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EMP 33(12) #64353 [EIC ID No. 6817]

Electric Power Components and Systems, 33:1385–1402, 2005

Copyright © Taylor & Francis Inc.

ISSN: 1532-5008 print/1532-5016 online

DOI: 10.1080/15325000590964425

An Assessment of the Methods for Calculating

Ampacity of Underground Power Cables

FARUK ARAS

Kocaeli University

Technical Education Faculty

Department of Electrical Education

Kocaeli, Turkey

CÜNEYT OYSU

Kocaeli University

Engineering Faculty

Mechatronics Engineering

Kocaeli, Turkey

GÜNE¸S YILMAZ

Uluda

˘

g University

Engineering Faculty

Electronics Engineering

Bursa, Turkey

Endurance of an insulation material to high temperatures determines the maximum

current-carrying capacity (ampacity) of an underground power cable. Cable ampacity

is calculated conventionally using the installation conditions and maximum steady-

state operation temperature according to IEC-60287 standard. In this article, ﬁnite

element method results are compared with IEC-60287. A further veriﬁcation has

also been made with experiments. In this work, ampacity analyzes of 154 kV high

voltage XLPE underground power cable are made using ANSYS 5.61 ﬁnite element

analysis software. An experimental set-up is developed to measure conductor and

surface temperature of the cable in underground conditions. The results of experiments

and numerical calculations are compared to the IEC-60287 standard. Additionally,

the thermal regions of three cables in ﬂat installation are investigated to show the

versatility of the ﬁnite element method. The effects of insulation thickness and external

thermal source on ampacity are also analyzed using the ﬁnite element method.

Keywords ampacity, ﬁnite element, thermal rating, underground power cables

Manuscript received in ﬁnal form on 17 March 2005.

Address correspondence to Dr. Faruk Aras, Kocaeli University, Dept. of Electrical Education,

Anitpark, Izmit, Kocaeli, TR-41100, Turkey. E-mail: arasfa@kou.edu.tr

1385

1386 F. Aras et al.

Introduction

Underground power cables are more expensive to install and maintain than overhead lines.

The greater cost of underground installation reﬂects the high cost of materials, equipment,

labor, and time necessary to manufacture and install the cable. The large capital cost

investment makes it necessary to use their full capacity. On the other hand, conductor

temperature of a power cable limits its ampacity (i.e., maximum allowable temperature

for XLPE insulated cables is 90

◦

C). Also, the operating temperature adversely affects the

useful working life of a cable. Excessive conductor temperature may irreversibly damage

the cable insulation and jacket.

A successful model was proposed for calculating ampacity of underground cable by

Neher-McGrath in 1957 [1]. The Neher-McGrath Model has been widely accepted for

over 40 years. Today, the greater majority of utilities and cable manufacturers have been

using the IEC-60287 standard [2]. This method employs a lot of simpliﬁcations and has

its limitations. Thus, it is not reliable for the analysis of complex conﬁgurations.

The ﬁnite element method (FEM) is more powerful and precise in terms of geomet-

rical modeling complexity. The ﬁnite element method solves problems that are described

by partial differential equations, using numerical techniques. A domain to be analyzed is

represented as an assembly of ﬁnite elements. Approximating functions in ﬁnite elements

are deﬁned in terms of nodal values of a physical ﬁeld. A continuous physical problem

is transformed into a discretized ﬁnite element problem with unknown nodal values. For

linear problems, developed system of linear algebraic equations are solved numerically.

Values inside ﬁnite elements can be obtained using nodal solutions.

Although the idea of dividing a continuum into small ﬁnite pieces had been ﬁrst

suggested by Courant in 1943 [3], the development of the ﬁnite element method coin-

cided with major advances in computers technology and programming languages. Under

operation circumstances, the measurement of the conductor temperature of the HV un-

derground cable is very difﬁcult. Thus, the usual conditions of installation and operation

can be simulated in laboratory environment. A heating test bed according to IEC62067,

which is an extension of IEC60840 for higher voltages, is set up to measure temperature

distribution. The cable loaded with various currents can be investigated. Consequently,

the thermal rating of the cable can be determined.

