nonlinear-coupled electric-thermal modeling of underground cable systems

4 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 1, JANUARY 2006

Nonlinear-Coupled Electric-Thermal Modelingof Underground Cable Systems

Niksa Kovac, Ivan Sarajcev, and Dragan Poljak, Member, IEEE

AbstractAn original nonlinear-coupled electric-thermalmodel of underground cables with the solid sheaths is proposed.The model deals with the numerical evaluation of losses, heating,and ampacity.

The computation of the current dependent losses is under-taken by means of the filament method, where conductors andsheaths are represented by a number of smaller subconductorsor filaments. Furthermore, heat-transfer phenomena through aninfinite domain beneath the soil surface are modeled combiningthe finite and the mapped infinite elements, respectively. The cor-responding finite-element meshes are generated by the advancingfront method.

The numerical results presented throughout this work suggestthat the International Electrotechnical Commission relation con-cerning the external thermal resistance for touching cables, placedin flat formation, having appreciable sheath losses, should be re-ex-amined.

Index TermsAmpacity, numerical modeling, underground ca-bles.

I. INTRODUCTION

LOSSES, heating, and ampacity are unavoidable parame-ters in underground cable design depending on cable ma-terials, laying condition of the cable system, thermal propertiesof the media, bonding arrangement, etc.

Generally, numerical methods provide more accurate mod-eling of underground cable systems than purely analytical oranalytical/numerical techniques. However, numerical methodsare sometimes too complex to be handled by engineers. Someof the papers dealing with a numerical approach to the anal-ysis of underground cable losses [1][4], external or internalthermal resistances [5][10], as well as heating and/or ampacitycalculations [11][30] are highlighted in this work. On the otherhand, rare are the papers dealing with the complete numericalapproach for the evaluation of all quantities of interest (i.e.,losses, heating, and ampacity). Thus, losses have often been as-sumed to be known values, or computed by means of Interna-tional Electrotechnical Commission (IEC) or NeherMcGrathrelations, while heat-transfer phenomena have been handled vianumerical methods using losses as input data [11][17], [19],[21][23], [27][30].

Manuscript received April 7, 2004; revised September 4, 2004. Paper no.TPWRD-00173-2004.

N. Kovac and I. Sarajcev are with the Department of Electrical Engineering,Faculty of Electrical Engineering, Mechanical Engineering and Naval Archi-tecture, Split University, Split 21000, Croatia (e-mail: [email protected]; [email protected]).

D. Poljak is with the Department of Electronics, Faculty of Electrical Engi-neering, Mechanical Engineering and Naval Architecture, Split University, Split21000, Croatia (e-mail: [email protected]).

Digital Object Identifier 10.1109/TPWRD.2005.852272

Hwang [18] has proposed the finite-element modeling of bothlosses and heating for underground cable systems. The numer-ical computation of these parameters always requires consider-ably high computational effort. In [18], the cable cross-sectionhas been approximated by an octagon. Also, the shield and di-electric losses have been incorporated into the model as heatsources distributed at the cable surface. However, this simpli-fied modeling of the cable cross-section is not quite satisfactory,since the temperature drop between conductor and cable sur-face could be significant. Moreover, by posing the Dirichletsboundary conditions at a certain distance from the cable struc-ture to solve the heat conduction equation, an infinite domainbeneath the soil surface has been truncated. The truncation of thedomain size represents a compromise between opposite require-ments: the higher accuracy of numerical results and the lowercomputational cost.

It has been shown in [27] that the cable domain tempera-tures have been influenced by the truncation, even in the casewhen the truncated domain boundaries have been placed quitefar away from the cables. The more accurate treatment of theinfinite domain containing underground cables can be carriedout by the integral or integro-differential equation formulation.Thus, the application of the boundary-element method (BEM)and the coupled finite/boundary-element approach (FEM/BEM)have been reported in [28], [29], and [30], respectively.

The present work deals with an original nonlinear coupledelectric-thermal model of underground cables with the solidsheaths. The proposed model provides numerical evaluationof losses, heating, and ampacity. The computation of cur-rent dependent losses is performed by means of the filamentmethod [4], [31], [32], where conductors and sheaths arereplaced by a number of smaller subconductors or filaments.Furthermore, heat-transfer phenomena through an infinitedomain beneath the soil surface are modeled combining thefinite elements and the mapped infinite elements [33], [34].In particular, the finite-element meshes are generated by theadvancing front method [35]. The accurate thermal modelingof both cable cross-sections and surroundings is performedby the second-order isoparametric elements. Some illustrativecomputational results, presented throughout this work, clearlydemonstrate the efficiency of the proposed approach.

