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Estimation of Thermal Useful Life of MV/LV Cables

in Presence of Harmonics and Moisture Migration

P. Caramia G. Carpinelli A. Russo P. Verde

Abstract: The problem of the evaluation of the buried cable thermal

useful life in presence of harmonic distortion and taking into

account the moisture migration is analysed. The evaluation is

effected in non-deterministic scenario with a probabilistic

approach. Numerical applications referred to medium voltage

cables are presented and discussed in order to show the influence of

the harmonics and moisture migration on the cable useful life.

Keywords:Cables, Useful life, Harmonics, Moisture migration

I.INTRODUCTION

The cable thermal useful life is usually evaluated with

reference to deterministic scenarios, in which both the time

varying nature of cable current and the random changes of the

thermal conditions that occur during the cable service are

neglected.

Indeed, the heat transfer is certainly random in nature [1]

so that the most adequate way to evaluate the cable thermal

useful life is to introduce random variables and apply

probabilistic approaches.In a preceding paper [2], the effects of randomly varying

temperature on cable useful life in MV/LV distribution systems

have been examined, employing both Monte Carlo procedure

and closed form solutions. One of the key finding of this study is

that the soil thermal resistivity, the ambient temperature and the

cable loading are crucial factors in estimating the cable thermal

useful life; however, the moisture migration phenomenon was

not taken explicitly into account.

In [3] the approach developed in [2] is extended to the case

in which the moisture migration is explicitly taken into account;

in such a way the proposed approach allows a more realistic and

accurate representation of the buried cable thermal ambient

conditions.In both papers [2] and [3] the presence of harmonics is

neglected and only sinusoidal currents flowing in the cables are

considered.

However, the presence of harmonics in distribution

systems is well known; the harmonics are mainly due to the

presence of static converters and can damage the electric system

components. In the case of cables, the harmonics can cause so

relevant additional losses that not acceptable cable life reduction

arises, if they are neglected in the cable thermal sizing.

In this paper, the presence of moisture migration and

harmonic distortion are considered together, taking into account

the time varying nature of both the cable loading (fundamental

plus harmonics) and the thermal conditions that occur during the

cable service. A probabilistic approach is employed and applied

to the case of medium voltage cables.

II. CABLES THERMAL USEFUL LIFE IN NON-DETERMINISTIC

SCENARIOS

Electrical and thermal stresses are, in general, the most

significant for insulation in MV/LV cables so that the most

adequate cable life modelings take into account both of them.

Neglecting moisture migration, while the cables seem able to

withstand even large current harmonic distortions, they suffer of

very large loss of life in presence of even limited supply voltage

distortions [4]. Taking into account moisture migration, the

thermal stress can assumes a significant role [3]. In the

following, the influence of current harmonics on MV/LV buried

cables is analysed taking into account the moisture migration

phenomenon; since the moisture migration influences only the

cable heat transfer, without loss of generality and for sake of

clarity, we constraint our interest to the cable thermal life model.

The extension to the case of the electrothermal life model is very

easy.

The heat transfer is a non-deterministic phenomenon so

that the cable temperature and hence the cable thermal life is

a random quantity, whose randomness is linked to the

probabilistic behaviour of the involved quantities in the heat

transfer process. So, in the frame of the cumulative damage

theory, reference can be done to the expected value of the

thermal loss of cable life proposed in [5]:

[ ]( )

=0

c dfTRE (1)

being f the probability density function (pdf) of the cable

temperature in Tc and () the reaction rate equation ofArrhenius. The successive estimation of the cable useful life can

be effected summing the expected values of the thermal losses of

life, which come in succession until reaching the unity.

The application of (1) requires the knowledge of the cable

temperature pdf; to do this the relation between the cable

temperature and the variables it depends on has to be known.

Among these variables, in case of buried cables, the thermal

resistivity of the environment surrounding the cable assumes a

fundamental role; in fact, results reported in literature [6] show

___________________________________

P. Caramia, A. Russo and P. Verde are with the Dipartimento di Ingegneria

Industriale - Universit degli Studi di Cassino, Via G. Di Biasio, 43 Cassino

(FR), Italy G. Carpinelli is with the Dipartimento di Ingegneria Elettrica

Universit degli Studi di Napoli, Via Claudio 21 Napoli, Italy

0-7803-7967-5/03/$17.00 2003 IEEE

Paper accepted for presentation at 2003 IEEE Bologna Power Tech Conference, June 23th-26th, Bologna, Italy


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that the sensitivity of cable temperature to variations in thermal

conductivity of the surrounding medium is at least one order of

magnitude greater than sensitivity to variations in other

parameters affecting the heat transfer.