Ampacity analysis of cables with FEM has been studied by many researchers [4–7].

Lately, an IEC technical report, namely IEC TR 62095 [8], has been published for

current ratings calculation using the ﬁnite element method when IEC 60287 [2] can-

not be applied. Nahman and Tanaskovic [9] have measured and modeled thermal re-

gions outside the cable using FEM. Even if they used linear triangular elements in the

FEM analysis, their results were in good agreement with their measurements. If they

had used quadratic quadrilateral elements in their method, improved results could be

obtained.

In this project, current rating calculations for power cables in steady-state conditions

were performed using IEC 60287, ﬁnite element method and experiments. The models

developed for the ﬁnite element analysis were veriﬁed with IEC 60287 and experiments.

A numerically efﬁcient ﬁnite element model is obtained with sensitivity analysis. Since

the domain under the cable is not an isotherm, the control volume to be analyzed is very

effective in the degree of accuracy of the analysis. Fundamental terms and boundary

conditions in the FEM analysis were clearly identiﬁed in the results section, which can

help engineers to develop their models for ratings calculations.

An Assessment of the Methods for Ampacity 1387

Ampacity Calculation

The ampacity calculation of a power cable using IEC 60287 formulation can be written

in the following form:

I =

θ

c

− θ

a

− θ

d

(T

1

+ (1 + λ

1

)(T

3

+ T

4

))R

ac

(1)

where

θ

c

maximum operating conductor temperature

◦

K

θ

a

ambient temperature

◦

K

θ

d

temperature rise due to dielectric loss

◦

K

λ

1

sheath loss factor

R

ac

AC resistance of conductor at temperature θ

c

T

1

, T

3

, T

4

thermal resistances of insulation, covering and soil (

◦

Km/W), respectively.

The external thermal resistance T

4

, depends on the diameter of the cable, the depth of

laying, the thermal characteristics of the soil, mode of installation and on temperature

rise generated by neighboring cables.

Finite Element Method

A basic equation of heat transfer for an isotropic body with temperature dependent heat

transfer has the following form:

−

∂q

x

∂x

+

∂q

y

∂y

+ Q = ρc

∂θ

∂t

(2)

where q

x

and q

y

are components of heat ﬂow through the unit area; Q is the inner heat

generation rate; q is density; c is thermal heat capacity; θ is temperature, and t is time.

Fourier’s law describes the heat ﬂow equations as

q

x

=−k

∂θ

∂x

,q

y

=−k

∂θ

∂y

(3)

Here k denotes the thermal coefﬁcient of the material. Substitution of the above relations

gives

∂

∂x

k

∂θ

∂x

+

∂

∂y

k

∂θ

∂y

+ Q = ρc

∂θ

∂t

(4)

Then the variation of the temperature and temperature gradients inside an element can

be expressed in terms of nodal temperatures using shape functions N

i

as

{θ}=[N]{θ

e

} (5)

1388 F. Aras et al.

Differentiation of the temperature ﬁeld gives

∂θ

∂x

=

∂N

∂x

{θ

e

} (6)

where θ

e

are the nodal temperatures for the eth element. N represents the quadratic shape

functions for 8-node quadrilateral elements (serendipity elements)

While shape functions are expressed through the local coordinates ξ,η, the matrix

contains derivatives in respect to the global coordinates x,y. Derivatives can be easily

converted from one coordinate system to the other by means of the chain rule of partial

differentiation:

∂N

i

∂ξ

∂N

i

∂η

=

∂x

∂ξ

∂x

∂η

∂y

∂ξ

∂y

∂η

(7)

Using Galerkin method, we can rewrite Eq. (4) in the following form:

=

1

2

{θ

n

}

T

k

∂

2

[N]

T

∂x

2

∂

2

[N]

∂x

2

+

∂

2

[N]

T

∂y

2

∂

2

[N]

∂y

2

{θ

n

}−

{θ

n

}

T

[N]

T

q

dxdy (8)

Equations are rearranged to give stiffness matrix of [K] and force vector of {f } as

[K]=

∂

2

[N]

T

∂x

2

∂

2

[N]

∂x

2

+

∂

2

[N]

T

∂y

2

∂

2

[N]

∂y

2

kdxdy (9)

{f }=

[N]

T

qdxdy (10)

FEM equations are found by minimization of functional in terms of nodal temperatures.

