parametros de cables subterraneos

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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 17, NO. 4, OCTOBER 2002 1085

Calculation of Frequency-Dependent Parameters of Power Cables: Matrix Partitioning Techniques

Richard A. Rivas , Member, IEEE, and José R. Martí , Fellow, IEEE

Abstract—In this paper, a new algorithm for the calculation of the frequency-dependent parameters of arbitrarily shaped powercable arrangements is presented. The algorithm uses digitalimagesto discretize the cable geometry and partial subconductor equiv-alent circuit method to estimate the cable parameters. The pro-posed methodology adaptsitselfto thephysical memory of thecom-puter, allowing the program to partition the partial subconductorimpedance matrices when their sizes exceed the available physicalmemory. Coaxial cables and sector-shaped cables are studied, andthe results are compared with those obtained from analytical for-mulae and the finite-element method.

Index Terms—Conductor, EMTP, frequency-dependent param-

eters, inductance, PEEC, power cables, proximity effect, PSEC, re-sistance, skin effect.

I. INTRODUCTION

S IGNIFICANT effort has been made to model power ca-bles in EMTP-type studies [1], [2]. However, to perform

detailed transients simulations, an accurate calculation of thefrequency-dependent parameters of the conductors is required.

As far as the frequency-dependent parameter calculation of power cables is concerned, very useful work has been carriedout for the analysis of cylindrically shaped arrangements [3],[4]. Difficulties arise, nonetheless, when arbitrarily shaped con-

figurations without closed-form solutions are studied. Typicalcases are oval- and sector-shaped cables, cables with concentricneutral conductors, square busbars, and power rails of transitsystems, where numerical techniques must be applied to deter-mine the parameters.

To include skin and proximity effects into the analysis, one of the alternatives of solution is to represent the cable system withequivalent coupled circuits of partial subconductors [5]–[9]. In[10], we proposed that the cable geometry discretization for ar-bitrarily shaped cases can be automatically obtained fromdigitalimages [11].

In the present paper, we discuss the details of the proposedpartial subconductor equivalent circuit (PSEC). The method and

its implementation in a general-purpose parameter calculationprogram are also observed.

Manuscript received March 30, 2001; revised February 8, 2002. R. A.Rivas was supported by “La Universidad Simón Bolívar” and “El ConsejoNacional para el Desarrollo de las Investigaciones Científicas y Tecnológicas(CONICIT),” both in Caracas, Venezuela, Apart. Postal 8900.

R. A. Rivas is with the Departamento de Conversión y Transporte de Energía,Universidad Simón Bolívar, Caracas, Venezuela.

J. R. Martí is with the Department of Electrical and Computer Engineering,The University of British Columbia, Vancouver, BC, Canada V6T 1Z4.

Digital Object Identifier 10.1109/TPWRD.2002.803827

The basic idea of the proposed technique is to draw the cablegeometry, or scan its photograph, and to use this digital image(pixel map) to automatically identify the spatial coordinatesof the square-shaped (pixel-shaped) partial subconductors intowhich the different conductors can be subdivided. Memoryrequirements and solution time, however, increase noticeablyas the frequency and the number of subconductors rise.

To overcome the memory problems, an algorithm to partitionand reduce the impedance matrices is presented. The proposedalgorithm minimizes the interaction between hard disk and cen-tral processing unit (CPU) and allows the PSEC program to cal-

culate the frequency-dependent parameters of power cable ar-rangements in reasonable time.

This paper is organized as follows: Section II presents thePSEC method, describing the derivation of the equations formulticonductor systems. Section III explains the proposed par-tition methodology. Section IV studies a single-phase coaxialcable and a three-phase sector-shaped cable and compares theresults obtained with the PSEC method with those obtainedfrom analytical solutions, the finite-element (FE) method, andapproximate formulae. Finally, Section V states some conclu-sions.

II. PSEC METHOD

A. Method Formulation

Let us suppose that an arbitrarily shaped cable geometry withconductors is subdivided into subconductors or filaments

andthat a fictitious, lossless, andcircularly shaped ring enclosesthe system under study (Fig. 1).

Let us also consider that the subconductors are sufficientlysmall and that there is mutual coupling among subconductors.Under these conditions, constant current density canbe assumedin each subconductor with different subconductors having dif-ferent current densities.

For subconductors and , for example, with a ring-returnpath , the self impedance in ohms per meter of the loop “sub-conductor /ring return” is , and the mutualimpedance in ohms per meter between the loops “subconductor /ring return” and “subconductor /ring return” is .

