Validación de la Síntesis de Laplace a través de Mediciones en Sistemas de Cables Subterráneos

Uribe-Campos Felipe Alejandro

Departamento de Mecánica Eléctrica, División de Ingenierías, Universidad de Guadalajara, CUCEI, Jalisco. E-mail: fauribe@ieee.org

Abstract

Underground cable electrical parameters Zϒ as well as their modal propagation characteristics are highly frequency dependent which in certain cases turns its analysis difficult. To perform electromagnetic transient studies of cables the calculation of electrical parameters is essential to obtain the waves propagation solution through the multiconductor system. At the same time this requires to solve the inverse Laplace transform on a numerical form. Although the analytic Laplace transform has an indisputable accuracy, the application of its numerical version up-to-date has not been completely accepted. A complete methodology is developed in this work to guide analyst engineers or graduate students in the calculation of electromagnetic transients of underground cable systems. Finally, to help the validation of the numerical inverse Laplace transform a scaled prototype experiment is performed in the laboratory in which a transient step-response at the remote end of an energized conductor is measured.

Keywords: transient analysis, underground power cables, ground return, Laplace Transform, laboratory measurements.

Resumen

Los parámetros eléctricos Zϒ de cables subterráneos y sus características de propagación modal son altamente dependientes de la frecuencia lo que en ciertos casos dificulta su análisis. Al realizar estudios de transitorios electromagnéticos en cables el cálculo de parámetros es primordial para obtener la solución de la propagación de ondas a través del sistema multiconductor. Esto a su vez requiere resolver la transformada inversa de Laplace en forma numérica. Aunque la transformada analítica de Laplace tiene una indiscutible precisión, la aplicación de su versión numérica no ha sido, hasta la fecha, totalmente aceptada. En este trabajo se desarrolla un metodología completa para guiar al ingeniero analista o estudiantes de posgrado al cálculo de transitorios electromagnéticos en sistemas de cables subterráneos. Finalmente, para ayudar a validar la transformada numérica inversa de Laplace se desarrolla en el laboratorio un experimento escalado prototipo, en el cual se mide la respuesta transitoria a un escalón de voltaje en el extremo receptor de un cable energizado.

Introduction

An accurate calculation of electromagnetic (EM) transients on buried cable systems may take into account the Skin Effect in the ground and cable conductors, while relaxation effects on cable insulation layers (Wedepohl and Wilcox, 1973; Dommel, 1986; Marti, 1982; Marti, 1988; Uribe et al., 2002; Schellkunoff, 1934; Pollaczek, 1926; Uribe et al., 2004; Semlyen, 1985; Saad et al., 1996; Wedepohl, 1983). Consequently, the modal propagation functions of the cable system are highly influenced by the inductive loops formed between the power cables through the ground (Wedepohl and Wilcox, 1973). Thus, due to electromagnetic inductions phenomenon, the ground model is evaluated in this paper by means of two approaches for a qualitative comparison: the numerical solution of the exact Pollaczek’s integral and using the classical approximate formulas previously issued by Wedepohl, Ametani and Semlyen (Wedepohl and Wilcox, 1973; Dommel, 1986; Semlyen, 1985; Saad et al., 1996).

A benchmark model for calculating the voltage transient step-responses at the remote-end of a buried cable system is proposed in this paper (Wedepohl and Wilcox, 1973; Dommel, 1986). The transient step-response is synthesized in this paper through the Numerical Laplace Transform (NLT) (Uribe et al., 2002; Wedepohl, 1983).

Finally, the calculated responses are qualitative, validated here through a laboratory measurement performed on a scaled prototype experiment.

A Benchmark case for the Electromagnetic Transient Analysis of Underground Cables

The voltage and current wave propagation in underground cable transmission systems is described by (Wedepohl and Wilcox, 1973; Dommel, 1986; Marti, 1982; Marti, 1988; Uribe et al., 2002):

C₁ and C₂ are the integration constant vectors determined by the boundary conditions, H_± is the transmission system propagation function, Y_C and Z_C are the characteristic admittance and impedance matrices, respectively (Uribe et al., 2002).

