evaluation of transmitted over-voltages through a power
TRANSCRIPT
2014 International Conference on Lightning Protection (ICLP), Shanghai, China
Evaluation of Transmitted Over-Voltages through a
Power Transformer Taking into Account
Uncertainties on Lightning Parameters
B. Jurisic, A. Xemard
Electricité de France
Clamart, France
I. Uglesic*, F. Paladian, S. Lallechere, P. Bonnet
*University of Zagreb,
Zagreb, Croatia
Université Clermont Auvergne, Université Blaise Pascal,
Institut Pascal, France
Abstract— Lighting striking an overhead line is at the origin of
high frequency over-voltages, which propagate along the line and
through the power transformers located at its ends. To study the
transmission of over-voltages through transformers, low
frequency transformer models are not adapted. Therefore, in the
past few decades, many high frequency transformer models were
developed. Recently, in the CIGRE brochure 577, an overview of
such models is given.
In this paper the over-voltages caused by lightning strikes to a 20
kV overhead line are simulated to calculate the amplitudes of
over-voltages in a 6.8 kV distribution grid, transmitted through a
Yd11d11, 64 MVA, 24/6.8/6.8 kV transformer unit. A high
frequency black box transformer model is used. This model is
based on frequency response measurements, rational
approximation and state space equations. Uncertainties on
lightning parameters are taken into account by using the Monte
Carlo and Stochastic Collocation methods. The aim of the paper
is to present a methodology to evaluate the statistical variations
of transmitted over-voltages.
Keywords-Transformer; Black Box Model; Monte Carlo Method;
Stochastic Collocation Method; Uncertainty Quantification;
Electromagnetic Transient Program (EMTP)
I. INTRODUCTION
Lightning strikes generate high over-voltages in the electric
power network. Consequently, apparatuses in the power
network have to be protected against them. To determine the
lightning withstand voltage of the power network’s equipment
as well as surge arrester installations needs, it is necessary to
conduct insulation coordination studies. These studies are
usually based on Electromagnetic transient simulations
(EMTP) and require a precise model for each component
inside the power network.
Traditional, low frequency models cannot be used for
lightning studies since electrical devices like transformers at
high frequencies exhibit a different behavior than at low
frequencies. To represent high frequency electromagnetic
transformer behavior in simple studies, capacitance divider
models can be used, while for more detailed studies, special
high frequency transformer models should be developed. In
the case of a power transformer, it is very important to have a
precise high frequency model which will allow an accurate
evaluation of the transmitted over-voltages. However, high
frequency transformer models are often too complex or
require confidential information on transformer geometry. The
lack of knowledge on transformer geometry data led to the
development of black box transformer models which as an
input require only data measured from the transformer
terminals. That makes them suitable for usage in insulation
coordination studies, especially for power utilities’ engineers.
The transformer model, developed for the purpose of this
paper is a state of the art black box model built from a
transformer admittance matrix, measured with a standard
sweep frequency response analyzer (FRA). It is based on a
rational approximation with passivity enforcement and a state
space representation compatible with an EMTP-like software
program. The model is suitable to represent accurately enough
transmitted over-voltages in the lightning frequency range, for
which a power transformer behaves as a linear component.
In this paper the over-voltages caused by lightning strikes to
a 20 kV overhead line are simulated in order to evaluate the
amplitudes of the over-voltages in the distribution grid,
transmitted through a Yd11d11, 64 MVA, 24/6.8/6.8 kV
transformer unit. Uncertainties regarding lightning parameters
may be taken into account by using the well-known Monte
Carlo method (MC) or more sophisticated stochastic
approaches such as the Stochastic Collocation methods (SC)
[1]-[3]. It should be noticed that the probability distributions
of input parameters are taken into account based on data given
in [4]. The aim of this paper is to calculate the mean and
standard deviation of the amplitude of the transmitted over-
voltages through the power transformer.
II. TRANSFORMER HIGH FREQUENCY
MODELLING
In this section, the black box high frequency transformer
model is presented. It is based only on measurements data.
Transformer’s scattering parameters, admittance parameters,
impedance parameters or transfer function can be measured.