Resorting to the definition of geometric mean distance( ) [12], the self and mutual inductances and inH/m can be calculated as

(1)

0885-8977/02$17.00 © 2002 IEEE



RIVAS AND MARTÍ: CALCULATION OF FREQUENCY-DEPENDENT PARAMETERS 1089

TABLE IHARD-DISK TIME VERSUS RECORD LENGTH

binary data to a direct access file than to write it to a sequentialaccess file. Also, one can conclude that writing 50 MB of datawith five 10-MB records is faster (approximately 0.17 s 50.85 s) than writing 50 MB with only one record (160.49 s).

IV. CASE STUDIES

A. Coaxial Cable

Fig. 2 depicts the geometry of a coaxial cable. Its parameters,whichcan be obtained analytically through Bessel functions, areused to validate the PSEC algorithm. The sheath is consideredthereturnpath andthegroundis not included in thecalculations.

Applying segmentation [11], the cable geometry can be di-vided into twoimages—one for thecore conductor andtheotherfor the conducting sheath. Using square-shaped subconductors,subconductor densities (resolutions) of 42.9 and 29.9 subcon-ductors/in yield area discretization errors of 1.51 and 2.20% for

core and sheath, respectively. With the resolutions from before,the core is subdivided into 4337 subconductors, the sheath issubdivided into 712 subconductors, and the cable geometry isthus represented with a total of 5049 partial subconductors.

The subconductors must have sizes with the same order of magnitude as the penetration depth ( ) to take skin effects intoconsideration[14]. Given the existing circular symmetry, the hy-potenuse of every square subconductor should be at least equalto , where can be obtained from , , and arethe conductivity and permeability of conductor, and is the fre-quency in Hertz.

From the chosen resolutions, the lengths of the hypotenusesof the square subconductors of core and sheath are 0.84 and 1.2

mm, respectively, so the cutoff frequencies [6] associated withthe internal resistances should be close to 10.61 kHz for the coreand 36.56 kHz for the sheath.

Fig. 3 compares the loop parameters obtained with the twomethods using a scale proportional to along the abscissae.Four points per decade (e.g., 10, 40, 70, and 100 kHz) are con-sidered up to a frequency of 100 kHz. As shown in the plots,both models predict the right behavior of the frequency-depen-dent parameters described in [14].

The error in the loop resistance is positive (calculated resis-tance greater than actual resistance) at 10 kHz, and becomesnegative (smaller calculated resistance) at 40 kHz. The errorsare 0.12 and 1.92% at 10- and 40-kHz frequencies, respec-

tively, which bound the 10.61- and 36.56-kHz values predictedfor the cutoff frequencies of core and sheath. The maximum andminimum differences between the values of the PSEC methodand those of the analytical method are 6.49 and 0% on the re-sistance curve, occurring at 100 kHz and 0.1 Hz, and 1.11%and 0.78% on the inductance curve, taking place at 10 kHzand 0.1 Hz, respectively.

The averagesimulation timewas 59.2 min/frequency using anIntel Pentium III, 733-MHz computer with 256 MB of physicalmemory, three partitions for matrix modification, and 10-MBsubpartitions (records). Setting limits of 192 MB for physicalmemory and 1 MB for each subpartition, the simulation timewas 1 h 38 m 32 s/frequency, with two partitions for matrix con-struction, three partitions for matrix modification, and a total of 222 partitions for matrix reduction (74 for multiplication factorsand 148 for the rest of the elements).

As shown, the computation time is high, especially when thememory resources are low. However, it should be mentionedthatthese calculations need to be made only once for a given cable.After obtaining the parameters, they can be stored in data bases

for future use.Frequencies above 100 kHz could also be studied if the reso-

lutions are increased. This would result in more subconductorsand longer computation times. However, frequencies up to 100kHz are sufficient for power cable transient calculations.

In addition, there are many system studies where very highfrequencies are not needed (e.g., harmonic studies). In thosecases, one can use fewer but larger subconductors to reducethe computation time per frequency. For example, for systemstudies up to 1 kHz, resolutions of at least 13.3 and 17.7 subcon-ductors/in should be used for core and sheath, respectively. Inthat case, core and sheath are subdivided into 422 and 264 sub-conductors, respectively, and the cable geometry is representedwith a total of 686 subconductors. Under these circumstances,the average simulation time using the computer described be-fore was 10.16 s/frequency.

B. Sector-Shaped Cable

Fig. 4 shows the geometry of a sector-shaped cable studiedwith the FE method in [15] and with approximate formulaein [16]. Its parameters, which cannot be obtained throughclosed-form formulae, are calculated with the PSEC algorithmand compared with those obtained with the other two methods.The area of each sector-shaped conductor of the core is 299.76mm , the sheath is considered the return path, and the ground

is not included in the calculations. Also,and due to symmetry.Using square-shaped subconductors, resolutions of 65.5 sub-

conductors/in and 46 subconductors/in result in area discretiza-tion errors of 0.02% and 0.01% for the conductors and the con-ducting sheath, respectively. With the resolutions mentioned be-fore, each sector-shaped conductor is subdivided into 1994 sub-conductors, the sheath is subdivided into 1055 subconductors,and the cable geometry is thus represented with a total of 5043partial subconductors.