II. NONLINEAR-COUPLED ELECTRIC-THERMAL MODEL

The electric-thermal model is outlined in a few steps: 1) lossesevaluation; 2) heating evaluation; 3) electric-thermal coupling;and 4) ampacity evaluation.

0885-8977/$20.00 2006 IEEE

KOVA C et al.: NONLINEAR-COUPLED ELECTRIC-THERMAL MODELING OF UNDERGROUND CABLE SYSTEMS 5

Fig. 1. Representation of an electrical section of a cable transmission line.

A. Losses EvaluationIn order to determine heating of underground cable system,

the system losses have to be known. The current dependentlosses of cables with the solid sheaths are computed by meansof the filament method [4], [31], [32]. The calculation of the di-electric losses is considered as a rather straightforward task, anddetails can be found elsewhere [31]. Sheaths are predominantlymade of aluminum, lead, or lead alloy. The filament method pro-vides conductors and sheaths to be represented by a number ofsmaller subconductors or filaments, sufficiently small to assumethe uniform current density. Moreover, the governing equationsare formulated using the additional assumptions: 1) cables arearranged in parallel, and 2) cable line is longitudinally homoge-nous. The skin and proximity effect are taken into account bythe filament method, as well. An electrical section of a transmis-sion line composed of three single-core cables is considered, asshown in Fig. 1.

The line is assumed to be a part of an earthed circuit. Thesheaths are solidly bonded and earthed at both ends. The con-ductors and sheaths are divided into and filaments, re-spectively. Hence, the total number of filaments is

. The earthed circuit can be replaced by a balanced oper-ating voltage system , as well as a balanced loadimpedance system (Fig. 1). Thissimplified circuit modeling is satisfactory for the losses evalua-tion, since the systems are used only to set up the correspondingcurrents flowing through the filaments.

The filament currents are calculated by the mesh-currentmethod. Each mesh is represented by a loop consisting of:1) the associated conductor or sheath filament; 2) both theassociated voltage and the load impedance phase, if the loopcontains the conductor filament; and 3) the ground return path.The corresponding matrix equation can be written as follows:

(1)where

vector of the loop voltages;matrix of self and mutual impedances of the

loops;vector of the filament currents.

Equation (1) can be written in a more convenient matrix form

(2)

where

.

.

.

.

.

.

.

.

.

are vectors of phase voltages concerning the loops withthe conductor filaments; stand forself and mutual impedance submatrices of the loops withthe filaments of the conductors , and , respectively;

denote mutual impedance sub-matrices between the loops with the filaments of the differentconductors; is mutual impedance submatrixbetween the loops with the conductor filaments and thosecontaining the sheath filaments; stands forself and mutual impedance submatrix of the loops with thesheath filaments.Elements of the submatrices are given by

(3)

wheremutual impedance between th and th filamentin the presence of the ground return path;self impedance of the filament in the presence ofthe ground return path.

The mathematical details regarding the assessment of the im-pedances and , as well as their testing procedure, can befound in the Appendix.

The elements of the other impedance submatrices are

(4)

The vector of the filament currents can be obtained by simplyinverting the matrix in (2).

If the cables are a part of a circuit with the insulated neutrals,there is a voltage difference between these points, providingthe following condition equation is to be posed:

(5)

The resulting matrix equation then can be written as follows:

.

.

.

.

.

.

.

.

.

(6)

The matrix equation (6) contains equations withunknowns (i.e., filament currents and the voltage difference

). Solving the matrix (6), the unknown currents and voltagedifference are obtained.


Fig. 2. Boundary conditions.

Knowing the filament currents, the losses of the conductorsand sheaths are simply computed from the relations

(7)

(8)

wherefilaments of conductors;filaments of sheaths;resistances of conductor and sheath fila-ments, respectively.

The resistance of each filament is computed on the basis ofthe average temperature of its associated metallic partmp(i.e., the conductor or sheath).B. Heating Evaluation

Two-dimensional steady-state temperature distribution in theinfinite domain beneath the soil surface, generated by un-derground cables (assuming negligible moisture migration inthe cables vicinity) is governed by the heat conduction equation

(9)where is the thermal conductivity coefficient and is the heatgenerated per-unit time and volume. The following boundaryconditions [40], associated with the heat conduction equation,are:

on (10)on (11)

wheretemperature at infinity;portion of the domain boundary at infinity;heat flux on the ground surface;portion of the domain boundary with the heat flux ;convective heat-transfer coefficient by which the av-erage radiation is included [14], [18];air temperature, as shown in Fig. 2.

