Earth Return Path Impedances of Underground Cables for ... - LabPlan

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J z = Jez + Jsz (12) where J ez is the eddy current density and J sz the source current density, given by (13) and (14) respectively. J =− jωσ A (13) ez J sz z =−σ∇Φ (14) FEM is applied for the solution of (9) and (10) with the boundary conditions of (11). Values for J on each conductor i of conductivity σ i are then obtained and equation (8) takes the form V J sz σ i i i Zij = = ( i, j = 1, 2, …, n) (15) I I j j linking properly electromagnetic field variables and equivalent circuit parameters. A distinct advantage of the FEM is its capability of handling cases of geometrical or electromagnetic ground irregularities of almost any kind, where conventional methods fail. V. NUMERICAL RESULTS The proposed numerical integration scheme was applied in the following cases of SC cable arrangements in homogeneous ground. First the case of a shallowly located SC cable system of Fig. 2 was examined [12] in a horizontal (a) and a vertical (b) cable arrangement. Cable is in a depth h=1,2 m with a spacing −3 s=0,25m. The core radius is = 23,4 ∗10 m, while the outer radius is r S = 48,4 ∗10 −3 sz i m. The conductance of the cable core is σ = 5, 88235∗10 7 S/m and µ = 1 for the ground. r C r air earth h r c r c r c h r c air earth compared to the corresponding obtained by EMTP and FEM for earth resistivities from 2 Ωm up to 1000 Ωm and for a frequency range of 1 Hz to 10 MHz. For each case the relative differences of the form of (16) were calculated ZNum − ZEMTP Relative difference (%) = ∗ 100 (16) Z EMTP In Fig. 3 the relative differences for the magnitude of the self impedance of the horizontal cable arrangement of Fig. 2 are shown. Differences up to 12% occur, especially for low earth resistivities and high frequencies. difference(%) 40% 30% 20% 10% ρ=2 Ωm ρ=10 Ωm ρ=50 Ωm ρ=100 Ωm ρ=200 Ωm ρ=300 Ωm ρ=500 Ωm ρ=1000 Ωm 0% 1,E+00 1,E+01 1,E+02 1,E+03 1,E+04 1,E+05 1,E+06 1,E+07 frequency(Hz) Fig. 3: Differences in Z 11 between numerical integration and EMTP. In Fig. 4 the corresponding differences for the magnitude of the mutual impedance of the cable arrangement are shown. The differences in this case are due only to the different approaches in earth return impedance calculation and are higher reaching up to 25%. difference(%) 40% 30% 20% 10% ρ=2 Ωm ρ=10 Ωm ρ=50 Ωm ρ=100 Ωm ρ=200 Ωm ρ=300 Ωm ρ=500 Ωm ρ=1000 Ωm r s r s r s s a s Fig. 2: Horizontal (a) and vertical (b) SC cable arrangements in homogeneous ground. Series impedances were calculated for this configuration using the proposed integration scheme and they were s s b r s r c r c r s r s 0% 1,E+00 1,E+01 1,E+02 1,E+03 1,E+04 1,E+05 1,E+06 1,E+07 frequency(Hz) Fig. 4: Differences in Z 12 between numerical integration and EMTP. Next the case of a SC core cable with core and sheath, as in Fig. 5 is considered. Original cable data from [3] are reproduced in Table I. The cable arrangements are similar to those in Fig. 1 with h=0,75m, s=0,15m and µ r = 1 for both media. The differences of (16) were calculated for the above cable arrangement using the previous earth resistivities and frequencies. In Fig. 6 and Fig. 7 these differences are shown for the case of vertical configuration, for the real and the imaginary parts of the mutual impedance between the cable
sheaths respectively. Differences of almost 25% are recorded The results of the numerical integration were also checked against the corresponding by FEM for all above cases. Differences calculated by a formula similar to (16) were less than 1,5% in all cases over the whole range of frequencies sheath conductor r 1 r 2 r 3 r 4 impedance matrices of cables buried in a two-layered earth. The two horizontal arrangements, discussed previously, were considered. difference (%) 40% 30% 20% 10% ρ=2 Ωm ρ=10 Ωm ρ=50 Ωm ρ=100 Ωm ρ=200 Ωm ρ=300 Ωm ρ=500 Ωm ρ=1000 Ωm . difference (%) insulation Fig. 5: Single Core cable with core and sheath. Core radius 1 TABLE I. DATA OF THE SC CABLE OF FIG. 5 Main insulation radius r 2 Sheath radius r 3 Outer insulation radius 4 r 2 Dc resistance of copper core Dc resistance of lead sheath 40% 30% 20% 10% ρ=2 Ωm ρ=10 Ωm ρ=50 Ωm ρ=100 Ωm ρ=200 Ωm ρ=300 Ωm ρ=500 Ωm ρ=1000 Ωm 1, 27 ∗10 − m 2, 28∗10 −2 m 2,54∗10 −2 m r 2 2,79∗10 − m 0,034 Ω/km 0,436 Ω/km 0% 1,E+00 1,E+01 1,E+02 1,E+03 1,E+04 1,E+05 1,E+06 1,E+07 frequency (Hz) Fig. 6: Real part of the mutual impedance of cable sheaths. Differences between numerical integration and EMTP. . The numerical integration scheme proved to be absolutely numerically stable. The computation time for the numerical integration was less than 5s for a set of 280 resistivity and frequency combinations using an Intel Pentium IV PC at 1,7 GHz. The case of a multi-layered earth was considered next. Sunde’s extension [8] refers only to conductors over, or on the surface of a multi-layered earth. For the case of underground conductors no solution for the electromagnetic field exists. Therefore the FEM was used for the calculation of the 0% 1,E+00 1,E+01 1,E+02 1,E+03 1,E+04 1,E+05 1,E+06 1,E+07 frequency(Hz) Fig. 7: Imaginary part of the mutual impedance of cable sheaths. Differences between numerical integration and EMTP. Six different two-layered earth models were investigated, based on actual grounding parameter measurements [16]. The corresponding data for the resistivities of the first ρ 1 and second ρ 2 layer and for the depth d of the first layer are shown in Table II. The second layer is of infinite extend. TABLE II. TWO-LAYERED EARTH MODELS ρ 1 (Ωm) ρ 2 (Ωm) d(m) CASE I 372,729 145,259 2,690 CASE II 246,841 1058,79 2,139 CASE III 57,344 96,714 1,651 CASE IV 494,883 93,663 4,370 CASE V 160,776 34,074 1,848 CASE VI 125,526 1093,08 2,713 Cable impedances obtained from FEM for the frequency range of 50 Hz to 1 MHz were compared to the corresponding from the proposed numerical integration scheme, under the assumption of homogeneous earth. The differences calculated by a formula similar to (16) are shown in Fig. 8 concerning the magnitude of the mutual impedance between the cable sheaths of the SC cable in Fig. 5. The homogeneous earth model was assumed to have the resistivity ρ 1 of the first earth layer. Differences of up to 40% are encountered, even at power frequency, especially in cases of great divergence between the resistivities of the two layers. The differences seem to be minimized near the frequencies, for which the penetration depth p given from (17) approaches the depth of the first layer. 1 p = (17) π f µσ 0
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Page 3: Equations (4) and (7) differ in two
Page 7: special interests are power systems

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