∂

∂θ

n

= 0 and [K]{θ

n

}={f } (11)

Stiffness matrix [K] and force vector {f } integrals are calculated on each element numer-

ically (using Gaussian quadrature), and then total stiffness matrix is set up by summation

of each equation system. The total equation system is then solved for each node.

Numerical and Experimental Work

154 kV underground cable in single installation is analyzed using FEM and IEC-60287.

These results are used for determining the insulation thickness representing the soil’s

thermal resistance in a thermal rating experiment. Further further loading cycles are then

applied to the test cable, and the values obtained from a full heating of a cycle is compared

with FEM and IEC 60287 results. The effects of insulation thickness on current ratings

An Assessment of the Methods for Ampacity 1389

were also analyzed using both methods. Another current rating calculation deﬁned in

IEC 60287, three cables in ﬂat formation, were also analyzed and compared with FEM

analysis. A further application of this problem in the case of a heat pipe installed parallel

to the cable was studied as well.

154 kV Single Underground Cable Analysis with FEM and IEC

The ampacity of the underground cable, whose sectional view is illustrated in Figure 1,

is analyzed using the ﬁnite element method. Under the operation conditions, the amount

of heat generated from the cable should be calculated to determine ampacity. Since the

limiting operation temperature of XLPE cables is 90

◦

C, the heat generated from the

charged cable should be transferred to the environment to reside under this temperature.

That’s why the correct calculation of heat transfer from the cable affects the ampacity

analysis directly. The soil temperature is 20

◦

C for northern Turkey. 1.2 m is the depth

the underground cables are normally buried under. However, the temperature and thermal

properties of soil depend on various parameters like the moisture content, the seasonal

variations and cable depth. These were reported by Brakelmann [10]. In steady-state oper-

ation conditions, moisture around the cable can be ignored since the heat generation from

the cable dry out the moisture nearby. IEC-60287 standard assumes that the properties

of soil and boundary conditions for the calculations are constant. Using these conditions,

and assuming the soil is homogenous, heat transfer and ampacity are calculated both by

IEC-60287 and the ﬁnite element method. The boundary conditions for FEM analysis is

shown in Figure 2.

The steady-state calculation of the heat transfer gives general operating ampere ca-

pacity. For shorter times, this current can be exceeded under cable manufacturer’s limiting

values. In this analysis, transient effects are not considered, thus time is not a parameter.

Since the heat transfer coefﬁcients of XLPE and the semi-conductor material are

very close to each other, they are assumed to be same and are considered as a single

layer. The effects of coating layers and very thin metallic screens are omitted as their

thermal resistivities are very small and have very little effect on the results. Generalized

Figure 1. Sectional view of the cable.

1390 F. Aras et al.

Figure 2. The layout position of the 154 kV cable is given above.

layer dimensions of a 154 kV underground cable and the dielectric losses generated on

XLPE are as follows:

D

Conductor

= 37.7mm

D

XLPE+Semi Conductor

= 81.7mm

D

Screen

= 98.7mm

D

Cover

= 106.7mm

h

depth

= 1200 mm

q

dielectric

= 3.57 W/m

k

soil

= 1.2 W/Km

k

XLPE

= 0.2857 W/Km

k

screen

= 384.6 W/Km

For the underground cables, the earth surface can be assumed as isotherm as described

in IEC 62095. The domain under the cable is earth, which is an inﬁnite medium. The

soil temperature also increases gradually as it gets deeper underneath the earth’s surface.

However, the soil temperature below 5 meters converges to a certain value and does not

ﬂuctuate seasonally [10]. That’s why earth underneath the cable can also be taken as

isotherm at a certain depth. This effective control volume can be found using sensitivity

analysis. Various control volume sizes are analyzed, and heat transfer rates for the half of

the region are shown in Figure 3. At the depth of approximately 10 meters, heat transfer

rate converges to a value within the 1% range. Heat dissipation increases by 0.6% if the

region to be discretized is increased from 5m to 10 m in depth. Thus, a control volume

of 10 m depth and 18 m width is taken for all other cases as well.