An underground cable transmission system of length, l, as the one shown in Figure 1, can be represented by a two port network. A nodal admittance representation, relates the voltage and current responses from the sending end at z = 0 to the remote end at z = l (Wedepohl and Wilcox, 1973; Dommel, 1986; Marti, 1982; Marti, 1988; Uribe et al., 2002; Schellkunoff, 1934):

where V₀(s)and I₀(s) are the voltage and current vectors at the sending-end of the line z = 0. V_l(s) and I_l(s) are vectors of voltages and currents at remote-end of the transmission line or, in this case, cable system z = l.

Accurate calculation of ZY

The flat buried cable system shown in Figure 1a formed three coupled loops between cables through the ground. In addition, each cable (as shown in Figure 1b) formed internal and external loops between the nucleus and the sheath, respectively. The internal loop is due to the impedances of the nucleus, EP insulation and the internal sheath, while the external loop if due to the impedances of the external sheath, the PVC jacket and the ground (Wedepohl and Wilcox, 1973). ]]>

A well known expression for calculating the nucleus impedance of a cylindrical cable conductor is (Wedepohl and Wilcox, 1973):

Where I₀ and I₁ are the zero and first order modified Bessel functions, p is the skin-effect layer thickness and the other variables are defined at the nomenclature.

The insulation impedances between two contiguous cylindrical cable conductors with radii r_ext and r_int as shown in Figure 1b are given by (Wedepohl and Wilcox, 1973):

The internal, external and mutual impedances of the tubular sheath conductor are given by the Schellkunoff theory for cylindrical formulae (Schellkunoff, 1934):

where ς_int = r_s-i / p, ς_ext = r_s-e / p and the Wronskian is W = K₁(ς_int) x I₁(ς_ext) – K₁(ς_ext) I₁(ς_int).

The ground return impedances Z_G(ω) are calculated in this paper through two different approaches for a qualitative comparison purpose.

First, the Pollaczek’s integral is solved with the efficient, accurate and reliable algorithmic strategy proposed in Uribe et al. (2004). For comparison, the direct numerical integration is implemented here by using the adaptive Gauss-Lobatto quadrature routine, quadl, available from Matlab^® v7 (Gander and Gautschi, 2000).

Then the closed-form solutions previously issued by Wedepohl, Ametani and Semlyen are also implemented in this paper to verify their application accuracy ranges (Wedepohl and Wilcox, 1973; Dommel, 1986).

On assuming a Quasi-TEM_Z propagation mode, the self and mutual ground-return impedance Z_G(ω) of the cable system are given by (Wedepohl and Wilcox, 1973; Pollaczek, 1926; Uribe, 2004):

where J is the Pollaczek’s integral, β is the dummy variable and the other variables are listed in the nomenclature. The combination of regular and irregular oscillations due to the complex exponential factors in (5b) provokes convergence problems when applying generic quadrature routines at certain physical variable application ranges (Uribe et al., 2004).

On introducing in (5b) the variable change b = u/|p| and the following defined dimensionless parameters, we obtain

and after some algebraic manipulations, we have

To implement the direct numerical integration of Pollaczek’s integral (5b) may be transformed into:

The truncation criterion for (7a) is based on the properties of the damping exponential factor in (6a) when F(u ) → u. Thus, for the second factor in (6a) the entire range of u can be split-up into:

where ε _rel is the relative error defined in this paper as:

with a truncation limit

After Pollaczek’s mathematical statement in 1926, electrical engineering researchers have looked forward for many years of closed-form approximations to Z_G(ω) (Wedepohl and Wilcox, 1973; Pollaczek, 1926). Some of the most often used in EMTP applications are the ones proposed by Wedepohl, Ametani and Semlyen.

where γ is the Euler constant.

Another important approximation is the one implemented in the cable constants routine of the EMTP program (Dommel, 1986). Basically here, Ametani replaces Pollaczek integral by the one of Carson, assuming in (5b) the following consideration (Dommel, 1986):

Around 1982, Wedepohl conjectured a formula for calculating the self Z_G of a buried conductor. The formula is based on the complex depth penetration of the electromagnetic fields in the ground. Subsequently, in 1985 Semlyen reported the following formula (Semlyen, 1985):

where r_PVC is the external cable radius (Figure 1b), over the outer insulation PVC (Polyvinyl chloride) jacket.

