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Topological measurement and characterization of substation grounding grids based on derivative method
Electrical Power and Energy Systems 63 (2014) 158–164
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Topological measurement and characterization of substation grounding grids based on derivative method Li Chunli a,1, He Wei a, Yao Degui b, Yang Fan a, Kou Xiaokuo b, Wang Xiaoyu a,⇑ a b
State Key Laboratory of Power Transmission Equipment & System Security and New Technology, The Electrical Engineering College, Chongqing University, Chongqing 400044, China State Grid Henan Electric Power Research Institute, Zhengzhou 450052, China
a r t i c l e
i n f o
Article history: Received 25 June 2013 Received in revised form 19 May 2014 Accepted 20 May 2014
Keywords: Grounding grids Derivative method Topology Magnetic field
a b s t r a c t Derivative method, which avoids pathological solution of the magnetic field inverse problem, was proposed to solve the problem of drawing loss or unknown of grounding grids. First, a shape function was introduced to describe the distribution of magnetic field which is perpendicular to the surface of grounding grids. Since its odd-order derivatives contain main peaks, the 1st-, 3rd- and 5th-order derivatives were chosen to solve the topology of substation grounding grids and measure the branch current of the grid by determining the position and value of main peak, respectively. Numerical example and experimental result show that the 1st- and 3rd-order derivative method works well with low error and less computation. Ó 2014 Elsevier Ltd. All rights reserved.
Introduction The main objectives of grounding systems are to guarantee personal safety, equipment protection and power supply continuity [1]. A grounding system comprises all the interconnected grounding facilities in a specific area, but its main element is the grounding grids. In general, most grounding grids of electrical substations consist of a mesh of interconnected cylindrical conductors, which are horizontally buried and supplemented by vertically thrusting ground rods in certain places of the substation [2]. Typically the conductors made of bare copper or iron are buried 0.3–2 m deep underground. The meshes are usually spaced 3–7 m apart and the ratio of the sides of a mesh is 1:l–1:3 [3]. The electromagnetic field theory (EMF) and the electric circuit theory (with lumped or distributed parameters) are both widely used to diagnose the corrosion status of the grounding grids in a power substation [4]. The electric circuit theory establishes and solves the diagnosis equation that describes the nonlinear relationship between branch resistance and node voltage based on the application of Tellegen’s Theorem [5,6]. However, the method needs to know the topology of grounding grids. The electromagnetic field theory, which was proposed by Dawalibi [7], measures the surface electromagnetic field and establishes the magnetic field
⇑ Corresponding author. Tel.: +86 15823311160. E-mail addresses: [email protected] (L. Chunli), wangxiaoyucqu@gmail. com (W. Xiaoyu). 1 Tel.: +86 15123856702. http://dx.doi.org/10.1016/j.ijepes.2014.05.046 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
inverse equations to judge the topology and corrosion status of the grounding grids [8,9]. However, the solution to the inverse problem usually is not unique and stable, the solution process is complex. In geophysical prospecting, the topology and burial depth of the target can be detected by the resistivity prospecting [10], electrical prospecting [11], seismic refraction tomography [12], gravity and magnetic prospecting [13], gamma radiometric prospecting [14], electromagnetic prospecting Hedley et al. [15]. The geophysical prospecting methods are mainly for large size and deeply buried targets, for example, the faults shear zones, iron ore deposit, buried cavity, water table, and have shortcoming to detect the small size (cross section less than 1 cm 10 cm) and shallow buried (0.3–2 m deep underground) branch conductors of grounding grids. In the paper, a method is developed for calculating the topology and branch current of grounding grids and avoids pathological solution of the magnetic field inverse problem. First, the theoretical analysis of the method is discussed. Then a numerical example is used to test the viability of the method. Finally experimental result is used to test the reliability and precision of the method by a small grounding grid model in the lab. Modeling of single current-carrying conductor by using derivative method Most grounding grids are composed by a lot of branches in finite length. We can analyze the characterization of every single branch, and draw the topology of grounding grids by measuring
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Fig. 1. The magnetic flux density caused by a finite-length current filament on the x axis is B = lI/4pq(sin u1 + sin u2)e/; the ground surface is at z = h plane. Fig. 3. Grid current-carrying model. The grounding grid model depicts the topology, branch length, point position, branch name, branch current and current direction of the grid, then the model is applied to test the viability of the Derivative method.
The application of derivation method in magnetic field distribution analysis
ð3Þ ð5Þ Fig. 2. Graph of Bð1Þ z ðyÞ, Bz ðyÞ and Bz ðyÞ.
the position of the each branch. The process of derivation method is introduced by analyzing the current-carrying conductor in finite length. Derivative method The derivative method (or differential technique) is mainly used to improve the resolution ratio of seismic records, and compensate the frequency loss and amplitude loss of seismic waves during propagation [16,17]. Let f(x) be a differentiable function, and let f(n)(x) be its n-order derivative. n
F ðnÞ ðxÞ ¼ ðjxÞ FðxÞ with n 2 N
ð1Þ
where
FðxÞ ¼
Z
1
f ðxÞejxx dx;
F ðnÞ ðxÞ ¼
1
Z
1
f ðnÞ ðxÞejxx dx
ð2Þ
1
In Eq. (1), the n-order derivative of the function is equivalent to filtering it with a filter characteristic (jx)n.