Using the weighted residual approach and applying theGalerkinBubnov scheme of the finite-element method, thefollowing matrix equation is obtained:

(12)where

global conductivity matrix;vector of the unknown values of the nodal tempera-tures;global load vector.

The global conductivity matrix as well as the load vectorare assembled from each element

(13)

(14)

wheretotal number of elements;area of an element;number of nodes assigned to any ele-ment;vector of the shape functions of anelement;heat generated within an element areaper unit time and volume;portion of the boundary of whichlies on .

Heat sources associated with the current dependent losses areconductors and sheaths (i.e., the metallic partsmp). As theyare not placed alongside a domain boundary , the last term in(14) concerning the load vector of the metallic parts equals zero.The heat generated per-unit time and volume within an element

of the cable metallic part is determined by

(15)

where denotes the losses generated within element ,is the area of element , and stands for the length

of a cable section. Due to the rather high thermal conductivityvalues of the metallic parts, the uniform losses distribution canbe assumed over a particular conductor or sheath through theheating evaluation. On the other hand, the nonuniform distribu-tion is taken into account through the losses evaluation by meansof the filament method, since the nonuniformity can affect thetotal value of a particular sheath losses and, consequently, thecable heating. Therefore, it follows:

(16)

wheretotal losses of a particular conductor or sheath con-taining element ;cross-section of a particular conductor or sheath con-taining element .


Fig. 3. Two-dimensional mapping.

Hence, the load vector of the metallic part element can bewritten in the form

(17)

The infinite domain beneath the ground surface is treatedvia the mapped infinite elements [33], [34]. Applying themethod of images, the heat transfer of underground cables canbe considered as the dipole source-type problem. Two-dimen-sional (2-D) dipole-source-type problems have been solvedvery accurately using the mapped elements [33]. Consequently,the application of the mapped infinite elements is expectedto be well suited for the treatment of underground cables.Furthermore, if the finite/infinite approach is applied, one canretain the differential equation formulation (9).

Since the detailed theoretical background of the mapped infi-nite elements can be found in [33], for the sake of brevity, onlythe basic concepts are given in this work. The algorithm is basedon a simple mapping of the global infinite element into the localfinite element. The mapping of a 2-D quadratic infinite element,Fig. 3, can be written as

(18)

(19)

whereglobal coordinates;local coordinates;global nodes coordinates of an infi-nite element;mapping functions;standard Lagrange shape functions.

The mapping functions are given in the form

(20)

(21)

The mapped elements retain the finite-element integrationweights and abscissae and shape functions as well.

C. Electric-Thermal CouplingThe nonlinear behavior of the electric-thermal model is

caused by an electrical conductivity of a material, whichdepends on the corresponding temperature value. The iter-ative procedure of the temperature matching starts with theevaluation of losses according to Section A, by using thearbitrarily chosen conductor and sheath temperatures as wellas the load impedance . Assembling the global equationsystem (12) yields the first iteration of nodal temperatures

, where denotes the total number ofnodes. Subsequently, the average temperature of a particularconductor and sheath is calculated by the following formula:

(22)where

.

.

.

and is the total number of elements associated with a partic-ular conductor or sheath, while stands for the total numberof conductors and sheaths.

Using thus obtained average temperatures, the losses arecomputed again. The procedure goes on repeatedly as long asthe prescribed permissible temperature discrepancy throughsuccessive iterations is achieved.

It is to be mentioned that the noncoupled model incorpo-rates the computation of the conductor and sheath resistances onthe basis of the arbitrarily assumed temperatures, thus causingthe non-negligible current and losses calculation errors. For ex-ample, the discrepancy of 15 C between the assumed and actualtemperature of aluminum conductor results in the correspondingresistance error of around 5%.

D. Ampacity EvaluationIf the ampacity evaluation is of interest, as well, a new ex-

ternal iterative procedure of the load impedance matching isrequired. As a matter of fact, it is necessary to determine theimpedance and, hence, the corresponding currents, whichgive the temperature rise to cable insulation up to the permis-sible value . Cable losses are not affected by the load powerfactor , for the constant absolute value of . Thus, theproblem can be reduced to the assessment of the load resistance

only. The load resistance in the th iteration is obtainedusing the linear interpolation of the maximum temperature re-sults obtained for cable insulation within the iterations and

(Fig. 4).The external iterative procedure ends up when the prescribed

permissible discrepancy between and is achieved.The complete internal iterative procedure of the temperaturematching is performed through each external iteration. The


Fig. 4. Assessment of the load resistance.

