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3
Electrical Safety
UDC 621.316.9 https://doi.Org/10.20998/2074-272X.2021.5.09
D.G. Koliushko, S.S. Rudenko, A.N. Saliba
METHOD OF INTEGRO-DIFFERENTIAL EQUATIONS FOR INTERPRETING THE RESULTS OF VERTICAL ELECTRICAL SOUNDING OF THE SOIL
The paper is devoted to the problem of determining the geoelectric structure of the soil within the procedure of testing the grounding arrangements of existing power plants and substations to the required depth in conditions of dense development. To solve the problem, it was proposed to use the Schlumbergers method, which has a greater sounding depth compared to the Wenner electrode array. The purpose of the work is to develop a mathematical model for interpreting the results of soil sounding by the Schlumberger method in the form of a four-layer geoelectric structure. Methodology. To construct a mathematical model, it is proposed to use the solution of a particular problem about the field of a point current source, which, like the observation point, is located in the first layer of a four-layer soil. Based on this expressions, a system of linear algebraic equations of the 7-th order with respect to the unknown coefficients ai and bi was compiled. On the basis of its analytical solution, an expression for the potential of the electric field was obtained for conducting VES (the point current source and the observation point are located only on the soil surface). Results. Comparison of the results of soil sounding by the Schlumberger installation and the interpretation of its results for the same points shows a sufficient degree of approximation: the maximum relative error does not exceed 9.7 % (for the second point), and the average relative error is 3.6 %. Originality. Based on the obtained expression, a test version of the program was implemented in Visual Basic for Applications to interpret the results of VES by the Schlumberger method. To check the obtained expressions, the interpretation of the VES results was carried out on the territory of a 150 kV substation of one of the mining and processing plants in the city of Kriviy Rih. Practical significance. The developed mathematical model will make it possible to increase the sounding depth, and, consequently, the accuracy of determining the standardized parameters of the grounding arrangements ofpower stations and substations. References 13, figures 3.
Key words: electrical substation, grounding arrangements, vertical electrical sounding, Schlumberger method, method of integro-differential equations.
Робота присвячена проблематиц визначення геоелектричноХ структури грунту в межах випробування заземлювальних пристроив дючих електричних станщй та тдстанцш на необхiдну глибину в умовах щтьног забудови. Для вирШення проблеми запропоновано використати установку Шлюмберже, яка мае бтьшу глибину зондування у порiвняннi з установкою Веннера. За допомогою методiв штегро-диференцшних рiвнянь було отримано аналтичш вирази для ттерпретаци результатiв зондування грунту установкою Шлюмберже у випадку чотиришарового грунту. Для перевiрки отриманих виразiв була проведена ттерпретащя результатiв вертикального електричного зондування на територи тдстанцй 150 кВ одного з гiрничо-збагачувальних комбiнатiв: максимальна вiдносна похибка не перевищуе 9,7 %, а середня - 3,6 %. Бiбл. 13, рис. 3.
Ключовi слова: електрична пщстанщя, заземлювальний пристрш, вертикальне електричне зондування, установка Шлюмберже, метод штегро-диференцшних рiвнянь.
Formulation of the problem. The procedure for determining the soil resistivity as a component of testing of the grounding arrangement (GA) for power stations and substations is regulated in the IEEE standards [1, 2]. In this case, it is recommended to use the Wenner installation for conducting vertical electrical sounding (VES) of the soil. Although, in the general case, the soil is a multilayer structure with many anisotropic inclusions, the expressions to interpret VES curves in the form of a two-layer geoelectric space with plane-parallel interfaces between layers are mainly used. The quality of VES and the interpretation of its results significantly affect the accuracy of calculating the parameters of GA, and, consequently, on the electrical safety of personnel and the reliability of the substation equipment.