As shown in Fig. 1, a current carrying conductor MN in length L is buried in the uniform monolayer soil with permeability l. The conductor is on the x-axis. The ground surface is at z = h plane and the current I along the positive x-axis flows through the conductor. For point P at position (0, y0, h), its vertical distance from the conductor is q. The angles between segment OP with the z-axis, segment NP, and segment MP are h, u1 and u2, respectively. The length of OM and ON is L1 and L2, respectively. The conductor’s leakage current in the soil is ignored. Based on Biot–Savart law, the magnetic flux intensity B at point P is
B¼
lI ðsin u1 þ sin u2 Þe/ 4p q
ð3Þ
At the point P the unit vector e/ is in the direction perpendicular to the eq. Since q2 = h2 + y2, the magnetic flux density Bz in the ez direction is
Bz ðyÞ ¼
lI y ðsin u1 þ sin u2 Þ 4p h2 þ y2
L1 And sin u1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 2
ð4Þ
L2 sin u2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2 2
h þy2 þL1
h þy2 þL2
Magnetic field distribution of the single current-carrying conductor in the ez direction can be described by Eq. (4), called the shape function. The magnetic field distribution of the grounding grids is equivalent to the superposition of the shape functions of each current-carrying branch. Because the higher order derivative of Eq. (4) has a more complex expression, the odd-order derivatives contain main peaks, the
Table 1 The comparison table of the three shape functions’ graph characteristics. Shape function
Graph characteristics
Widess Resolution
Influence sphere (m) (1%)
Main peak width (m)
Side peak width (m)
Total number of peaks
19.72
2
8.86
3
Bzð3Þ ðyÞ
4.052
0.8284
1.5858
3
2.0308
Bzð5Þ ðyÞ
3.8
0.2679
0.7321
5
2.5699
Bzð1Þ ðyÞ
Annotation: I = 1 A; h = 1 m.
1.2796
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Fig. 4. Graph of (a) zero-order, (b) 1st-order, (c) 3rd-order, and (d) 5th-order field derivative respectively.
Fig. 5. Contour map of (a) zero-order, (b) 1st-order, (c) 3rd-order, and (d) 5th-order field derivative respectively.
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1st-, 3rd- and 5th-order derivatives are employed while ignoring P2 0 the o i¼1 ðsin ui Þ :
Bzð1Þ ðyÞ
lI h2 y2 ðsin u1 þ sin u2 Þ 4p ðh2 þ y2 Þ2
Bzð3Þ ðyÞ
3lI h y4 þ 6y2 h ðsin u1 þ sin u2 Þ 4 2 2p ðh þ y2 Þ
4
2
6
Bzð5Þ ðyÞ
ð5Þ
2
4
30lI h y6 þ 15y4 h 15h y2
p
2
ðh þ y2 Þ
ð6Þ
6
ðsin u1 þ sin u2 Þ
ð7Þ
Simplify the model and let L ? 1, then sin u1 + sin u2 ? 2. Set ð3Þ ð5Þ h = 1 m and I = 1 A, the characteristics of Bð1Þ z ðyÞ, Bz ðyÞ and Bz ðyÞ are shown in (Fig. 2). Definition that, the influence sphere is the range of the maximum distance between the two points, where the value is 1% of the extreme value of the main peak; The main peak width is the width between the two zero-value points of the main peak; The side peak width is the width between the two zero-value (or 1% of the extreme value of the main peak) points of the side peak near the main peak. The Widess Resolution P (Eq. (8)) is the ratio 2 between the extreme value energy bM of the main peak and the total energy E of the shape function [18,19]. 2
P¼
bM E
Among E ¼
Z
1
2
b ðyÞdy
ð8Þ
1
Table 1 shows that, while the 1st-, 3rd- and 5th-order derivatives of Bz(y) are consecutively obtained, the influence sphere, main peak width and side peak width will decrease, especially the influð1Þ ence sphere of Bð3Þ z ðyÞ decreases to 20.55% of that of Bz ðyÞ, the main ð3Þ peak width of Bz ðyÞ becomes 20.44% of its influence sphere. The number of peaks shown for each field derivative in Fig. 2 are 3,3 and 5 respectively. The Widess Resolution P is gradually improved and the ability of signal identification is enhanced. According to Ref. [3], the meshes are usually spaced 3–7 m apart. ð3Þ ð5Þ While the half of the influence sphere of Bð1Þ z ðyÞ, Bz ðyÞ and Bz ðyÞ is
ð3Þ less than the grid space, the mutual influence of Bð1Þ z ðyÞ, Bz ðyÞ and ð5Þ Bz ðyÞ of two parallel branches can be ignored, respectively. According to Eqs. (5)–(7), the position of the main peaks of ð3Þ ð5Þ Bð1Þ z ðyÞ, Bz ðyÞ, Bz ðyÞ and the position of the current-carrying conductor are the same and at y = 0, respectively. Therefore, the position of grounding grid branch in the measurement area can be determined by calculating the position of the main peak of ð3Þ ð5Þ Bð1Þ z ðyÞ, Bz ðyÞ or Bz ðyÞ. The length L of grounding grid branch can be determined in the same way. The distance between the main peak and the side peak of Bð1Þ z ðyÞ, ð5Þ Bð3Þ ðyÞ and B ðyÞ is L , L , L , respectively: z1 z3 z5 z z
Lz1 1:732 h
ð9Þ
Lz3 0:7265 h
ð10Þ
Lz5 0:4816 h
ð11Þ
so the burial depth h can be calculated by solving Lz1, Lz3 or Lz5. According to Eq. (4), the limits of the 1st-, 3rd- and 5th-order derivatives at y = 0 point are employed while not ignoring the P 0 2 o i¼1 ðsin ui Þ :
lim Bð1Þ z ðyÞ ¼ I y!0
2 X i¼1
lim Bð3Þ z ðyÞ ¼ I y!0
lim Bð5Þ z ðyÞ ¼ I y!0
lLi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Ikz1 2 4ph ðL2i þ h Þ
2 X i¼1
2
ð12Þ
2
3lLi ð2L2i þ 3h Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Ikz3 4 2 2 4ph ðL2i þ h Þ ðL2i þ h Þ
2 4 2 X 15lLi ð8L4i þ 20L2i h þ 15h Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Ikz5 6 2 2 2 2 i¼1 4ph ðL þ h Þ ðL2i þ h Þ i
ð13Þ
ð14Þ
limy!0 Bzð1Þ ðyÞ, limy!0 Bzð3Þ ðyÞ and limy!0 Bð5Þ z ðyÞ mean the value of the main peak of Bzð1Þ ðyÞ, Bzð3Þ ðyÞ and Bð5Þ z ðyÞ, respectively, that F i ¼ limy!0 BzðiÞ ðyÞði ¼ 1; 3; 5Þ.