Fig. 5. Cable system domain.

Fig. 6. Boundary between the finite and infinite elements.

ampacity equals the conductor currents flowing in accordanceto the load resistance value evaluated in the last iteration.

III. MESH GENERATIONLosses, heating, and ampacity are evaluated for the un-

derground cable system in a flat arrangement. Three 35-kV,direct-buried 400 mm copper conductor cables having im-pregnated paper insulation and aluminum sheath are shownin Fig. 5. An automatic mesh generation, as a part of the fi-nite-element approach, is an important step for the accuracy ofsolution and reduction of computational time. The advancingfront method [35] has been widely used for the meshing ofarbitrarily shape domains. This meshing procedure starts withthe polygonal [polyhedral in three-dimensional (3-D)] dis-cretization of material boundaries. The only input data requiredfor the mesh generation are the boundary nodes arising fromthe discretization. The complete mesh generation algorithm hasbeen promoted in [35].

The soil meshing is performed using a software packagebased on the advancing front method. Additional software isdeveloped to mesh the cable cross-section, consisting of thespecific shape materials, such as cylindrical conductor andtubular insulation, sheath, and covering. The boundary betweenthe finite and infinite elements is chosen in accordance toFig. 6 to avoid the high-temperature gradients inside the infiniteelements. The finite-element mesh of the cable cross-section isshown in Fig. 7. The curved edges of the six-noded elements aresomewhere drawn by two lengths connecting the correspondingnodes due to the graph determination with less difficulty. The

Fig. 7. Mesh of the cable cross-section.

(a)

(b)Fig. 8. (a) Mesh of the surrounding soil, cable spacing s = 0 mm. (b) Anenlarged section in the cable vicinity, cable spacing s = 0 mm.

surrounding soil meshes concerning the different cable spac-ings (touching mm; double brick thickness140 mm) are shown in Figs. 8 and 9, respectively.

IV. NUMERICAL RESULTS

The cable system, Fig. 5, having 1.5-mm-thick aluminumsheaths bonded at both ends of an electrical section, is assumedto be a part of a 35-kV circuit with the insulated neutrals. Thestranded 400 mm copper conductor consists of five layers rep-resenting filaments. The layers are composed of

wires with the diameter mm.The sheath is divided into filaments, as it is visiblefrom Fig. 10. Hence, the total number of filaments related to thethree-phase system is . In general, one deals with a


(a)

(b)Fig. 9. (a) Mesh of the surrounding soil, cable spacing s = 140 mm.(b) An enlarged section in the cable vicinity, cable spacing s = 140mm.

Fig. 10. Sheath filaments.

significantly lower number of unknowns than is required for thelosses evaluation via the finite-element method [1].

The operating voltage system can be written as follows:kV, kV,kV. Furthermore, the elements , (3)

and (4), of the submatrices associated with (2), are computedusing the mutual and self impedances and of thefilaments with the ground return path. The impedances aredetermined by (A.2) and (A.7), presented in the Appendix.The geometric mean distances, incorporated in impedanceexpressions, are calculated via (A.8) and (A.9) concerning theconductor filaments, while the distances of the sheath filamentsare computed using (A.10)(A.12).

The ampacity results computed using the coupled non-linear electric-thermal model are compared with the resultsobtained via the IEC relations [41]. The relations are derived

TABLE IAMPACITY COMPARISON

in accordance with the assumption of the isothermal tempera-ture value of the actually convective and radiative soil surface.Hence, the results calculated via the coupled model are obtainedusing the same assumption. Otherwise, the discrepancy betweenthe results would be affected by the different boundary condi-tion. The temperature values at both the soil surface and in-finity, respectively, are chosen to be 20 C. In general, the con-vective and radiative heat transfer through the soil surface canbe taken into account by the coupled model via (11). The dielec-tric losses W/m are calculated in a straightforwardmanner and included in the thermal calculations.

It is worth mentioning that the effective cross-section areaof the conductor considered is mm (i.e., slightlydiffers from the value of 400 mm covered by IEC 228, wherethe maximum dc resistances of stranded conductors at 20 C aregiven). Therefore, the following equation [31] is used:

where 1.02 is the empirical factor taking the strands into ac-count, while is the electrical resistivity of conductors. Thefurther IEC procedure for the ampacity evaluation follows thestandard one.

