 2. Selection of mother wavelet 
III. MODIFICATIONS IN IMPLEMENTATION OF
ALGORITHMS 
Considering that the phasor estimation is a continuous 
process, it is necessary to consider algorithms which will 
update the phasor estimate as newer data samples are acquired. 
For example, in non-recursive DFT algorithm, unlike recursive 
DFT algorithm, all the calculations are performed fresh for 
each window without using any data from the earlier estimates. 
It may be noted that recursive DFT is only computational 
efficient. It still requires full cycle window for phasor 
estimation. However, speed in estimation might be achieved, if, 
all the non-fault samples of window were removed on 
detection of fault and a new window of only fault samples was 
considered for phasor estimation. With this in view, both DFT 
and DWT algorithms were modified in their implementations 
for updating phasor estimates and results were compared. The 
faults were detected using DWT as given below. 
A. Detection of fault  
Measured current signals were decomposed using db1 
wavelet. This level contains the high frequencies that are 
associated with faults. Norm is a measure of amount of energy 
content of a signal, which increases with fault. By calculating 
the norms of detail coefficients at level 1 for all the current 
signals, fault was detected. This detection was used to adjust 
the samples in the data window. To illustrate how a fault can 
be detected using DWT, a single line to ground fault AG on 
400 KV line was simulated using MATLAB. The norm values 
of detail coefficients at level 1 for three phase currents are 
shown in Fig. 3. It can be seen that norm for AG fault, ‘A’ 
phase current is much higher than the corresponding values for 
the other two phases. 
77
Proc. of the International Conference on Science and Engineering (ICSE 2011)
20 30 40 50 60 70 80 90 100
0
5
10
15
20
25
Sample number
N
or
m
Fault Detection
phase a
phase b
phase c
Fig. 3. Norm of detail coefficients 
         before and after AG fault 
IV. SOURCES OF ERRORS IN PHASOR ESTIMATION
A. DC offset 
The major component of a power system signal is its 
fundamental frequency component. However, post fault 
current and voltage signals fail to be pure fundamental 
frequency sinusoids for a variety of reasons. The most 
predictable non-fundamental term is the decaying exponential, 
which can be present in the current signal. For the series RL 
model of the line, under some assumptions, the instantaneous 
current for a fault at a time t is given by: 
   
   
where, parameter α  controls the instant on the voltage wave 
when short circuit occurs, and θ is the impedance angle of 
transmission line. The second term in (4) decays exponential 
with the time constant of the line and is the main cause of 
transient overreach in high speed relays. Some algorithms 
require that the offset be removed prior to processing while 
some algorithms do not require its elimination. Traditional 
method of reducing the dc component in current signal is with 
the use of mimic impedance which can be implemented as   
r+j x burden on CT secondary or can be implemented in 
software. 
B. High frequency noise 
The contributors to the high frequency noise or 
harmonics include (i) transient response of capacitive voltage 
transformer to the abrupt change in voltage (ii) non linear 
behavior of the fault arc (iii) reflections of fault generated 
traveling waves between bus and fault. The anti-aliasing low 
pass filter which is a part of numerical relay will remove most 
of these error signals. It will add a small delay to the overall 
operating time of the relay. However, it can be seen from the 
fig.4 that phase lag at the fundamental frequency of 50 Hz is 
about 11o, which corresponds to the phase delay of 0.75 ms. 
Considering that the low pass filter has been designed for a 
sampling frequency of 1 KHz, the phase delay produced by it 
is less than the sampling period of 1 ms. 
0 50 100 150 200 250 300
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
Frequency (Hz)
D
el
al
y 
(m
s)
Group delay of LP filter
Fig. 4. Group delay of low pass second 
order Butterworth filter 
V. RESULTS AND ANALYSIS
A. Results of Phasor Extraction for equation based signals 
The results of phasor extraction by applying 
algorithms with and without the modifications as mentioned in 
section 2 and using equation based signals are discussed 
below. A typical fault current waveform, assuming no 
harmonics, can be modeled using equation (4), which is an 
expression of current in time domain when ac voltage is 
imposed on a R-L circuit representing transmission line. 
Taking α =θ, the waveform was devoid of dc offset as shown 
in fig.5. The waveform was sampled at 1 KHz and its phasor 
estimation was obtained using four algorithms viz. (i) DFT, 
(ii) modified DFT (MDFT), (iii)DWT and (iv) modified DWT 
(MDWT). As explained in section III, modification was done 
in their implementation to update the phasor estimation when 
fault occurs. From the fig. 5, it can be seen that DWT output 
rises faster than DFT and MDFT, but output of remaining 
three methods stabilizes at the same time, exactly after one 
cycle. However, MDWT reaches the final value in less than 
half cycle and its performance looked promising. 
0 10 20 30 40 50 60 70 80
-100
-80
-60
-40
-20
0
20
40
60
80
100
Sample number
C
ur
re
nt
 (
A
)
Signal without DC offset
(a) 
(4)        e )-(sin 
|  Z|
V
  )-(wtsin 
|  Z|
V
(t) i (R/L)t-mm αθθα ++=
78
Proc. of the International Conference on Science and Engineering (ICSE 2011)
0 10 20 30 40 50 60 70 80
0
10
20
30
40
50
60
70
80
Sample number
C
ur
re
nt
 m
ag
ni
tu
de
 (
A
)
Phasor Estimation
DFT
MDFT
DWT
MDWT
(b) 
Fig. 5. (a) Fault current without dc offset (b) its phasor estimation 
Now consider a second fault signal with dc offset with 
fault occurring at 30 ms, and the results of phasor estimation 
using all the four algorithms as shown in Fig. 6. It can be 
observed that all the algorithms performed poorly. Both DFT 
and MDFT remain oscillatory over a narrow band around the 
true phasor magnitude, while DWT and MDWT overshoot and 
produced large errors in addition to being oscillatory. There 
was hardly any difference between the performances of DFT 
and MDFT in both the cases described above. 
0 20 40 60 80 100 120
-80
-60
-40
-20
0
20
40
60
80
100
120
Sample number
C
ur
re
nt
 (
A
)
Signal with dc offset
(a) 
0 20 40 60 80 100 120
0
10
20
30
40
50
60
70
80
Sample number
C
ur
re
nt
 M
ag
ni
tu
de
 (
A
)
Phasor Estimation
DFT
MDFT
DWT
MDWT
(b) 
Fig. 6. (a) Fault current with dc offset (b) its phasor estimation 
  