The insulation thickness of a cable affects the permissible maximum current and heat

transfer from the cable. Higher insulation thickness also blocks the heat generated in the

conductor. Various insulation thicknesses for a 154 kV underground cable are analyzed

An Assessment of the Methods for Ampacity 1391

Figure 3. Sensitivity analysis between the control volume depth and heat dissipation.

in this example. For each cable, the heat ﬂux is applied on the conductor found by an

iterative approach. The heat ﬂux is increased gradually until a maximum temperature

of 90

◦

C on the conductor is achieved. For the cable with 22 mm XLPE insulation, the

thermal ﬂux generated on the conductor surface is found to be 0.585 W/m

2

.

Since the model is symmetrical about y-axis, only half of the domain is modeled. As

a general rule of ﬁnite element method, element shapes are kept close to an aspect ratio of

unity. Heat generation by dielectric losses is applied on the XLPE domain. 1074 elements

are found to be dense enough for the ﬁnal calculation. FEM analyzes with different mesh

densities, such as the number of elements 557, 690, 1074, and 1486, have shown that

the total heat transfer on the conductor converges to a certain value by increasing mesh

density. This sensitivity analysis regarding mesh density is necessary for evaluating the

correctness of the FEM analysis as described in IEC TR 62095. The mesh chosen after

the sensitivity analysis are shown in Figures 4 and 5.

Figure 6 show that the ampacity for the cable has a difference of around 1% be-

tween the FEM and IEC 60287 models. The results also illustrated that the reduction in

the insulation thickness by 22% (from 22 to 17 mm) has an effect of 2.9% increase in

the ampacity. Since the heat transfer mechanism is conduction, and this is directly pro-

portional to the contacting surface area, reduction in the diameter reduce both thermal

barrier and contacting surface.

Thermal Rating of 154 kV XLPE Power Cable

The experiment conditions are devised to comply with IEC-62067, which is an extension

of IEC-60840 for higher voltages section 12.4.7 on power cable systems [11, 12]. Thermal

rating standards involve a series of heating cycles by conductor current and cooling cycles

with natural cooling. In a heating cycle range from approximately 310 K to 380 K,

conductor and shield temperatures measured on experiments can be compared to the

FEM and IEC-60287 results to observe the performance of the both methods.

For this purpose, a 15 m of 154 kV XLPE power cable is used in the experiments.

A heating system is devised using thermocouples and thermal insulators for the temper-

ature measurements on the conductor and shield of the cable. Thermal resistance of the

soil is represented with a paper layer which has a thermal conductivity of 0.475 W/Km.

1392 F. Aras et al.

Figure 4. FEM mesh around the cable.

Figure 5. The mesh modeling both the cable and the surrounding soil with 1074 elements.

An Assessment of the Methods for Ampacity 1393

Figure 6. The ampacities for different insulation thickness by both FEM and IEC 60287 model.

The thickness of the paper layer wrapped on the cable is found by a trial and error

technique. In steady-state conditions at 363 K conductor temperature, the shield tem-

perature was measured and compared to the analytical values while the paper thickness

increased gradually. As the same values of conductor and shield temperatures are ob-

tained from both analytical calculations and experiments, the wrapped paper thickness

gives the thermal resistance of the soil. Three thermocouples are used to measure con-

ductor temperature. One is located at the center and two others are located at a distance

of one meter from the ends. A precise temperature measurement is achieved by drilling

the cables till 3 mm in conductors, as shown in Figure 7. The space at the tip of the

thermocouple and the conductor is ﬁlled with a highly conductive material.

Figure 7. Experimental layout.

1394 F. Aras et al.

Table 1

Experimental results

Cycle 10 Cycle 11

T-conductor (

◦

K) T-shield (

◦

K) T-conductor (

◦

K) T-shield (

◦

K)

319 298 327 304

321 300 330 307

335 313 339 316

351 322 355 324

369 330 365 329

381 337 375 332

382 338 377 333

382 338.5 377 333

377 333

Four thermocouples are positioned evenly in length on the outer surface of the

cable. All of the thermocouples are connected to a temperature recorder (Siemens-

Kompensograph C1015).

