This is the author’s version of a work that was submitted/accepted for publication in the
Electric Power Systems Research journal in the following source:
Božidar Filipović-Grčić, Dalibor Filipović-Grčić, Petar Gabrić, Estimation of load
capacitance and stray inductance in lightning impulse voltage test circuits, Electric
Power Systems Research, Volume 119, February 2015, Pages 439-446, ISSN 03787796, http://dx.doi.org/10.1016/j.epsr.2014.11.007.
Changes resulting from the publishing process, such as peer review, editing, corrections,
structural formatting, and other quality control mechanisms may not be reflected in this
document.
Changes may have been made to this work since it was submitted for publication. A
definitive version was subsequently published in Electric Power Systems Research,
[Vol. 119, 2015].
© Copyright 2015 Elsevier S.A.
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http://dx.doi.org/10.1016/j.epsr.2014.11.007
URL: http://www.sciencedirect.com/science/article/pii/S0378779614004064
1
Estimation of Load Capacitance and Stray Inductance in Lightning
Impulse Voltage Test Circuits
Božidar Filipović-Grčić a,*, Dalibor Filipović-Grčić b, Petar Gabrić b
a,
* Corresponding author: B. Filipović-Grčić (e-mail: bozidar.filipovic-grcic@fer.hr)
is with the Faculty of Electrical Engineering and Computing, University of Zagreb,
10000 Zagreb, Croatia, tel.: +385 1 6129 714; fax: +385 1 6129 890.
b
D. Filipović-Grčić (e-mail: dfilipovic@koncar-institut.hr) and P. Gabrić (e-mail:
pgabric@koncar-institut.hr) are with the Končar Electrical Engineering Institute, 10000
Zagreb, Croatia.
Abstract: In order to obtain the lightning impulse voltage waveshape regarding front
time, time to half and relative overshoot magnitude within the limits prescribed by IEC
60060-1, it is useful to accurately estimate the test circuit parameters, e.g. load
capacitance and circuit inductance. A stray inductance consists of the inductance of
impulse generator and the inductance of connecting leads. Load capacitance consists of
voltage divider capacitance, test object and parasitic capacitances. In practice, the test
object capacitance is often unknown. Capacitance measurement takes time and makes
testing procedure more complex. Also, it is very difficult to estimate parasitic
capacitances although their influence can sometimes be significant.
This paper presents a new genetic algorithm (GA) based method for fast and accurate
estimation of load capacitance and circuit inductance during lightning impulse voltage
testing of a capacitive load. Computational and experimental verification of the method
is successfully performed for standard and non-standard lightning impulse waveforms
with various relative overshoot magnitudes.
Keywords: lightning impulse voltage testing, genetic algorithms, load capacitance,
stray inductance
2
1. Introduction
High voltage equipment has to be tested with lightning impulse (LI) voltage in order
to prove the capability against such overvoltages. In order to simulate the effect of
transient overvoltage on high voltage equipment the various national and international
standards define the impulse voltages and their appliance to a test object. Time
parameters of lightning impulse voltage are shown in Fig. 1 according to IEC standard
[1].
Fig 1. Lightning impulse voltage time parameters [1]
Tolerances of 1.2 µs ± 30 % for front time T1 and 50 µs ± 20 % for time to half-value T2
are permitted. The test circuit has an inductance which consists of the inductance of
impulse generator, ground leads and the connecting leads. In some cases inductance
causes overshoot and oscillation at the crest of the lightning impulse voltage waveform.
Overshoot usually occurs when the connecting leads from impulse generator to test
object are very long and the inductance is comparably high. In case of a test object with
high capacitance, low values of the impulse generator front resistors are used which in
some cases can lead to oscillations occurrence. Fig. 2 shows the overshoot β which
3
represents the increase of amplitude of an impulse voltage due to a damped oscillation
(frequency range usually 0.1 MHz to 2 MHz) at the peak caused by the inductance of
the test circuit and the load capacitance.
Fig 2. Determination of overshoot β magnitude from the recorded lightning impulse
voltage and base curve
Overshoot magnitude β is the difference between the extreme value of the recorded
impulse voltage curve and the maximum value of the base curve. The base curve is an
estimation of a full lightning impulse voltage without a superimposed oscillation. The
relative overshoot magnitude β′ represents the ratio of the overshoot magnitude to the
extreme value and it is defined by expression (1).