F i ¼ Ikzi ðL1 ; L2 Þ L ¼ L1 þ L2
i ¼ 1; 3; 5
Fig. 6. Longitudinal cross section of (a) zero-order, (b) 1st-order, (c) 3rd-order, and (d) 5th-order field derivative respectively.
ð15Þ
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L. Chunli et al. / Electrical Power and Energy Systems 63 (2014) 158–164
From what we have discussed above, the value of the branch current I can be determined by calculating the value of the main ð3Þ ð5Þ peak of Bð1Þ z ðyÞ, Bz ðyÞ , Bz ðyÞ and the length L. Numerical example The numerical example features a 10 m 10 m square grid (copper conductors with 5 mm radius) with 5 m 5 m meshes. The current 1 A is injected from node 4 and flows out from node 9. We assume that the current value and current direction of the each branch are as shown in Fig. 3. The ground surface is at z = h = 1 m. The conductor’s leakage current in the soil is ignored. The magnetic flux density on the ground surface in the ez direction is Bz(x, y). With the derivative method, the 1st-, 3rd- and 5thð3Þ ð5Þ order derivatives of Bz(x, y) were Bð1Þ z ðx; yÞ, Bz ðx; yÞ and Bz ðx; yÞ, respectively. We can have:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 @Bz ðx; yÞ @Bz ðx; yÞ ¼ þ @x @y vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 !2 u u @ 3 B ðx; yÞ @ 3 Bz ðx; yÞ z t ðx; yÞj ¼ þ jBð3Þ z @x3 @y3 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! !2 u u @ 5 B ðx; yÞ 2 @ 5 Bz ðx; yÞ z ð5Þ t jBz ðx; yÞj ¼ þ @x5 @y5
jBð1Þ z ðx; yÞj
ð16Þ
ð17Þ
ð18Þ
As showed in Figs. 4 and 5, while obtaining the consecutive derivatives of Bz(x, y), the mutual influence between two parallel branches will decrease, the topology of the grids will be more clear. The two cross section x = 2.5 m and x = 0 m of jBð1Þ z ðx; yÞj, ð5Þ jBð3Þ z ðx; yÞj and jBz ðx; yÞj, are shown in Fig. 6. ð3Þ ð5Þ For the cross section x = 2.5 m, Bð1Þ z ðyÞ, Bz ðyÞ and Bz ðyÞ have the main peak at y = 5 m, y = 0 m and y = 5 m respectively, so the position of branches R7 9 and the length of branches R5 6
Fig. 7. (a) Grounding grid model. (b) Laboratory experiment.
Fig. 8. (a) Zero-order, (b) 1st-order, (c) 3rd-order, (d) 5th-order field derivative of Magnetic field Bz(x) on the line B1 respectively.
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L. Chunli et al. / Electrical Power and Energy Systems 63 (2014) 158–164 Table 2 Comparison the simulation results with the theoretical values in derivative method.
Theoretical value Derivative method
Errors
Grid space (L/m)
Burial depth (h/m)
R5 branch current (mA)
R6 branch current (mA)
R7 branch current (mA)
R9 branch current (mA)
1st-Order 3rd-Order 5th-Order
5 4.9565 4.9962 4.9963
1 0.9296 1.0008 1.0058
500 404.3 497.8 492.0
350 303.18 351.8 344.71
300 232.9 297.1 295.27
550 540.0 548 542.3
1st-Order 3rd-Order 5th-Order
0.87% 0.07% 0.07%
7.04% 0.08% 0.58%
19.14% 0.44% 1.6%
13.38% 0.51% 1.51%
22.37% 0.97% 1.58%
1.82% 0.36% 1.4%
Table 3 Comparison the actual values with the measurement results in derivative method. Branch
Line B1
Line B3
R10
R5
R1
R9
R8
R7
Actual value
Branch position (mm) Branch current (A) Burial depth (h/mm)
36 9.58 30
176 14.52
316 5.56
45 14.19
185 6.04
325 9.58
1st-Order derivative
Branch position (mm) Position error Branch current (A) Branch current error Burial depth (h/mm) Burial depth error
34.8 0.857% 10.8615 13.38%
176.2 0.143% 15.9001 9.50% 30.4434 1.47%
314.7 0.929% 6.7489 21.38%
45 0 20.2447 42.67%
183.6 1% 7.0264 16.33% 31.0352 3.45%
324.1 0.643% 11.0842 15.70%
3rd-Order derivative
Branch position (mm) Position error Branch current (A) Branch current error Burial depth (h/mm) Burial depth error
35.4 0.43% 11.7467 22.62%
176.3 0.21% 16.1521 11.24% 27.9421 6.86%
313.3 1.93% 8.2785 48.89%
45.9 0.64% 23.1520 63.16%
183.4 1.14% 6.4063 6.06% 30.0757 0.25%
324.6 0.286% 11.0305 15.14%
