Abstract. Waveforms obtained from a superconducting Fault Current Limiter are investigated using the Prony modelling technique. Comparable measurements with the use of standard tools of spectral analysis (like FFT) fail because of short data available and low levels of higher spectral components

Jacek Rezmer

Yang Wang

Ying Xin

Zbigniew Leonowicz

Zhengjian Cao

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Investigations of Influence on Power Quality of Superconducting Fault Current Limiter

Waveforms obtained from a superconducting Fault Current Limiter are investigated using the Prony modeling technique. Comparable measurements with the use of standard tools of spectral analysis (like FFT) fail because of short data available and low levels of higher spectral components

Leonowicz, Z

Xin, Y

Rezmer, J

Cao, Z

Wang, Y

Repozytorium Eny Politechnika Wrocławska

Zbigniew Leonowicz2, Ying Xin1, Jacek Rezmer2, Zhengjian Cao1,Yang Wang1 
Innopower Superconductor Cable Co., Ltd., Beijing (1), Wroclaw University of Technology (2) 
 
 
Investigations of Influence on Power Quality of Superconducting  
Fault Current Limiter 
 
 
 
Abstract. Waveforms obtained from a superconducting Fault Current Limiter are investigated using the Prony modelling technique. Comparable 
measurements with the use of standard tools of spectral analysis (like FFT) fail because of short data available and low levels of higher spectral 
components.  
 
Streszczenie. W pracy przedstawiono wyniki analizy widmowej prądów i napięć zarejestrowanych w obwodach z nadprzewodzącym dławikiem 
przetężeniowym. Zastosowano metodę parametrycznego modelowania za pomocą metody Prony. Klasyczne metody oparte na FFT zawodzą w tym 
przypadku z powodu krótkich sekwencji danych, niestacjonarności przebiegów i niskiego poziomu wyższych harmonicznych. 
 
Keywords: Prony method, Superconducting Fault Current Limiter, harmonic analysis, power quality. 
Słowa kluczowe: metoda Prony, nadprzewodzący dławik przetężeniowy, analiza widmowa, jakość energii. 
 
 
Introduction 
 The quality of voltage waveforms is nowadays an issue 
of the utmost importance for power utilities, electric energy 
consumers and also for the manufactures of electric and 
electronic equipment. The voltage waveform is expected to 
be a pure sinusoidal with a given frequency and amplitude. 
Modern electric apparatus generate a wide spectrum of  
components which can deteriorate the quality of the 
delivered energy, increase the energy losses as well  as  
decrease the  reliability of a power system. Investigated 
signals contain multiple harmonic components and present 
often non-stationary behaviour [1]. 
 The standard method for studying time-varying signals 
is short-time Fourier transform (STFT) where the Fourier 
transform is applied to a windowed signal to obtain the 
energy distribution along the frequency direction at the time 
corresponding to the centre of the window. The crucial 
drawback of this method is that the length of the window is 
related to the frequency resolution. Increasing the window 
length leads to improving frequency resolution but it means 
that the non-stationarities occurring during this interval will 
be smeared in time and frequency. This inherent 
relationship between time and frequency resolution 
becomes more important when one is dealing with signals 
whose frequency content is changing. [2] 
 In this paper the time-varying characteristics of power 
system signal components are estimated using the Prony 
model [3].  
 Prony method is a technique for modeling sampled data 
as a linear combination of exponentials [4]. Although it is 
not a spectral estimation technique, Prony method has a 
close relationship to the least squares linear prediction 
algorithms used for AR and ARMA parameter estimation. 
Prony method seeks to fit a deterministic exponential model 
to the data in contrast to AR and ARMA methods that seek 
to fit a random model to the second-order data statistics. 
The paper [3] presents a new method of real-time 
measurement of power system frequency based on Prony 
model. 
 The paper contains detailed analysis of the fundamental 
component waveforms of the currents and higher harmonic 
analysis of the voltages recorded at the test setup with a 
superconducting Fault Current Limiter (FCL) using the 
Prony method, recalled below. 
  
Fault Current Limiter 
For the large-scale power system, the development of FCLs 
for suppressing the over fault currents is strongly desired. 
FCLs with diverse principles, which apply superconducting 
techniques, have been proposed, such as: resistive, 
inductive, magnetic shielding and saturated core FCLs, etc 
[5]. The structure of a prototype saturated core FCL 
incorporating the DC HTS (High Temperature 
Superconductor) bias winding is shown in Fig. 1.  
 Because of the nonlinear influence of the saturated 
magnetic core, it is expected that the currents and voltages 
measured from the testing circuit shown in Fig. 2 can show 
a certain level of harmonic and interharmonics components 
as well as non-characteristic components.  
 The DC bias winding for FCL is expected to use HT 
superconductors, which will drive the core into saturation 
with a low DC power supply voltage. In this case of 
investigated Saturated Core High Temperature 
Superconducting Fault Current Limiter (FCL) [5] the spectral 
contents of the currents and voltages and, therefore, 
periodicity intervals are unknown. Detection of additional 
frequency components can be used for assessment of the 
impact of such devices on the quality of energy and for fault 
identification. 
 