Since the lengths of thehypotenuses of thesquaresubconduc-tors of core and sheathare 0.55 and 0.78 mm, respectively, resis-tance values that are smaller than the actual resistances should



1090 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 17, NO. 4, OCTOBER 2002

Fig. 2. Geometry of coaxial cable.

Fig. 3. Resistance and inductance of coaxial cable.

start to appear after 14.44 and 416.34 kHz on the core and sheathcurves, respectively.

Fig. 5 depicts the results obtained for the loop parametersusing a scale proportional to along the abscissae and fre-quencies up to 600 kHz. The results of the PSEC method areplotted with ten points per decade, whereas the results of theapproximate method and the FE method are only plotted for thefollowing six points: 6 Hz, 60 Hz, 600 Hz, 6 kHz, 60 kHz, and600 kHz.

As shown, the PSEC method and the FE method display sim-ilar curves, anda cutoff effect is hardlynoticed at approximately490 kHz. This result can be explained by analyzing the con-ductivities of the conductors. The conductivity of the sheath ismuch smaller than the conductivity of the core, so the internalresistance of the sheath is always much greater than the internal

resistance of the core. As a result, the cutoff frequency of theloop resistances is given by that of the internal resistance of thesheath.

For the self parameters, substantial differences between theresults of the approximate method and those of the numericalmethods are only visible above 60 kHz, where the approximatemethod yields much greater values of resistance, and below 60Hz, where it produces much smaller values of inductance. Forthe mutual parameters, the approximate method yields resultstoo different from those of the numerical methods.

For the self resistance, the maximum and minimum differ-ences between the values of the PSEC method and those of theFE method are 7.12 and 1.06%, occurring at 600 kHz and 600Hz, respectively. Between the values of the PSEC method and

those of the approximate method, the maximum and minimumdifferences are 27.03% and 1.55%, taking place at 600 kHzand 6 Hz, respectively.

For the self inductance, the maximum and minimum differ-ences between the results of the PSEC method and those of theFE method are 12.48 and 0.52%, occurring at 6 kHz and 6 Hz,respectively. Between the results of the PSEC and those of theapproximate method, the maximum and minimum differencesare 17.72 and 5.98%, taking place at 6 and 600 Hz, respectively.

For the mutual parameters, the maximum and minimum dif-ferences between the values of the PSEC method and those of the FE method are 6.26 and 1.22% on the resistance curve, oc-curring at 60 kHz and 600 Hz, and 9.15 and 4.01% on the

inductance curve, taking place at 60 Hz and 600 kHz, respec-tively.The average simulation time was 58.32 min/frequency with

three partitions for matrix modification.As previously mentioned, for power cable transient studies,

a frequency of 100 kHz suffices. However, [15] reported resultsup to a frequency of 600 kHz when using the FE method. Tocompare allof themethods under thesame conditions, thePSECprogram calculated the cable parameters up to 600 kHz as well.

For symmetrical geometries, the accuracy of the proposedmethod was verifiedby analytical solutions (Fig. 3). For an arbi-trarily shaped geometry, it would be desirable to compare withmeasurements. The measured impedances provided in [16] for



RIVAS AND MARTÍ: CALCULATION OF FREQUENCY-DEPENDENT PARAMETERS 1091

Fig. 4. Geometry of sector-shaped cable.

Fig. 5. Resistance and inductance of sector-shaped cable.

the cable depicted in Fig. 4 correspond to a case with magneticpipe, but our PSEC routines at this time can still not considercases with nonuniform permeabilities.

In general, the FE element method is a formulation whose

results compare well with those of field tests. Therefore, the FEmethod was used in this case as an indication of the expectedvalues.

V. CONCLUSIONS

A methodology to calculate frequency-dependent parametersof arbitrarily shaped power cable arrangements based on the au-tomatic spatial discretization of their cross-sections and the par-tial subconductor equivalent circuit method has been presented.

The proposed methodology partitions the partial sub-conductor impedance matrices when their sizes exceed theavailable physical memory, thus allowing the PSEC program

to estimate the parameters in reasonable time even in lowrandom-access-memory (RAM) computers.

The developed algorithm can be used to calculate the series

parameters of oval- and sector-shaped cables, cables with con-centric neutral conductors, square busbars, power rails of transitsystems, and interconnections of integrated circuits.

Standard graphic formats, such as bitmap, are used to set upsubconductor coordinates of cable configurations, given eithertheir manufacturer drawings or their pictures of actual cross-sections. The use of these graphic format files is a novel ap-proach which overcomes the difficulties associated with arbi-trarily shaped power cables.

