International Journal of Pure and Applied Mathematics
Volume 114 No. 12 2017, 583-592
ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)
url: http://www.ijpam.eu
Special Issue
ijpam.eu
Calculation of Multistage Impulse Circuit and Its
Analytical Function Parameters
G. Ramarao1, K. Chandrasekaran2
1
Department of Electrical Engg., National Institute Of
Technology, Raipur, Chhatisgarh, India.
grr231@gmail.com
2
Department of Electrical Engg., National Institute Of
Technology, Raipur, Chhatisgarh, India.
vrkchandran@yahoo.com
Abstract
A new approach has been proposed for the evaluation of multi stage impulse circuit
parameters from known rise or peak time and tail time (physical parameters) using NelderMead algorithm (NMA). Results of five stage (n=5) impulse circuit parameters such as C1‫ ׳‬, C2‫׳‬,
R1‫׳‬, R2‫ & ׳‬Rs are reported for generating full lightning impulse (LI) and full switching impulse
(SI) as given in standards (IEEE 4-2013 and IEC 60060-1). The determined circuit parameters
are validated for practical applications through the impulse circuit developed in
MATLAB/Simulation for 150kV peak impulse signal. The presented method enables accurate
estimation of unknown impulse circuit parameters from the known physical parameters.
Impulse wave (LI & SI) attained using double exponential expression and through simulation
circuits are reported. The closeness in peak time, rise time, tail time and peak voltage between
the output of double exponential function and Simulink model for both standard LI (1.2/50µs)
and standard SI (250/2500µs) are also described.
Key Words:- Analytical function parameters, circuit parameters, full lightning impulse (LI),
Nelder-Mead algorithm (NMA), full switching impulse (SI).
1 Introduction
In recent years, lightning and switching over voltages have been considered
to be the major reason of damage or destruction to power system apparatus. Lightning
impulse and transient switching over voltages can cause great damage to insulation of
power apparatus used in its network [1-2]. Hence, the power apparatus used in power
system network should be designed and constructed to withstand against such lightning
583
International Journal of Pure and Applied Mathematics
Special Issue
and switching over voltages. Power system network mainly concern with reliability of
each apparatus used in it. In order to maintain the reliability of power system against
lightning and switching over voltages, it is important to test and verify the electrical
breakdown strength or capability of insulation of high voltage power apparatus before
introducing into power system network [3-6]. International standards such as IEC
60060-1 [7] and IEEE std 4-2013 [8] describes both full lightning impulse (LI) and
switching impulse (SI) voltage tests normally performed on insulation of power system
apparatus, in order to check the dielectric strength. According to these standards
1.2/50µs and 250/2500µs impulses are considered for full LI and SI. Hence, it is
essential to check the shape and properties of the both LI and SI before applied to the
test object in the laboratory. [9].
Standard full LI and SI waveforms shown in Fig. 1 and Fig. 2 are characterized
by three parameters: peak value (Vpeak), time to rise (trise) (for full LI) or time to peak
(tpeak) (for full SI), and time to half peak value (ttail). According to IEEE standards IEEE
std 4-2013, full LI has trise = 1.2µs and ttail = 50µs with tolerances of ±30% for trise and
±20% for ttail [10]. According to IEC standards IEC 60060-1, full SI has tpeak = 250µs and
ttail = 2500µs with tolerances of ±20% for tpeak and ±60% for ttail [11]. These parameters
can be termed as physical parameters because these are generally determines from the
impulse waveforms (both full LI and SI).
Fig.1 Double exponential wave with trise and
ttail of the standard full lightning impulse
voltage [12]
Fig.2 Double exponential wave with tpeak
and ttail of the standard full switching
impulse voltage wave [12]
Standard full LI and SI waveforms are mathematically expressed by double
exponential function (1).