can be determined according to Eqs. (5)–(7). Similarly, the topology of the grounding grids can be accurately drawn. The burial depth h can be calculated according to Eqs. (9)–(11). For the cross section x = 0 m, the current value of branches R5 6 can be determined according to Eqs. (12)–(14). Similarly, the value of the branch current of the grounding grids can be obtained. With the derivative method, the grid topology, the burial depth h, and the current value are solved in Table 2. The 3rd-order derivative method works best, the error of the topology and the branch current is less than 0.1% and 2%, respectively. Experimental result In the experiment, an F.W. BELL Model 7010 gauss/tesla meter is used to measure magnetic flux density by utilizing a Hall Effect probe. And a Fluke 319 clamp meter is used to measure the branch current of the grid model, with an accuracy of ±1.6% at 40 A. Then a 3D numerical control motor platform is used to move the Hall Effect probe, with a spatial movement range of 400 400 400 mm (X, Y, Z) and a positioning accuracy of 0.01 mm. As showed in Fig. 7(a), the small model of grounding grid features a 280 mm 280 mm square grid (Ø3.5 mm copper wire) with 140 mm 140 mm meshes. The probe height from the experimental grid is h = 30 mm. A current of 29.84 A is injected from node 4 and flowed out from node 9. In order to shorten the measurement time, only the magnetic flux density in the ez direction on line B1, B2, B3 and B4 are measured, and the four lines pass the midpoint of the corresponding branches. For measurement on a line, the total number of measurement points is 361, the point spacing is 1 mm, the stabilization time of probe at each point is 2 s, the moving speed of the probe is 0.5 mm/s and the total measurement time is 28 min 50 s. Fig. 7(b) shows the lab experiment results of the model.
Considering the lab electromagnetic interference, the original magnetic field B0(x) of line B1 is measured at first, then a current is injected to the model and measure the magnetic field Bz0(x). Thus the actual magnetic field Bz(x) on line B1 is Bz(x) = Bz0(x) B0(x) (Fig. 8(a)). With the derivative method, the 1storder derivative of Bz(x) is obtained (Fig. 8(b)). A low-pass filter is used to filter noise signal. The 3rd-order and 5th-order derivatives of Bz(x) are shown in Fig. 8(c and d) in the same way, respectively. There are three main peaks in Fig. 8(b and c), so the branch position and branch current can be obtained according to Eqs. (5), (6), (12), (13). The burial depth h can be obtained according to Eq. (10). The 5th-order derivative of Bz(x) in Fig. 8(d) is not clear enough to show the positions of the main peaks, so it is not used. Table 3 shows the difference between the actual values and the measured results with the derivative method. Table 3 shows that, the error of branch position and burial depth is less than 2% and 7%, respectively; the error of branch current can be less than 17% while the branch is placed in the center of the measurement area. The results show that the 1st- and 3rd-order derivatives can be used to find the branch position, branch current and burial depth in the problem of drawing loss or unknown of grounding grids. Conclusion Based on the electromagnetic field theory, the derivative method for measuring topology of the grounding grids is developed. The theoretical analysis, numerical example and experimental result shows that the method can measure branch position and branch current of the grounding grids, with low error and less computation. The method can be used to solve the problem of drawing loss or unknown of grounding grids.
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Acknowledgement This work was supported by the Scientific Research Foundation (2007DA10512713403) of State Key Laboratory of Power Transmission Equipment & System Security and New Technology (Chongqing University). References [1] Colominas I, Gómez-Calviño J, Navarrina F, Casteleiro M. Computer analysis of earthing systems in horizontally or vertically layered soils. Electr Power Syst Res 2001;59:149–56. [2] Sverak JG, Dick WK, Dodds TH, Heppe RH. Safe substation grounding – Part I. IEEE Trans Power Apparatus Syst 1981;100:4281–90. [3] IEEE Guide for Safety in AC Substation Grounding. IEEE Std 80-2000; 2000. [4] Celli G, Ghiani E, Pilo F. Behaviour of grounding systems: a quasi-static EMTP model and its validation. Electr Power Syst Res 2012;85:24–9. [5] Yugen L, Leishi X, Jinhu T. Optimized corrosion diagnosis of large-scale grounding grid. IEEE Power Energy Soc Gen Meet 2010;2010:1–6. [6] Gang X, Zhenhua Z, Yanping B. Optimization algorithm of corrosion diagnosis for grounding grid. In: 2010 2nd International conference on mechanical and electronics engineering (ICMEE); 2010. p. 42–6. [7] Dawalibi F. Electromagnetic fields generated by overhead and buried short conductors Part 2 – Ground networks. IEEE Trans Power Deliv 1986;1:112–9. [8] Yan M, Karady GG, Kucuksari S. Testing continuity of grounding grid using the AC current injection method. IEEE Power Energy Soc Gen Meet 2010;2010:1–6.