β ' = 100 ⋅
Ue −Ub
%
Ue
(1)
According to [1], the relative overshoot magnitude shall not exceed 10 %.
In high voltage laboratories, lightning impulse voltages are most commonly produced
using the Marx lightning impulse generator [2]. Equivalent circuit of the impulse
generator is shown in Fig. 3.
4
Fig. 3. Single-stage impulse generator circuit
The generator capacitance C1 is slowly charged from a DC source until the spark gap
G breaks down. Resistor R1 primarily damps the circuit and controls the front time T1,
while resistor R2 discharges the capacitors and controls the time to half T2. C2 represents
the capacitance of test object and all other capacitive elements which are in parallel to
the test object (e.g. capacitor voltage divider used for measurement, additional load
capacitor, sometimes used for avoiding large variations of T1 and T2 if the test objects
are changed, and parasitic capacitances). L represents the inductance of impulse
generator and the connecting leads.
Available values of R1 and R2 are limited in practice and therefore the standardized
nominal values of T1 and T2 are difficult to achieve. Changing these resistors on the
generator usually requires a trial-and-error process or accumulated experience with
previous impulse tests on similar equipment. For this reason it is obvious that the simple
and easy-to-use method for generator parameter determination would make lightning
impulse testing procedure less complicated and less time-consuming.
Many published papers deal with the calculation of impulse generator parameters:
Thomason [3] determined circuit formulas of the most commonly used impulse
generators circuits; Feser [4], Kannan and Narayana [5] and Del Vecchio et al. [6]
investigated circuit design for the lightning impulse testing of transformers; Khalil and
Metwally [7] developed a computerized method to reconfigure the impulse generator
5
for testing different types of objects. Methods described in the previously mentioned
papers use C2 as an input parameter which means that the load capacitance should be
known or measured before testing. In practice, test object capacitance is often unknown
and the measurement of it takes time and makes testing procedure more complex. Also,
parasitic capacitances cannot be taken into account by this approach although their
influence can be significant especially when testing low capacitance objects of large
dimensions.
Genetic algorithm [8,9] and other optimization methods [10,12] have already been
used for curve fitting based estimation of lightning impulse parameters such as peak
value, front time and time-to-half-value.
However, the aim of this paper is to introduce a new genetic algorithm [13] (GA)
based method for obtaining test circuit parameters in case of a capacitive load testing.
The main advantage of this method is a fast estimation of the load capacitance and test
circuit inductance from the recorded lightning voltage impulse and from the known
values of generator capacitance, front and discharge resistance. Once when all circuit
parameters are known it is less complicated to determine circuit elements which will
provide T1, T2 and β’ that are within limits prescribed by [1]. Hence, the presented
method saves time and makes the reconfiguration of impulse generator easier.
2. Analysis of the lightning impulse voltage test circuit
The method presented in this paper estimates lumped stray inductance and load
capacitance in a test circuit. In the real situation, a stray inductance consists of the
inductance of impulse generator and the inductance of connecting leads while load
capacitance consists of voltage divider capacitance, test object capacitance and their
parasitic capacitances. By taking this into account a more realistic equivalent scheme of
the test circuit would be obtained. However, this model is more complex and solving it
6
would take more time and the algorithm has to be fast in order to be applied in practice.
The presented method cannot differentiate all individual inductances and capacitances
mentioned above. However, estimation of lumped inductance and capacitance in a test
circuit proved sufficient for practical application. In [2], [14] and [3] it is demonstrated
that the calculations can be made much more easily if certain approximations are used,
and these are found not to introduce appreciable errors in practice. Even more
complicated circuit representations have been examined, particularly by Thomason [3],
but the resulting expressions are of little more than academic or mathematical interest,
especially as the stray capacitances and inductances are distributed throughout the
circuit and no precise numerical values can be assigned to them. Therefore in practice it
is convenient to simplify the calculations and use equivalent circuit shown in Fig. 3.