The thermal resistivity of the soil surrounding the cable is

strongly dependent on its moisture content so that an accurateevaluation of the cable temperature can not be effected without

taking into account the moisture migration phenomenon [1,7].

Moreover, in presence of static converters the additional

losses due to harmonics have to be considered too.

In the following, the problem of the cable temperature pdf

evaluation in presence of moisture migration and harmonics is

analysed, in order to estimate via (1) the expected value of the

thermal loss of life.

III. ESTIMATION OF CABLE THERMAL LOSS OF LIFE IN

PRESENCE OF HARMONICS AND MOISTURE MIGRATION

The soil thermal resistivity is related to physical andhydrological characteristics of the soil, its density, porosity and

moisture content; the last parameter is one of the most important

and complex to be taken into account. In fact, moisture

migration results in a zone with thermal resistivity higher than

the original one with consequent non-negligible cable

temperature variations; moreover, the heat transfer itself can

contribute significantly to considerable changes of moisture

content in the soil around the cable.

In this paper, assuming the cable subject to successive steady

state operating conditions so long to neglect the transients

between subsequent conditions, the cable temperature is

evaluated by a simplified procedure, which is based on the

method proposed by Donazzi et al. [7]. Although it can only

approximate real situations, the proposed simplified procedure

appears the most useful for the probabilistic approaches being

quick and easy to be applied.

In [7], under appropriate hypotheses, three different

hydrological states are considered: state 1 (Fig. 1a), in which

the soil will be dried out; state 2 (Fig. 1b), in which the soil

is resulting in two-zone, one dry and one moist, and state 3

(Fig. 1c), in which the soil is resulting moist.

Dry soilMoist soil

(a) (b)

Dry soil

Ground surface Ground surface

Moist soil

Ground surface

(c)

Fig. 1: Hydrological states: (a) state 1; (b) state 2; (c) state 3

The natural soil conditions that define the existence of each

state are the following:

the state 1 is characterised by a saturation degree h lowerthan a critical value crh ;

the state 2 is characterised by a saturation degree h greaterthan the critical value crh but lower than limit values (i.e. the

values defining a limit curve);

the state 3 is characterised by a saturation degree h greaterthan the aforementioned limit values.

The critical value of the saturation degree crh depends on

both soil chemical characteristics and granulometry, and in

practice it does not depend on temperature. This quantity can be,

then, evaluated "a priori" on the basis of laboratory

measurements; its value is already known for the most common

classes of available soils.

The prediction of the existence of state 2 or 3 is not so easy;

in fact, the critical conditions are dependent not only on thevalues of the saturation degree but also on temperature gradient.

A simple but approximate criterion to make this prediction is

based on the introduction of the critical temperature rise above

ambient x [8]. The critical temperature rise above ambient

x is defined as the value of the temperature rise of the cablesurface above which a dry zone will form around it, the outer

boundary of this zone being on an isotherm related to that

particular temperature rise; within the dry zone, the soil has a

uniformly high value of thermal resistivity while outside the dry

zone the soil has a uniform thermal resistivity corresponding to

the site moisture. The essential advantage of the introduction of

this quantity is that the value of the critical temperature riseabove ambient is dependent on the type of soil, compaction and

ambient moisture content, but not dependent on the ambient

temperature or the heat flux from the cable. The critical

temperature rise x is related to two experimentallydetermined quantities:

the critical degree of saturation hcr a migration parameterand to the degree of saturation h of the soil controlled by

ambient moisture at the site; it is given by the following formula:

( )

( ) ( )( )crcr2cr2

2

crx

h21hh1

hh2

1

h1

h1lnh1

1cr

+

=

(2)

An experimental method of deriving the values of crh and

is described in [7].

Once known the critical temperature rise x , theapproximate criterion to make the prediction of the existence of

either state 2 or state 3 can be based on the comparison between

the temperature rise of the cable surface evaluated assumingthe soil uniformly moist and x : if x> , the state 2arises, otherwise the state 3 arises.


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When the cable feeds static converters, the evaluation of the

temperature rise of the cable surface has to be effectedconsidering the presence of harmonics.

The following formula (3) for state 1, (4) for state 2 and (5)

for state 3 can lead the evaluation of the cable temperature:

( )[ ]d43212 1,Ba TTCBTATI ++++= (3)

( )[ ] ( ) xm43212 1,Ba 1TTCBTATI +++++= (4)

( )[ ]m43212 1,Ba TTCBTATI ++++= (5)where:

2

i

maxH

1iiRA =

=;

( ) 2imaxH

1iii,1 R1NB +=

=;

( ) 2imaxH

1iii,2i,1 R1NC ++=

=;

1,B

i,B

iI

I= .