VES is carried out by injecting a test current by a generator between current electrodes A and B and measuring the voltage drop at a certain area of the soil surface at potential electrodes M and N. The value of the apparent resistivity is equal to the product of the ratio of the measured voltage and current by the geometric factor of the installation [3]:
Pk = yk, (1)
where U is the voltage drop across the potential electrodes M and N (see Fig. 1); I is the current flowing through the current electrodes A and B; k is the geometrical
coefficient of installation. For the Wenner installation, k = 2nL, where L - distance between the electrodes.
Fig. 1. The principle of conducting VES
The ratio of the spacing lengths of the current and potential electrodes depends on the choice of the VES installation, and the maximum distance between the current electrodes is determined by the required sounding depth. For the Wenner installation, the sounding depth is equal to 1/3 of the spacing length of the current electrodes [3]. Its advantages include:
• poor sensitivity to profile inclusions;
• direct relationship between electrode spacing and sounding depth;
© D.G. Koliushko, S.S. Rudenko, A.N. Saliba
• relatively simple expressions for calculating the apparent resistivity due to the equality of the interelectrode spacing between the current and potential electrodes.
In [4], based on the analysis of the experimental VES curves in the locations of more than 600 energy objects in Ukraine, it was shown that the vast majority of soils at the locations of power stations and substations have a three-layer structure (72,7 %), and another part (19 %) has more than three layers (usually four). Therefore, it is relevant to use interpretation tools with at least four layers, this will cover more than 90 % of energy objects in Ukraine. The authors in [5] based on the solution of the basic problem of the field of a point current source in a four-layer conducting half-space, expressions were obtained for interpreting the results of soil sounding by the Wenner installation. However, carrying out of VES for operating substations, as a rule, has to be performed in conditions of dense industrial or urban development, which does not allow providing the required sounding depth, which is several times greater than the largest diagonal of the GA [6]. Analysis of the literature shows that in the world, as a rule, models for the interpretation of VES curves built on numerical methods [7, 8] or using the method of images model [9] have found application. It was shown in [10] that the calculation error using such models can reach 20 %.
One of the ways to increase the sounding depth while maintaining the spacing of the current electrodes can be the use of a symmetric Schlumberger installation, which is a common case of the Wenner installation. However, a significant drawback is the lack of analytical expressions for interpreting the sounding results.
The purpose of the work is to develop a mathematical model for interpreting the results of soil sounding by the Schlumberger method in the form of a four-layer geoelectric structure.
Research materials. Interpretation of VES results is an inverse problem of electrical prospecting and, in the general case, is an ill-posed problem with many existing solutions that differ from the true one [5]. In this case, the relationship between the measured values of the apparent resistivity and the parameters of this model is expressed by integral equations.
To construct a mathematical model, it is proposed to use the solution of a particular problem about the field of a point current source, which, like the observation point, is located in the first layer of a four-layer soil [5]. When solving the problem, the following assumptions were made: the current does not pass through the boundary of the earth and the atmosphere, the interfaces between the layers are plane-parallel, and within each of them the electrical resistivity pt is uniform. It was assumed that a point current source j is located in the first layer of a four-layer conducting half-space with plane-parallel interfaces (see Fig. 2). The electrical resistances of the first, second, third and fourth layers are denoted by p1, p2, p3 and p4, respectively. The depths of the interfaces of the first and second layers - h1, the second and third - h2, the third and fourth - h3.
The formulation of the problem under consideration consists of the Laplace equation and additional conditions. The electric field of a point current source in a
four-layer medium has axial symmetry, and the potential does not depend on the ^--coordinate; therefore, the Laplace equation in a curvilinear orthogonal cylindrical coordinate system takes the form:
d2 m 1 dm d2 m
- +--— + —j- = 0 .
dz2
r dr 0
dr2
(2)
V777777777777777777777-
5Kj(o,Ti) .Pi(r"Zi) pl
7777777777777 77777
--
V77777777777777777.
P2
777777777777777777
P3
Y77777777777777777,'
P4
Fig. 2. A point current source j located in the first layer of the four-layer structure; P1 (r1, z1) - observation point
Solution (2) is found by the Fourier method for separation of variables [11]:
m(r, z) = J J 0 (Xr ) • ( eX + bi e~X )dX, (3) 0
where at and b are constants determined by soil parameters, coordinates of a point current source and observation point; X is a separation parameter of variables; J0 is a zero-order Bessel function of the first kind.