Since in this paper excellent results were obtained by using equivalent circuit shown in
Fig. 3 it is not convenient to take into account a more realistic equivalent scheme of the
test circuit because it would not significantly improve the results. For that reason, a
simplified circuit represented in Fig. 3 was used. The capacitances of the test object and
of the voltage divider (and their stray capacitances) were lumped together in C2, while
the total inductance within the generator – load circuit is combined to a single
inductance L. Laplace transform of the circuit for lightning impulse voltage testing form
Fig. 3 is shown in Fig. 4.
U0
s
1
sC1
1
sC2
Fig. 4. Laplace transform of the circuit for lightning impulse voltage testing
7
Voltage U1 is determined by using the expression (2).
U 1 (s) =
U0 ⋅Z2 ,
s (Z 1 + Z 2 )
(2)
where:

1 

R 2 ⋅  R1 + sL +
sC
1
2 

Z1 =
; Z2 =
1
sC 1
R1 + R 2 + sL +
sC 2
.
(3)
The output voltage U2 is calculated in frequency domain using the expression (4).
U 2 (s) =
U 1 (s) ⋅ Z 4
Z3 + Z4
,
(4)
where:
Z 3 = R 1 + sL ; Z 4 =
1
sC 2
.
(5)
The output voltage is expressed in the time domain using the inverse Laplace transform:
U 2 (t ) = L−1 (U 2 ( s ) )
(6)
3. Method for estimation of circuit inductance and load capacitance
Exact values of all circuit parameters are unknown in practice. Nominal test circuit
parameters and their tolerances are given by manufacturers or can be more or less
accurately measured, but always with measurement uncertainty. The charging voltage
U0, also, is not exactly known. GA based method determines circuit inductance L and
load capacitance C2 for standard and non-standard lightning impulse waveforms. GA
selects L and C2 in a wide boundary range of values and R1, R2, C1 and U0 in a narrow
boundary range around nominal values. All these parameters form a vector Vi, an
individual solution, while a group of vectors forms a population in GA terminology.
Fig. 5 shows the flowchart of the described method.
8
Fig. 5. Method flowchart
The first step is to obtain the input data consisting of the recorded impulse voltage
from the measuring system, R1, R2, C1, U0 and their tolerances. GA generates the initial
population of vectors Vi, i=1…n. Population size n specifies how many individuals there
are in each generation. Initial population is created randomly and it satisfies the defined
bounds of parameters R1, R2, C1, U0, L and C2. After the creation of the initial
population, the output voltage U2 is determined in time domain for each Vi.
Since the GA performs many calculations finding an acceptable solution, it is very
important to minimize the execution time. To achieve this, it is helpful to reduce the
number of calculations by comparing measured waveform with GA results in several
representative points only. Therefore, characteristic points (tcp, Ucp) are selected on
measured waveform front (at 10 %, 30 %, 50 %, 75 %, 90 % and 100 % of the
9
amplitude), tail (at 90 %, 75 %, 60 %, 50 %, 40 % and 30 % of the amplitude) and at the
local extremes in case of oscillatory impulses. For each tcp, voltage U2 is calculated with
circuit parameters selected by GA and compared to measured values. Describing the
waveform by characteristic points is a great advantage because it significantly reduces
calculation time and enables this method to be used during laboratory impulse testing.
The fitness function is the objective function minimized by the GA [15]. In this case,
the fitness function takes into account the voltage percentage error for each
characteristic point. Fitness function is calculated using the expression (7).
ε=
U 2 calculated (t cp ) − U cp (t cp )
U cp (t cp )
⋅ 100 (%)
(7)
The algorithm stops when the voltage percentage errors of all characteristic points are
lower than the user defined limit value or when a certain time elapsed. All calculations
are performed using Matlab software on PC Intel core i5 CPU, 2.53 GHz with 4 GB
RAM and a selected time period of 1 min was enough to reach the stopping criteria.