In the above relations it is:

Hmax = maximum harmonic order;

IB,i = current harmonic of orderi flowing in the cable;

N = number of conductors;

Ri = alternating current conductor resistance at the

frequency corresponding to the ith

harmonic order and

at the temperature ;T1 = thermal resistance between conductor and sheath orscreen;

T2 = thermal resistance between sheath or screen and

armour;

T3 = thermal resistance of the external serving;mT4 ,

dT4 =thermal resistance of the moist and dry soil;

a = ambient temperature; = cable operating temperature;1,i, 2,i = ratio of the losses in the metal sheath or

screen and in the armour at the

frequency corresponding to the ith harmonic order to the

total losses in all conductors; = 2/, ratio of the thermal resistivity of the dry zone tothe moist one.

In relations (3), (4) and (5) the dielectric losses are neglected

since this paper is referred to buried cables in MV/LV energy

systems whose dielectric materials and voltage levels do not

require their consideration. The evaluation of cable resistances

as a function of frequency is reported in Appendix.

It should be noted that the thermal resistancesm

4T ,d

4T of the

soil in moist and dry conditions depend, via well known

relationships, on the thermal resistivities of the moist and dry

surrounding medium, these last valuable by the following

relation given in [7]:

( ) ( ) Gh108.3exp7.1 2G10G = (6)

where 0 is the thermal resistivity of the bulk material and G theporosity factor.

The analysis of the relations from (2) to (6) shows that the

cable temperature depends on many parameters showing randomvariations.

The expected value of the thermal loss of life (1) can be

evaluated applying classical probabilistic techniques of analysis

as the Monte Carlo procedure. This procedure allows the

evaluation of temperature probability density function starting

from the knowledge of the probability density functions of all the

variables on which the cable temperature depends and taking

into account relations from (2) to (5).

In this paper the ambient temperature, the degree of

saturation, the fundamental and harmonic currents flowing in the

cable are considered as random input variables.

IV. NUMERICAL APPLICATION

The probabilistic technique described in Section III is

applied to evaluate the thermal useful life of distribution system

cables.

Several tests have been performed, considering different

cables. The results obtained with and without the harmonic

presence are compared.

As an example, in the following the results obtained

considering the 95 mm2

20 kV EPR cable described in Table II

of [9] are shown1. The cable feeds both linear loads and six pulse

static converters.

The cable is buried in sandy soils, then, the critical saturation

degree hcris 0.3 [7]; it is also assumed that:

the migration parameter is equal to 110-4 K-1; the thermal resistivity 0 of the bulk material is equal to 0.25

mK/W;

the porosity factor G is equal to 0.42.

In Tab. I the expected values of thermal useful life,

neglecting the harmonic presence (sinusoidal conditions) and

with the harmonic presence (non sinusoidal conditions) are

reported. It is assumed that:

the linear load current is Gaussian distributed with the meanvalue and standard deviation equal to 149 A (about 45% of

the cable ampacity in sinusoidal conditions) and 5%,

respectively;

the ambient temperature is Gaussian distributed with themean value and standard deviation equal to 20 C and 5%,

respectively.

Moreover, the cable is assumed to feed a 5.15 MVA six-

pulse rectifier with the firing angle uniformly distributedwithin the range [10, 30]. The harmonic pdfs are derived

applying the well known relations proposed in [10].

Finally, the saturation degree has been considered Gaussian

distributed too with two different mean values (0.46 and 0.48)

and with the same standard deviation value (2.5%).

The expected value of useful life obtained neglecting the

moisture migration phenomenon in sinusoidal (non sinusoidal)

1 It should be noted that the cable ampacity in sinusoidal and deterministiccondition is equal to 319 A while the rated useful life is equal to 30 years.


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conditions is equal to 113 (77) years. These values have been

evaluated assuming the thermal resistivity of the soil constant

and equal to 1 Km/W, that is an usual value assumed in

determistic conditions.