The form of function (3) is common for all layers of the conducting half-space. However, in each layer, depending on the relative position of the point current source and the observation point, the constants take on their particular values. To find the constants at and b in the first layer, we use the additional conditions:
• with an unlimited increase in the z-coordinate the potential q> tends to zero, therefore
a4 = 0 ; (4)
• in accordance with the principle of electric current continuity at the interface between the i-th and (i+1)-th layers the normal components of the vectors of the electric current density are equal to each other:
_L dm=_L dm ,+1 . (5)
Pi dz Pi+1 dz '
• from the condition of equality of the tangential components of the electric field strength vector at the boundaries of adjacent layers, at the interface between the i-th and (i+1)-th the potentials are equal:
m =m+1; (6)
• the condition on the boundary of the conducting half-space has the form
dm1
dz
= 0.
(7)
z=0
Based on expressions (4) - (7), a system of linear algebraic equations of the 7-th order with respect to the unknown coefficients a1 and b was compiled. On the basis of its analytical solution, an expression for the potential of the electric field was obtained in [5]. Considering that when conducting VES, the point current source and the observation point are located only on the soil surface (i.e. z = 0 and n = 0), this expression will take the form:
F4 (X) =
Fz X) '
(9)
where FZ (X) is
Fz (X) = 1 - K%xe-2)
1 - K
-2)h
3,2e
2 - K
-2xh3
4,3e
+ K 2,1K32e - 2)(h2- M + k 2i1K 43e - 2)( - hi) + + K3,2K43e-2)(- h2) - K21K3,2K43e- 2)(-h2- hi )
V\,l(r ,0) = P
2n
, m
- + 2K2,\ X
Kn
=o{r 2 + (2h- + Hn )2
-2K3,2 £
Kn
n2 + (2h2 + Hn )2
(8)
-2K4,3 £
Kn
=0^ 2 + (2A3 + Hn )2 where is the coefficient of soil heterogeneity equal
to Ki+1i =PP+1—P ; Kn and Hn are the coefficients
' P+1 + Pi-
obtained as a result of the expansion of the function characterizing the multilayer medium; n is the number of the term of the series; m is the number of terms of the series.
The values of Kn and Hn are found by the least squares method [12] when approximating the function F4(X), which characterizes a four-layer soil at X^- œ:
Thus, a basic expression was obtained for the development of a mathematical model for interpreting the VES results in the form of a four-layer geoelectric structure.
To develop a model that allows us to interpret the results obtained using the Schlumberger method, we will use the expression for determining the apparent resistivity (1), the geometric configuration of the installation itself (see Fig. 1) and the expression for determining the potential on the soil surface (8).
Based on the principle of superposition, the voltage at the potential electrodes M and N will be determined as (10), where çAM, ÇbM, 9aN and çBN are the values of potential at electrodes M and N, induced from current electrodes A and B, respectively.
Substituting the expression for potential (8) into (10) and taking into account the symmetry of the Schlumberger installation, the voltage drop will have the form (11).
Taking into account (11), the geometric coefficient of installation and transformations, we obtain expression (1) in the following form (12).
(
U = Pm -Vn =
\ (
Vam,
~Pbm,
U =
¡PL
2n
-AM
^AN
■2£ k
i+1,i
i=1
£-
r=LBM
Kn
Van
Vbn
r=lan
BN
Kn
n=°ij Lam 2 + (2hi + h n )2 n=^ Lan 2 + (2ht + hn )2
Pk =
lamlan
(lam - lan )
L
AM
L
AN
2£ Ki+1,i
i=1
£-
Kn
Kn
n=°yl Lam 2 + (2hi + hn )2 n=^ Lan 2 + (2h, + hn )2
(10)
(11)
(12)
Based on the obtained expression, a test version of the interpretation of the VES results (see Table 1) was
the program was implemented in Visual Basic for carried out on the territory of a 150 kV substation of one
Applications to interpret the results of VES by the of the mining and processing plants in the city of Kriviy
Schlumberger method. To check the obtained expressions, Rih (see Fig. 3).