[9] Wei H, Qingyan S, Fan Y, Ruiqiang Z. Computation method of magnetic field inverse problem on grounding grids fault diagnosis. J Chongqing Univ 2012;35:80–5. [10] Apparao A, Rao TG, Sastry RS, Sarma VS. Depth of detection of buried conductive targets with different electrode arrays in resistivity prospecting. Geophys Prospect 1992;40:749–60. [11] Apparao A, Sastry RS, Sarma VS. Depth of detection of buried resistive targets with some electrode arrays in electrical prospecting. Geophys Prospect 1997;45:365–75. [12] Cardarelli E, Cercato M, Cerreto A, Di Filippo G. Electrical resistivity and seismic refraction tomography to detect buried cavities. Geophys Prospect 2010;58:685–95. [13] Vincent A, Charles M, Willis A, Gerald M. Geophysical exploration of iron ore deposit in Kimachia area in Meru county in Kenya, using gravity and magnetic techniques. Int J Sci Res (IJSR) 2013;2:104–8. [14] Buchanan S, Triantafilis J. Mapping water table depth using geophysical and environmental variables. Ground Water 2009;47:80–96. [15] Hedley CB, Roudier P, Yule IJ, Ekanayake J, Bradbury S. Soil water status and water table depth modelling using electromagnetic surveys for precision irrigation scheduling. Geoderma 2013;199:22–9. [16] Yu H, Ying X. Derivative seismic processing method for GPR data. IEEE Int Geosci Rem Sens 1997;1(1997):145–7. [17] Liu Q. Sensitivity and Hessian matrix analysis of power spectral density functions for uniformly modulated evolutionary random seismic responses. Finite Elem Anal Des 2012;48:1370–5. [18] Widess MB. Quantifying resolving power of seismic systems. Geophysics 1982;47(1982):1160–73. [19] Puryear CI, Castagna JP. Layer-thickness determination and stratigraphic interpretation using spectral inversion: theory and application. Geophysics 2008;73:37–48.
Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Topological measurement and characterization of substation grounding grids based on derivative method Li Chunli a,1, He Wei a, Yao Degui b, Yang Fan a, Kou Xiaokuo b, Wang Xiaoyu a,⇑ a b
State Key Laboratory of Power Transmission Equipment & System Security and New Technology, The Electrical Engineering College, Chongqing University, Chongqing 400044, China State Grid Henan Electric Power Research Institute, Zhengzhou 450052, China
a r t i c l e
i n f o
Article history: Received 25 June 2013 Received in revised form 19 May 2014 Accepted 20 May 2014
Keywords: Grounding grids Derivative method Topology Magnetic field
a b s t r a c t Derivative method, which avoids pathological solution of the magnetic field inverse problem, was proposed to solve the problem of drawing loss or unknown of grounding grids. First, a shape function was introduced to describe the distribution of magnetic field which is perpendicular to the surface of grounding grids. Since its odd-order derivatives contain main peaks, the 1st-, 3rd- and 5th-order derivatives were chosen to solve the topology of substation grounding grids and measure the branch current of the grid by determining the position and value of main peak, respectively. Numerical example and experimental result show that the 1st- and 3rd-order derivative method works well with low error and less computation. Ó 2014 Elsevier Ltd. All rights reserved.
Introduction The main objectives of grounding systems are to guarantee personal safety, equipment protection and power supply continuity [1]. A grounding system comprises all the interconnected grounding facilities in a specific area, but its main element is the grounding grids. In general, most grounding grids of electrical substations consist of a mesh of interconnected cylindrical conductors, which are horizontally buried and supplemented by vertically thrusting ground rods in certain places of the substation [2]. Typically the conductors made of bare copper or iron are buried 0.3–2 m deep underground. The meshes are usually spaced 3–7 m apart and the ratio of the sides of a mesh is 1:l–1:3 [3]. The electromagnetic field theory (EMF) and the electric circuit theory (with lumped or distributed parameters) are both widely used to diagnose the corrosion status of the grounding grids in a power substation [4]. The electric circuit theory establishes and solves the diagnosis equation that describes the nonlinear relationship between branch resistance and node voltage based on the application of Tellegen’s Theorem [5,6]. However, the method needs to know the topology of grounding grids. The electromagnetic field theory, which was proposed by Dawalibi [7], measures the surface electromagnetic field and establishes the magnetic field
⇑ Corresponding author. Tel.: +86 15823311160. E-mail addresses: [email protected] (L. Chunli), wangxiaoyucqu@gmail. com (W. Xiaoyu). 1 Tel.: +86 15123856702. http://dx.doi.org/10.1016/j.ijepes.2014.05.046 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
inverse equations to judge the topology and corrosion status of the grounding grids [8,9]. However, the solution to the inverse problem usually is not unique and stable, the solution process is complex. In geophysical prospecting, the topology and burial depth of the target can be detected by the resistivity prospecting [10], electrical prospecting [11], seismic refraction tomography [12], gravity and magnetic prospecting [13], gamma radiometric prospecting [14], electromagnetic prospecting Hedley et al. [15]. The geophysical prospecting methods are mainly for large size and deeply buried targets, for example, the faults shear zones, iron ore deposit, buried cavity, water table, and have shortcoming to detect the small size (cross section less than 1 cm 10 cm) and shallow buried (0.3–2 m deep underground) branch conductors of grounding grids. In the paper, a method is developed for calculating the topology and branch current of grounding grids and avoids pathological solution of the magnetic field inverse problem. First, the theoretical analysis of the method is discussed. Then a numerical example is used to test the viability of the method. Finally experimental result is used to test the reliability and precision of the method by a small grounding grid model in the lab. Modeling of single current-carrying conductor by using derivative method Most grounding grids are composed by a lot of branches in finite length. We can analyze the characterization of every single branch, and draw the topology of grounding grids by measuring
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Fig. 1. The magnetic flux density caused by a finite-length current filament on the x axis is B = lI/4pq(sin u1 + sin u2)e/; the ground surface is at z = h plane. Fig. 3. Grid current-carrying model. The grounding grid model depicts the topology, branch length, point position, branch name, branch current and current direction of the grid, then the model is applied to test the viability of the Derivative method.
