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Artículo Científico / Scientific Paper |
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https://doi.org/10.17163/ings.n24.2020.01 |
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pISSN: 1390-650X / eISSN: 1390-860X |
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RESISTANCE EFFECT ESTIMATION IN THE SHORT CIRCUIT
CURRENT THROUGH A SENSITIVITY ANALYSIS |
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ESTIMACIÓN DEL EFECTO DE LA RESISTENCIA EN LA CORRIENTE DE CORTOCIRCUITO MEDIANTE UN ANÁLISIS DE SENSIBILIDAD |
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Moronta R. José A.1,*, Rocco Claudio M.2 |
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Abstract |
Resumen |
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The assessment of the current in an electrical power system (EPS)
after a fault, is generally termed short circuit
analysis. The magnitude of those currents is used
for dimensioning the protection equipment of the EPS. Short-circuit analysis
assumes that the electrical resistances of the components can be neglected,
since they do not significantly affect the magnitude of the short-circuit
currents. This work quantifies the effect of the electrical resistance of the
elements of the EPS on the magnitude of the short-circuit current, by means
of sensitivity (SA) and uncertainty (UA) analyses. The SA is
based on the variance decomposition of an output variable, and can
quantify the main effects (importance) and the interactions of the variables
considered. On the other hand, The UA allows assessing how the variations in
the variables considered affect the output. The proposed approach is illustrated on two networks from the literature,
considering three-phase and single-phase faults. The results of such proposed
approach numerically show that the effects due to taking into account the
electrical resistance are indeed negligible, when compared to the rest of
variables considered in the short-circuit analysis. This result coincide with
the assumptions reported in the literature for the calculation of the fault
currents. |
El cálculo de los valores de corriente que fluyen en un sistema eléctrico de potencia (SEP) posterior a una falla, se denomina análisis de falla o de cortocircuito. Los valores de la corriente de cortocircuito son empleados para el dimensionamiento de los equipos de protección del SEP. Los análisis de cortocircuito tienen como una de sus premisas despreciar la resistencia eléctrica de los elementos del sistema, pues esta no afecta en mayor medida las magnitudes de las corrientes de cortocircuito. En este trabajo se propone cuantificar el efecto de la resistencia eléctrica de los elementos del SEP en la magnitud de la corriente de cortocircuito, mediante un análisis de sensibilidad (AS)e incertidumbre (AI). El AS se basa en la descomposición de la varianza de una variable de salida y puede cuantificar los efectos principales (importancia) y las interacciones de las variables consideradas. Por otro lado, el AI permite evaluar cómo las variaciones en las variables consideradas afectan la salida. La propuesta se ilustra sobre dos redes de la literatura, considerando fallas trifásica y monofásica. El resultado de nuestra propuesta muestra numéricamente que los efectos debidos a considerar la resistencia eléctrica son de hecho insignificantes, en comparación con el resto de los factores que intervienen en el análisis de cortocircuito. El resultado coincide con las premisas de cálculo de las corrientes de falla supuestas en la literatura. |
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Keywords: Short circuit current, electrical resistance, sensitivity analysis. |
Palabras clave: corriente de
cortocircuito, resistencia eléctrica, análisis de sensibilidad.
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1,*Departamento de Tecnología Industrial, Universidad Simón Bolívar, Venezuela. Corresponding author ✉: jmoronta@usb.ve. 2Facultad de
Ingeniería, Universidad Central de Venezuela, Venezuela.
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Received: 14-01-2020, accepted after review: 08-04-2020 Suggested citation: Moronta R. José A. and
Rocco Claudio M. (2020). «Resistance effect estimation in the short circuit
current through a sensitivity analysis». Ingenius.
N.◦ 24, (july-december). pp. 9-16. doi: https://doi.org/10.17163/
ings.n24.2020.01. |
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1. Introduction The short
circuit studies are analyses that are used to determine
the magnitude of the electrical currents that go through the electrical power
systems (EPS), during a fault. Afterwards, such magnitudes are
utilized to specify or validate the characteristics of the components
of the system, such as breakers, buses, among others [1]. Short circuit studies in the EPS generally assume that the electrical
resistance of the elements of the system is negligible, and exclusively
consider their electrical reactance for calculating the magnitude of the
short circuit current. According to [2], for most calculations of short
circuit currents in medium and high voltage, and in some cases in low
voltage, when the reactances are «much greater»
than the resistances, is sufficiently accurate and simpler neglecting the
resistances and considering only the reactances.
