VARATON OF LOW VOLTAGE POWER CABLES ELECTRCAL PARAMETERS DUE TO CURRENT FREQUENCY AND EARTH PRESENCE G.T. Andreou, D.P. Labridis, F.A. Apostolou, G.A. Karamanou, M.P. Lachana Aristotle University of Thessaloniki Dept. of Electrical & Computer Engineering Power Systems Laboratory P.O. Box 486, GR-5414, Thessaloniki, GREECE Tel: +30310996374, Fax: +3031099630 e-mail: labridis@auth.gr Abstract Many models proposed in the literature to describe Low Voltage power distribution networks in consumer premises as communication media require knowledge of the electrical parameters of the cables comprising these networks. These parameters are nevertheless affected by a large number of factors which may vary greatly from case to case, making it thus very difficult to achieve an exact estimation about them. n this work a finite element approach is used to study the variation of the resistance and inductance per unit length of cables usually installed in the Low Voltage networks of interest. The main parameters under study were current frequency, cable distance from earth and earth resistivity. 1. ntroduction Modeling of Low Voltage (LV) power distribution networks in consumer premises as communication media is a matter of great importance for powerline communications. Many of the models proposed in the literature describe the cables which comprise these networks by using their lumped or distributed electrical parameters, i.e. their resistance R, inductance L, conductance G and capacitance C per unit length [1]- [3]. An exact theoretical estimation of these parameters is nevertheless very difficult for most residential networks due to the large number of factors that affect them, which cannot be accounted for in all cases. The resistance per unit length R is mainly affected by the material and cross-sectional area of the conductors comprising the cable, their geometry (due to the proximity effect), the current frequency (due to the skin effect), and the presence of the earth. Nevertheless, these factors can be accounted for only in the case of a continuous cable, as any discontinuity (such as a unction or an insulation fault) may lead to problems difficult to overcome. The inductance per unit length L is generally affected by two factors, the geometry of the conductors comprising the cable and the presence of the earth. The latter produces the most problems, as its contribution may vary greatly from case to case. Moreover, the inductance of a cable will alter in the presence of current carrying conductors running in parallel with the cable, which introduces an additional factor of uncertainty, as LV power distribution networks inside consumer premises usually include cables from different circuits which run along the same routes. Concerning the cable capacitance per unit length C, its value depends mainly on the dielectric materials, as well as on the presence of the earth. Although there is a substantial amount of work in the literature about the characteristics of various dielectric materials in the frequency range up to several tens of MH, it applies in most cases in transient stability and has not yet been satisfactorily connected to the capacitance per unit length of cables operating in normal conditions. Another problem is caused by the absence of experimental measurements in the frequency range of interest. The above reasons also affect the accurate estimation of a cable s conductance per unit length G, as its value is usually calculated from the product of the cable s capacitive susceptance per unit length and the loss tangent of the dielectric material (1). G = ω C tan δ (1) Moreover, there is a significant contradiction in the literature about the values of the loss tangent for various dielectric materials [4]. As can be seen from the above, factors such as current frequency and presence of the earth affect all four electrical parameters. n this work a finite element approach is used to evaluate the influence of these factors on the resistance and inductance per unit length of cables usually installed in residential LV power
distribution networks. Presence of the earth was taken into account by two factors, namely the distance of the cables under study from the ground and the earth resistivity.. Finite element approach A system of N parallel conductors, carrying rms currents i ( i = 1,,..., N) is considered. The nonuniform current distribution inside the conductors influences the effective impedance of the conductors at a given frequency. The following matrix equation links voltages and currents in any conductor of the cable, V = Z(ω) where V is the voltage vector with respect to a reference, and is the current vector. The elements of matrix Z(ω) are the frequency dependent series impedances per unit length, depending on the geometric configuration, skin and proximity effect and eddy currents flowing in other conducting paths. The problem itself 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 impedance Z between conductor i and another conductor carrying current, where all other conductors are forced to carry ero currents, is then given by: Z i i () Vi = ( i, = 1,, K, N ) (3) The self impedance of a conductor may also be calculated from (), by setting i =. n such a case, the following procedure may be used for the calculation of the cable impedance matrix Z(ω): o A sinusoidal current excitation of arbitrary magnitude is applied sequentially to each conductor, while the remaining conductors are forced to carry ero currents. The o corresponding voltages are recorded. Using (), the -th column of Z(ω) may be calculated. This procedure is repeated N times, in order to calculate the N columns of Z(ω). The problem is then 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 Finite Element Method (FEM) formulation of the electromagnetic diffusion equation. The previously described cable, 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 direction, the problem becomes two-dimensional, confined in the x-y plane, in which the conductors cross sections lie. The linear electromagnetic diffusion equation is described by the following system of equations [5]: where 1 A µ 0µ r x A + + ωσa s y s = 0 (4) ωσ A + = (5) ds = i, i = 1,,..., N (6) S i A is the direction component of the magnetic vector potential (MVP). n (5) the total current density two components, e is decomposed in = + (7) where e is the eddy current density and s the source current density, given by (8) and (9) respectively. s = ωσ (8) e A = σ Φ (9) s FEM is applied for the solution of (4) and (5) with the boundary conditions of (6). Values for on each conductor i of conductivity σ i are then obtained and equation (3) takes the form [6], Z i Vi = = s i σ i s i ( i, = 1,, K, N ) (10) linking properly electromagnetic field variables and equivalent circuit parameters. Finally, positive, negative and ero sequence impedances may be easily obtained. The positive sequence series impedance matrix leads to the calculation of the operational cable resistance R' and inductance L' per unit length [6].