The above analysis was based on signals generated from 
equation and these signals do not represent the fault signals in 
terms of decaying dc offset, transients and harmonics. 
Therefore, the results of simulated waveform based on the 
double end fed power system model are discussed next. 
B. R esults of Phasor Extraction fo rsimulated signals 
The results of phasor extraction by applying DFT and 
MDWT algorithms as mentioned in section 2 and using 
MATLAB/Simulink based signals are discussed below.A 
double end fed 400 KV power system model consisting of 100 
Km transmission line section was selected for fault 
simulations. The transmission line was simulated using 
distributed parameter model with parameters as follows. 
Z0 = 0.151 + j 1.1655 Ω / km and C0 =0.0009 μF/km 
Z1=0.0134 + j 0.3144 Ω / km and C1 =0.0017 μF/km 
Phasor for the fault current in phase ‘a’ for AG fault was 
estimated using DFT and MDWT methods. As seen from   
Fig. 7, for Fault Inception Angle (FIA) of 0 deg., MDWT 
output rises faster to final value but overshoots to a greater 
extent as compared to output of DFT method due the effect of 
dc offset. Both the methods take longer time (more than a 
cycle) to settle. In Fig. 8, fault current had no dc offset and 
both methods stabilize at the same time with DWT output 
rising faster. The graph of error in the estimation as a function 
of fault inception angle for both the methods was plotted in 
Fig. 9 for AG fault. Thus, the method based on wavelet 
analysis does not seem have any advantage over traditional 
Fourier based method. 
20 30 40 50 60 70 80 90 100
0
10
20
30
40
50
60
Sample number
C
ur
re
nt
 (
A
)
Phasor Estimation (FIA 0 deg)
DFT
MDWT
Fig. 7 Phasor estimation ( FIA 0 deg.) 
20 30 40 50 60 70 80 90 100
0
10
20
30
40
50
Sample number
C
ur
re
nt
 (
A
)
Phasor Estimation ( FIA 90 deg)
DFT
MDWT
Fig. 8 Phasor estimation ( FIA 90 deg.) 
0 18 36 54 72 90
0
5
10
15
20
FIA
%
 e
rr
or
Fault: AG
DFT method
MDWT method
Fig. 9. Test results for AG fault 
79
Proc. of the International Conference on Science and Engineering (ICSE 2011)
VI. CONCLUSIONS 
The protective relays based on principles such as over-
current protection, differential protection, distance protection 
and directional protection require that the phasors of current 
and/or voltage signals be extracted. DC offset which may be 
present in the post fault current signal can adversely affect this 
estimation. 
It was found that DWT method performed better in absence 
of dc offset, in terms of speed of measurement. With the 
existence of exponentially decaying DC offset, the amplitude 
of the fundamental frequency component calculated by both 
DFT and DWT methods, deviated from the expected value. 
However, DWT method performed more badly in presence of 
dc offset and harmonics. It can not be assumed that post fault 
signals will be devoid of non fundamental frequency 
components and the error in estimation would subsequently 
reduce the accuracy of protective relays. Therefore, though 
wavelet analysis has many other useful applications in power 
system protection, it has limited role in phasor estimation. 
REFERENCES
[1] Paithankar, Y.G., Bhide, S.R., Fundamentals of power system protection, 
Prentice-Hall of India, 2004. 
[2] Phadke, A.G., Thorp, J.S., Computer relaying for power systems, Second 
Edition, John Wiley, 2009. 
[3] Kim C.H., Aggarwal, R., “Wavelet transforms in power systems,” IET 
Power Engineering Journal, vol. 15, pp. 193-200, Aug. 2001. 
[4] Chi-kong Wong, Ieng-tak Leong, Chu-san Lei, Jing-tao Wu, Ying-duo 
Ham, A novel algorithm for phasor calculation based on wavelet analysis, 
IEEE Power Engineering Society Summer Meeting, 2001, vol. 3, pp. 1500-
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[5] Osman A.H., Malik O.P., Transmission line distance protection based on 
wavelet transform, IEEE Trans. on Power Delivery, vol. 19, no. 2, April 2004, 
pp.515-523. 
[6] Liang F., Jeyasurya B., Transmission line distance protection using 
wavelet transform algorithm, IEEE Trans. on Power Delivery, vol. 19, no. 2, 
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[7] Sang-Hee Kang, , Dong-Gyu Lee, , Soon-Ryul Nam, Peter A. Crossley, 
and Yong-Cheol Kang, ‘Fourier Transform-Based Modified Phasor 
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[8] Zhu T.X., Effective value calculation in wavelet domain, IEEE Trans. on 
Power Deliver, vol. 19, no. 1, Jan. 2004, pp. 400-4004. 
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[10] MATLAB 7.1 User’s Guides for SimPowerSystem, Wavelet Toolbox.
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Comparison of Wavelet Transform and Fourier Transform based methods of Phasor Estimation for Numerical Relaying

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Comparison of Wavelet Transform and Fourier Transform based methods of Phasor Estimation for Numerical Relaying

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