Note that the rule is not specific, only suggests when the reactance is much
greater than the resistance. The same assumption for calculating the short circuit currents is made in the standard [3], where it is indicated that
the calculation is much simpler, but with a loss of accuracy if the
resistances are neglected, when the reactance/resistance (X/R) ratio is
greater than 3.33. The literature related to the analyses of EPS suggest similar
procedures, regarding the possibility of neglecting the resistance of the
elements. For example, it is indicated in [1] that
it is possible to neglect the resistances in the fault study, because «it is
not likely that they significantly influence in the level of the fault
current». A group of assumptions is presented in [4] for short circuit
calculation, where it is suggested to neglect all the resistances of the
elements (generators, transformers and transmission lines) to simplify the
calculation. Other works [5, 6], establish 5 % as a maximum error in the value of
the short circuit current, if the resistance of the elements of the system is
neglected. This suggestion is derived from the general expression to
determine the fault current, see equation (1):
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Where: Icc = Short circuit
current (A) Vfalla = Pre-fault
voltage (V) Rfalla = Resistance at
the fault point (Ω) Xfalla = Reactance at
the fault point (Ω) Where Rfault and Xfault correspond to the equivalent Thevenin impedance at the fault point. When the ratio
(X/R) is greater than 4, the error made by neglecting the resistance is
smaller than 4 %. This is valid except for distribution or industrial systems
where this ratio is smaller than 4 [6]. In power systems, the value of the resistance of the elements is
usually very small compared to the value of their reactance. This
consideration is the reason in which standards and authors are
based for neglecting the resistances and their influence in the
determination of the fault currents. In real power systems, the ratio X/R
between the reactance and the resistance at the fault point is usually of an
order between 15 and 120 times [7]. The references agree that the electrical resistance may
be neglected in short circuit analyses, but there is no consensus
regarding their effect in the magnitude of the short circuit currents. In this work, the sensitivity analysis (SA) is used
to quantify the effect of varying the resistance in the value of the short
circuit current, for two networks in the literature. In addition, it is
defined the relationship of these variations with the remaining parameters
that enable determining the short circuit current (voltage and electrical
reactance of the elements). The analysis is based on the use of the theory of
SA and UA, and as a result of these analyses the same conclusion presented in
the literature is directly reached, but from a different perspective. The structure of the work is as follows: the first section presents
some fundamental definitions associated to the sensitivity analysis; the
second describes the procedure for determining the uncertainty associated to
the short circuit current and the test electrical power systems; the third
presents and discusses the results obtained and, finally, the fourth presents
conclusions and future works. 1.1. Sensitivity
and uncertainty analyses According to
[8], the uncertainty analysis (UA) is defined as the study of the amount of
uncertainty contributed to the output of a model, by the different sources of
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uncertainty in the input.
On the other hand, the sensitivity analysis (SA) evaluates the importance of
the input variables of a model. Such importance is measured
as a function of how much variability in the output of the model is due to
the variability in the input variables. In this case, the uncertainty in the
input variables is modeled through distribution
functions with known parameters. According to [8], the steps to carry out the SA/UA approach are
defined as: • Establish the objective
of analysis, and accordingly define the form of the outputs of the model. • Decide which
input factors will be included in the analysis. • Choose a
probability distribution function for each of the input factors. • Choose a
method for SA, according to the characteristics of the problem under study. • Generate the
sample of the input factors. The sample is generated
according to the specifications of the known parameters and the selected size
of the sample. • Evaluate the
sample generated in the model and produce the corresponding outputs, which
contain the values according to the form specified in step 1. • Analyze the
outputs of the model, determine the sensitivity indices (importance) and
establish the conclusions. The methods of SA can be classified according
to the output of their measures: quantitative or qualitative, local (they do
not allow varying all factors simultaneously) or global (they allow varying
all factors simultaneously) and dependent or independent of the model [9]. Given a model Y = F(X1, X2, X3, . . .