In addition, Saad, Gaba and Giroux published a very interesting closed-form approximation to Pollaczek integral based in the complex ground return plane and in the Cauchy’s integral theorem solution (Saad et al., 1996). The derivation process of this formula is very similar to the early one employed to obtain a simplified model of the Carson’s integral for calculating Z_G of aerial lines (Dommel, 1986):

It can be seen in Figures 2 and 3, that in both impedance calculations the closed-form approximations as well as the direct numerical quadrature of Gauss/Lobatto are in good agreement with the algorithmic solution developed by Uribe et al. (2004) used in this paper to establish a benchmark for calculating Z_G(ω).

However, there are practical engineering cases when the distance between cables x (Figure 1) is considerably long, or when the cable trench is surrounded by soil with a very low resistivity (Dommel, 1986).

Appendix I shows the calculated mutual external loop of Z_G(ω) for the buried cable transmission system shown in Figure 1, but with a separation distance between cables of x = 30 m and with an homogeneous ground conductivity of σ = 1 S/m.

The dispersive dielectric effects of each cable insulation layer (better known as relaxation effects), are introduced into the electromagnetic transient analysis by the shunt admittance matrix, based in the following relation (Dommel, 1986):

Y(ω)= G (ω) + J · ω C(ω)

where ε _r(ω ) = ε _r’(ω) – j · ε _r´´(ω) is the frequency dependent complex permittivity, which has a vector relation to the loss tangent loss factor through the electrical conductances as follows (Dommel, 1986):

where tan δ is the insulation loss factor. Another option to evaluate the complex permittivity is to synthesize ε_r through a multi-term order Debye or Cole-Coles model (Dommel, 1986):

where ε_∞ is the very high frequency permittivity value, N is the number of relaxation terms, τ _i, used for the fitting, Δε  = ε _s –  ε_∞  and ε _s is the static frequency permittivity value.

Modal propagation properties

The characteristics of wave attenuation and velocities are thus explained by The Theory of Natural Modes of Propagation by Wedepohl (Wedepohl and Wilcox, 1973; Dommel, 1986). The voltage and current wave propagation modes of the system are characterized by H_±(ω) and Z_C(ω) in equiations (2c) and (2e), respectively.

Figure 4 shows the decreasing monotonic behavior of |Z_C| and |e ^{±√(Zϒ)x·l}| which have been evaluated with two different Z_G(ω) models (Pollaczek and Wedepohl). The mode switching effect on H_±(ω) has been removed by using the alternate method proposed by Wedepohl in (Wedepohl et al., 1996) for calculating transformation matrices tracking the order of eigenvectors and eigenvalues with their previous one corresponding frequency. The Z_C of the system directly depends on the relation between frequency dispersion conductor effects and on the insulation relaxation effects of the cable.

In Figure 4a the influence of the ground-modes is noticeably greater for the metallic conductor loops according to the high inductive Z_G(ω) at the low frequency range as seen in Figures 2b and 3b. It can be noticed from Figure 4b, that the propagation function depends on the product of the cable parameters Zϒ. Where the influence of the high inductive Z_G(ω) is inversely related to the cable system modes. This can be corroborated from the Wedepohl model for Z_G(ω) shown in Figures 2b and 3b, both according behavior in Figure 4b.

Frequency response

The accurate calculation of Zϒ has been performed in the evaluation of parameter matrices A and B in (Equations 3a and 3b) with the Pollaczek and Wedepohl ground-return models. The nodal representation in (3a) is shown in Figure 5 where the cable system length l = 40 km. The open-circuit voltage and short-circuit current frequency responses are shown in Figure 6.

The network elements connected to the transmission cable system are represented by generalized admittances in the sending end with Ys and in the remote end with ϒ _R.

The voltage at the remote end, V_l, and the injected current at the sending end, I_s, are related to the boundary conditions as shown in Figure 5 (Uribe et al., 2002):

H(s) is the transfer function of the network system. Thus, the open-circuit voltage responses for the cable system in Figure 5 are calculated in the following paper section.