Table 1
The results of experimental measurements by the Schlumberger method
1
+
+
n
+
r=L
1
1
1
1
LmN2, m 0,1 0,13 0,17 0,22 0,27 0,33 0,4 0,5 0,6 0,8 1
Lab/2,m 0,3 0,39 0,51 0,66 0,81 0,99 1,2 1,5 1,8 2,4 3
Uli, Q 56,01 49,03 34,69 27,91 23,07 19,72 16,39 13,53 11,87 9,491 8,231
pk, Q-m 70,38 80,1 74,11 77,16 78,27 81,78 82,39 85,01 89,5 95,41 103,43
Lmn/2, m 1,3 1,7 2,2 2,7 3,3 4 5 6 8 9
Lab/2,m 3,9 5,1 6,6 8,1 9,9 12 15 18 24 27
U/i, Q 6,834 5,199 4,046 3,342 2,861 2,162 1,539 1,176 0,658 0,537
pk, Q-m 111,64 111,07 111,86 113,39 118,64 108,67 96,7 88,67 66,15 60,73
Comparison of the results of soil sounding by the Schlumberger installation and the interpretation of its results for the same points shows a sufficient degree of
approximation (see Fig. 3): the maximum relative error does not exceed 9.7 % (for the second point), and the average relative error is 3.6 %. The results obtained can
be used to determine the electrical properties of the soil, including the propagation of an electromagnetic wave with a short front, created by a special generator [13] that simulates the lightning current.
s
g 100,00 O
^AM> m
Fig. 3. Interpretation of VES results obtained by the Schlumberger method
Conclusions.
1. Based on the analytical solution of the problem of the field of a point current source located in the first layer of a four-layer geoelectric structure, a mathematical model has been developed for interpreting the results of soil sounding by the Schlumberger installation in the form of a four-layer geoelectric structure.
2. Based on experimental studies carried out at the existing 150 kV substation, the correctness of the developed mathematical model was confirmed. A test computer program has been developed for the interpretation of soil sounding results in an interactive mode.
3. The developed mathematical model will make it possible to increase the sounding depth, and, consequently, the accuracy of determining the standardized parameters of the grounding arrangements of power stations and substations.
Conflict of interest. The authors declare that they have no conflicts of interest.
REFERENCES
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12. Rencher A.C., Christensen W.F. Methods of multivariate analysis, Third Edition. John Wiley & Sons, Inc., 2012. 768 p. doi: https://doi.org/10.1002/9781118391686.
13. O. Rezinkin, M. Rezinkina, A. Danyluk, R. Tomashevskyi, Formation of high-voltage pulses with nanosecond fronts in low-impedance loads. IEEE 2nd Ukraine Conference on Electrical and Computer Engineering (UKRCON), 2019, pp. 464-467 doi: https://doi.org/10.1109/UKRC0N.2019.8880015.
Received 15.06.2021 Accepted 30.08.2021 Published 26.10.2021
D. G. Koliushko1, PhD, Senior Research Scientist,
S.S. Rudenko1, PhD, Senior Research Scientist,
Abdel Nour Saliba2, General Manager, Postgraduate Student,
1 National Technical University «Kharkiv Polytechnic Institute», 2, Kyrpychova Str., Kharkiv, Ukraine, 61002,
e-mail: nio5_molniya@ukr.net (Corresponding author)
2 TMC Group,Beirut, Horsh Tabet, Sin el Fil, Lebanon, e-mail: abdel.nour.saliba@gmail.com
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How to cite this article:
Koliushko D.G., Rudenko S.S., Saliba A.N. Method of integro-differential equations for interpreting the results of vertical electrical sounding of the soil. Electrical Engineering & Electromechanics, 2021, no. 5, pp. 67-70. doi: https://doi.org/10.20998/2074-272X.2021.5.09.