That makes the model suitable for practical applications in
power utility companies because they usually do not have
access to detailed transformer’s geometry data. To interact
with an EMTP-like software program, an admittance matrix is
calculated from the named measured parameters. Among
many different approaches to input the model’s admittance
matrix in an EMTP-like software program, the most used one
is to approximate the matrix elements with rational
expressions [5], [6]. As a final representation, usually the state
space equations are used [7]. State space equations are used to
represent a linear network. Therefore, they can be used to
represent a transformer, since transformer’s behavior can be
considered as linear at high frequencies.
For the purpose of this paper, to measure the admittance
matrix elements, a sweep frequency response analyzer (FRA)
measuring equipment is used. This is a standard equipment for
measuring the frequency response of a transformer as
suggested in the IEC standard [8]. The measurement
procedure is similar to the one described in [9]. A frequency
response analyzer, is capable of measuring the ratio (H)
between the input (Vin) and the output (Vout) voltages:
)f(V
)f(V)f(H
in
out
, where f stands for the frequency. Note that the
measurements are done at discrete frequency points. Since the
FRA measurement’s equipment is not normally used for
measuring Y matrix, a specific procedure for measuring is
established.
The measuring method stems from the following expression:
)f(U
)f(U
)f(U
)f(U
)f(Y)f(Y
)f(Y)f(Y
)f(I
)f(I
)f(I
)f(I
N
NNNN
N
N
N 1
2
1
1
111
1
2
1
Expression (2) is valid for a transformer with N terminals.
The measuring procedure includes N*N measurements as it is
shown in [10] and [11]. Note that the measuring methods
differ for off-diagonal and diagonal matrix elements.
Since the transformer model has to be built in an EMTP-like
software program, the results of the measurement have to be
prepared accordingly. This can be done by using a fitting
method to approximate each admittance matrix element Yij(f)
with a rational expression Yij,fit(s) of the type given below [5],
[12] :
ij
Np
n ij,n
ij,nfit,ijij d
as
c)s(Y)f(Y
1
In (3) an,ij represents the poles which can be either real or
complex conjugated pair, cn,ij represents the residues which
can also be either real or complex conjugated pair, dij are real
value constants. s stands for j2πf. Np is the number of poles
used for the approximation of each matrix element.
The rational functions (3) have to be both stable and passive
since the transformer is a passive component of the electric
grid. Stability is obtained by keeping only the poles which are
stable. Passivity is enforced by perturbation of the residues
and constants values in order to match the passivity criterion:
0 v)s(YvReP fit*
, in which Yfit(s) represents the matrix of the fitted rational
functions. Expression (4) means that the model will not
produce power for any complex vector v. The expression
above will be positive only if all the eigenvalues of the real
part of the Yfit(s) are positive:
0)))s(Y(Re(eig fit
By introducing the passivity criterion (4), the fitting problem
can be formulated as follows:
minimize i j
ij,fitij )s(Y)s(Y 2
passive )s(Y fit
The problem formulated with (6) and (7) is a non-convex
one (may have multiple feasible regions and multiple locally
optimal points within each region). To solve such a problem,
many algorithms have been developed [13]-[16]. These
algorithms can be separated into two groups: the ones using an
unconstrained minimization combined with a post processing
perturbation to enforce passivity such as Vector Fitting (VF)
combined with the Fast Residue Perturbation method (FRP
method) [13], [14] and the ones simultaneously enforcing
passivity during the fitting process by formulating a convex
optimization problem as the semi-definite programming
method (SDP method) [15], [16].
When fitted, the rational expression (3) allows using the
state space equations as shown below:
)s(UB)s(XA)s(sX
)s(UD)s(XC)s(I
Matrices A, B, C and D for the state space representation can
be input directly into the state space block in the EMTP-RV.
These matrices are obtained by using the values of poles and
residues of the rational functions (3) and constructing the
function given below:
)s(UDAIs
BC)s(U)s(Y)s(I
Expression (10), in which [I] is the identity matrix, can be
obtained from (8) and (9). It represents the relation between
the terminal’s currents and voltages of the transformer,
suitable to represent the rational functions given by (3).
The black box model in the EMTP-RV, built for the
transformer unit described in this paper has 10 terminals: 3
terminals of the HV windings, the neutral of the HV windings,
3 terminals per LV winding.