Numerical Laplace Inversion

The voltage waveform response at the remote end of the cable system in Figure 5 is synthesized in this paper through the inverse NLT (Wedepohl and Wilcox, 1973; Wedepohl, 1983):

The discretization of (11a) leads to the numerical solution of Vz(t) at z = l where T = mΔt and Ω = nΔs as follows (Uribe et al., 2002; Wedepohl, 1983):

where N is the number of time samples and σ _n is the data window which is used for attenuating Gibbs phenomena errors (Wedepohl, 1983). The following Vonn Hann window is applied:

where Ω is the frequency truncated range. The frequency domain discretization of V_ℓ(s) provokes frequency leaking in the time domain. The damping Laplace factor, c, is thus used to quench frequency leaking errors. However, since Gibbs error is not completely eliminated by data windows, it would be amplified by the un-damping function, exp(cmΔt), in (11b). Thus, the selection of a value for c is a trade off. The following criterion has been proposed by Wedepohl (1983):

where T = NΔt is the observation time for the transient and ε is the error level whose lower bound is determined by N. The discretization frequency step Δω at (11b) is implicitly considered as follows:

where NΔω/2 is the truncation frequency for the discrete representation of V_ℓ(s). Consider now from (11c):

The left-hand side of (11f) is the sampling frequency while the right hand side is twice the truncation frequency. In addition (11f) agrees with the Nyquist sampling criterion.

Scaled prototype laboratory measurements

An initial scaled experimental setup has been implemented in this paper to perform a qualitative comparison between the obtained voltage measurements in the laboratory and the NLT methodology. A thin wire of 1.302 mm² (16 AWG) for 600V with r = 1.25 mm, h = 0.1 m, ρ_CU = 1.72 × 10^-4 Ω · m, ρ_g = 1000Ω · m (it is assumed that the laboratory has a solid rocky soil without moisture), and a cable length of l = 35 m has been used for the scaled test.

A voltage source of 5V is switched at the sending end of the cable conductor as is shown in channel-1 of the oscilloscope obtained in Figure 7a. The transient step voltage response is measured with a TDS2024 oscilloscope at the remote end of the cable as is shown in channel-2 in the same figure.

In Figure 7b, the comparison between both measured voltages and the synthesized Laplace voltage response is illustrated having a good agreement. However, a small attenuation between both measured and synthesized voltage responses can be noticed in this figure. It is probably that the attenuation difference between both curves is due to a mistaken measurement taken from the ground resistivity of the laboratory.

Conclusions

An accurate methodology for calculating Zϒ parameters for buried cable systems is developed in this paper. Ground-return impedanceshave been calculated here by solving the Pollaczek integral through direct numerical integration and with an algorithmic strategy proposed by the author.

Both, the measured and the synthesized voltage transient step-responses are in good agreement. Thus, it is possible to suggest a qualitative validation for the NLT technique in power transient analysis applications.

Nomenclature

References

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Citation for this aricle:

Chicago citation style
Uribe-Campos, Felipe Alejandro. Laplace Synthesis Validation Through Measurements on Underground Transmission Cables. Ingeniería Investigación y Tecnología, XV, 04 (2014): 561-584.

ISO 690 citation style
Uribe-Campos F.A. Laplace Synthesis Validation Through Measurements on Underground Transmission Cables. Ingeniería Investigación y Tecnología, volume XV (issue 4), October-December 2014: 561-584.

Felipe Alejandro Uribe-Campos.  Received the B. Sc. and M. Sc. degrees of Electrical Engineering, both from the State University of Guadalajara, in 1994 and 1998, respectively. During 2001 he was a visiting researcher at the University of British Columbia, B.C., Canada. In 2002 he received the Dr. Sc. degree of Electrical Engineering from the Center for Research and Advanced Studies of Mexico. The dissertation was awarded with the Arturo Rosenblueth prize. From 2003 to 2006 was a full professor with the Electrical Graduate Program at the State University of Nuevo Leon, México. From May 2006. He joined the Electrical Engineering Graduate Program at the State University of Guadalajara, México, where he is currently a full time researcher. Since 2004, he is a member of the National System of Researchers of Mexico. His primary interest is the electromagnetic simulation of Biological tissues for early Cancer detection and power system harmonic and transient analysis.