Fig.  1. Schematic view of the structure of HTS current limiter. 
 
 The  frequencies of signal components are estimated 
using the Prony model. Prony method is a technique for 
modelling sampled data as a linear combination of 
exponentials. Although it is not a spectral estimation 
technique, Prony method has a close relationship to the 
least squares linear prediction algorithms used for AR and 
ARMA parameter estimation. Prony method seeks to fit a 
deterministic exponential model to the data in contrast to 
AR and ARMA methods that seek to fit a random model to 
the second-order data statistics [6] . 
 
Prony Method 
 Assuming the M data samples [x1, x2, … xM] the 
investigated function can be approximated by p exponential 
functions: 
(1)                     ( )( )1
1
[ ] k k s k
p
j n T j
k
k
y n A e α ω ψ+ − +
=
= ∑  
 
where    1, 2,...,n M=  
 sT – sampling period, Ak – amplitude, 
 αk – damping factor, ωk – angular velocity, 
 ψk – initial phase. 
 
 The following vectors can be defined : 
 
(2)                 0 1 1w p px x x−⎡ ⎤= ⎣ ⎦x "              
(3)                      0 1 1
T
k p p Mx x x+ −⎡ ⎤= ⎣ ⎦x "   
(4)              1 1 2
T
k p p Mx x x+ +⎡ ⎤= − − −⎣ ⎦x "  
(5)                      2 1 2
T
k px x x⎡ ⎤= ⎣ ⎦x "  
 
where p is the assumed number of exponential 
components, which can approximate the investigated 
signal. 
 The Toeplitz matrix: 
 
(6)                  
1 1
1 2
1 2
p p
p p
M M M p
x x x
x x x
x x x
−
+
− − −
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
t
"
"
# # #
"
 
 
created from (2) and (3), makes it possible to determine the 
vector of coefficients a of the characteristic polynomial: 
 
(7)                         1k⋅ =t a x ,   
where:  
(8)                        1 2
T
pa a a⎡ ⎤= ⎣ ⎦a "   
 
The roots of the characteristic polynomial: 
 
(9)                    11 1 0
p p
p pz a z a z a
−
−+ + + =…   
 
written in a vector form: 
(10)                     1 2
T
p⎡ ⎤= ⎣ ⎦z z z z"  
 
define a Vandermonde matrix: 
 
(11)               
⎥⎥
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢⎢
⎢
⎣
⎡
=
−−
−
−
−
−
11
1
1
1
11
1
1
1
00
1
0
1
p
p
p
p
p
pp
pp
zzz
zzz
zzz
v
"
####
"
"
.            
 
Vector of complex values h can be calculated from (5) 
and (11) solving the system of equations: 
(12)                                2k⋅ =v h x   
 
where: 
(13)        1 2
T
p⎡ ⎤= ⎣ ⎦h h h h" .   
 
Using the equations (10) and (13), parameters of 
exponential components for k=1, 2 ,…  ,p can be calculated 
from: 
(14) k kA = h  amplitude, 
(15) ( )argk kφ = h  initial phase, 
(16) lnk p kfα = ⋅ z  damping factor, 
(17) ( )argk p kfω = ⋅ z    angular velocity. 
 
Investigation results and discussion 
 Because of the short data records available (two periods 
of the fundamental component of 50 Hz), only limited extent 
of investigations was possible. The measurement setup is 
shown in Fig. 2. The signals under investigation were phase 
load currents sampled at the frequency of 5 kHz. 
 
 
Fig.  2. Test measurement setup with FCL. 
 
 Initial investigations included the computation of current 
spectra using classical Welch spectrum. As expected, the 
presence of 3rd 5th and 11th harmonic was detected. Some 
additional basic results are presented in Table 1. 
 
Table 1. Parameters of phase currents 
 without FCL with FCL 
phase A B C A B C 
IRMS [A] 39,7 35,0 34,7 38,3 34,3 33,9 
THD % 9,87 6,20 5,65 10,69 5,94 5,66 
 
 Above presented results suggest a slight increase of 
harmonic distortion in the circuit with FCL and quite strong 
asymmetry between phases.  
 In the next step the analysis of fundamental frequency 
(50 Hz) was performed using the Prony method.  
 Fig. 3 shows the current in the circuit without FCL and 
Fig. 4. – current with FCL. Figures represent the 
fluctuations in time of frequency and of amplitude of the 
fundamental component of the current. 
 There is quite a strong evidence that FCL causes 
important fluctuations of parameters of the fundamental 
component. Nature of these fluctuations is periodical (when 
comparing results without FCL where fluctuations are lower 
and of stochastic nature with results with FCL- where 
fluctuations are much stronger and periodical. 
 When roughly comparing the fluctuations (over 2 
periods) it appears that FCL causes 2.2 Hz and 1.016 (pu) 
of fluctuations compared to 1.3 Hz and 1.0086 pu without 
FCL. Results based on shorter data sequences (not 
reported in this paper) show higher values although these 
results can be less accurate and more affected more by 
noise. 
 