The ampacity values related to the various cable spacingsand soil conductivities are shown in Table I. The negligibledifferences between the phase currents , and , obtainedby the model, arise from the unequal temperatures of the phaseconductors as well as the mutual electromagnetic (EM) cou-pling. The average phase current is assigned to as the ampacityvalue .


TABLE IILOSSES COMPARISON

When single-core cables with appreciable sheath losses areinstalled in flat formation, the losses nonlinearly increase to-gether with the cable spacing , providing the greater gener-ated heat per-unit time and volume. The whole procedure ofthe sheath losses evaluation can be found in [31], [42]. On theother hand, the external thermal resistance decreases for thegreater spacing [31]. Hence, the optimal spacing, for whichthe maximum ampacity is obtained, stands in a balance betweenthese effects. Since the actual cable has a rather thick aluminumsheath and, thus, the low value of the associated resistance, avery rapid increase in the sheath losses is achieved formm and mm, compared with the spacing ofmm. Consequently, the greatest ampacity value, taking into ac-count the spacings shown in Table I, refers to the cable touching.

If the cables are touching each other, the discrepancy be-tween the ampacity results evaluated via different approachesis approximately %. On the other hand, if the cables arespaced, very close agreement is achieved.

The analysis of the ampacity discrepancy for the touchingcables starts up with the comparison of the system losses ob-tained via the coupled model and IEC relations (Table II). TheIEC losses are calculated for the coupled model currents ,shown in Table I, since the losses comparison should be basedon the same conductor current through both of the approaches.The phase and sheath losses computed by the coupled model are

and .The conductor and sheath losses obtained via the IEC rela-tions are and . Obviously,the very close agreement is achieved for the spaced cables.The discrepancy of the sheath losses for the touching cables(particularly for the middle onephase ) arises from thenonuniform distribution of a particular sheath losses (Fig. 11),which is not taken into account by the IEC relations. Such adistribution originates from the proximity effect. Particularsheath losses for the spaced cables do not possess such anonuniformity (Figs. 12 and 13). Also, the discrepancy arisesfrom the linkage simplifications undertaken in the evaluation ofthe inductances via the IEC relations, according to the approachpresented by Arnold [42].

In order to eliminate the influence of the losses discrepancyto the value of the ampacity discrepancy , the losses for thetouching cables, obtained by the proposed model, are used

Fig. 11. Sheath losses distribution s = 0 mm, k = 1W/m C.



within the IEC procedure of the ampacity evaluation. The novelampacity value , as well as the discrepancy value areshown in Table III. It can be noticed that is around %.Consequently, the use of the same losses evaluation procedurethrough both of the approaches decreases the ampacity discrep-ancy of approximately 3%.

The rest of the ampacity discrepancy originates from the IECformula of the external thermal resistance for touching cablesin flat formation, based on Symms paper [43], since the com-putation of the internal resistances for the single-core cables israther straightforward. The integral equation method assumingthe isothermal cable surfaces has been used in [43]. The nu-merical results presented in Table III show that the IEC relationshould be re-examined for touching cables with unequal appre-ciable sheath losses. The temperature distribution of the cable


TABLE IIIAMPACITY COMPARISON FOR THE TOUCHING CABLES (s = 0 mm)THE

LOSSES COMPUTED BY THE MODEL ARE USED WITHIN THE IECPROCEDURE OF THE AMPACITY I EVALUATION

Fig. 14. Temperature distribution of the cable surfaces s = 0 mm, k = 1W/m C.

TABLE IVAMPACITY COMPARISON FOR THE TOUCHING CABLES (s = 0 mm)BOTH

THE LOSSES COMPUTED BY THE MODEL AND VAN GEERTRUYDENS FORMULAARE USED WITHIN THE IEC PROCEDURE OF THE AMPACITY I EVALUATION

surfaces cannot be assumed as an isothermal one, as can be no-ticed in Fig. 14. For example, the temperature variation for theouter cables (phases and ) exceeds 5 C.

Van Geertruyden [44] has also developed a relation for theexternal thermal resistance, based on the finite-element anal-ysis by which the infinite domain beneath the soil surfacehas been truncated. However, the cable temperatures are influ-enced by the domain truncation, even in the case when the do-main boundaries have been placed quite far away from the ca-bles [27]. Hence, the appropriate relation taking into account theinfinite domain influence should be developed. Nevertheless,the expression given in [44] is better suited for the computation

Fig. 15. Temperature values through the cable cross section, phase B, s = 0mm, k = 0:5 W/m C, # = 0 .