If stopping criteria is not fulfilled, then selection, crossover and mutation are
performed. The stochastic uniform selection function chooses the parents for the next
generation based on fitness results. The elite count specifies the number of individuals
that are guaranteed to survive to the next generation (in this case 50). Crossover fraction
specifies the fraction of the next generation, other than elite individuals, that are
produced by crossover (in this case 80 %). The remaining individuals are produced by
mutation. The scattered crossover function creates a random binary vector. It then
selects the genes for which the vector value is a 1 from the first parent, and the genes for
which the vector value is a 0 from the second parent, and combines the genes to form
the child. Mutation functions create small random changes in individuals from a
population, and they provide genetic diversity and enable the GA to search a broader
10
space. Adaptive feasible mutation was used which randomly generates directions that
are adaptive with respect to the last successful or unsuccessful generation. A step length
is chosen along each direction so that bounds are satisfied.
If T1, T2 or β’ are not acceptable, it is necessary to modify the test circuit parameters.
This usually requires changing R1 and/or R2 and/or introduction of additional capacitor
Cad in parallel with the test object. Now, with known L and C2, one can easily calculate
output voltage using the finite pool of R1, R2 and Cad available in a high voltage
laboratory instead of trial-and-error voltage applications in actual test circuit.
In case of a purely capacitive test objects, maintaining the T1 is much more
complicated than maintaining the T2. Stray inductance L and front resistance R1 tend, in
general, to retard the length of the wave front. The inductance also introduces
oscillations in the wave [3]. It may be impossible to realize standard waveforms within
the standard tolerances for certain test circuits and test objects. In such cases extension
of front time T1 or overshoot may be necessary (guidance for such cases should be given
by the relevant Technical Committee).
Of all parameters used in the GA based method it is noticed that only the population
size significantly affects the GA convergence rate and execution time, while the
influence of other parameters is small. At first, a smaller population sizes were used but
the convergence rate was slow and the execution time was long. Execution time of this
algorithm should not be longer than a few minutes (in this paper a 1 min criterion
adopted) in order to be applied in practice. It is noticed that the increase of population
size to a certain value improves the convergence rate and shortens the execution time to
an acceptable value. Satisfactory convergence rate and acceptable execution time were
11
achieved with the population size of 500 in case of computational verification and with
the population size of 2000 in case of experimental verification.
4. Verification of the presented method
The presented method is verified in the following sections:
4.1 Computational (numerical) test.
4.2 Tests with recurrent surge generator.
4.3 Tests in high voltage laboratory.
4.1 Computational verification
All circuit parameters are exactly known in this example and the method’s ability to
determine L and C2 is tested. The following cases are examined:
a) U2 without oscillations, L=0, GA selects only C2;
b) U2 with oscillations L=26.5 - 1950 µH, relative overshoot β’=9.09 - 33.33 %, GA
selects L and C2.
In both cases other parameters are C1=50 nF, C2=1 nF while R1 and R2 are varied to
provide standard and non-standard waveforms with T1=0.6 - 1.8 µs and T2= 30 - 70 µs.
GA input values are R1, R2, C1, U0 and characteristic points from U2. The size of GA
population in each generation is n=500. GA task is to choose L and C2 in a fairly wide
range of values (boundaries 0.1 - 20 nF for C2 and 10 - 4000 µH for L) and to find the
output voltage which best fits the inputted U2. Table 1 shows estimated C2 for case a).
Tables 2-6 show estimated C2 and L for case b).