3. Cable configurations A NYM 3x,5 mm cable was used for the evaluation of the effects the aforementioned factors have on resistance and inductance per unit length. This cable type is widely used in residential power installations, and its geometrical and physical data are fully described in [7]-[8]. A schematic representation of the cable is shown in Figure 1. 4. Results The following figures show the results concerning the series impedance matrices of the above cases. 0 16 - R1 (Ω/m) 1 8 4 R1 Figure 1: Schematic representation of a NYM 3x,5 mm cable Two different cases were examined. n the first one the cable was considered to be located at different distances from the ground surface, with a constant value of ρ = 100 Ω m being used for the earth resistivity. n the second case the cable was considered to be located 0,5 m above ground surface, while a range of values was used concerning the earth resistivity. n both cases the cable s series impedance matrix was calculated in a frequency range from 0 to 30 MH. Both cases are shown in Table 1 and Table. Table 1 Configurations for Case 1 Earth Resistivity Height from ground surface Configuration 1.1 100 Ωm -0.5 m Configuration 1. 100 Ωm 0.05 m Configuration 1.3 100 Ωm 0.5 m Configuration 1.4 100 Ωm 1 m Configuration 1.5 100 Ωm 1.5 m Table Configurations for Case Earth Resistivity Height from ground surface Configuration.1 10 Ωm 0.5 m Configuration. 100 Ωm 0.5 m Configuration.3 500 Ωm 0.5 m Configuration.4 1000 Ωm 0.5 m Configuration.5 5000 Ωm 0.5 m Configuration.6 10000 Ωm 0.5 m - (µh/m) 0 0,00E+00 1,00E+07,00E+07 3,00E+07 f (H) Figure : Self and mutual resistance of a conductor vs. frequency (Configuration 1.) - R1 (Ω/m),50,00 1,50 1,00 0,00E+00 1,00E+07,00E+07 3,00E+07 f (H) Figure 3: Self and mutual inductance of a conductor vs. frequency (Configuration 1.) 0,07 0,06 0,05 0,04-1 -0,5 0 0,5 1 1,5 R1 Figure 4: Self and mutual resistance of a conductor vs. distance from earth (Case 1, f = 50 kh)
, 0,07 - (µη/m),1 1,9 - R1 (Ω/m) 0,06 0,05 R1 1,8-1 -0,5 0 0,5 1 1,5 Figure 5: Self and mutual inductance of a conductor vs. distance from earth (Case 1, f = 50 kh) 0,04 0 4000 8000 1000 Figure 8: Self and mutual resistance of a conductor vs. earth resistivity (Case, f = 50 kh) 35 3 - R1 (Ω/m) 5 15 5 R1-1 0 1 Figure 6: Self and mutual resistance of a conductor vs. distance from earth (Case 1, f = 30 MH) - (µh/m),5 1,5 0 4000 8000 1000 Figure 9: Self and mutual inductance of a conductor vs. earth resistivity (Case, f = 50 kh) 1,8 30 - (µh/m) 1,6 1,4 1, - R1 (Ω/m) 0 10 1-1 0 1 Figure 7: Self and mutual inductance of a conductor vs. distance from earth (Case 1, f = 30 MH) 0 R1 0 4000 8000 1000 Figure 10: Self and mutual resistance of a conductor vs. earth resistivity (Case, f = 30 MH)
- (µh/m) 1,75 1,5 1,5 1 0 4000 8000 1000 (30 MH) is the main reason for the great variation of both the resistance and inductance per unit length of our cable, as well as for the notches appearing in their variation with the cable distance from the earth and the earth resistivity. The above are presented schematically in Figures 1 and 13, where the magnetic vector equipotentials in the vicinity of a NYM 3x,5 mm cable are shown. n Figure 1 the cable current frequency is 50 H, while in Figure 13 it is 30 MH. n both cases the cable is located 1,5 m above ground surface, and the earth resistivity has a value of 100 Ωm. Figure 11: Self and mutual inductance of a conductor vs. earth resistivity (Case, f = 30 MH) n all cases two elements of the series impedance matrix are shown. The first one corresponds to the resistance per unit length of the cable s first conductor (indexed in the figures), and its self inductance (indexed in the figures). The second one corresponds to the mutual resistance per unit length