Fig. 16. Temperature values through the cable cross section, phase B, s = 0mm, k = 1:5W/m C, # = 0 .

of external thermal resistance than the relation accepted in IECstandards, as can be noticed in Table IV. The ampacity results

, obtained using both the losses computed by the model andVan Geertruydens formula within the IEC procedure of the am-pacity evaluation, are compared with the coupled model results

. The absolute value of the discrepancy is below 3%.Upon the whole, it is worth underlining how important the

accurate modeling of the cable cross-section is, as can be seenin Figs. 15 and 16. The temperature drops between conductorand cable surface for the touching cables are around 6 C for

W/m C as far as 14 C for W/m C.

V. CONCLUDING REMARKSThis paper proposes an original nonlinear coupled electric-

thermal model of underground cables with the solid sheaths,thus providing the numerical evaluation of losses, heating, andampacity. The model is based on the filament method as wellas the FEM, including the infinite domain modeling carriedout by the mapped infinite elements. The corresponding finite-element meshes are generated by the advancing front method.The principal contributions of the approach presented so far areas follows.

1) The cable systems, which cannot be handled via simpli-fied analytical or empirical equations, can be treated bythe filament/FE model. Moreover, the proposed modelshows a certain advantage compared to the fully FE-basedmodel, since the use of the filament method for the lossesevaluation requires rather lower computational cost thanthe FEM.

2) The finite/infinite element approach provides the differ-ential equation formulation of the thermal problem, con-trary to, up to now used, more complex integral or in-


tegro-differential formulations featuring the boundary-el-ement method (BEM) and coupled FEM/BEM approach,respectively.

However, the thermal part of the model should be modified inaccordance to the equations given in [24] and [25] to solve theproblem of moisture migration in the cable vicinity.

The numerical results presented throughout this work sug-gest that the IEC formulation concerning the external thermalresistance for touching cables, laid in flat formation, having ap-preciable sheath losses, should be re-examined. Future workwill deal with an extensive numerical analysis to develop a newequation related to the external thermal resistance for touchingcables, by which the infinite domain influence will be takeninto account.

APPENDIX

The mutual and self impedances and are derived fromCarsons infinite integral solution, where the ground influenceis taken into account [36]. Well-known simple closed-form ap-proximations for the overhead wires in the low-frequency rangeare used in the present work. This range is relevant for the eval-uation of underground cable losses.

The mutual impedance per-unit length of two solid circularconductors in the presence of the ground return path is deter-mined by [36]

(A.1)

whereoperating frequency;angular frequency;absolute permeability;distance between the conductor centers;electrical resistivity of the soil.

If conductors are arbitrarily shaped, the mean geometric dis-tance should be used. The mutual impedance of the filamentswith the ground return path, related to the underground cablesystem in Fig. 1, is written by

(A.2)

wherelength of an electrical section;geometric mean distance between the th and thfilament.

The self ground return impedance per-unit length is given by[36]

(A.3)

where is the conductor radius. The self impedance of thecircular conductor with the ground return path is obtained byadding the internal impedance to (A.3)

(A.4)

Fig. 17. ith filament associated with the conductor.

If the internal impedance is derived under dc conditions, it fol-lows:

(A.5)

where is the resistance of the circular nonmagnetic conductorper-unit length. Hence, the self impedance can be written as

(A.6)

where is the self geometric mean distance of the solidcircular conductor.

Consequently, the self impedance of the th filament of anelectrical section of the cable system with the ground return path(Fig. 1) is

(A.7)

whereresistance of the th filament;self geometric mean distance of the th filament.

It is to be mentioned that , concerning the conductor fila-ments, is multiplied by the empirical factor 1.02, by which thestrands are taken into account.

A. ConductorEach layer of the stranded conductor is considered to be a

filament. The ith filament with the radius associated withconductor is shown in Fig. 17.

It consists of wires with the radius . According to [37],the self mean geometric distance can be approximated by

(A.8)The mutual mean geometric distance between the th and thconductor filament, where the second is placed inside the thfilament, is determined by

(A.9)If the th filament is located outside the th one, the mutual meangeometric distance equals the distance between their centers.

B. SheathThe nonuniform losses distribution of the solid sheath, arising

from the proximity effect, can be taken into account using thesheath partitioning. Thus, the sheath is replaced by a number


Fig. 18. Sheath partioning.

Fig. 19. Configuration with two underground conductors.

Fig. 20. Relative discrepancy in the resistive part of Z ; f = 50 Hz.

