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

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surface irregularities and non-homogeneous, stratified soil, where most classical methods usually fail. Scope of this paper is to present a systematic comparison of results obtained by different approaches for the earth return impedances of underground single core cable arrangements. First a novel numerical integration technique is proposed and used for the direct calculation of the infinite integrals for the case of homogeneous earth. This technique is based on a combination of Gauss-Legendre and Gauss-Laguerre methods with the Lobatto rule [14], in order to overcome efficiently the problems arising from the oscillative form of the infinite integrals. Results are compared to those obtained by EMTP for several cable arrangements, earth resistivities and frequencies. The results by the novel numerical integration technique are also compared against those obtained by the application of the FEM in the case of homogeneous earth for further justification. Finally the FEM formulation is used for the analysis of two different cases of single core cable arrangements in a twolayered earth. Six cases of two layer earth models, based on actual electrical ground measurements are investigated. Results are compared to those of the homogeneous earth case. II. EARTH RETURN IMPEDANCES OF UNDERGROUND CABLE SYSTEMS A transmission line is generally described by the two matrix equations (1) and (2), linking the voltages and currents of the line, ∂ v = −Z′ (ω) i (1) ∂z ∂ i = −Y′ (ω) v (2) ∂z where v is the voltage vector with respect to a reference conductor, i is the current vector and z is the longitudinal direction along the transmission line. Matrices Ζ΄(ω) and Υ΄(ω) are the frequency dependent series impedance and shunt admittance per unit length matrices, respectively. For the case of an underground cable system Z΄(ω) may be considered to consist of two components [4], ( ω) = Z′ i ( ω) + Z ( ω) Z ′ ′ (3) where Z i΄(ω) represents the internal impedances of the conductors in the cable system and Z e΄(ω) accounts for the influence of the earth return path. The exact form and the dimensions of the above matrices depend on the number of conducting elements of each cable, e.g. core, sheath, armor and on the type and the actual configuration of the cable system. In the case of single core (SC) cables, the diagonal elements of Z e΄(ω) represent the impedances of the loops formed by the outermost tubular conductor of the cables, either sheath or armor and the earth, while the non-diagonal elements account for the mutual impedances between the outermost tubular conductors of each pair of cables and the e earth. Next the cable arrangement of Fig. 1 is considered. It consists of two SC cables i and j, buried in the ground, which is considered to be semi-infinite, homogeneous having an earth resistivity ρ and relative permeability and permittivity of µ r and ε r respectively. h Cable i d x D Image of cable j x y y Cable j air earth ρ, µ r , ε r Fig. 1: Geometric configuration of two SC underground cables. According to [2], [3], the mutual earth return impedance between the two underground cables i and j is where { K ( mD) − K ( md ) } ρm Z + 2π 2 ′ mutual = 0 0 J m J m ∞ 2 2 −( h+ y) a + m e jax = ∫ e da 2 2 −∞ a + a + m 2 = + ( + ) 2 , d x 2 + ( h − y) 2 D x h y and K 0 (4) (5) = , m= jωµ rµ 0 ρ is the modified complex Bessel function of the second kind with zero order. The self-impedance formula is derived from (4) and (5) by replacing x = r and y = h , where r is the radius of the outermost surface of the cable. By applying some simple transformations in (5) the following relation results [8]: 2 2 ∞ −( h+ y) a + m e J m = 2∫ cos( xa) da (6) 2 2 0 a + a + m III. NUMERICAL EVALUATION OF THE INFINITE INTEGRAL The corresponding expression for the mutual earth return impedances of conductors above ground is [1]: 2 ∞ − ( h+ y) a ρm e Zm′ = cos( xa) da π ∫ (7) 2 2 a+ a + m 0