Both ends of the cable are connected to secondary probes of the voltage transformer.

Then the cable is placed as the secondary winding of a heating transformer. The cable

is gradually heated up in steps and awaited in each step until the difference between

each conductor thermocouples are less than 1

◦

K. Each experimental cycle took at least

8 hours to reach maximum temperature. Twenty experiments are performed consequently.

The experimental results in 10th and 11th cycles, when stable periodic conditions were

achieved, are given in Table 1.

The results from two cycles construct a continuous path, even if the recorded con-

ductor temperatures are different in each experiment. The heat dissipation from the cable

is calculated from the measured conductor and shield temperatures.

While the experimental setup was calibrated with analytical calculations at 363 K,

the heating experiments ranged from 320 K to 383 K. Consequently the comparison

of the FEM and IEC-60287 with experiments gives an indication about both methods’

performances. The currents necessary to generate the heat versus conductor temperature

are plotted on Figure 8. There is a difference between the FEM and experiment results

at lower temperatures. Since the speed of the heating is too high at the beginning of the

experiment, the converged temperature values have not been reached. However on the

rest of the graph, experimental results are very close to FEM results.

At the beginning of the experiment, the current was increased so much that cal-

culations under steady-state conditions would not be expected to conform. The heating

process after 340

◦

K became more stable and the ratio changed linearly. FEM and IEC-

60287 analysis were performed using homogenous and isotropic material properties.

Nonhomogenous effects in cable layers, like the screen and ﬁlling materials, deteriorate

the analysis conditions. Therefore, small differences between the results are very obvious.

Analysis of 154 kV Three Cables in Flat Installation Using FEM

In the analysis of three cables in ﬂat installation, the distances between the cables are

considered to be twice the cable’s diameter, as given in IEC-60287. Thermal and geo-

An Assessment of the Methods for Ampacity 1395

Figure 8. Conductor temperature while current increases.

metrical properties of the cable are as given in single cable layout analysis with 22 mm

XLPE insulation. The cable installation is shown in Figure 9. The earth is assumed to

be homogenous and isotropic for the FEM analysis.

The heat generation boundary conditions are applied at each conductor and increased

gradually until the center conductor temperature reaches 90

◦

C. The heat transfer is

44.4 W/m while the conductor temperature at the center is 90

◦

C and at the sides are

85.08

◦

C. Besides the thermal boundary conditions applied on the cable, the dielectric

losses are also estimated (wd = 3.57 W/m) and applied as a thermal source over the

XLPE.

Figure 9. 154 kV cable installed in ﬂat conﬁguration.

1396 F. Aras et al.

Figure 10. FEM model and boundary conditions.

Cable layout in Figure 9 shows that the domain to be considered is symmetrical

in y-axis. Utilization of symmetrical boundary conditions as shown in Figure 10 on the

analysis saves modeling and computation time signiﬁcantly, especially in large problems.

Symmetric boundary condition can be used by assuming the heat transfer on the symmetry

lines are adiabatic. The mesh around the three cable installations are shown in Figures 11

and 12.

Thermal contours of the region nearby the cables are given in Figures 13 and 14.

These ﬁgures illustrate that the heat transfer from the cable at the center is lower than the

others. Thus, the ampacity analysis should be made for the center cable, as it is the most

critical one. IEC-60287 also considers the center cable as the signiﬁcant cable on its cal-

culations. In the IEC calculations, the ampacity is 2.4% less than the numerical analysis.

Heat transfer from the cables and the ampacity values estimated using two methods

are given below.

Heat transfer (IEC-60287) Q

AN

= 42.35 W/m

Heat transfer (ﬁnite element method) Q

FEM

= 44.4 W/m

Ampacity (analytical) I

AN

= 1296 A

Ampacity (ﬁnite element method) I

FEM

= 1327 A

An Assessment of the Methods for Ampacity 1397

Figure 11. FEM model with 948 elements and 3001 nodes.

Figure 12. Mapped mesh around the cables.

1398 F. Aras et al.

Figure 13. Temperature distribution between the cable and surrounding.