Fig. 21. Relative discrepancy in the inductive part of Z ; f = 50 Hz.

of filaments having cross-section shape similar to a rectangle(Fig. A2).

The self geometric mean distance of the th sheath filamentaccording to [38] can be written as

(A.10)where and are assigned in Fig. 18. is given by

(A.11)where

distance between the th sheath filament and cablecenter;central angle of the th filament.

Fig. 22. Relative discrepancy in the resistive part of Z ; f = 950 Hz.

Fig. 23. Relative discrepancy in the inductive part of Z ; f = 950 Hz.

The mutual mean geometric distance between the sheath fila-ments is approximated by

(A.12)Although Carsons method is widely applied to the overhead

lines, the underground cable analysis in this work is undertakenby the approximate relations for and due to their sim-plicity. In order to test the accuracy of the approximations forburied cables in the low-frequency range, the comparison toSaads closed-form solution is performed [39]. Saads approachto the impedances of underground cables seems to be the veryaccurate for frequencies up to 1 MHz. For the comparison pur-poses, the configuration with two underground conductors isconsidered as it is shown in Fig. 19. The relative discrepanciesfrom the Saads solution in the resistive and inductive part ofthe impedance , for the fundamental frequency Hzare shown in Figs. 20 and 21, respectively. The cable distancevaries from the 0.02 m (touching) to 1 m. The same data for the19th harmonic ( Hz) are presented in Figs. 22 and 23.Moreover, the relative discrepancies in the resistive and induc-tive part of , for Hz are % and %, respec-tively. The values corresponding to the 19th harmonic frequencyare % and %.

Therefore, the satisfactory agreement between the differentapproaches is achieved. It can be also concluded that the simpleclosed-form relations arising from Carsons integral solutioncan be utilized for the computation of underground cable losses.

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Niksa Kovac was born in Split, Croatia, on December 28, 1968. He receivedthe Ph.D. degree from the University of Split, Split, Croatia.

Currently, he is an Assistant Professor of Faculty of Electrical Engineering,Mechanical Engineering, and Naval Architecture, University of Split. His re-search interests are numerical analysis related to the underground cables andthermal effects of human exposure to electromagnetic fields as well.

Ivan Sarajcev was born in Split, Croatia, on October 28, 1947. He received thePh.D. degree from the University of Zagreb, Zagreb, Croatia.

Currently, he is Associate Professor of Faculty of Electrical Engineering, Me-chanical Engineering, and Naval Architecture, University of Split. His primaryinterest is in the fields of power cables, overvoltage protection, and electromag-netic compatibility (EMC).

Dragan Poljak (M96) was born in Split, Croatia, on October 10, 1965. Hereceived the Ph.D. degree from University of Split, Split, Croatia.

Currently, he is Associate Professor of the Faculty of Electrical Engineering,Mechanical Engineering, and Naval Architecture, University of Split. His re-search interest is in computational methods in electromagnetics, particularlyin the numerical modeling of wire antennas and related electromagnetic-com-patibility (EMC) problems using both frequency- and time-domain techniques.He also deals with the numerical modeling applied to the environmental as-pects of electromagnetic (EM) fields. He is a Series Editor of Advances in Elec-trical and Electronic Engineering, Wessex Institute of Technology (WIT) Press,Southampton, U.K. and was a Guest Editor of the International Journal of En-gineering Analysis with Boundary Elements (EABE) Special Issue on Electro-magnetics. He is author of five books, published by WIT Press. He is a reviewerfor IEEE TRANSCATIONS ON ELECTROMAGNETIC COMPATIBILITY. He has pub-lished five papers in several IEEE TRANSACTIONS.

tocNonlinear-Coupled Electric-Thermal Modeling of Underground CableNik a Kova, Ivan Saraj ev, and Dragan Poljak, Member, IEEEI. I NTRODUCTIONII. N ONLINEAR -C OUPLED E LECTRIC -T HERMAL M ODEL

Fig.1. Representation of an electrical section of a cable transA. Losses Evaluation

Fig.2. Boundary conditions.B. Heating Evaluation

Fig.3. Two-dimensional mapping.C. Electric-Thermal CouplingD. Ampacity Evaluation

Fig.4. Assessment of the load resistance.Fig.5. Cable system domain.Fig.6. Boundary between the finite and infinite elements.III. M ESH G ENERATION

Fig.7. Mesh of the cable cross-section.Fig.8. (a) Mesh of the surrounding soil, cable spacing ${\rm s}IV. N UMERICAL R ESULTS