III. STATISTICAL ANALYSIS OF THE
TRANSMITTED OVER-VOLTAGES
In this section the case of a direct lightning strike to a 20 kV
overhead line, without shield wire is described. Uncertainties
regarding the lightning parameters are taken into account in
the time domain simulations. In this framework, two different
methods to include uncertainties are used: the Monte Carlo
and Stochastic Collocation techniques [1]-[3]. Since the
EMTP-RV does not have an inbuilt module for uncertainty
studies, one was developed in MATLAB. The interaction
between these two software packages is managed by using an
EDF’s in-house software program code as a link. This
approach allows to adjust the simulation parameters from
MATLAB, running the EMTP-RV simulations in parallel and
post-processing the simulation results in the MATLAB.
A. Case of the lightning strike to the 20 kV overhead line
In this section we consider that the Yd11d11, 64 MVA,
24/6.8/6.8 kV transformer unit is connected to the 20 kV
overhead line without shield wire. This is a purely academic
case aiming at presenting the method for taking lightning
parameters uncertainty into account in the time domain
simulations.
The case of a direct lightning strike to the second tower of
the 20 kV overhead line, observed from the transformer
primary side, is simulated. The primary side of the transformer
is connected to the 20 kV overhead line. The neutral of the
primary side of the transformer is grounded with a 1 kΩ
resistor. The transformer is intended to be connected inside a
plant for powering its distribution system. A surge arrester,
which is normally connected before the transformer, is not
represented in order to concentrate the study on the influence
of the transformer. It is assumed that the transformer internal
insulation will withstand over-voltages. The secondary
transformer windings are grounded through 250 nF capacitors
per phase, which represent the influence of the distribution
grid. These capacitances have a beneficial influence on the
transmitted over-voltages amplitude [17].
The transformer is simulated using the black box model. For
the fitting of the measured admittance matrix coefficients SDP
method with 40 poles is used. The fitting is done for the
frequency range from 10-500 kHz. Simulated transformer has
2 secondary 3 phase windings and their terminals names are
respectively: a1, b1, c1, a2, b2, c2. The 20 kV overhead line is
simulated using a frequency dependent line model, from the
EMTP-RV library. The line is simulated with 4 towers and 5
spans in order to avoid any unphysical reflections of the
travelling wave across the overhead line, as it can be seen in
figure 1. Data used for the line simulation is shown in table 1.
The line is not equipped with a shield wire and has
ungrounded reinforced concrete poles. Consequently the tower
footing is not considered in the simulation.
TABLE I. OVERHEAD LINE DATA
Span length
[m]
Conductor
height [m]
Horizontal
distance between
2 adjacent
conductors [m]
Vertical distance
between the
conductors [m]
100 8 1.5 0
Grounding
resistance [Ω]
Ground
return
resistivity
[Ωm]
External
diameter of the
conductor [mm]
Conductor’s
resistance
[Ω/km]
10000 (i.e. ungrounded)
1000 19.6 0.146
Each tower of the overhead line is simulated as one constant
parameter line, with a surge impedance equal to 106.35 Ω and
the propagation wave speed equal to 2.4*108 m/s. Tower arms
are modeled as inductances, whose lineic inductance is equal
to 1 µH/m, while the air gaps are simulated by using the equal
area criterion model [18], with the parameters given in table 2.
Span 1Span 2Span 4 Span 3Span 5
Tower 4 Tower 3 Tower 2 Tower 1Transformer
Capacitances (250 nF per phase)
Lighting strike
20 kV line to line sinus voltage
source
10 k tower grounding resistance
Air gap
Tower s arm
Figure 1. Direct lightning strikes the 20 kV overhead line simulation in the EMTP-RV.
TABLE II. AIR GAP PARAMETERS
10% flashover voltage of the insulation [kV] Air gap distance [mm]
112.5 305
Lightning, for the case described above, strikes the tower 2.
The lightning parameters of the CIGRE lightning current
source are respectively lightning current amplitude, If,
lightning front time, tf, lightning time to half, th and lightning
maximum slope, Sm. For the explanation of these parameters,
see figure 2.
I [kA]
t [μs]
If
tf
0.9*If
0.3*If
Sm
Figure 2. CIGRE concave shape [18].
