IEEE TRANSACTIONS ON MAGNETICS, VOL. 51, NO. 3, MARCH 2015
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Fast Calculation of the Green Function of a Point Current Source in a Horizontal Layered Soil With a New Complex Path and Its Application in Grounding System Jun Zou, Xuelong Du, and Chenglong Zhou Department of Electrical Engineering, Tsinghua University, Beijing 100084, China It is very important to efficiently evaluate the Green function of a point current for analyzing a large grounding system. In this paper, a very efficient approach to calculate the Green function of a point current source in a horizontal layered soil with a new complex integration path is presented. As can be found, the highly oscillatory integrand has been changed to be a quickly damped one that can be evaluated using the built-in quadrature routine in MATLAB. The numerical examples show that it is practical to evaluate the parameters of a large ground system within a reasonable computational time with the proposed approach. Index Terms— Complex integration path, Green function, grounding system, moment method.
I. I NTRODUCTION
A
S IS WELL known, a well-designed grounding system is definitely necessary in a modern substation to ensure adequate safety for people and to protect the installation. In past decades, various numerical methods have been utilized to analyze the grounding system with a complex geometry [1]. The method of moment (MoM) was one of the most widely used approaches due to the fact that the grounding system is made up of a set of interconnected cylindrical thin conductors. For the homogeneous soil, the Green function of a point current source has a closed form expression [2]; however, for a horizontal multilayered soil, the Green function, expressed by the form of general Sommerfeld integral (GSI), must be calculated repeatedly, when the MoM is used. Generally speaking, the GSI has a highly oscillatory and slowly damped integrand, which results in numerical obstacles if the direct numerical quadrature is used. In the past, to evaluate the GSI efficiently, the most popular and successful approach has been the so-called complex image method (CIM) [3]. The CIM needs to fit the integrand with the sum of several exponential terms, which is a rather complicated procedure involving the singular value decomposition of matrix. Another numerical difficulty of the CIM is to determine a proper truncation point for fitting the kernel, which might bring the uncertain error for evaluating GSI. II. D EVELOPMENT OF THE P ROPOSED A PPROACH
A. MoM-Based Circuit Model of the Grounding System The grounding system is assumed to be a network made up with a set of interconnected cylindrical thin conductors buried in a horizontal three-layered soil, as shown in Fig. 1. The parameters of the soil model are: the soil conductivities, σ1 , σ2 , and σ3 and the thickness of each layer, h 1 and h 2 . Manuscript received May 19, 2014; revised July 23, 2014; accepted August 18, 2014. Date of current version April 22, 2015. Corresponding author: X. Du (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2014.2350486
Fig. 1.
Grounding system buried in a horizontal multilayered soil.
First of all, the grounding system is divided into a number of straight segments that server as elemental units. Hereafter, in this paper, the so-called thin wire approximation that means the length of each segment is much greater than its cross section diameter is assumed. An interconnected rectangle grid, being a part of the whole grounding system, as shown in Fig. 2(a), is chosen to elaborate the MoM-based circuit model of the grounding system. In each conductor segment, the current is supposed to uniformly drain to the surrounding ground. A current, as an excitation, is injected into the grounding grid at Point A. To model the influence of the resistance of the grid conductors, each segment is split into two parts, and the leaky current is assumed to drain at the middle point of each segment, as shown in Fig. 2(b). With Fig. 2(b) in mind, the grounding system in Fig. 1 can be easily modeled to an equivalent circuit network with unknown leaky currents. Using the conventional nodal voltage analysis (NVA) [4] approach in circuit theory, one can have Uc Jc Ycc Ycg = (1) Ygc Ygg Ug Jg where Y cc , Y cg , Y gc , and Y gg , respectively, are the admittance matrices in the NVA approach, and Uc = [V1 , V2 , V3 , V4 ]T , Ug = [V5 , V6 , V7 , V8 ]T
(2)
Jc = [I1 , 0, 0, 0] , Jg = [I5 , I6 , I7 , I8 ] .