TABLE I

EXPECTED VALUES OF THE THERMAL USEFUL LIFESaturation degree

Condition

Expected Value of

Useful life

Mean

value

Standard

deviation

(%)

[years]

0.46 2.5 sinusoidal

non sinusoidal

19.1

9.3

0.48 2.5 sinusoidal

non sinusoidal110

59.9

The results in Tab. I confirm that the influence of the

saturation degree on the useful life is significant. When the

saturation degree assumes the value of the first row in Tab. I, itis very likely that the state 2 arises (soil results in two zone: one

dry and one moist) while in the other case it is more frequent the

occurrence of the state 3 (soil results moist); so, in this last case,

lower soil thermal resistivities, lower cable temperature and then

higher useful life arise.

Moreover, the influence of harmonics is significant, also due

to a great contribution to the harmonic losses due to the sheath.

It should be noted that the same incremental harmonic losses

cause different increment of cable temperature in dependence of

the soil state; in particular, when the state 2 arises the influence

on cable temperature of the harmonic losses is more significant

than when state 3 arises, because of the greater soil thermal

resistivity. However, in spite of this, when the state 2 arises, theexpected useful life in non sinusoidal conditions can be

characterized by a lower variation in comparison with the one in

sinusoidal conditions; in fact, even if the state 2 is characterized

by a greater cable operating temperature (due to a greater

thermal resistivity), there is a lower sensitivity of the reaction

rate equation of Arrhenius () to the cable temperature (due tothe higher operating temperature).

Finally, in Fig. 2 the expected value of the cable useful life is

reported as a function of the ratio between the fundamental of

the static converter current and the total fundamental current; the

mean value and standard deviation of the saturation degree are

the ones of the first row of Tab. I.

The analysis of Fig. 2 confirms the significant influence of

harmonics, also with not particular high percentage of static

converters.

V. CONCLUSIONS

In this paper the influence of the soil moisture migration and

of the harmonics caused by static converters upon the cable

useful life has been considered.

The numerical results on medium voltage cables have

shown that in presence of the moisture migration phenomenon

the harmonic content can significantly affects the cable

temperature and, consequently, the cable useful life.

Fig. 2 Expected useful life versus percentage of static converters

The influence is particularly significant in presence of high

sheath harmonic losses. The influence of the harmonic losses on

the useful life depends on the soil thermal resitivity and on the

sensitivity of the reaction rate equation of Arrhenius to the cable

temperature.

Studies are in progress to compare the influence of the

thermal stress and of the multistress (thermal and electrical) in

order to outline the most adequate life model.

VI. APPENDIX

The cable ac resistance at the ith

harmonic, Ri, is affected by

the skin effect, xs(i), and proximity effect to other conductors,

xsp(i).

It can be evaluated starting from the knowledge of the dc

resistance,Rdc, by the following relation:

)i(x)i(x1RR spsdci ++= . (A.1)

The contributions to the ac resistance due to skin and

proximity effects are given by [11]:

+

+

=

=

2

c

p

2

c

psp

o1

1

os

s

Di312.0

27.0)x(F

18.1

s

D)x(F)i(x

14

)a'k()a'k(sin)a'k(M

)a'k(M

2

a'k)i(x

(A.2)

where:

14

)x()x(sin)x(M

)x(M

2

x)x(F

kR

kx

i602k

kk'k

o1

1

o

p

dc

p

s

=

=

=

=

(A.3)

a conductor radius; conductor permittivity;

30 40 50 60 7002468

1012

percentage of non linear load [%]

expected

usefullife

[ye

ars]


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conductor conductivity;Dc conductor diameter;

kp,ks empirical factor from Table 2 of reference [11];

s conductor separation distance;

Mo(x),M1(x),0(x),1(x) are Bessel functions from Table 1 of

reference [11].The ratio of the losses in the metal sheath or screen at the

frequency corresponding to the ith

harmonic order to the total

losses in all conductors, 1,i , are given by the following relations[8]:

( )( )

++

++

+=

22i,s

22i,s

ms

22

i,s

2

22

i,s

2

i

i,s

i,1

PRQR3

PQXR2

PR

P75.0

QR

Q25.0

R

R

+=

22

i,s

2

i

i,s

i,1QR

Q

R

R;

( )( )

+++

++

+=

22i,s

22i,s

ms

22 i,s

2

22 i,s

2

i

i,s

i,1

PRQR3

PQXR2

PR

P75.0

QR

Q25.0

R

R

(A.4)

respectively for the leading, the middle and the lagging phase

conductors.

Rs,i and Ri are the resistances of the sheath and of the conductor

at the ith harmonic; the P and Q coefficients are auxiliary

quantitites defined by :

3XXQ

XXP

m

m

=+=

(A.5)

where:

Xm is the mutual reactance between the sheath of the outer cable

and the conductors of the outher two given by:

Xm =4 f10-7 ln(2) (A.6)X is the sheath reactance given by:

=

s

7

D

s2ln10f4X (A.7)

with Ds equal to the sheath diameter.