, Xn) where Y is its output and Xi
represent the input variables modeled as random variables (i.e., its
uncertainty is modeled as a probability density function (pdf)), the variance
V (Y) of the output Y may be described as in equation (2) [8]:
Where: Vi = V (E(Y lXi))
is the main effect (or of first order) due to xi Vij = V (E(Y lXi ,
Xj)) − Vi − Vj is the second order effect due to the
interaction between xi and xj,
and so forth. |
The main sensitivity
(Si) and total (STi) effects,
may be defined as given in equations (3) and (4), respectively, according to
[8]:
Where: X−i = (x1, x2,
. . . xi−1, xi+1, ..., xk)
and E−i(Y | Xi) is the
expected value of Y conditioned to xi, and, therefore, is only a
function of xi. The main index Si is the fraction of the variance V (Y) of
the output that can be attributed to xi
only, while STi corresponds to the
fraction of V (Y) that can be attributed to xi, including all its
interactions with the other input variables. The main index Si is the measure employed to determine the
input variables that mainly affect the output uncertainty, while STi is utilized to
identify the subset of not influential input variables, i.e., those variables
that can be fixed at any value in their uncertainty range, and they do not
significantly affect the variance of the output [10]. The estimates of Si y STi
are approximated: 1) assuming (statistical) independence between the input variables;
2) using particular sampling techniques to generate samples of the input
variables; and 3) evaluating the group of samples obtained in 2) from the
model under study [8]. There are different techniques for sensitivity analysis based on the
decomposition of the variance; several of these techniques are
mentioned in [10]. These techniques differ with respect to their
computational complexity, as well as in the effects that they evaluate (main
and/or total). Among these techniques, it should be mentioned: Sobol [11] that enables evaluating the main and total
effects, and EFAST (Extended Fourier Amplitude Sensitivity Test) [12], an
extension of FAST (Fourier Amplitude Sensitivity Test) [13], that also
evaluates the main and total effects (Si and STi),
but with less computational complexity than the Sobol
method. |
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2. Methodology The following approach is proposed to estimate the effect of the
electrical resistance of the elements (input variables of the model) on the
short circuit current (output of the model): A uniform distribution U[0 − 1, 2 ×
valor base] is assumed for the input variables (pre-fault voltage, resistance
and impedance of the elements of the power system). The distribution is
asymmetrical and enables quantifying the effect of neglecting (values close
to zero) the resistance of the elements of the power system. After the evaluation of the described procedure (SA), a Monte Carlo
simulation [14] is carried out considering only the
variables of interest (electrical resistance of the elements). The Monte
Carlo simulation is a method employed to evaluate the propagation of
uncertainty through the generation of random variables. In this way, the
propagation of the uncertainty is quantified at the
output of the model, i.e., the variation of the magnitude of the short
circuit current is quantified. The short circuit calculation and the SA/UA were
carried out in the R free software [15]; specifically, the algorithms
from the Sensitivity library were utilized for the SA. 2.1. Test electrical
systems 2.1.1. Test
power system 1 (TPS1) The power system
employed [16] is a nonmeshed network with two
sources, as shown in Figure 1. It is constituted by an
external system, two transmission lines, one transformer and one generator.
It is assumed a solid three-phase to ground fault in
k3. Figure 1. Radial test power system (TPS1) The elements of the radial test power system 1 (TPS1) are modeled; the
values are shown in Table 1. These values represent the input variables of
the model of SA. |
Table 1. Impedance of the elements of the test power system
1 (TPS1) 2.1.2. Test
power system 2 (TPS2) The meshed
electrical network taken from [16], for which it is assumed a solid one-phase
to ground fault in k1 (see Figure 2). The values of the elements are shown in Table 2, which are the input factors for the
sensitivity analysis. Figure 2. Meshed test power system (TPS2) Since it is a one-phase fault, it is solved using the method of the
sequence networks [1], and the elements should be modeled with their
corresponding values of positive, negative and zero sequence. In order words,
the parameters that take part in the SA are increased by 3. For example, for
a line there will be a resistance of positive, negative and zero sequence. |
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Table 2. Impedance of the elements of the test power system
2 (TPS2) |
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3. Results 3.1. Test power
system 1 (TPS1) Figure 3 shows
the sensitivity indices of first order (Si) (white part of the
bars) and total (STi) (complete bar,
white and gray parts) for the TPS1. The variables that most affect the short
circuit current are, in order of importance: the pre-fault voltage (S1
= 0.456), the reactance of the external system (S3 = 0.252) and
the reactance of line 1 (S5 = 0.108). The remaining factors,
including the resistance of the elements, have very small values of
importance STi, and consequently their
effects may be considered negligible. Figure 3. Main (Si) and total (STi)
effects of the input variables in the short circuit current of the TPS1 |
As it was mentioned, the pre-fault voltage has
a very high importance (0.456); due to this, it is subsequently fixed as a
constant in the model, in order to only evaluate the effects of the reactances and resistances of the elements; likewise very
small effects were obtained for the resistances. For comparison purposes, all the main effects of the resistances and