(3)
T
T
In (1)–(3), the subscript g denotes the nodes with the leaky currents, and the subscript c denotes the junction points of
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 51, NO. 3, MARCH 2015
point can be evaluated by an infinite number of point spheres of an infinitesimal diameter dln . The point current contribution for the potential can be evaluated using the local cylindrical coordinate system, (r , z ), as shown in Fig. 3. With these two coordinate systems, the entry of the matrix R can be formulated to be 1 G(r , z )dlm dln (6) Rmn = l m l n lm ln Fig. 2. Model of an interconnected rectangle grid. (a) Grid conductors with uniformly leaky current. (b) Equivalent circuit model of the grid conductors.
where r = (x 2 + (y − Y )2 )1/2 , and z = z . lm and ln , respectively, are the length of the mth and the nth segment. If the mth and the nth segments are placed in any positions and orientations, Rmn in (6) has to be evaluated using the numerical quadrature. The Green function of a point current source in a horizontal layered soil typically has the following form [5]: +∞ F(λ)e−λz J0 (λr )dλ (7) G(r , z ) = 0
r
Fig. 3.
Conductor segment with its global and local coordinate system.
the conductor segments. In accordance with the current field theory [5], the voltages at the middle point of each conductor can be expressed as follows: RJg = Ug
(4)
where R is the matrix that stands for the mutual ground coupling of the conductor segments, and its entry will be determined in the following section. Substituting (4) into (1), one can have Ycc Ycg Uc J = c . (5) Ug 0 Ygc Ygg − R−1 Obviously, the nodal voltages can be solved with (5) without difficulties, and the parameter of grounding system, such as the grounding resistance, the step voltage, and the ground potential rise, can be calculated accordingly. The model described above is similar to the ones in [6] and [7]; however, the concept and the procedure are pretty much simplified. For a large sized grounding system, the total number of the conductor segments will be large as well, and the computational cost for evaluating R−1 is high. In this situation, the iterative solver should be utilized. To accelerate the convergence, a suitable preconditioner of R−1 can be −1 Diag (R11 , R2−1 , . . . , R −1 N N ), where N is the total number of segments of the grounding system. For the sake of conciseness, the detailed iteration procedure is omitted on purpose. B. Calculation of the Green Function With a New Path Fig. 3 is the configuration to calculate the mutual resistive coupling between the mth and the nth segment. To simplify the calculation, the global rectangular coordinate system (x, y, and z) is established, and its origin is at the middle point of the nth segment. The potential at any field
where denotes the distance between the source and the field point, z is the equivalent depth, and J0 is the zeroth-order Bessel function of the first kind; F(λ) reflects the impact of the soil with different structure, which can be calculated using an analytical expression or a recursive formulation. In this paper, the recursive formulation presented in [5]–[8] is used to evaluate the value of F(λ) without needing the explicit expression. Let T = 10/(4r ). When λ > T , the following approximation holds [9]: 2 1 1 √ cos λr − π J0 (λr ) ≈ πr λ 4
2 1 j λr − 14 π Re e . (8) √ = πr λ Splitting (7) into two parts, one can have G(r , z ) = G 1 (r , z ) + G 2 (r , z ) T ≈ F(λ)e−λz J0 (λr )dλ 0 +∞
2 F(λ)e−λz j λr − 1 π 4 e + Re √ dλ. (9) πr λ T The first term of (9), G 1 (r , z ), has a damped integrand with a definite interval [0, T ], which has no numerical difficulties in evaluating it. G 2 (r , z ) is an improper integral with infinite upper limit, and is generally time-consuming if the integration path is along the real axis. The situation will be even worse if the horizontal distance r is larger. Define a new path C, from a to b, as shown in Fig. 4 C : λ = T + j ξ, 0 ≤ ξ < ∞ .
(10)
Within the contour abc, the integrand in (7) is analytic. With this fact, if the point b and c goes to infinity, respectively, the integral along the arc bc can therefore be discarded. In accordance with the residual theorem in complex function theory, it is easy to know that the integral along the segment ab has the same value as the integral along the segment ac.