These parameters are varied with the distributions given in
the CIGRE brochure 63 for the first negative downward stroke
[4]. Distributions used are the ones for the backflash domain
since the simulated overhead line does not have a shield wire.
These distributions are shown in table 3.
TABLE III. DISTRIBUTION OF THE LIGHTNING PARAMETERS
If [kA] tf [µs] Sm [kA/µs] th [µs]
log-N(33.3,
0.605)
log-N(0.906 If 0.411,
0.494)
log-N(6.5If 0.376,
0.554)
log-N(77.5,
0.577)
It is to be noted that the distributions for the lightning front
time and the lightning maximum slope value are conditional
ones. Therefore, in the scope of this paper we consider them as
deterministic variables and as such their values are calculated
from the expressions for their medians as a function of
lightning current amplitude value.
CIGRE brochure 63 does not include any condition which
should be fulfilled in order for the shape given in figure 2 to
exist:
f
fm
t
IS
However, condition (11) holds for the following study
without influencing the distribution of the lightning current
parameters.
Three other parameters which should be taken into account
in our uncertainty analysis are the angles of the sinus phase 20
kV voltage source. Since this source is considered as
symmetrical, only one angle parameter is independent, while
the other two are dependent. The probabilistic distribution law
considered for the phase voltage angle is U[-180, 180] without
any further information about this parameter.
To recap, in the simulation 3 parameters are considered as
independent, random variables (the lightning current
amplitude, the lightning time to half value and the phase
voltage angle) and 4 parameters are considered as dependent,
deterministic variables (the lightning front time, the lighting
maximum slope and the other two voltage phase angles). The
lightning current dependent parameters are determined by
taking condition (11) into account.
As an output data the maximum value of the induced over-
voltages of the two three phase 6.8 kV systems located on the
secondary side of the transformer are observed.
B. Statistical analysis with Monte Carlo method
Monte Carlo (MC) technique is based on repeatedly random
sampling and the application of the law of the large numbers.
According to this law the mean value of the results will
converge with the number of samples. Input parameters are
varied by taking into account the probability distributions
given in table 3. Due to hard drive space limitation (used for
storing the results data), the number of MC realizations was
imposed (no more than 10000 inputs). The log normal
distribution of the input parameter If is shown in figure 3. Each
column in the figure shows the number of the parameters If in
spans of approximately 10 kA.
Figure 3. Distribution of input parameter If.
The convergence of the output data is checked for the
maximum transmitted overvoltage in each phase of both 6.8
kV systems located on the secondary side of the transformer.
The convergence is obtained for the two first statistical
moments (i.e. mean and variance of each output parameter).
C. Statistical analysis with Stochastic Collocation method
In comparison to the Monte Carlo method, a great advantage
of the Stochastic Collocation (SC) method is to require fewer
input data to achieve a similar level of convergence. Another
key factor relies on the non-intrusiveness of SC method,
allowing a straightforward use of existing deterministic
calculation codes. Depending on the number of points
(collocation points) which are used to simulate the input data
distributions, levels of accuracy will differ. In this study
convergence is expected with only a few collocation points.
Consequently 3, 5, 7 and 9 points per independent input
parameter (i.e. lightning current amplitude, lightning time to
half, and phase voltage angle) are used. The number of
simulations needed can be calculated as follows: 3
independent parameters with 9 points per each leads to 93=729
simulations.
IV. RESULTS AND DISCUSSION
First of all, it is necessary to make a comment about the
convergence of the MC and SC methods. Figures 4 and 5 are
respectively devoted to the mean and variance of the
maximum of transmitted over-voltages in the phase a1, for the
MC method and the SC method with 3, 5, 7 and 9 points per
independent input parameter.
From the figures the MC method seems to converge when
the number of samples is around 10000. The same behavior
was observed for most of other output parameters.
Furthermore, it can be seen that the SC method needs
significantly smaller number of simulations (with 9 points per
input parameter, 729 in comparison of 10000 for MC) to
converge to the same value as MC method. To conclude, SC
methods reveal to be far more efficient than MC techniques
(decreasing the required number of realizations) with
comparable levels of accuracy for the studied case.
Figure 4. Convergence of MC method for mean value of maximum in the
phase a1.