Analysis of 154 kV Cable by a Heat Source

In many cases, cable routes cross other cables or steam pipes. Due to the thermal in-

terference in the crossing region, the cable ampacities can be decreased noticeably [13].

Analytical methods give poor results because of the geometrical complexity. However,

FEM can easily analyze this kind of problem. In this numerical analysis, a heat pipe with

insulation buried 600 mm below the surface is considered as shown in Figure 15. The

steam temperature is normally higher than 100

◦

C, and under steady-state conditions, the

temperature on the surface of insulation is assumed to be a 60

◦

C isotherm. Cable and

pipe crossing at an angle involves a three-dimensional analysis. However, the worst case

occurs when they are run in parallel in a trench. That’s why the cable properties and

installation dimensions are as given in the three cables installation case previously. The

heat generation on the conductor is again estimated by increasing the heat generation

Figure 14. Temperature distribution around the cable.

An Assessment of the Methods for Ampacity 1399

Figure 15. Cables parallel to steam pipe with boundary conditions.

gradually, until 90

◦

C conductor temperature is reached. The mesh for FEM analysis is

given in Figure 16.

When three cables are installed parallel to a heat pipe at 60 cm distance, the ampacity

of the cable becomes 1236 A. Three cable installations without steam pipes have an

ampacity of 1327 A, and a derating factor is necessary for application of crossing heat

sources. For this case, the derating factor is 0.92. When the same boundary conditions

given in the Finite Element Method section is applied to the case with steam pipe, a

temperature rise of 9.4

◦

C on the conductor is observed. Figure 17 shows the temperature

distribution after the FEM analysis.

Ampacity of 154 kV cable with 22 mm XLPE insulation under different conditions

are calculated by FEM as:

FEM (A) Analytical (A) % difference

Single cable 1657 1635 1.3

Three cables 1327 1296 2.4

Three cables and heat source 1236 1130 9.4

1400 F. Aras et al.

Figure 16. FEM mesh with 1215 elements.

Figure 17. Temperature distribution around the cables and steam pipe.

An Assessment of the Methods for Ampacity 1401

Conclusions

The analytical modeling of heat transfer mechanism by IEC-60287 works well in simple

cable installations. However, simplifying assumptions and empirical correlation to obtain

the analytical method can be signiﬁcant in complex installations like crossing cable

ducts, cables on trays, cables near buildings, cable splices, etc., thus making the solution

become impossible. Today’s computer technology gives ﬁnite element method a capability

to solve any of these cases with very complex geometrical conﬁguration. It can solve

complex installations in any environment and subjected to any type of load condition

together with transient analysis efﬁciently. When the cable surrounding is composed of

various materials with different thermal resistivities, IEC-60287 formulation fails to get

an acceptable result.

The numerical solutions for the underground cables showed that the ﬁnite element

method gives a solution in any domain free from any geometrical complexities. Conse-

quently, an improved calculation of ampacity saves the investment and operation expenses

considerably. Currently, used power cables can be operated with higher capacity. In three

cable installation analysis using FEM, the temperature distribution inside and around the

cable is also determined, which is not possible using the analytical approximations. This

is useful when the temperature ﬁeld and speciﬁc isotherms around the cable are to be

found out. The difference in the ampacity analysis using FEM and analytic solution of

single cable increases in the three cable layout case.

Cables installed near the other heat sources have to be treated carefully. FEM results

showed that the ampacity should be decreased by 8%. Derating factors of 0.92 found

empirically could be used in crossing cable cases. However, FEM gives almost exact

solutions to these kinds of problems. The mutual heating of the cables are more accurately

handled in FEM.

In this study, FEM and IEC methods are assessed for some common problems. The

boundary conditions and control volume determined are similar for a wide range of

problems. Models for the problems with more geometric complexity can be developed

using the similar approach given in this publication. Additionally the various soil thermal

properties depending on the moisture content can be easily analyzed by FEM.

References

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of cable systems,” AIEE Trans., vol. 76, pp. 752–772, 1957.

2. IEC Publication 60287, Calculation of the Continuous Current Ratings of Cables, 1982.

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Bulletin of American Mathematical Society, vol. 49, pp. 1–43, 1943.

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