ZOU et al.: FAST CALCULATION OF THE GREEN FUNCTION OF A POINT CURRENT SOURCE
Fig. 5.
Fig. 4.
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Potential of a point current source buried in a two-layered soil.
New complex path on the λ plane.
Using the new path defined in Fig. 4, one can reformulate the integral to be 1 2 G 2 (r , z ) = Re e−T (z − j r )− j 4 π πr +∞ F(T + j ξ ) − j ξ z −ξr e e × j dξ . √ T + jξ 0 (11) In most cases, for analyzing the grounding system, r z , so the exp(−ξr ’) makes (11) achieve the convergence very quickly even using the quadrature routine. The larger the horizontal distance r is, the faster convergence of (11) can have. If the MATLAB platform is used, the function quadgk, which implements adaptive quadrature based on a Gauss–Kronrod pair (14th- and sixth-order formulas), supports infinite intervals and can handle moderate singularities at the endpoints, is a qualified candidate to calculate (11). It should be pointed out the exp(− j ξ z ) is an oscillatory function. When r < z , the integral along the new path should not be used, and (7) should be integrated along the real axis. This circumstance will be demonstrated in the numerical example in the following section. III. N UMERICAL E XAMPLES A. Validation of the Proposed Approach Using a Point Current Source Buried in a Two-Layered Soil To validate the proposed approach, the potential of a point current source buried in a horizontal two-layered soil, which can be calculated using either the Sommerfeld integral or the sum of the infinite series, is chosen an example. The geometry of the point current source and the coordinate system are shown in Fig. 5. The potential in the first layer can be expressed [5] as ∞ 1 1 √ + F1 (λ)e−λz J0 (λr )dλ V1 (r, z) = 2 + z2 4πσ1 r 0 ∞ +λz + F2 (λ)e J0 (λr )dλ (12) 0
where K = (σ1 − σ2 )/(σ1 + σ2 ), and K e−2λh 1 + e−2λt 1 − K e−2λh 1 K e−2λ(h 1 −t ) + e−2λh 1 F2 (λ) = K . 1 − K e−2λh 1 F1 (λ) =
(13)
Fig. 6. Comparison of the potential of a point current source using two different approaches.
Using Taylor’s expansion in (13), one can have the series expression of (12) V1 (r, z) ∞
1 1 1 n √ + K = 2 2 2 4πσ1 r +z r + (2nh 1 + z)2 n=1 1 1 + + 2 2 2 r + (2nh 1 + 2t + z) r + (2nh 1 − 2t − z)2 1 . (14) + 2 r + (2nh 1 − z)2 Fig. 6 shows the error comparison of the potential in the first layer between the results calculated using the proposed approach with (12) and the ones using (14), respectively, where the results using (14) are thought to be accurate. The necessary parameters are as follows. t = 1.0 m; h 1 = 3 m; σ2 = 1/50 S/m; z = 1.0 m.r and σ1 are two variables to plot the curves in Fig. 6. For calculating (14), an adaptive procedure that the infinite series is truncated till the series sum reaches the predefined error is used. The predefined error, in this example is 10−3 . The results calculated by the proposed approach can reach excellent agreement with the ones using (14). To demonstrate the high efficiency of the proposed approach, the direct numerical quadrature, called DirectInte, is adopted. The entire real axis is divided into a set of subintervals in terms of zero points of the Bessel function J0 (λr ); and the adaptive Gauss–Lobatto quadrature is carried out in each subinterval. The calculation process will be continued till the error of two successive integrations is less than the predefined value, which is set up to be 10−3 . Fig. 7 shows the computing time using the proposed approach and the DirectInte method with the parameters σ1 = 1/300 S/m and σ2 = 1/50 S/m. The DirectInte method has the strength for a short horizontal
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 51, NO. 3, MARCH 2015
TABLE III C OMPARISON OF THE G ROUNDING R ESISTANCE IN D IFFERENT S OILS W ITH S16 G ROUNDING G RID
Fig. 7. Comparison of the computing time using the proposed approach and the DirectInte method. TABLE I PARAMETER OF D IFFERENT S OIL S TRUCTURES