VII. REFERENCES

[1] S.M. Foty, G.J. Anders and S.J. Croall: Cable Environmental Analysis

and the Probabilistic Approach to Cable Rating, IEEE Trans. on Power

Delivery, vol.5, n. 3, 1990, pp.1628-1633.

[2] P. Caramia, G. Carpinelli, P. Verde, F. Vitali: Probalistic Evaluation ofthe Cable Thermal Useful Life in MV/LV Energy Systems, 5 th

International Conference on Probabilistic Methods applied to Power

Systems (PMAPS97), Sept. 97, Vancouver (BC), Canada, pp. 379-384.

[3] P. Caramia, A. Losi, A. Russo, P. Verde: Estimation of thermal loss of

life of MV/LV cables taking into account moisture migration, JICABLE

99 , Paris.

[4] P. Caramia, G. Carpinelli, A. Cavallini, G. Mazzanti, G.C. Montanari, P.

Verde: An Approach to Life Estimation of Electrical Plant Components

in the Presence of Harmonic Distortion, IEEE PES International

Conference ICHQP, Orlando (USA), October 2000, pp. 887-891.

[5] P. Caramia, G. Carpinelli, E. Di Vito, A. Losi and P. Verde: Probabilistic

Evaluation of the Economical Damage due to Harmonic Losses in

Industrial Energy Systems, IEEE Trans. on Power Delivery, vol. 11, n.2,

1996, pp. 1021-1031.

[6] M.A. El-Kady: Calculation of the Sensitivity of Power Cable Ampacity to

Variation of Design and Environmental Parameters, IEEE Trans. onPower Apparatus and Systems, vol. PAS-103, n.8, 1985, pp. 2043-2050.

[7] F. Donazzi, E. Occhini and A. Seppi: Soil Thermal and HydrologicalCharacteristics in Designing Underground Cables, IEE Proc., vol. 126, n.

6, 1979, pp. 506-516.

[8] G.J. Anders: Rating of Electric Power Cables, IEEE Press, NY , 1997.

[9] P. Caramia, G. Carpinelli, A. La Vitola, P. Verde,: On the EconomicSelection of Medium Voltage Cable Sizes in Nonsinusoidal Conditions,

IEEE Trans. on Power Delivery, vol. 17, n.1, 2002, pp. 1-7.

[10] Y.J. Wang, L. Pierrat, L. Wang: Summation of Harmonic Currents

Produced by AC/DC Static Converters with Randomly Fluctuating

Loads, IEEE Trans. on Power Delivery, vol. 9, n.2, 1994, pp. 1129-1135.

[11] S. Meliopoulos, M.A. Martin: Calculation of Secondary Cable Losses and

Ampacity in the Presence of Harmonics - IEEE Trans. on Power

Delivery, n. 22, April 1992, pp. 451- 459.

Biographies

Pierluigi Caramia was born in Naples, Italy, in 1963. He received his

degree in electrical engineering from the Universit degli Studi di Cassino in

1991. Currently, he is Associate Professor of Electrical Power Systems at

Universit degli Studi di Cassino. His research interest concerns mainly

power electronics in power systems.

Guido Carpinelli was born in Naples, Italy, in 1953. He received his

degree in Electrical Engineering from the Universit degli Studi di Napoli in

1978. He became Professor in Industrial Energy Systems in 1990 at

Universit degli Studi di Cassino, Italy; currently, he is Professor at

Universit degli Studi di Napoli, Italy. His research interest concerns

electrical power systems. Guido Carpinelli is a member of IEEE, component

of the IEEE Task Force on Probabilistic Aspects of Power System Harmonics

and member of CIGRE WG 36.07.

Angela Russo was born in Cassino, Italy, in 1972. She received her

master degree in Electrical Engineering and her Ph.D. degree in Industrial

Engineering from the Universit degli Studi di Cassino in 1996 and in 1999,

respectively. She is Assistant Professor of Electrical Power Systems at

Universit degli Studi di Cassino, since 1999. She is a member of IEEE.

Paola Verde (M92) was born in Benevento, Italy, in 1964. She received

the degree in Electrical Engineering Universit degli Studi di Napoli, Naples,

Italy, 1988. Currently, she is Full Professor of Electrical Power Systems atUniversit degli Studi di Cassino. Her research interest concerns mainly

power electronics in power systems. Prof. Verde i s a member of the IEEE,

component of the Working Group on Power System Harmonics

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