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
48.5
49
49.5
50
50.5
51
51.5
52
Time [s]
Fr
eq
ue
nc
y 
[H
z]
 
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
0.98
0.985
0.99
0.995
1
1.005
1.01
1.015
1.02
1.025
Time [s]
M
ag
ni
tu
de
 [-
]
 
Fig.  3.  Analysis of fundamental frequency of the current without 
FCL (50 Hz) using Prony method. (Upper –frequency, Lower- 
Magnitude). 
 
 In the next step the voltages recorded during tests with 
FCL were investigated. The Fig. 5 shows the voltage 
waveform in phase B with superposed approximated 
waveform using parameters obtained using the Prony 
method.  
 
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-10
-8
-6
-4
-2
0
2
4
6
8
10
Time [s]
M
ag
ni
tu
de
 
Fig.  4. Original (black) and approximated (using Prony method) 
(red) waveform of the voltage in phase B. 
 
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
48.5
49
49.5
50
50.5
51
51.5
Time [s]
Fr
eq
ue
nc
y 
[H
z]
 
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
0.985
0.99
0.995
1
1.005
1.01
1.015
Time [s]
M
ag
ni
tu
de
 [-
]
 
Fig.  5. Analysis of fundamental frequency of the current with FCL 
(50 Hz) using Prony method. (Upper –frequency, Lower- 
Magnitude). 
 
0 500 1000 1500 2000 2500
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Frequency [Hz]
M
ag
ni
tu
de
 
Fig.  6. Spectrum FFT of the voltage waveform. 
 The spectra of voltage waveforms are shown in Figs. 6 
and 7. The FFT spectrum shows the presence of 3 
harmonic components (fundamental, third and fifth 
harmonic) which is confirmed by Prony parameter 
estimation which yields the values of main spectral 
components as: 49,95Hz, 149,8Hz and 248,4Hz. It was also 
observed an expressed asymmetry between phases which 
needs further investigations.  
0 500 1000 1500 2000 25000 
0.5 
1 
1.5 
2 
2.5 
3 
3.5 
4 
4.5 
5 
Frequency [Hz] 
M
ag
ni
tu
de
 
Fig.  7. Estimated spectral components of the voltage using Prony 
method. 
 
 Conclusions 
 Higher harmonics generated within the circuits with non-
linear devices (such as Fault Current Limiter) can affect the 
power quality and disturb the function of connected 
apparatus. Therefore it is important to accurately know the 
spectral contents of investigated waveforms. The 
investigations show the advantages of parametric modelling 
using the Prony model, which delivers highly accurate 
results, even for short data records.  
 
This work was supported in part by the Ministry of Science 
under Grant No. 3T10A 04030. 
 
REFERENCES 
 
[1] Bo l len  M.  H .  J ,  What is power quality, Electric Power 
System Research, 66 (2003), 5-14.  
[2]  Kay S .M. ,  Modern spectral estimation: theory and application 
(Englewood Cliffs, NJ: Prentice-Hall, 1988). 
[3]  Lobos  T . ,  Rezmer  J ., Real Time Determination of Power 
System Frequency, IEEE Trans. on Instrumentation and 
Measurement, (1997), 46, No. 4, 877-881. 
[4]  Lobos  T . ,  Rezmer  J . ,  Schegner  P ., Parameter 
estimation of distorted signals using Prony method. 2003 IEEE 
Bologna PowerTech Proceedings. (2003), ref. BPTO3-45. 
[5]  Zhao  C . ,  Wang  S . ,  Q iu  J . ,  Guo  Z ,  Guo  Y. ,  Gong  
W. ,  Cao  Z . , Transient Simulation and Analysis for Saturated 
Core High Temperature Superconducting Fault Current Limiter, 
12th Biennial IEEE Conference on Electromagnetic Field 
Computation, (2006), 503 – 503. 
[6]  Meun ie r  M. ,  B rouage  F ., Fourier transform, wavelets, 
Prony analysis: tools for harmonics and quality of power, Proc. 
8th Int. Conf. on Harmonics and Quality of Power, Athens, 
Greece, (1998), 71-76. 
 
 
Authors: Dr Ying Xin Zhengjian Cao,Yang Wang, Innopower 
Superconductor Cable Co., Ltd., Beijing, 100176, China, E-mail: 
yingxin@innopower.com, dr inż. Zbigniew Leonowicz i dr inż. Jacek 
Rezmer, Politechnika Wrocławska, Instytut Podstaw Elektrotechniki 
i Elektrotechnologii, Wyb. Wyspianskiego 27, 50-370 Wrocław, E-
mail: leonowicz@ieee.org, jacek.rezmer@pwr.wroc.pl. 
 
 


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Investigations of Influence on Power Quality of Superconducting Fault Current Limiter

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