𝑉(𝑡) = 𝑉0 𝐾(𝑒 −𝛼𝑡 − 𝑒 −𝛽𝑡 )
(1) Where V0 is amplitude, α and β are the analytical (double exponential) function
parameters. To keep the pulse positive polarity, the analytical (double exponential)
function constant parameters α and β should satisfy the condition that β > α > 0, t ≥
0 [13]. K is the amplitude modifying factor [14].
Proper physical parameters can be achieved through the selecting the suitable
values of circuit parameters. In the same manner, it is also important to determine
proper analytical function parameters (α, β) from known physical parameters. Hence
this paper, a new approach for aforementioned transformation from physical
parameters to multi stage impulse circuit parameters for both full LI and SI has been
attempted and explained in Section 3 and reported in section 4. Results of circuit
584
International Journal of Pure and Applied Mathematics
Special Issue
parameters are validated through simulation impulse circuit designed in the MATLAB
is explained in Section 5.
2 Details of Multi Stage Impulse Circuit and Its Parameters
Simulation and practical multi stage impulse circuit consisting of parameters
R1‫׳‬, R2‫ & ׳‬Rs) as shown in fig.3 has been considered for generation of standard
full LI and SI [15-16]. Double exponential function (1) is commonly used for generation
of both full LI and SI voltages. The shape properties of standard full LI and SI voltages
using (1) are given in Fig. 1 and Fig. 2 respectively.
(C1‫׳‬,
C2‫׳‬,
Generation of full lightning and switching impulses as per the standards in
laboratory, it is necessary to determine the suitable multi stage impulse circuit
parameters (C1‫׳‬, C2‫׳‬, R1‫׳‬, R2‫ & ׳‬Rs) corresponds to physical parameters (trise or tpeak and
ttail) rather than single stage impulse circuit parameters because single stage impulse
circuit generates voltage up to certain limit, beyond that voltage, a single capacitor and
its charging unit used in single stage impulse circuit become too costly. In order to
production of 150kV (peak) impulse voltage (full LI or SI), charging capacitor (C 1‫ )׳‬of
each stage in multi stage impulse generation circuit (5 stages) is charged to K×30kV in
parallel and discharge them in series. In this regard, the charging resistance (Rs) is
chosen for charge the capacitors in parallel to supply voltage in about 1 minute and
limits the charging current to about 50 to 100mA [17]. Initially, charging capacitor (C1)
of single stage impulse generation circuit is selected based on maximum charging
voltage (V0max) and impulse energy (W) of the generator as given by (2). Then, single
stage charging capacitor (C1) is converted into multistage charging capacitor (C1‫)׳‬
according to [12]. In this paper, impulse energy of 150kV (peak) impulse generator (full
LI or SI) is chosen as 225J.
C1 = (2×W)/ (V20max)
(2)
R1'
G
R2'
s
R
C1
'
R 1'
s
R
G
C1'
R 2'
R 1'
s
R
G
C1'
R 2'
R 1'
Rs
G
D.C.
C1'
R 2'
Fig.3. Multi stage impulse generator circuit (n=5)
585
C 2'
V0
International Journal of Pure and Applied Mathematics
Special Issue
3 Proposed Impulse Circuit Parameter Estimation Technique
The proposed technique for estimating multi stage (n=5; where n is the number
of stages) impulse circuit parameters (C1‫ ׳‬, C2‫׳‬, R1‫׳‬, R2‫ & ׳‬Rs) from known physical
parameters has two major parts, one is Nelder-Mead algorithm (NMA) to determination
of analytical (double exponential) function parameters (α, β) from the initially defined
physical parameters (trise and ttail) and second one is determination of multi stage
impulse circuit parameters from α, β for the multi stage (n=5) impulse generation circuit
is shown in Fig. 3.