the results in [1] are thought to be accurate, is less than 1%. As a matter of fact, an extensive comparison, including the resistance of grid, the ground potential rise, and so on, has been conducted, and those results are also satisfactory. A large-sized ground grid, 300 × 300 m2 , is used as an example to demonstrate the total computing time, which is less than 1 min with an ordinary personal computer. Most of computing time is cost to calculate the entry of the matrix R, so the proposed approach plays an essential role in this paper. IV. C ONCLUSION
TABLE II C OMPARISON OF THE G ROUNDING R ESISTANCE IN D IFFERENT S OILS W ITH S4 G ROUNDING G RID
The obvious advantage of the proposed approach is that it is a completely general method and does not have to change or fit the kernel like the CIM does. The proposed approach can be generalized to evaluate other Sommerfeld type integral in other applications, for example, evaluating the mutual impedance in the analysis of microwave circuit; or the radiation field of a lightning stroke. The proper integral contour is defined. As pointed out, the analytic property of the integrand must be ensured with the contour. ACKNOWLEDGMENT
distance but suffers with a large one. The reason is as simple as more subintervals are involved when the horizontal distance increases. The computational time of the proposed approach is almost independent with the horizontal distance. As explained earlier, when r z, the proposed approach is capable of saving the computational time, and Fig. 7 just proves that statement, though, in a numerical way. B. Examples of Some Grounding Systems In [1], the parametric analysis of different grounding systems, which are buried in horizontal multilayered soils, has been conducted. For convenience, the notions are the same as those defined in [1]. There are two types of grounding grids, S4 and S16, which means the 20 × 20-m grounding grid with 4 or 16 meshes, respectively. Two set of soil structures, Types C and D, are defined, and its detailed parameters are listed in Table I. The diameter of the grid conductor is 2 cm. In Tables II and III, the resistance of grounding grid are compared between results in this paper and the ones in [1] with different grids and soil structures. As can be found, two results reach agreement very well. The relative error, where
This work was supported by the National Natural Science Foundation of China under Grant 51177087. R EFERENCES [1] F. P. Dawalibi, J. Ma, and R. D. Southey, “Behaviour of grounding systems in multilayer soils: A parametric analysis,” IEEE Trans. Power Del., vol. 9, no. 1, pp. 334–342, Jan. 1994. [2] F. Freschi, M. Mitolo, and M. Tartaglia, “An effective semianalytical method for simulating grounding grids,” IEEE Trans. Ind. Appl., vol. 49, no. 1, pp. 256–263, Jan./Feb. 2013. [3] Y. L. Chow, J. J. Yang, and K. D. Srivastava, “Complex images of a ground electrode in layered soils,” J. Appl. Phys., vol. 71, no. 2, pp. 569–573, 1992. [4] W. N. James and A. R. Susan, Electric Circuits. New York, NY, USA: McGraw-Hill, 2000. [5] J. Wait, Electromagnetic Wave in Stratified Media. New York, NY, USA: Pergamon, 1962. [6] A. F. Otero, J. Cidras, and J. L. del Alamo, “Frequency-dependent grounding system calculation by means of a conventional nodal analysis technique,” IEEE Trans. Power Del., vol. 14, no. 3, pp. 873–878, Jul. 1999. [7] J. Yuan, H. Yang, L. Zhang, X. Cui, and X. Ma, “Simulation of substation grounding grids with unequal-potential,” IEEE Trans. Magn., vol. 36, no. 4, pp. 1468–1471, Jul. 2000. [8] Q. Z. Ma, “The boundary element method for 3-D dc resistivity modeling in layered earth,” Geophysics, vol. 67, no. 2, pp. 610–617, 2002. [9] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards (Applied Mathematics). New York, NY, USA: Dover, Jan. 1965.