the main effects of the less important reactances were grouped (see Figure 4), keeping constant the
pre-fault voltage. The sum of the main effect of the resistances is
negligible compared to the main effect of the reactances,
in this system and, for this particular fault, the
most important variable is the reactance of the external system. Figure 4. Main effects of the reactances and sum of
the main effects of the resistances. |
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After evaluating the SA, and only considering the resistances of the
elements, the evaluation of the Monte Carlo method [14] is
carried out, to obtain the approximate probability distribution of the
short circuit current for 5000 evaluations. The short circuit current for the
values of Table 1 is 7196 A, which corresponds to the normalized value of 1
unit, in the approximate histogram of the short circuit current presented in
Figure 5. The minimum and maximum values obtained for the short circuit
current were 0.995 and 1.031, respectively. The average value of this
distribution is 1.022, and the shape of the distribution is asymmetrical,
with a bias to the upper end. Figure 5. Approximate histogram of the short circuit current
(TPS1) The literature
refers to the X/R ratio to neglect the electrical resistance in the short
circuit calculations. For this reason, Figure 6 shows the short circuit
current normalized for the values of X/R obtained in the Monte Carlo
simulation. The values of the X/R ratio vary from approximately 4 to 400;
considering this wide range, the normalized short circuit current does not
exhibit variations achieving at least 4 %. Figure 6. Short circuit current normalized for the values of X/R (TPS1). |
3.2. Test power
system 2 (TPS2) The results obtained
for the meshed power system TPS2 are similar to those obtained for the TPS1.
The pre-fault voltage turns out to be the most important variable (see Figure
7). In this system, there is more uncertainty of superior order associated to
the interaction of variables (ST i - Si), which is due to the fact that the system is meshed, and various
equivalent impedances (successive sums and products) should be calculated to
obtain the short circuit equivalent impedance. Figure 7. Main (Si) and total (ST i)
effects of the input variables in the short circuit current of the TPS2. The numbers of the variables (x-axis) of Figure 7,
are in accordance with the numbering of Table 2. Note that the variable 37
(zero sequence reactance of transformer 2) appears as the second most
important variable, even though with a very small contribution. Figure 8 shows a comparison of the sum of the total indices of the
resistances and of the reactances of the system
TPS2, considering constant the pre-fault voltage; the percentage represented
by the index of the sum of the resistances is slightly smaller than 7 %. Figure 8. Sum of the main effects of the resistances and reactances
of the TPS2 |
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Figure 9 shows the approximate histogram for the short circuit
current, obtained using the Monte Carlo technique disturbing only the
resistances of the elements of the system TPS2. For the values in Table 2 this current is 13679 A, which corresponds to the normalized
value of 1 unit, in the histogram of Figure 9. The minimum and maximum values
obtained for the short circuit current were 0.9999 and 1.0037, respectively.
Note that carrying out variations in the resistances, the effect on the short
circuit current is negligible. This distribution is more symmetrical than the
one in the previous example, and has an average value of 1.002. Figure 9. Approximate histogram of the short circuit current
(TPS2) Figure 10 shows the short circuit current normalized for the values of
X/R obtained in the Monte Carlo simulation, for the TPS2 system. In this case the variation is much smaller, since the normalized
short circuit current does not exhibit variations achieving 1 %. Figure 10. Short circuit current normalized for the values of
X/R (TPS2) |
4. Conclusions In this work, it
is estimated the uncertainty of the short circuit current due to the
electrical resistance of the elements, through an approach of sensitivity and
uncertainty. This analysis can be applied to any
other power system with any location and/or type of fault. For the two cases under consideration, the results are in accordance
with what is suggested by different authors: it is
possible to neglect the resistance of the elements, whenever their reactance
is much greater than their resistance (X/R ratio). The variation of the
magnitude of the short circuit current in the two cases evaluated does not
exceed 4 %, which coincides with the value of 4-5 % reported in some
consulted works [5, 6]. This analysis not only considers the uncertainty in the short circuit
current, due to the uncertainty in the resistances and reactances,
but also enables quantifying the total variation, i.e., the percentage due to
the uncertainty of the resistances of the components of the EPS. In these two
cases, the effect of the resistance of the elements is approximately 7 % (in
the test power system 2 (TPS2)); the rest is associated to the reactances, if the value of the pre-fault voltage is
considered constant, for the sensitivity analysis carried out. The results obtained in these two systems, could be
extrapolated to real power systems in medium or high voltage, because
in such systems the reactance is usually much greater than the resistance (X≫R). As a
future work, it is intended to establish ratios
and/or critical values between the reactance/resistance factor (X/R) at the
fault point of the system, and the main indices (Si) of the uncertainty of
the short circuit current. References
[1]
J. Grainger and W. Stevenson, Análisis de Sistemas
de Potencia. Mc Graw
Hill, 1996. [Online]. Available: https://bit.ly/3bsZqDF [2] IEEE, IEEE
Recommended Practice for Electric Power Distribution for Industrial Plants, 1994.