Nelder-Mead algorithm is used for determination of analytical (double
exponential) function parameters (α and β) from known (target) physical parameters
(trise and ttail).The initial approximations given in [18] by (3) are used to initiate the
algorithm
1
1
𝛼𝑖𝑛𝑖𝑡𝑖𝑎𝑙 = 𝑡
and 𝛽𝑖𝑛𝑖𝑡𝑖𝑎𝑙 = 𝑡
(3)
𝑡𝑎𝑖𝑙
𝑟𝑖𝑠𝑒
Enter tail and rise (or) peak
times
(ttail & trise (or) tpeak)
If trise
Is
trise (or) tpeak
If tpeak
Calculate analytical function parameters (α, β)
from trise and ttail using Nelder-Mead Algorithm
Plot the impulse voltage waveform using α and β
Determine peak value of the waveform
&
Time corresponds to peak (tpeak)
Calculate parameter Y using peak and tail times
Y= ttail/tpeak
Calculate x value from below equation [20]
Y = -ln {0.5 (exp ( -ln(x)/(x-1)) – exp (-xln(x)/(x-1)))} / (ln(x)/(x-1))
Determine the parameter a, b and c by using x value [21]
a = ln(x) / ((x-1)×tpeak), b = a × x, c = 2 × a
Calculate the circuit parameters by using a, b and c [21]
C2 = 1, C1 = (a + b – c)^2 / (ac + bc – c^2 – ab)
R2 = 1 / (a + b – c), R1 = (ac + bc - c^2 _ ab) / (ab (a + b – c)
Calculate α and β values by using circuit parameters [15]
α = 1 / (R2 × C1) & β = 1 / (R1 × C2)
Construct impulse voltage waveform using α and β values &
Determine rise time and tail time from the impulse voltage waveform
(trise_cal, ttail_cal)
No
trise_cal == trise &
ttail_cal == ttail
Recalculate the circuit parameters by using above α &β values
C1new = (2×W)/(V20max), C2new = C1new /100,
R1new = 1 / (β × C2new) & R2new = 1 / (α× C1new)
Yes
Conclude that C1, C2, R1
and R2 are the circuit
parameters
Calculate new values of α, β
αnew = 1 / (R2new × C1new), βnew = 1 / (R1new × C2new)
Reconstruct the impulse voltage waveform by using αnew and βnew &
Determine the new rise and tail times (trise_new, ttail_new)
No
Yes
trise_new == trise &
ttail_new == ttail
Conclude that
C1new,C2new , R1new and
R2new are the circuit
parameters
Convert above circuit parameters into
multistage impulse circuit parameters
[12]
Fig. 4. Flowchart for determination of multi stage impulse circuit parameters from physical
parameters using Nelder-Mead algorithm
586
International Journal of Pure and Applied Mathematics
Special Issue
Then, the algorithm is continued for searching the close value of analytical
function parameter α and β corresponding to the given ttail and trise respectively. Values
of α and β are moderately changed in a specific interval upward or downward in each
step, then ttail and trise are calculated corresponding changes in the α and β. These
calculated tail and rise times are compared to the known physical parameters for each
changes of α and β. At the end of each step, the α and β are taken and chosen as the
start of next step. If the values of α and β are same as that of the previous α and β, then
specific interval is decreased by half. This process is repeated until the desired target
parameters ttail and trise are achieved with smaller relative error (between tail and rise
time of final α, β and given target) than the preset accuracy [19]. This algorithm is quite
simple and gives more precise results near the end. This process offers the possibility
for transform physical parameters of the impulse wave to analytical function
parameters. Complete steps involving to find the multi stage (n=5) impulse circuit
parameters from the known rise or peak time and tail time (physical parameters) using
Nelder-Mead algorithm (NMA) are given in the form of flow chart in Fig. 4.