Figure 5. Convergence of MC method for variance value of maximum OV in the phase a1.
In addition the convergence for the SC method versus the
number of points per independent parameter is shown.
Figure 6. Convergence of SC method for mean value of maximum OV at each phase.
From figure 6, it can be seen that the convergence of the SC
method is obtained with a small number of collocation points
per independent parameter.
The aim of this paper is to evaluate the probable amplitude
of the transmitted overvoltage through the power transformer
for all possible over-voltages caused by a lightning based on
data distributions given in table 3. Therefore, the results for
the maximum amplitude in all the phases of the transformer’s
secondary side are given in table 4, in terms of mean value, µ
and 2 standard deviations, σ both for the MC method and the
SC method with 9 points.
TABLE IV. RESULTS OF TRANSMITTED OV MAXIMUM AMPLITUDE IN
TERMS OF 2
a1
[kV]
b1
[kV]
c1
[kV]
a2
[kV]
b2
[kV]
c2
[kV]
µ+2σ with SC
16.69 18.51 18.37 17.74 19.31 19.33
µ-2σ with SC
9.46 9.95 11.44 10.76 10.84 12.10
µ+2σ with MC
16.67 18.47 18.47 17.72 19.27 19.43
µ-2σ with MC
9.40 10.03 11.36 10.71 10.91 12.03
Knowing the ranges shown in table 4, it is clear that the
transmitted overvoltage amplitude in all the phases of the
secondary side of the transformer will not exceed 19.43 kV (in
respect to 2 standard deviations), which is below the withstand
voltage of the 6.8 kV distribution grid (withstand voltage is 40
kV [19]).
Finally, the wave shapes of the transmitted over-voltages in
the phase c2 (in which the highest value of the overvoltage is
possible with the highest possibility among phases located on
transformer’s secondary side) are given in figure 7. The first
0.1 ms of time domain responses are given for each input set
of data (by using the 729 deterministic results from SC
points).
Figure 7. Wave shape of the transmitted OV amplitude in the phase c2.
It is interesting to note that the shapes of the transmitted
over-voltages are not significantly influenced by the lightning
impulse parameters. Therefore, we can conclude that the
shapes of the over-voltages are mostly a function of the
frequency behavior of the electric grid components such as the
transformer.
In the future, lightning data from lightning location systems
(LLS) could be used in studies similar to the one presented in
this paper. An experience with LLS has shown that lightning
strokes tend to have a lower crest value than the ones given in
the CIGRE brochure 63 [4]. Additionally, by using the LLS
data the local characteristics of the lightning can also be taken
into account.
V. CONCLUSION
In this paper an application of the state of the art high
frequency black box transformer model is shown. The case
studied in the paper simulates direct lightning striking the 20
kV overhead line and the maximum amplitudes of transmitted
over-voltages through the transformer are observed. Lightning
is a random phenomenon. Nowadays, several tools are
available to take these uncertainties into account when
performing insulation coordination studies. For the purpose of
the paper, a link between MATLAB and EMTP-RV is used to
adjust the simulation parameters in MATLAB, to run EMTP-
RV simulations in parallel and to post-process the simulation
results in MATLAB. Uncertainties regarding the lightning
parameters are considered by taking into account the
probabilistic distributions laws given in the CIGRE brochure
63 [4].
Two different methods are used to take lightning parameters
uncertainties into account: the Monte Carlo method and the
Stochastic Collocation method. Both methods gave similar
results. However, the Stochastic Collocation method requires
fewer simulations than the Monte Carlo method to achieve the
same accuracy for the studied case. This paper shows the
interest of taking into account uncertainties of the lightning
impulse and especially the advantage of the SC method for the
prediction of the induced over-voltages in the power network.
Additional simulation are necessary to confirm the higher
efficiency of the SC method comparing to the conventional
MC method.
ACKNOWLEDGMENT
The authors express their thanks to Siemens Končar Power
Transformers for providing the measurement results which
were used for the development and validation of the black box
transformer model presented in this paper. Furthermore, the
authors would like to thank Mr. Ali El-Akoum and Mr.
Manuel Martinez for allowing them to use their codes as the
link between MATLAB and EMTP-RV.
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