Fig.9. (a) Mesh of the surrounding soil, cable spacing ${\rm s}Fig.10. Sheath filaments.TABLE I A MPACITY C OMPARISONTABLE II L OSSES C OMPARISON

Fig.11. Sheath losses distribution ${\rm s}_{0} = 0$ mm, ${\rm Fig.12. Sheath losses distribution ${\rm s}_{0} = 70$ mm, ${\rmFig.13. Sheath losses distribution ${\rm s}_{0} = 140$ mm, ${\rTABLE III A MPACITY C OMPARISON FOR THE T OUCHING C ABLES ( ${\rFig.14. Temperature distribution of the cable surfaces ${\rm s}TABLE IV A MPACITY C OMPARISON FOR THE T OUCHING C ABLES ( ${\rmFig.15. Temperature values through the cable cross section, phaFig.16. Temperature values through the cable cross section, phaV. C ONCLUDING R EMARKS

Fig.17. $i$ th filament associated with the conductor.A. ConductorB. Sheath

Fig.18. Sheath partioning.Fig.19. Configuration with two underground conductors.Fig.20. Relative discrepancy in the resistive part of ${Z}_{m},Fig.21. Relative discrepancy in the inductive part of ${Z}_{m},Fig.22. Relative discrepancy in the resistive part of ${Z}_{m},Fig.23. Relative discrepancy in the inductive part of ${Z}_{m},D. Labridis and P. Dokopoulos, Finite element computation of fieJ. Kuang and S. Boggs, Pype-type cable losses for balanced and uA. Konrad, Integro-differential finite element formulation of twI. Saraj ev, M. Majstrovi, and I. Medi, Calculation of losses G. J. Anders, A. K. T. Napieralski, and W. Zamojski, CalculationG. J. Anders, A. Napieralski, and Z. Kulesza, Calculation of intM. A. El-Kady, J. Motlis, G. A. Anders, and D. J. Horrocks, ModiK. E. Saleeby, W. Z. Black, and J. G. Hartley, Effective thermalE. Tarasiewicz, M. A. El-Kady, and G. J. Anders, Generalized coeM. A. El-Kady and D. J. Horrocks, Extended values for geometric N. Flatabo, Transient heat conduction problems in power cables sJ. K. Mitchell and O. N. Abdel-Hadi, Temperature distribution arM. A. Hanna, A. Y. Chikhani, and M. M. A. Salama, Thermal analysW. Z. Black and S.-I. Park, Emergency ampacities of direct burieC. C. Hwang, Calculation of thermal fields of underground cable D. Mushamalirwa, N. Germay, and J. C. Steffens, A 2-D finite eleG. J. Anders, M. Chaaban, N. Bedard, and R. W. D. Ganton, New apM. A. Kellow, A numerical procedure for the calculation of the tM. A. El-Kady, Calculation of the sensitivity of power cable ampM. Liang, An assessment of conductor temperature rises of cablesG. J. Anders and H. S. Radhakrishna, Computation of temperature N. Kova, I. Saraj ev, D. Poljak, and B. Jajac, Electromagnetic-tN. Kova, B. Jajac, and D. Poljak, Domain optimization in FEM modG. Gela and J. J. Dai, Calculation of thermal fields of undergroE. Tarasiewicz and J. Poltz, Mutually constrained partial differE. Tarasiewicz, E. Kuffel, and S. Grzybowski, Calculation of temG. J. Anders, Rating of Electric Power Cables . New York: IEEE PP. de Arizon and H. W. Dommel, Computation of cable impedances bO. C. Zienkiewicz, C. Emson, and P. Bettess, A novel boundary inF. Damjani and D. R. J. Owen, Mapped infinite elements in transN. Kova, S. Gotovac, and D. Poljak, A new front updating solutioJ. R. Carson, Wave propagation in overhead wires, with ground rrB. Stefanini, Power Transmission (in (in Croatian)) . Zagreb, CrD. Oeding and K. Fesser, Geometric mean distances of rectangularO. Saad, G. Gaba, and M. Giroux, A closed-form approximation forH. C. Huang and A. S. Usmani, Finite Element Analysis for Heat T

Electric Cables Calculation of the Current Rating, 1993/94. IEC,A. H. M. Arnold, Theory of sheath losses in single-conductor leaG. T. Symm, External thermal resistance of buried cables and troA. Van Geertruyden, External Thermal Resistance of Three Buried

nonlinear-coupled electric-thermal modeling of underground cable systems

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