4 Results of Proposed Technique
The circuit parameters (C1‫׳‬, C2‫׳‬, R1‫׳‬, R2‫ & ׳‬Rs) for multi stage impulse generator
and analytical (double exponential) function parameters (α, β) are obtained from Section
3 and reported in section 4, are validated through simulation using impulse circuit
implemented in MATLAB/Simulation [23]. The circuit parameters have been derived for
the generation of standard full lightning impulse (IEEE std 4-2013) voltage signal
(1.2/50µs) with 150kV peak results are given in table 1. Similarly circuit parameters are
also derived for the generation of 150kV peak standard full switching impulse (IEC
6060-1) voltage signal (250/2500µs) is tabulated in table 2.
Table 1. Analytical (Double Exponential) Function Parameters (α and β) & Multi Stage
Impulse Circuit Parameters of Standard Full Lightning Impulse Voltage
Standard full
lightning impulse
physical
parameters(rise &
tail time)
(µs)
1.2/50
Result of α and β for the
physical parameters using
NMA
Multistage impulse circuit parameters
corresponds to physical parameters
α
β
C1‫׳‬
C2‫׳‬
R1‫׳‬
R2‫׳‬
RS
(×104)
(×106)
(µF)
(nF)
(Ω)
(Ω)
(MΩ)
1.4912
1.6335
0.1
0.2
606.6
676.8
600
The circuit parameters for standard full lightning and standard full switching
impulse voltage shown in table 1 and table 2 are validated through the multi stage
impulse generator circuit as given in [15]. The percentage error of peak time, rise time,
tail time and peak voltage between results of simulation and analytical function are
calculated and reported. The charging resistance Rs is chosen here for charging the
each stage capacitors in parallel is calculated using (4).
587
International Journal of Pure and Applied Mathematics
Special Issue
Rs × C1‫ = ׳‬1 min.
(4)
Table 2: Analytical (Double Exponential) Function Parameters (α and β) & Multi Stage
Impulse Circuit Parameters of Standard Full Switching Impulse Voltage
Standard full
switching impulse
physical
parameters peak &
tail time
Result of α and β for the
physical parameters using
NMA
(µs)
α
250/2500
377.54
β
6000.1
Multistage impulse circuit parameters
corresponds to physical parameters
C1‫׳‬
C2‫׳‬
R1‫׳‬
R2‫׳‬
RS
(µF)
(nF)
(kΩ)
(kΩ)
(MΩ)
0.1
0.2
155.5
28.4
600
5 Results Validation and Discussion
The multi stage impulse generator circuit implemented in MATLAB/Simulink is
shown in Fig. 5. The five stage (n=5) impulse circuit parameter (C1‫( ׳‬C1s=C1s=C1s
=C1s=C1s), C2‫׳‬, R1‫( ׳‬Rd1=Rd2=Rd3=Rd4=Rd5), R2‫( ׳‬Rg1=Rg2=Rg3=Rg4=Rg5) and RS given in table
1 and table 2 for standard lightning impulse voltage and standard switching impulse
voltage signals are substituted in respective elements shown in Fig. 5 and its results are
reported in Figs. 6 and 7.
Fig.5 Multi stage impulse circuit implemented in MATLAB/Simulink
The standard full lightning impulse (LI) and switching impulse (SI) voltage
waveforms obtained using both analytical function (double exponential) and multi
stage impulse generation circuit (Fig. 5) are illustrated in Fig. 6 and Fig. 7
respectively for both full LI & SI voltage, peak is chosen to be 150kV and
588
International Journal of Pure and Applied Mathematics
Special Issue
corresponding correction factor k is calculated and incorporated in Simulink model.
The percentage error is calculated for both full LI and SI waveforms by taking four
parameters i.e. rise time, tail time, peak time and peak voltages. These errors are
reported in table 3 and table 4 respectively. From the Figs. 6 and 7 it is observed
that, the simulation output is matched well with the analytical function waveforms
obtained using NMA algorithm. Hence the proposed method is validated through
simulation.
150
Multistage impulse circuit
Analytical function
Multistage impulse circuit
Analytical function
Amplitude (kV)
Amplitude (kV)
150
100
50
0
0
10
20
Time (s)
30
40
Fig.6. Standard full lightning impulse
voltage waveform(1.2/50µs) using
multistage impulse generation circuit
and analytical function parameters.