Equations (4) and (7) differ in two points. First (4) includes the Bessel functions, which seem to be no serious problem as they can be evaluated easily using standard library functions. The key difference though is the fact that the exponential function in the infinite integral in (6) is complex, while in (7), it is real. As a result the real and imaginary parts of the infinite integral of (7) are monotonic and non-oscillatory functions [1] and can be calculated by infinite series, which converge rapidly [5]. In the approach of Wedepohl [3] the infinite integral of (5) is divided in parts, each of which is approximated using infinite series. Simplified expressions of the general formulas are also proposed for low frequencies and usual cable configurations. In the supporting routines CABLE CONSTANTS/PARAMETERS of EMTP, the earth return impedances of cables are calculated using Carson’s formula (7) for overhead conductors, which is evaluated by infinite series. This approximation is based on the assumption that a >> m . This is a highly uncertain approximation, which is obviously false for high frequencies and low earth resistivities. This approximation is used in [5] together with the approach of Wedepohl [3] for a single case of two underground conductors. The results are compared with those obtained by the numerical integration of (5), using Romberg extrapolation [14]. The approximation of EMTP shows significant differences for frequencies greater than 10 kHz, while Wedepohl’s approach is accurate enough at least up to 100 kHz. In the same reference, numerical efficiency problems concerning the application of the integration method are reported. Finally direct numerical integration is also used in [7]. Several integration methods, such as Simpson rule and Romberg rule [14] are tried. Problems concerning the numerical efficiency of all methods are reported, leading to the adaptation of artificial intelligence techniques for the evaluation of the infinite integrals. All above approaches either use approximations, which are valid only within certain limits, or suffer from numerical efficiency, which prohibits their generalized application. Therefore the problem of finding an efficient and accurate method for the numerical evaluation of the infinite integral in (6) is still of research interest. Direct numerical integration of the infinite integral is the method used in this contribution for the evaluation of (6). Instead of using the previously mentioned integration methods, a novel integration technique is used, based on combinations of integration methods. More specifically, the Gauss- Legendre method, a highly accurate numerical integration method applicable in finite intervals of functions is combined with two other methods: the Gauss-Laguerre method, which is best suited for infinite integrals and the Lobatto rule, a very efficient method for oscillative functions [14]. The selective implementation of the different integration methods in the intervals between the roots of cos( ax) , for both the real and the imaginary part of J , leads to a quick and very efficient integration scheme for the evaluation of (6). m IV. THE FINITE ELEMENT APPROACH The problem of the calculation of the cable series impedance matrix in (1) could be greatly simplified, assuming that the per unit length voltage drop V i on every conductor is known for a specific current excitation. The mutual complex series impedance per unit length Z ′ between conductor i and another conductor j carrying current I j , where all other conductors are forced to carry zero currents, is then given by: ij Vi Zij ′ = ( i, j = 1, 2, …, n) (8) I j The self impedance of a conductor may also be calculated from (8), by setting i = j. In such a case, the following procedure may be used for the calculation of the cable series impedance matrix [13]: • A sinusoidal current excitation of arbitrary magnitude is applied sequentially to each cable conductor, while the remaining conductors are forced to carry zero currents. The corresponding voltages are recorded. • Using (8), the self and mutual impedances of the j cable conductor may be calculated. This procedure is repeated n times, in order to calculate the impedances of n conductors. Therefore, the problem is reduced to that of calculating the actual per unit length voltage drops, when a current excitation is applied to the conductors. This may be achieved by a suitable Finite Element Method (FEM) formulation of the electromagnetic diffusion equation. An underground power cable arrangement, consisting of n parallel conductors, is assumed to be long enough to ignore end effects. Furthermore, if the current density vector is supposed to be in the z direction, the problem becomes twodimensional and it is confined in the x-y plane, in which the conductors’ cross sections lie. The above two-dimensional diffusion problem is described by the system of equations [15], where 2 2 1 ⎡∂ Az ∂ A ⎤ z ⎢ + j A 2 2 ⎥− ωσ z + J µµ 0 r ⎣∂x ∂y ⎦ z sz z sz = 0 (9) − jωσ A + J = J (10) ∫∫ JdS z = Ii , i=1, 2, ... , n (11) Si A z is the z direction component of the magnetic vector potential (MVP). In (10) the total current density two components, J z is decomposed in
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