REFERENCES

[1] L. Martí, “Simulation of transients in underground cables with fre-

quency-dependent modal transformation matrices,” IEEE Trans. Power Delivery, vol. 3, pp. 1099–1110, July 1988.

[2] T. Noda, N. Nagaoka, and A. Ametani, “Phase domain modeling of fre-quency-dependent transmission lines by means of an ARMA model,”

IEEE Trans. Power Delivery, vol. 11, pp. 401–411, Jan. 1996.[3] L. M. Wedepohl and D. J. Wilcox, “Transient analysis of underground

power-transmission systems—System-model and wave-propagationcharacteristics,” Proc. Inst. Elect. Eng., vol. 120, pp. 253–260, Feb.1973.

[4] A. Ametani, “A general formulation of impedance and admittance of cables,” IEEE Trans. Power App. Syst., vol. PAS-99, pp. 902–910,May/June 1980.

[5] P. Graneau, Underground Power Transmission. New York: Wiley,1979.

[6] R. Lucas and S. Talukdar, “Advances in finite element techniques forcalculating cable resistances and inductances,” IEEE Trans. Power App.Syst., vol. PAS-97, pp. 875–883, May/June 1978.

[7] W. T. Weeks, L. L. Wu, M. F. McAllister, and A. Singh, “Resistive andinductive skin effect in rectangular conductors,” IBM J. Res. Develop.,vol. 23, pp. 652–660, Nov. 1979.

[8] P. de Arizón and H. W. Dommel, “Computation of cable impedancesbased on subdivision of conductors,” IEEE Trans. Power Delivery, vol.PWRD-2, pp. 21–27, Jan. 1987.

[9] Y.-J. Wang and J.-H. Wang, “Modeling of frequency-dependentimpedance of the third rail used in traction power systems,” IEEE

Trans. Power Delivery, vol. 15, pp. 750–755, Apr. 2000.[10] R. A. Rivas and J. R. Martí, “Calculation of frequency-dependent param-

eters of power cable arrangements using pixel-shaped conductor subdi-visions,” in Proc. Int.Conf. PowerSyst. Trans., Budapest,Hungary, June1999, pp. 335–340.

[11] R. González and R. Woods, Digital Image Processing. Reading, MA:Addison-Wesley , 1993.

[12] P. Oeding and K. Feser, “Geometric mean distances of rectangular con-ductors (in German),” ETZ-A, vol. 86, no. 16, pp. 525–533, 1965.



1092 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 17, NO. 4, OCTOBER 2002

[13] M. H. Hesse, “Electromagnetic and electrostatic transmission-line pa-rameters by digital computer,” IEEE Trans. Power App. Syst., vol. 82,pp. 282–291, June 1963.

[14] H. W. Dommel, Electromagnetic Transients Programs—Reference Manual, 2nd ed. Vancouver, BC, Canada: Microtran Power Syst.Anal. Corp., 1992.

[15] Y. Yin, “Calculation of frequency-dependent parametersof undergroundpower cables with finite element method,” Ph.D. dissertation, Univ.British Columbia, Vancouver, BC, Canada, Sept. 1990.

[16] A. Ametani and K. Fuse, “Approximate method for calculating the im-pedances of multiconductors with cross section of arbitrary shapes,” Elect. Eng. Japan, vol. 112, no. 2, pp. 117–123, 1992.

RichardA. Rivas (M’01)was born in Caracas, Venezuela,in 1965.He receivedthe electrical engineer degree and the M.Sc. degree in electrical engineeringfrom Universidad Simón Bolívar, Caracas, in 1988 and 1993, respectively, andthe Ph.D. degree in electrical engineering from The University of British Co-lumbia, Vancouver, BC, Canada, in 2001.

Currently, he is an Associate Professor at the Department of Energy Con-version and Delivery of Universidad Simón Bolívar, where he has taught since1991. From 1988 to 1991, he worked for La Electricidad de Caracas in the areasof power systems protection and substation design. His areas of interest includepower systems analysis, electromagnetic (EM) transients, and power systems

protection.

José R. Martí (F’02) was born in Spain in 1948. He received the electricalengineer degree from Universidad Central de Venezuela, Caracas, in 1971, theM.E.E.P.E. degree from Rensselaer Polytechnic Institute, Troy, NY, in 1974,and the Ph.D. degree in electrical engineering from The University of BritishColumbia (UBC), Vancouver, BC, Canada, in 1981.

Currently, he is a Professor in the Department of Electrical and ComputerEngineering at UBC, where he has taught since 1988. In Venezuela, he workedfor Exxon in the design of protective relaying and taught power systems analysisat UniversidadCentralde Venezuela.He has been involvedfor a number of years

in the development of models and solution techniques for EMTP-type real timesimulators.

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