Finite Element Characterization of LV Power Distribution Lines for High Frequency Communication Signals G.T. ANDREOU *, E.K. MANITSAS *, D.P. LABRIDIS *, P.L. KATSIS **, F-N. PAVLIDOU ** and P.S. DOKOPOULOS * * Aristotle University of Thessaloniki Dept. of Electrical & Computer Engineering Power Systems Laboratory P.O. Box 486, GR-54124, Thessaloniki, GREECE Tel: +302310996374, Fax: +302310996302 e-mail:
[email protected] Abstract The proper channel modeling is essential for reliable data transmission at high rates over the power grid. This paper presents a finite element approach for the calculation of the electrical parameters needed for the simulation of low voltage distribution cables carrying high frequency communication signals. The results obtained by our approach are validated through measurements performed on various cable types frequently installed in residential power distribution networks. 1. Introduction The conversion of networks designed to distribute electric power into communication media has been the objective of extensive research carried out over the last years. The growing demand on information exchange calls for high rate data transmission, which will in turn require the utilization of the power grid in the frequency range at least up to 30 MHz. Several problems are caused by the frequency dependent nature of the power grid, the presence of time varying loads, as well as by the structure of the grid itself, which is designed as a broadcast medium. One important aspect that may provide solutions for many problems present today is the proper modeling of the power grid as a communication medium. Models proposed in literature focus on the subject of in-home networks, i.e. the power distribution networks inside consumer premises, utilizing aspects of the transmission line theory and electromagnetic fields analysis ([1] – [3]). A common problem in such attempts is the accurate calculation of the distributed electrical parameters of the cables comprising these low voltage networks, especially in the frequency range up to 30 MHz. In this work a finite element approach is used to solve the equations that describe the electromagnetic field of the model, taking into account the skin and proximity effects.
** Aristotle University of Thessaloniki Dept. of Electrical and Computer Engineering Telecommunications Division e-mail:
[email protected]
The Finite Element Method (FEM) is a numerical method widely used for the solution of electromagnetic field equations in a region, regardless of the geometric complexity. In a recent work [4], electromagnetic field variables have been calculated using FEM and were properly linked with the transmission line equivalent circuit parameters. In our work we used the FEM to obtain the series impedances per unit length of various low voltage power distribution cables frequently installed in consumer premises. Moreover, we validated our results through laboratory measurements performed on various types of cables. These measurements were compared with similar work found in literature. During our investigation the problems caused by the topology of the power grid and the influence of the loads were not taken into account, as our main objective was the validation of the electrical parameters obtained by the FEM. 2. Distributed Cable Parameters The knowledge of the distributed parameters of a power cable (resistance R', inductance L', capacitance C' and conductance G' per unit length) permits us to calculate under specific circumstances its properties as a communication medium, using theoretical procedures. In our work we introduce the usage of a finite element approach for the calculation of the resistance and inductance per unit length of any given conductor geometry, and therefore of any given cable type. 2.1. Resistance and Inductance Calculation A system of N parallel conductors, carrying rms currents I i (i = 1 , 2 ,..., N ) is considered. The non-uniform 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 )I ∂z
(1)
another conductor j carrying current I j , where all other conductors are forced to carry zero currents, is then given by: (i, j = 1, 2, K, N )
z
i
,
i = 1, 2, ... , N
(5)
Si
where V is the voltage vector with respect to a reference, and I 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 Vi on every conductor is known for a specific current excitation. The mutual complex impedance Z ij between conductor i and
V Z ij = i Ij
∫∫ J dS = I
(2)
The self impedance of a conductor may also be calculated from (2), by setting i = j. In 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 zero currents. The corresponding voltages are recorded. o Using (2), the j-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 z 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] 1 ∂ 2 Az ∂ 2 Az + − jωσAz + J sz = 0 µ 0 µ r ∂ x 2 ∂ y 2
(3)
− jωσAz + J sz = J z
(4)
where Az is the z direction component of the magnetic vector potential (MVP). In (4) the total current density J z is decomposed in two components, J z = J ez + J sz (6) where J ez is the eddy current density and J sz the source current density, given by (7) and (8) respectively. J ez = − jωσAz (7)
J sz = −σ∇Φ
(8) FEM is applied for the solution of (3) and (4) with the boundary conditions of (5). Values for J sz i on each conductor i of conductivity σ i are then obtained and equation (2) takes the form [4], J sz i σ i V Z ij = i = (i, j = 1, 2, K, N ) (9) Ij Ij
linking properly electromagnetic field variables and equivalent circuit parameters. Finally, positive, negative and zero 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 [4]. 2.2. Capacitance and Conductance Calculation
Regarding the cable capacitance, there are formulas for its calculation known over the last decades. Smythe [6] has proposed the following formula for the calculation of the capacitance per unit length between two cylindrical conductors: D 2 − R12 − R2 2 C = 2πε cosh −1 ± 2 R1 R2
−1
(10)
where R1 and R2 the radii of the two cylindrical conductors, D the distance between their axes and ε the dielectric constant of the insulation between them. The lower sign in the formula is taken when one cylinder is inside the other and the upper sign when they are external to each other. Using this formula and taking into account the geometry and number of conductors of a given cable we can calculate its operating capacitance, i.e. its equivalent single-phase capacitance. For a geometry such as the one used in NYM cables for example, where there are three conductors placed in the vertices of an equilateral triangle (Fig. 1), the operating capacitance per unit length will be:
′ + C ′ = C12
′ ⋅ C23 ′ C13 ′ + C23 ′ C13
(11)
where Cij′ is the capacitance between conductors i and j, and the signal is injected between conductors 1 and 2 (Fig. 2). The capacitances Cij′ can be calculated with Smythe's
whereas Heinhold [9] suggests respectively a value of 0.1. Finally, Tsuzuki et. al. [10] propose values of tanδ that vary from 0.3 to 0.1 in the frequency range up to 30 MHz, based on their own measurements. For the calculation of G' we used (12) along with the values of tanδ proposed by Tsuzuki et. al. and our capacitance measurements.