12
Table 1
Estimated C2, circuit without stray inductance
T1/T2
R1
R2
Estimated Number of
(µs)
(Ω)
(Ω)
C2 (nF)
generations
0.6/30 203.9 812.7 0.9999998
7
0.6/50 198.7 1376.2 0.9999996
7
0.6/70 196.2 1940.6 0.9999999
7
1.2/30 432.4 781.5 1.0000003
7
1.2/50 413.0 1343.7 1.0000000
7
1.2/70 404.3 1906.8 0.9999998
8
1.8/30 683.8 751.5 1.0000000
7
1.8/50 641.5 1313.1 0.9999967
5
1.8/70 622.6 1875.2 0.9999924
5
Table 2
Estimated C2 and L (relative overshoot 9.09 %)
Exact Estimated Estimated Number of
T1/T2
(µs)
L (µH)
L (µH)
C2 (nF) generations
0.6/30
26.5
26.501
0.9999
11
0.6/50
26.0
25.977
1.0001
14
0.6/70
25.8
25.793
1.0002
11
1.2/30
110
109.99
1.0000
11
1.2/50
107
106.98
1.0001
10
1.2/70
105
105.01
0.9999
9
1.8/30
255
255.09
0.9995
9
1.8/50
248
248.00
1.0000
9
1.8/70
240
239.97
1.0001
10
Table 3
Estimated C2 and L (relative overshoot 16.67 %)
T1/T2 Exact Estimated Estimated Number of
(µs) L (µH)
L (µH)
C2 (nF) generations
0.6/30 45.0
44.995
1.0000
10
0.6/50 43.5
43.502
0.9999
10
0.6/70 43.5
43.506
1.0000
10
187
186.93
1.0002
8
1.2/30
1.2/50
181
180.96
1.0001
9
1.2/70
178
177.96
1.0001
9
1.8/30
447
446.85
1.0002
9
425
424.90
1.0004
8
1.8/50
1.8/70
410
410.06
0.9998
9
13
Table 4
Estimated C2 and L (relative overshoot 23.08 %)
T1/T2 Exact Estimated Estimated Number of
(µs) L (µH)
L (µH)
C2 (nF) generations
0.6/30 73.5
73.464
1.0005
10
0.6/50
71
71.025
0.9999
9
0.6/70 70.5
70.669
0.9984
9
1.2/30
310
310.27
0.9991
9
1.2/50
298
298.06
0.9999
10
1.2/70
290
290.06
0.9998
8
1.8/30
745
744.09
1.0007
8
1.8/50
695
694.49
1.0002
8
1.8/70
670
669.27
1.0006
8
Table 5
Estimated C2 and L (relative overshoot 28.57 %)
T1/T2 Exact Estimated Estimated Number of
(µs) L (µH)
L (µH)
C2 (nF) generations
0.6/30
121
121.13
0.9994
8
0.6/50
116
116.00
1.0005
8
0.6/70 115.5
115.57
0.9996
10
515.28
1.0006
8
1.2/30 515.5
1.2/50 490.5
490.19
1.0005
8
1.2/70
475
475.00
1.0001
8
1.8/30 1270
1269.8
1.0001
8
1.8/50 1160
1159.7
1.0005
7
1109.3
1.0011
7
1.8/70 1110
Table 6
Estimated C2 and L (relative overshoot 33.33 %)
T1/T2 Exact Estimated Estimated Number of
(µs) L (µH)
L (µH)
C2 (nF) generations
0.6/30
210
210.00
0.9999
8
0.6/50
195
195.18
0.9988
10
0.6/70
195
194.91
1.0007
8
900
899.24
1.0003
7
1.2/30
1.2/50
850
849.92
1.0001
8
1.2/70
820
820.64
0.9988
8
1.8/30 2340
2339.7
1.0001
8
2049.5
1.0001
7
1.8/50 2050
1.8/70 1950
1951.4
0.9995
7
Figs. 6-8 show comparison between input (original) waveform and calculated waveform
obtained with estimated L and C2.
14
160
140
120
U2 (kV)
100
80
60
40
20
0
0
1
2
3
4
5 6
t (µs)
7
8
9
10
Fig. 6. Comparison of original 0.6/30 µs impulse voltage waveform (colored lines) and
estimated waveform (black dotted lines)
160
140
120
U2 (kV)
100
80
60
40
20
0
0
1
2
3
4
5
6
t (µs)
7
8
9
10
Fig. 7. Comparison of original 1.2/50 µs impulse voltage waveform (colored lines) and
estimated waveform (black dotted lines)
160
140
120
U2 (kV)
100
β’=33.33 %
β’=28.57 %
β’=23.08 %
β’=16.67 %
β’=9.09 %
β’=0 %
80
60
40
20
0
0
Estimation
1
2
3
4
5 6
t (µs)
7
8
9
10
Fig. 8. Comparison of original 1.8/70 µs impulse voltage waveform (colored lines) and
estimated waveform (black dotted lines)
15
Fig. 9 shows the change of fitness value ε throughout generations for waveform
1.2/50 µs and relative overshoot β’=9.09 %.