The application of derivation method in magnetic field distribution analysis
ð3Þ ð5Þ Fig. 2. Graph of Bð1Þ z ðyÞ, Bz ðyÞ and Bz ðyÞ.
the position of the each branch. The process of derivation method is introduced by analyzing the current-carrying conductor in finite length. Derivative method The derivative method (or differential technique) is mainly used to improve the resolution ratio of seismic records, and compensate the frequency loss and amplitude loss of seismic waves during propagation [16,17]. Let f(x) be a differentiable function, and let f(n)(x) be its n-order derivative. n
F ðnÞ ðxÞ ¼ ðjxÞ FðxÞ with n 2 N
ð1Þ
where
FðxÞ ¼
Z
1
f ðxÞejxx dx;
F ðnÞ ðxÞ ¼
1
Z
1
f ðnÞ ðxÞejxx dx
ð2Þ
1
In Eq. (1), the n-order derivative of the function is equivalent to filtering it with a filter characteristic (jx)n.
As shown in Fig. 1, a current carrying conductor MN in length L is buried in the uniform monolayer soil with permeability l. The conductor is on the x-axis. The ground surface is at z = h plane and the current I along the positive x-axis flows through the conductor. For point P at position (0, y0, h), its vertical distance from the conductor is q. The angles between segment OP with the z-axis, segment NP, and segment MP are h, u1 and u2, respectively. The length of OM and ON is L1 and L2, respectively. The conductor’s leakage current in the soil is ignored. Based on Biot–Savart law, the magnetic flux intensity B at point P is
B¼
lI ðsin u1 þ sin u2 Þe/ 4p q
ð3Þ
At the point P the unit vector e/ is in the direction perpendicular to the eq. Since q2 = h2 + y2, the magnetic flux density Bz in the ez direction is
Bz ðyÞ ¼
lI y ðsin u1 þ sin u2 Þ 4p h2 þ y2
L1 And sin u1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 2
ð4Þ
L2 sin u2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2 2
h þy2 þL1
h þy2 þL2
Magnetic field distribution of the single current-carrying conductor in the ez direction can be described by Eq. (4), called the shape function. The magnetic field distribution of the grounding grids is equivalent to the superposition of the shape functions of each current-carrying branch. Because the higher order derivative of Eq. (4) has a more complex expression, the odd-order derivatives contain main peaks, the
Table 1 The comparison table of the three shape functions’ graph characteristics. Shape function
Graph characteristics
Widess Resolution
Influence sphere (m) (1%)
Main peak width (m)
Side peak width (m)
Total number of peaks
19.72
2
8.86
3
Bzð3Þ ðyÞ
4.052
0.8284
1.5858
3
2.0308
Bzð5Þ ðyÞ
3.8
0.2679
0.7321
5
2.5699
Bzð1Þ ðyÞ
Annotation: I = 1 A; h = 1 m.
1.2796
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Fig. 4. Graph of (a) zero-order, (b) 1st-order, (c) 3rd-order, and (d) 5th-order field derivative respectively.
Fig. 5. Contour map of (a) zero-order, (b) 1st-order, (c) 3rd-order, and (d) 5th-order field derivative respectively.
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1st-, 3rd- and 5th-order derivatives are employed while ignoring P2 0 the o i¼1 ðsin ui Þ :
Bzð1Þ ðyÞ
lI h2 y2 ðsin u1 þ sin u2 Þ 4p ðh2 þ y2 Þ2
Bzð3Þ ðyÞ
3lI h y4 þ 6y2 h ðsin u1 þ sin u2 Þ 4 2 2p ðh þ y2 Þ
4
2
6
Bzð5Þ ðyÞ
ð5Þ
2
4
30lI h y6 þ 15y4 h 15h y2
p
2
ðh þ y2 Þ
ð6Þ
6
ðsin u1 þ sin u2 Þ
ð7Þ
Simplify the model and let L ? 1, then sin u1 + sin u2 ? 2. Set ð3Þ ð5Þ h = 1 m and I = 1 A, the characteristics of Bð1Þ z ðyÞ, Bz ðyÞ and Bz ðyÞ are shown in (Fig. 2). Definition that, the influence sphere is the range of the maximum distance between the two points, where the value is 1% of the extreme value of the main peak; The main peak width is the width between the two zero-value points of the main peak; The side peak width is the width between the two zero-value (or 1% of the extreme value of the main peak) points of the side peak near the main peak. The Widess Resolution P (Eq. (8)) is the ratio 2 between the extreme value energy bM of the main peak and the total energy E of the shape function [18,19]. 2
P¼
bM E
Among E ¼
Z
1
2
b ðyÞdy
ð8Þ
1