formula (10). 3. Measurement configurations
During our research we used measurements in order to obtain the capacitance values for our cables in the frequency range up to 30 MHz, as well as to validate our theoretical results. In both cases we used various cable types that are frequently used in residential power distribution networks. 3.1. Capacitance measurements
Fig. 1: NYM-cable geometry
For our capacitance measurements we used a simple configuration with a signal generator (Marconi Instruments 2022D, 10kHz-1MHz AM/FM) and an oscilloscope (HP 54520A 500 Msa/s, 500 MHz). Our configuration can be seen in (Fig. 3).
Fig.2: Capacitances in a NYM-cable geometry
Another important issue concerning the cable capacitance is its variation with the frequency. Most of the work carried out regarding the basic cable parameters is valid in the vicinity of the fundamental frequency (50–60 Hz), as the idea of using the power grid for communication purposes is a fairly recent one. In our work we use therefore capacitance values obtained through extensive measurements carried out in the frequency range up to 30 MHz. The conductance per unit length G' of a given cable is usually calculated indirectly as the product of the cable capacitive susceptance and its loss tangent tanδ. G ′ = ωC ′ tan δ
(12)
The value of the loss tangent tanδ varies with the cable insulation. A value often used for PVC, an insulation material widely used in low voltage cables, is tanδ=0.01, which according to Paul [7] is an unrealistically large value in the frequency range up to 100 MHz (although the cable conductance in this case is still by two orders of magnitude less than the cable capacitive susceptance). Other authors seem to disagree in this point. Naidu and Kamaraju [8] propose a value of 0.015 – 0.02 for tanδ,
Fig. 3: Measurement configuration
Two of the cable's conductors are connected to the signal generator. The conductors are left open at the other end of the cable, so the circuit is closed via the cable's shunt capacitance. The circuit of (Fig. 3) is therefore equivalent to the one of (Fig. 4).
Fig. 4: Equivalent circuit
In (Fig. 4) VR is the voltage drop across the ohmic resistance R and VC is the voltage between the two connected cable conductors. VC will be equal to: VC =
I Cω
(13)
where I is the current running through R, C is the cable shunt capacity and ω is the angular velocity. We can obtain the value of the current I through the voltage drop across R.
Equation (13) yields: C=
VR VC ωR
(14)
In our experiments we used the oscilloscope to measure the voltages VR and VC. Subsequently we used equation (14) to obtain the cable capacity for any given angular velocity.
As stated already, the main purpose of our work was the validation of the values obtained by the FEM procedure. For this reason we only calculated the distributed parameters of the examined cables at specific frequencies. The results obtained by our measurements at these frequencies regarding the cable capacitance can be seen in (Fig. 7).
84 82
3.2. Attenuation measurements
We also conducted attenuation measurements using the configuration shown on (Fig. 5). Using BNC connectors we induced a signal of 0dBm to two of the three conductors of the cable under test and measured the scattering parameters (S21, S11) via a network analyzer (HP 8714ES 0.3MHz-3GHz). The third conductor was left open.