[Online]. Available: https: //doi.org/10.1109/IEEESTD.1994.121642 [3] IEC, IEC
Short-circuit currents in three-phase A. C. systems: Part 0: Calculation of
currents / Part 4: |
|
Examples for the calculation of short-circuit currents,, 2016. [Online]. Available: https://bit.ly/2WUfBEF
[4] J. Glover and M. Sarma, Sistemas de potencia: análisis y diseño. International Thomson Editores, 2003. [Online]. Available: https://bit.ly/2WOIW3s
[5]
R. Gil Bernal, Estudios en sistema de potencia, 2000. [Online]. Available: https://bit.ly/2Z27ZD4 [6]
F. M. González-Longatt, Cortocircuito simétrico,
2007. [Online]. Available: https://bit.ly/3dGVpgn [7] IEEE, IEEE Application
Guide for AC HighVoltage Circuit Breakers > 1000
Vac Rated on a Symmetrical Current Basis, 2017.
[Online]. Available: https://doi.org/10.1109/IEEESTD.2017. 7906465 [8] A. Saltelli,
S. Tarantola, F. Campolongo,
and M. Ratto, Sensitivity analysis in practice.
John Wiley and Sons, Ltd, 2013. [Online]. Available: https://bit.ly/2LD6oM5 [9] V. Schwieger,
“Variance-based sensitivity analysis for model evaluation in engineering
surveys,” INGEO 2004 and FIG Regional central and Eastern European conference
on engineering surveying, Bratislava, Slovakia, 01 2004. [Online]. Available:
https://bit.ly/3cw8IQy [10] S. Tarantola, “Variance-based methods for sensitivity analysis,” Int. Conf. on Sensitivity Analysis of Model Output. Joint Research Center of the European Commission, 2002. [Online]. Available: https://bit.ly/2y3Rzim |
[11] I. M. Sobol, “Sensitivity estimates for nonlinear mathematical
models,” Mathematica modelling and computational experiments, vol. 1, no. 4,
pp. 407–414, 1993. [Online]. Available: https://bit.ly/2Lq0M7F [12] A. Saltelli, S. Tarantola, and K.
P. S. Chan, “A quantitative model-independent method for global sensitivity
analysis of model output,” Technometrics, vol. 41,
no. 1, pp. 39–56, 2012. [Online]. Available: https://bit.ly/2Z28H2P [13] R. I. Cukier, C. M. Fortuin, K. E.
Shuler, A. G. Petschek, and J. H. Schaibly, “Study of the sensitivity of coupled reaction
systems to uncertainties in rate coefficients. i
theory,” The Journal of Chemical Physics, vol. 59, no. 8, pp. 3873–3878,
1973. [Online]. Available: https://doi.org/10.1063/1.1680571 [14] M. G.
Morgan and M. Henrion, “Uncertainty: A guide to
dealing with uncertainty in quantitative risk and policy analysis,” 1990.
[Online]. Available: https://bit.ly/2YZpJ1D [15] The R
Development Core Team, R: A language and environment for statistical
computing, 2013. [Online]. Available: https://bit.ly/3dMHNQV [16] M. Eremia and M. Shahidehpour, Handbook of Electrical Power System Dynamics: Modeling, Stability, and Control. John Wiley and Sons, 2013. [Online]. Available: http://doi.org/10.1002/9781118516072
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