Table 1 shows that, while the 1st-, 3rd- and 5th-order derivatives of Bz(y) are consecutively obtained, the influence sphere, main peak width and side peak width will decrease, especially the influð1Þ ence sphere of Bð3Þ z ðyÞ decreases to 20.55% of that of Bz ðyÞ, the main ð3Þ peak width of Bz ðyÞ becomes 20.44% of its influence sphere. The number of peaks shown for each field derivative in Fig. 2 are 3,3 and 5 respectively. The Widess Resolution P is gradually improved and the ability of signal identification is enhanced. According to Ref. [3], the meshes are usually spaced 3–7 m apart. ð3Þ ð5Þ While the half of the influence sphere of Bð1Þ z ðyÞ, Bz ðyÞ and Bz ðyÞ is
ð3Þ less than the grid space, the mutual influence of Bð1Þ z ðyÞ, Bz ðyÞ and ð5Þ Bz ðyÞ of two parallel branches can be ignored, respectively. According to Eqs. (5)–(7), the position of the main peaks of ð3Þ ð5Þ Bð1Þ z ðyÞ, Bz ðyÞ, Bz ðyÞ and the position of the current-carrying conductor are the same and at y = 0, respectively. Therefore, the position of grounding grid branch in the measurement area can be determined by calculating the position of the main peak of ð3Þ ð5Þ Bð1Þ z ðyÞ, Bz ðyÞ or Bz ðyÞ. The length L of grounding grid branch can be determined in the same way. The distance between the main peak and the side peak of Bð1Þ z ðyÞ, ð5Þ Bð3Þ ðyÞ and B ðyÞ is L , L , L , respectively: z1 z3 z5 z z
Lz1 1:732 h
ð9Þ
Lz3 0:7265 h
ð10Þ
Lz5 0:4816 h
ð11Þ
so the burial depth h can be calculated by solving Lz1, Lz3 or Lz5. According to Eq. (4), the limits of the 1st-, 3rd- and 5th-order derivatives at y = 0 point are employed while not ignoring the P 0 2 o i¼1 ðsin ui Þ :
lim Bð1Þ z ðyÞ ¼ I y!0
2 X i¼1
lim Bð3Þ z ðyÞ ¼ I y!0
lim Bð5Þ z ðyÞ ¼ I y!0
lLi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Ikz1 2 4ph ðL2i þ h Þ
2 X i¼1
2
ð12Þ
2
3lLi ð2L2i þ 3h Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Ikz3 4 2 2 4ph ðL2i þ h Þ ðL2i þ h Þ
2 4 2 X 15lLi ð8L4i þ 20L2i h þ 15h Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Ikz5 6 2 2 2 2 i¼1 4ph ðL þ h Þ ðL2i þ h Þ i
ð13Þ
ð14Þ
limy!0 Bzð1Þ ðyÞ, limy!0 Bzð3Þ ðyÞ and limy!0 Bð5Þ z ðyÞ mean the value of the main peak of Bzð1Þ ðyÞ, Bzð3Þ ðyÞ and Bð5Þ z ðyÞ, respectively, that F i ¼ limy!0 BzðiÞ ðyÞði ¼ 1; 3; 5Þ.
F i ¼ Ikzi ðL1 ; L2 Þ L ¼ L1 þ L2
i ¼ 1; 3; 5
Fig. 6. Longitudinal cross section of (a) zero-order, (b) 1st-order, (c) 3rd-order, and (d) 5th-order field derivative respectively.
ð15Þ
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From what we have discussed above, the value of the branch current I can be determined by calculating the value of the main ð3Þ ð5Þ peak of Bð1Þ z ðyÞ, Bz ðyÞ , Bz ðyÞ and the length L. Numerical example The numerical example features a 10 m 10 m square grid (copper conductors with 5 mm radius) with 5 m 5 m meshes. The current 1 A is injected from node 4 and flows out from node 9. We assume that the current value and current direction of the each branch are as shown in Fig. 3. The ground surface is at z = h = 1 m. The conductor’s leakage current in the soil is ignored. The magnetic flux density on the ground surface in the ez direction is Bz(x, y). With the derivative method, the 1st-, 3rd- and 5thð3Þ ð5Þ order derivatives of Bz(x, y) were Bð1Þ z ðx; yÞ, Bz ðx; yÞ and Bz ðx; yÞ, respectively. We can have:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 @Bz ðx; yÞ @Bz ðx; yÞ ¼ þ @x @y vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 !2 u u @ 3 B ðx; yÞ @ 3 Bz ðx; yÞ z t ðx; yÞj ¼ þ jBð3Þ z @x3 @y3 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! !2 u u @ 5 B ðx; yÞ 2 @ 5 Bz ðx; yÞ z ð5Þ t jBz ðx; yÞj ¼ þ @x5 @y5
jBð1Þ z ðx; yÞj
ð16Þ
ð17Þ
ð18Þ
As showed in Figs. 4 and 5, while obtaining the consecutive derivatives of Bz(x, y), the mutual influence between two parallel branches will decrease, the topology of the grids will be more clear. The two cross section x = 2.5 m and x = 0 m of jBð1Þ z ðx; yÞj, ð5Þ jBð3Þ z ðx; yÞj and jBz ðx; yÞj, are shown in Fig. 6. ð3Þ ð5Þ For the cross section x = 2.5 m, Bð1Þ z ðyÞ, Bz ðyÞ and Bz ðyÞ have the main peak at y = 5 m, y = 0 m and y = 5 m respectively, so the position of branches R7 9 and the length of branches R5 6
Fig. 7. (a) Grounding grid model. (b) Laboratory experiment.
Fig. 8. (a) Zero-order, (b) 1st-order, (c) 3rd-order, (d) 5th-order field derivative of Magnetic field Bz(x) on the line B1 respectively.