25
Fitness value (%)
20
15
10
5
0
0
1
2
3
4
5
6
Generation
7
8
9
10
Fig. 9. Change of fitness value ε throughout generations for waveform 1.2/50 µs and
β’=9.09 %
In all cases algorithm successfully estimates C2 and L with high precision within the
following stopping criteria: a time period of 1 min elapsed or the fitness value of 0.2 %
is achieved. The largest percentage differences between estimated and known C2 and L
are 0.16 % and 0.24 %, respectively.
4.2 Tests with RSG
RSG is a low voltage single stage equivalent of a high voltage impulse generator. It is
usually used to study the voltage distribution in high voltage windings during impulse
voltage stresses.
Parameters of impulse circuit are in fact parameters of RSG so, in this case, the exact
values of circuit parameters are unknown, but the nominal values are stated by RSG
manufacturer. Fig. 10 shows the test setup. The impulse voltages are recorded using a
digital oscilloscope (500 MHz, 1 GS/s) connected to PC.
16
Recorded
waveform
Data
acquisition
Digital
oscilloscope
Recurrent
surge generator
Fig. 10. Test setup for generation of low voltage impulses with RSG
RSG parameters are U0=250 V, R1=6.8 Ω, R2=100 Ω, C1=470 nF, C2=100 nF and L is
varied (10 µH, 20 µH and 30 µH). GA selects L from 0.1 µH to 2 mH and C2 from
10 nF to 200 nF. The size of GA population in each generation is n=2000. Fig. 11
shows the comparison between measured and calculated (GA obtained) waveforms.
Measured and GA obtained waveforms show excellent agreement.
320
280
240
U2 (V)
200
160
120
80
40
0
0
5
10
15
t (µs)
20
25
30
Fig. 11. Comparison of measured and estimated impulse voltage waveforms
17
Percentage differences between real and estimated C2 and L cannot be exactly
determined because the exact values of circuit parameters are unknown, but only the
nominal values are stated by RSG manufacturer. Nevertheless, comparison of estimated
and nominal values shows a good agreement (Table 7).
Table 7
Estimated C2 and L
Nominal Estimated Estimated Number of
L (µH)
L (µH)
C2 (nF) generations
10
10.8
107.7
11
20
20.6
103.7
7
30
30.7
103.3
7
4.3 Tests in high voltage laboratory
In Section 4.1 the exact values and in Section 4.2 the nominal values of all circuit
parameters were known. In this case L and C2 are unknown. The tests are performed on
oil-paper insulated inductive voltage transformer Um=245 kV and SF6 insulated current
transformer Um=765 kV. The measurement of impulse voltages was carried with
measurement systems consisting of digital impulse analyzing system and capacitor
voltage divider which fulfill the requirements of [1], [17] and [18].
4.3.1
Oil-paper insulated inductive voltage transformer Um=245 kV
Fig. 12 shows the test setup for lightning impulse voltage testing of an oil-paper
insulated inductive voltage transformer in high voltage laboratory. Parameters of the
test circuit are: U0 (618 to 798 kV), R2=1600 Ω, C1=50 nF and R1 is varied. GA selects
L from 0.1 µH to 1 mH and C2 from 0.5 nF to 10 nF.
18
Leads
Impulse
voltage
generator
Capacitor
voltage
divider
Inductive voltage
transformer
Fig. 12. Lightning impulse voltage test of oil-paper insulated inductive voltage
transformer (Um=245 kV) in high voltage laboratory
C2 consists of voltage divider capacitance (nominal value 817 pF), test object
capacitance (unknown) and parasitic capacitances (unknown). At first the impulse
voltage is recorded with R1=142 Ω (yellow curve in Fig. 13).
800
700
600
U2 (kV)
500
400
300
200
100
0
0
1
2
3
4
5 6
t (µs)
7
8
9
10
Fig. 13. Comparison of measured and estimated impulse voltage waveforms in case of
testing the oil-paper insulated inductive voltage transformer
This waveform contains some oscillations with front time T1=0.77 µs and tail time
T2=42.4 µs. According to [1] T1 is too short.
19
Increasing the external front resistor R1 value helps to damp oscillations while the value
of external leads inductance causes an oscillatory natural response of the system. The
critical serial resistance Rc for the circuit to be non-oscillatory is given by the following
well-known equation:
Rc = 2
L
C
(8)
where:
1
1
1
=
+
C
C1 C 2
(9)
This equation is suitable for determining the limiting values for the front resistor R1.