C' (pF/m)
80 78 76 74 72 70 68 0
5
10
15
20
25
30
Frequence (MHz)
Fig. 7: C' up to 30 MHz of a NYM 3x2,5 mm2 cable
Knowing a cable's distributed parameters we can calculate its attenuation constant α, which is defined as the real part of the propagation constant γ . Fig. 5: Attenuation measurements configuration
We tested various cable types. Both port 1 and port 2 were terminated with a resistor of 50Ω. The cables were spread out so that they were close enough only near the instrument, where they had to be connected to the ports. The cable lengths are such that enable us to observe some notches in the transfer function, probably arising due to resonance. 4. Results comparison
0,35
0,38
0,30
R' (Ω/m)
R'
0,34
0,20 0,15
0,32
0,10
L'
0,30
0,05 0,00
L' (µH/m)
0,36
0,25
a = Re{γ }
(16)
The attenuation constant α regarding the NYM, 3x2,5 mm2 cable as well as its attenuation in the examined frequency range (for a length of 30 m) are shown in (Fig. 8) and (Fig.9) respectively. Our theoretical results were very satisfactory at the selected frequencies, as can be seen in (Fig. 10), where results of the corresponding measurements are presented. Of course, the frequencies used weren't enough for a satisfactory overall depiction.
α (Np/m)
The FEM procedure we used provided us with the results shown in (Fig. 6) regarding the cable resistance R' and inductance L' per unit length. The results shown were obtained from a NYM 3x2,5 mm2 cable.
(15)
γ = ( R'+ jL' ω )(G '+ jC ' ω )
0,03 0,025 0,02 0,015 0,01 0,005 0 0
5
10
15
20
25
Frequency (MHz)
0,28 0
5
10
15
20
25
30
Frequence (MHz)
Fig. 6: R', L' up to 30 MHz of a NYM 3x2,5 mm2 cable
Fig. 8: Attenuation constant α of a NYM 3x2,5 mm2 cable
30
0
5
10
15
20
25
0 Attenuation (dB)
on Power-Line Communications and its Applications, Athens, Greece, March 27-29 2002
30
[2]
-2 -4 -6
[3]
-8 Frequency (MHz)
Fig. 9: Attenuation of a 30 m long NYM 3x2,5 mm2 cable
[4]
[5]
Fig. 10: Attenuation measurements for a 30 m long NYM, 3x2,5 mm2 cable
5. Conclusions
[7]
An important aspect of the research on powerline communications is the accurate calculation of the distributed parameters of the cables comprising the low voltage power distribution networks, especially in the frequency range up to 30 MHz. We propose a Finite Element procedure for the calculation of the resistance R' and inductance L' per unit length of these cables. Furthermore, we present analytical information on the calculation of the corresponding conductance G' and capacitance C' per unit length. Our theoretical procedures are validated through measurements performed on various cable types frequently installed in residential power distribution networks. More extensive measurements are essential for the overall depiction and the better understanding regarding the behavior of the power cables as communication media.
References
[1]
[6]
D. Anastasiadou, T. Antonakopoulos, "An Experimental Setup for Characterizing the Residential Power Grid Variable Behavior", Proceedings of the 6th International Symposium
[8]
F.J. Canete, L. Diez, J.A. Cortes, J.T. Entrambasaguas, "Broadband Modelling of Indoor Power-Line Channels", IEEE Transactions on Consumer Electronics, Volume: 48 Issue: 1, On page(s): 175 – 183, Feb. 2002 I.C. Papaleonidopoulos, C.G. Karagiannopoulos, N.J. Theodorou, C.E. Anagnostopoulos, I.E. Anagnostopoulos, "Modelling of Indoor Low Voltage Power-Line Cables in the High Frequency Range", Proceedings of the 6th International Symposium on Power-Line Communications and its Applications, Athens, Greece, March 27-29 2002 D.G. Triantafyllidis, G.K. Papagiannis and D.P. Labridis, "Calculation of Overhead Transmission Line Impedances: A Finite Element Approach," IEEE Transactions on Power Delivery, January 1999, Vol. 14, No. 1, pp. 287-293. D. Labridis, P. Dokopoulos, “Finite element computation of field, losses and forces in a three-phase gas cable with non-symmetrical conductor arrangement”, IEEE Trans. on Power Delivery. vol. PWDR-3, 1988, pp. 1326-1333. W.R. Smythe, Static and Dynamic Electricity, Mc Graw-Hill, p. 78, 1950. C.R. Paul, Analysis of Multiconductor Transmission Lines, Wiley-Interscience, p. 159, 1994. M.S. Naidu, V. Kamaraju, High Voltage Engineering, 2nd Edition, Mc Graw-Hill, p. 85
[9]
L. Heinhold, Power Cables and their Application, 3rd revised Edition, Pt. 1, p.333, 1990.
[10]
S. Tsuzuki, S. Yamamoto, T. Takamatsu, Y. Yamada, "Measurement of Japanese Indoor Power-line Channel", Proceedings of the 5th International Symposium on Power-Line Communications and its Applications, Malmö, Sweden, April 4-6, 2001, pp.79-84.