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Theoretical value Derivative method
Errors
Grid space (L/m)
Burial depth (h/m)
R5 branch current (mA)
R6 branch current (mA)
R7 branch current (mA)
R9 branch current (mA)
1st-Order 3rd-Order 5th-Order
5 4.9565 4.9962 4.9963
1 0.9296 1.0008 1.0058
500 404.3 497.8 492.0
350 303.18 351.8 344.71
300 232.9 297.1 295.27
550 540.0 548 542.3
1st-Order 3rd-Order 5th-Order
0.87% 0.07% 0.07%
7.04% 0.08% 0.58%
19.14% 0.44% 1.6%
13.38% 0.51% 1.51%
22.37% 0.97% 1.58%
1.82% 0.36% 1.4%
Table 3 Comparison the actual values with the measurement results in derivative method. Branch
Line B1
Line B3
R10
R5
R1
R9
R8
R7
Actual value
Branch position (mm) Branch current (A) Burial depth (h/mm)
36 9.58 30
176 14.52
316 5.56
45 14.19
185 6.04
325 9.58
1st-Order derivative
Branch position (mm) Position error Branch current (A) Branch current error Burial depth (h/mm) Burial depth error
34.8 0.857% 10.8615 13.38%
176.2 0.143% 15.9001 9.50% 30.4434 1.47%
314.7 0.929% 6.7489 21.38%
45 0 20.2447 42.67%
183.6 1% 7.0264 16.33% 31.0352 3.45%
324.1 0.643% 11.0842 15.70%
3rd-Order derivative
Branch position (mm) Position error Branch current (A) Branch current error Burial depth (h/mm) Burial depth error
35.4 0.43% 11.7467 22.62%
176.3 0.21% 16.1521 11.24% 27.9421 6.86%
313.3 1.93% 8.2785 48.89%
45.9 0.64% 23.1520 63.16%
183.4 1.14% 6.4063 6.06% 30.0757 0.25%
324.6 0.286% 11.0305 15.14%
can be determined according to Eqs. (5)–(7). Similarly, the topology of the grounding grids can be accurately drawn. The burial depth h can be calculated according to Eqs. (9)–(11). For the cross section x = 0 m, the current value of branches R5 6 can be determined according to Eqs. (12)–(14). Similarly, the value of the branch current of the grounding grids can be obtained. With the derivative method, the grid topology, the burial depth h, and the current value are solved in Table 2. The 3rd-order derivative method works best, the error of the topology and the branch current is less than 0.1% and 2%, respectively. Experimental result In the experiment, an F.W. BELL Model 7010 gauss/tesla meter is used to measure magnetic flux density by utilizing a Hall Effect probe. And a Fluke 319 clamp meter is used to measure the branch current of the grid model, with an accuracy of ±1.6% at 40 A. Then a 3D numerical control motor platform is used to move the Hall Effect probe, with a spatial movement range of 400 400 400 mm (X, Y, Z) and a positioning accuracy of 0.01 mm. As showed in Fig. 7(a), the small model of grounding grid features a 280 mm 280 mm square grid (Ø3.5 mm copper wire) with 140 mm 140 mm meshes. The probe height from the experimental grid is h = 30 mm. A current of 29.84 A is injected from node 4 and flowed out from node 9. In order to shorten the measurement time, only the magnetic flux density in the ez direction on line B1, B2, B3 and B4 are measured, and the four lines pass the midpoint of the corresponding branches. For measurement on a line, the total number of measurement points is 361, the point spacing is 1 mm, the stabilization time of probe at each point is 2 s, the moving speed of the probe is 0.5 mm/s and the total measurement time is 28 min 50 s. Fig. 7(b) shows the lab experiment results of the model.
Considering the lab electromagnetic interference, the original magnetic field B0(x) of line B1 is measured at first, then a current is injected to the model and measure the magnetic field Bz0(x). Thus the actual magnetic field Bz(x) on line B1 is Bz(x) = Bz0(x) B0(x) (Fig. 8(a)). With the derivative method, the 1storder derivative of Bz(x) is obtained (Fig. 8(b)). A low-pass filter is used to filter noise signal. The 3rd-order and 5th-order derivatives of Bz(x) are shown in Fig. 8(c and d) in the same way, respectively. There are three main peaks in Fig. 8(b and c), so the branch position and branch current can be obtained according to Eqs. (5), (6), (12), (13). The burial depth h can be obtained according to Eq. (10). The 5th-order derivative of Bz(x) in Fig. 8(d) is not clear enough to show the positions of the main peaks, so it is not used. Table 3 shows the difference between the actual values and the measured results with the derivative method. Table 3 shows that, the error of branch position and burial depth is less than 2% and 7%, respectively; the error of branch current can be less than 17% while the branch is placed in the center of the measurement area. The results show that the 1st- and 3rd-order derivatives can be used to find the branch position, branch current and burial depth in the problem of drawing loss or unknown of grounding grids. Conclusion Based on the electromagnetic field theory, the derivative method for measuring topology of the grounding grids is developed. The theoretical analysis, numerical example and experimental result shows that the method can measure branch position and branch current of the grounding grids, with low error and less computation. The method can be used to solve the problem of drawing loss or unknown of grounding grids.
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Acknowledgement This work was supported by the Scientific Research Foundation (2007DA10512713403) of State Key Laboratory of Power Transmission Equipment & System Security and New Technology (Chongqing University). References [1] Colominas I, Gómez-Calviño J, Navarrina F, Casteleiro M. Computer analysis of earthing systems in horizontally or vertically layered soils. Electr Power Syst Res 2001;59:149–56. [2] Sverak JG, Dick WK, Dodds TH, Heppe RH. Safe substation grounding – Part I. IEEE Trans Power Apparatus Syst 1981;100:4281–90. [3] IEEE Guide for Safety in AC Substation Grounding. IEEE Std 80-2000; 2000. [4] Celli G, Ghiani E, Pilo F. Behaviour of grounding systems: a quasi-static EMTP model and its validation. Electr Power Syst Res 2012;85:24–9. [5] Yugen L, Leishi X, Jinhu T. Optimized corrosion diagnosis of large-scale grounding grid. IEEE Power Energy Soc Gen Meet 2010;2010:1–6. [6] Gang X, Zhenhua Z, Yanping B. Optimization algorithm of corrosion diagnosis for grounding grid. In: 2010 2nd International conference on mechanical and electronics engineering (ICMEE); 2010. p. 42–6. [7] Dawalibi F. Electromagnetic fields generated by overhead and buried short conductors Part 2 – Ground networks. IEEE Trans Power Deliv 1986;1:112–9. [8] Yan M, Karady GG, Kucuksari S. Testing continuity of grounding grid using the AC current injection method. IEEE Power Energy Soc Gen Meet 2010;2010:1–6.
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