100
50
0
0
50
500
1000
1500
Time (s)
2000
2500
Fig.7. Standard full switching impulse
voltage waveform (250/2500µs) using
multistage impulse circuit and analytical
function parameters.
Table 3: Percentage Errors between Results of Analytical (Double Exponential)
Function & Multi Stage Impulse Generator Circuit of Standard Full Lightning Impulse
Voltage
Standard full lightning
impulse physical
parameters(rise & tail
time)(µs)
1.2/50
Percentage Error
Rise time
Tail time
Peak time
Peak voltage
1.48%
1.67%
1.79%
-0.02%
Table 4: Percentage Errors between Results of Analytical (Double Exponential)
Function & Multi Stage Impulse Generator Circuit of Standard Full Switching Impulse
Voltage
Standard full switching
impulse physical
parameters(rise & tail
time)(µs)
250/2500
Percentage Error
Rise time
Tail time
Peak time
Peak voltage
0.67%
4.06%
0.48%
-1.24%
589
International Journal of Pure and Applied Mathematics
Special Issue
From the table 3 and 4, it is observed that the percentage error in rise time, tail
time, peak time and peak voltage are within tolerance as provided in standards IEEE 42013 and IEC 60060-1 for both standard full LI and SI respectively. Also the maximum
percentage error observed from proposed method for determining the impulse circuit
parameter is 4%.
6 Conclusion
A new method for calculating analytical function and multi stage impulse circuit
parameters are attempted and reported. Results of impulse circuit parameters are
validated through simulation for both standard full LI and SI voltages. The percentage
error in rise time, tail time, peak time and peak voltage between the impulse waveforms
attained using circuit parameters and analytical function parameters are reported.
Maximum error observed in the proposed approach is ±4%. Proposed method enables an
accurate estimation of unknown impulse circuit parameters from known physical
parameters. Errors appeared in proposed approach are within the tolerance as given in
the standards. Hence the circuit parameters reported here can be useful in high voltage
laboratory or in related applications for generating standard high voltage lightning and
switching impulse signals with the peak of 150kV.
7 References
[1]. Xi Wang, Zhanqing Yu and Jinliang He, “Breakdown Process Experiments of 110-to
500-kV Insulator Strings Under Short Tail Lightning Impulse,” IEEE Trans. Power
del., vol. 29, no. 5, pp. 2394-2401, Oct. 2014.
[2]. Alejandro Ancajima, Alys Carrus, Enrico Cinieri and Carlo Mazzetti, “Behavior of
MV Insulators Under Lightning-Induced Overvoltages: Experimental Results and
Reproduction of Volt-Time Characteristics by Disruptive Effects Models,” IEEE
Trans. Power del., vol. 25, no. 1, pp. 221-230, Jan. 2010.
[3]. R. A. Fitch, “Marx-and Marx-Like – High-Voltage Generators,” IEEE Trans. Nucl.
Sci., vol. 18, no. 4, pp. 190-198, 1971.
[4]. J. Liang, L. Zhang, J. Li and Y. Li, “Study on Oscillating Switching Impulse Voltage
Generation for Power Transformer Onsite Test,” IEEE Trans. Power Del., vol. 29, no.
5, pp. 2223-2230, Oct. 2014.
[5]. P. Yutthagowith and N. Pattanadech, “Improved Least-Square Prony Analysis
Technique for Parameter Evaluation of Lightning Impulse Voltage and Current,”
IEEE Trans. Power Del., vol. 31, no. 1, pp. 271-277, Feb. 2016.
[6]. L. Satish and B. I. Gururaj, “Wavelet Analysis for Estimation of Mean-Curve of
Impuls Waveforms Superimposed by Noise Oscillations and Overshoot,” IEEE
Trans. Power Del., vol. 16, no. 1, pp. 116-121, Jan. 2001.