In this case Rc=264 Ω. Since high voltage laboratory has a finite pool of R1, the closest
value obtained was R1=295 Ω. L increases with R1 due to the stray inductance of the
resistors, so it is better to choose R1 slightly higher than calculated Rc.
Measurement and simulation results are shown in Fig. 13 and Table 8.
Nominal
R1 (Ω)
142
212
295
464
550
Measured
T1/T2 (µs)
0.77/42.4
0.88/51.6
1.07/58.4
1.88/60.1
2.20/62.8
Table 8
Estimated C2 and L
U0 Estimated Estimated Number of
C2 (nF) generations
(kV)
L (µH)
618
45.8
1.96
16
687
49.6
1.88
11
740
53.7
1.80
14
770
88.7
1.91
10
798
105.3
1.94
10
It can be seen that the oscillations are heavily damped when R1=295 Ω which is close to
calculated Rc, while T1 and T2 are within the limits of [1]. Other examples in Table 8
only show how front resistors introduce stray inductances and that this can be taken into
account.
Estimated C2 is quite stabile as expected and varies from 1.80 nF to 1.96 nF. Increase
of R1 damps the oscillation and reduces peak voltage without affecting wave tail and, as
20
a consequence, T2 increases. In Fig. 13, the charging voltage U0 of the impulse generator
was increased in order to compensate the influence of R1 on peak voltage reduction.
4.3.2
SF6 insulated current transformer Um=765 kV
Fig. 14 shows the test setup for lightning impulse voltage testing of SF6 insulated
current transformer in high voltage laboratory.
Current
transformer
Leads
Capacitor
voltage
divider
Impulse
voltage
generator
Fig. 14. Lightning impulse voltage test of SF6 insulated current transformer (Um=765
kV) in high voltage laboratory
Parameters of the test circuit are: U0=1640 kV, R2=1236 Ω, C1=53.6 nF and
R1=420 Ω. GA selects L from 0.1 µH to 1 mH and C2 from 0.5 nF to 10 nF. C2 consists
of voltage divider capacitance (nominal value 613 pF), test object capacitance (261 pF,
measured with Schering bridge and standard capacitor) and parasitic capacitances
(unknown value). GA simulation gives C2=1.69 nF and L=99.9 µH after 12 generations.
Fig. 15 shows measured and estimated impulse waveforms.
21
1600
1400
1200
U2 (kV)
1000
800
600
400
200
0
0
1
2
3
4
5 6
t (µs)
7
8
9
10
Fig. 15. Comparison of measured and estimated impulse voltage waveforms in case of
testing the SF6 insulated current transformer Um=765 kV
There is a significant difference between estimated C2 and the sum of capacitances of
the test object and voltage divider. This implies that parasitic capacitances have great
influence on C2 and cannot be disregarded.
Fig. 16 shows a comparison between measured and estimated waveforms when
parasitic capacitances are excluded from the simulation.
2000
1750
1500
U2 (kV)
1250
1000
750
500
250
0
0
1
2
3
4
5
6
t (µs)
7
8
9
10
Fig. 16. Comparison of measured and estimated impulse voltage waveforms in case of
testing the SF6 insulated current transformer Um=765 kV; parasitic capacitances are not
included in simulation
22
Load capacitance is set to a sum of voltage divider and test object capacitances while
circuit inductance is varied from 0 to 200 µH. It is obvious that even when test object
capacitance value is known it may not be enough for accurate estimation of impulse
voltage waveform.
According to the previous statement, it can be concluded that a test object capacitance
measurement may give insufficient information about a total load capacitance.
5
Conclusion
The paper describes a new GA based method for estimation of load capacitance and
circuit inductance in case of lightning impulse testing of capacitive loads.
The presented method enables a fast and accurate estimation of unknown circuit
parameters: the total load capacitance, including parasitic capacitances which cannot be
disregarded when testing low capacitance objects, and test circuit inductance. The
method is successfully verified on several examples.
Future work will be focused on expanding the capabilities of the method for
estimating the test circuit parameters in case of impulse testing of low inductance loads,
such as low voltage windings of power transformers.
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25