between conductors 1 and (indexed R1 in the figures), and the mutual inductance per unit length between the same conductors (indexed in the figures). The numbering of the conductors is as shown in Figure 1. The expected variation of resistance and inductance with the frequency can be seen in Figures and 3 for a random configuration. Moreover, in Figures 4 to 11 the influence of the earth on the parameters under study is presented. Specifically, the variation of resistance and inductance with the cable distance from the earth is shown in Figures 4 to 7 for two different frequencies (50 kh and 30 MH), whereas their variation with the earth resistivity can be seen in Figures 8 to 11 for the same frequencies. As the frequency rises, the penetration depth δ e of the return current through the earth falls rapidly according to (11). 1 δ e = (11) πµ µ σ f 0 where: f the current frequency µ 0 the vacuum magnetic permeability µ r the relative magnetic permeability, and σ the earth conductivity, which is the inverse of earth resistivity ρ. The concentration of the current returning through earth in a thin area close to the earth surface as the frequency approaches the upper limit of our study r Figure 1: Magnetic vector equipotentials (at ωt = 0) in the vicinity of a NYM 3x,5 mm cable (Configuration 1.5, f=50 H) Figure 13: Magnetic vector equipotentials (at ωt = 0) in the vicinity of a NYM 3x,5 mm cable (Configuration 1.5, f=30 MH)
5. Conclusion The series impedance matrix for a NYM 3x,5 mm cable has been obtained for a frequency range of 50 H 30 MH by the use of a finite element approach. The presence of the earth was taken into account by the study of the variation of resistance and inductance per unit length with the cable distance from earth and the earth resistivity. Besides the skin effect and proximity effect, earth presence proves to be a dominant factor in the variation of the electrical parameters under study, especially as the frequency approaches our upper limit of 30 MH, where the penetration depth of the current returning through the earth is relatively small compared to our geometry dimensions. [7] L. Heinhold, Power Cables and their Application Part 1, Berlin: Siemens Aktiengesellschaft, 1993, p. 56. [8] L. Heinhold, R. Stubbe, Power Cables and their Application Part 1, Berlin: Siemens Aktiengesellschaft, 1993, pp. 56-61. References [1] D. Anastasiadou, T. Antonakopoulos, "An Experimental Setup for Characteriing the Residential Power Grid Variable Behavior", in Proc. 00 of the 6th nternational Symposium on Power-Line Communications and its Applications, Athens, Greece. [].C. Papaleonidopoulos, C.G. Karagiannopoulos, N.. Theodorou, C.E. Anagnostopoulos,.E. Anagnostopoulos, "Modelling of ndoor Low Voltage Power-Line Cables in the High Frequency Range, in Proc. 00 of the 6th nternational Symposium on Power-Line Communications and its Applications, Athens, Greece. [3] M. Zimmermann, K. Dostert, "A Multipath Model for the Powerline Channel", EEE Trans. Comm., vol. 50, pp. 553-559, Apr. 00. [4] G.T. Andreou, E.K. Manitsas, D.P. Labridis, P.L. Katsis, F.-N. Pavlidou, P.S. Dokopoulos, Finite Element Characteriation Of LV Power Distribution Lines For High Frequency Communication Signals, in Proc. 003 of the 7th nternational Symposium on Power-Line Communications and its Applications, Kyoto, apan. [5] D. Labridis, P. Dokopoulos, Finite element computation of field, losses and forces in a threephase gas cable with non-symmetrical conductor arrangement, EEE Trans. on Power Delivery, vol. PWDR-3, 1988, pp. 136-1333. [6] D.G. Triantafyllidis, G.K. Papagiannis and D.P. Labridis, "Calculation of Overhead Transmission Line mpedances: A Finite Element Approach," EEE Transactions on Power Delivery, anuary 1999, Vol. 14, No. 1, pp. 87-93.