[7]. High-Voltage Test Techniques Part 1: General Definitions and Test Requirement ,
IEC Standard 60060-1, 2010.
[8]. IEEE Standard Technique for High-Voltage Testing, IEEE Standard 4-2013,
2013.
[9]. Gang Wu, “Shape Properties of Pulses Described by Double Exponential Function
and Its Modified Forms,” IEEE Trans. Electromagn. Compat., vol. 56, no. 4, pp. 923931, Aug. 2014.
590
International Journal of Pure and Applied Mathematics
Special Issue
[10]. Y. M. Li, J. Kuffel and W. Janischewskyj, “Exponential Fitting Algorithms for
Digitally Recorded HV Impulse Parameter Evaluation,” IEEE Trans. Power Del.,
vol. 8, no. 4, pp. 1727-1735, Oct. 1993.
[11]. J. Perez and J. Martinez, “Digitally Recorded Lightning Impulse with Overshoot
Parameter Evaluation by using the Kalman Filtering Method,” IEEE Trans. Power
Del., vol. 13, no. 14, pp. 1005-1014, Oct. 1998.
[12]. Vernon Cooray, “Lightning Protection,” London, United Kingdom, The Institute of
Engineering and Technology, 2010.
[13]. Feng Qin, C. Mao, Gang Wu and Hui Zhou, “Characteristic Parameter Estimation of
EMP Energy Spectrum,” 7th Asia Pacific International Symposium on
Electromagnetic Compatibility, pp. 11-13, May. 2016.
[14]. M. Camp and H. Garbe, “Parameter Estimation of Double Exponential Pulse (EMP,
UWB) With Least Squares and Nelder Mead Algorithm,” IEEE Trans. Electromagn.
Compat., vol. 46, no. 4, pp. 675-678, Nov. 2004.
[15]. M. S. Naidu and V. Kamaraju, “High Voltage Engineering,” New Delhi, Tata
McGraw-Hill, 1995.
[16]. C. Mao and Hui Zhou, “Novel Parameter Estimation of Double Exponential Pulse
(EMP, UWB) by Statistical Means,” IEEE Trans. Eelectromagn. Compat., vol. 50, no.
1, pp. 97-100, Feb. 2008.
[17]. A. Carrus and L. E. Funes, “Very Short Tailed Lightning Double Exponential Wave
Generation Techniques Based on Marx Circuit Standard Configurations,” IEEE
Trans. Power Apparatus and Systems, vol. PAS-103, no. 4, pp. 782-787, Apr. 1984.
[18]. Mathias Magdowski and Ralf Vick, “Estimation of Mathematical Parameters of
Double Exponential Pulses Using the Nelder Mead Algorithm,” IEEE Trans.
Electromagn. Compat., vol. 52, no. 4, pp. 1-3, Nov. 2010.
[19]. J. A. Nelder and R. Mead, “A Simplex Method for Function Minimization,” The
Computer Journal, vol. 7, no. 4, pp. 308-313, Jan. 1965.
[20]. S. C. Dutta Roy and D. K. Bhargava, “On the Design and Generation of the Double
Exponential Function,” IEEE Trans. Instrum. Meas., vol. 45, no. 1, pp. 309-312, Feb.
1996.
[21]. Joseph M. Googe, Paul D. Ewing and Richard A. Hess, “Parameter Correlation of
Impulse Shapes Using Two-Port Synthesis,” IEEE Trans. Instrum. Meas., vol. 41,
no.1, pp. 54-58, Feb. 1992.
[22]. E. Kuffel, W. S. Zaengl, and J. Kuffel, “Impulse Voltages,” in High Voltage
Engineering Fundamentals. Oxford, U. K., Butterworth Heinemann, 2000, pp. 48-66.
[23]. MATLAB–The Language of Technical Computing from Math works. inc‖.
(http://www.mathworks.com).
591
592