Cyclic Current Ratings

8/13/2019 Cyclic Current Ratings

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PRZEGL D ELEKTROTECHNICZNY, ISSN 0033-2097, R. 89 NR 5/2013 153

The variation of the cable conductor temperature duringa stepwise-constant load cycle can be determined bycalculating the thermal response of the HVAC cable and thesurrounding environment to each step change of loadcurrent. The two partial temperature transients are solvedseparately in a sequence and then combined, therebyfinding an analytical solution for the whole transient asfollows.

Transient temperature rise of the conductor above theambient temperature, due to the ith step of stepwise-constant load cycle is:(1) )()()()( ,, t t t t ieiici

where c,i(t ) is transient temperature rise of the conductorabove cable surface, e,i(t ) is cable surface temperature riseabove ambient temperature and i(t ) is attainment factor.Transient temperature rise of the conductor above cablesurface is calculated as:

(2) )]1()1([)( ,,bt

bat

ai J ic eT eT W t

where W J,i is power loss per unit length in a conductor dueto the ith current step, and T a, T b, a , b are thermal

resistances and corresponding coefficients of the first loopof CIGRE transient two loop network [5].For the case of three single-core cables buried directly inthe ground in a flat formation temperature rise of cablesurface above the ambient is:

(3)

t d

Eit

d Ei

t h

Eit

D Ei

W t citeie

16'

162

164)(

21

21

22,1

,

where: te is thermal resistivity of soil, is diffusivity, Dc isouter diameter of cable, h is laying depth, d 1 is center-to-

center distance between cables, 212

1 )2( d hd ,)1( 1,,1 i J i W W , 1 is sheath loss factor, and Ei is

exponential integral [9].The attainment factor in equation (1) is calculated as:

(4))(

)()(

,

,

bai J

ici T T W

t t

Finally, the temperature of conductor is obtained byadding the temperature rise for each step of the loaddiagram, calculated by (1), to temperature of the groundand temperature rise due to dielectric losses. Electricalresistance of conductor and corresponding Joule losses arecalculated with respect to temperature of conductor reachedin each step of daily diagram.

The cyclic rating factor is defined as the factor by whichthe rated current may be multiplied to obtain the permissiblepeak value of current during a daily load cycle such that theconductor attains, but does not exceed the ratedtemperature. For calculating cyclic rated factor, according tostandard IEC60853-2, only load cycle over a period of sixhours before the time of maximum temperature is needed,while earlier values are replaced with constant one,proportional to loss-load factor. Therefore, the cyclic ratingfactor is given by:

(5)2/1

5

0 )()6(1

)()(

)()1(

1

R

R

i R

R

R

Ri iiY

M

where:

(6) ))(1)(()()(

11 k t k t t

R

R

(7))()(

)(

441

4411 T T W T T W

T T W k

B A J

(8)

t d

t d

t h

t D

T T t te

21

21

22c

44

-Ei16

-Ei2

-Ei16-Ei)(4)(

Thermal resistance T 4 for cable buried directly in theground is:

(9)

122

ln2

2

4cc

te

Dh

Dh

T

while thermal resistance

T 4 in the case of cable lineconsisted of three single-core cables is:

(10)

1

14 ln d

d T te

Obviously, the method for cyclic rating factor calculationdoes not take into account the variation of conductortemperature and assumes that Joule losses are directlyproportional to square of current. The results obtained onthis way are therefore also conservative.

Electrothermal life modelElectrical stress and thermal stress are dominant factors

when considering the aging of HVAC cables. The most

popular electrothermal life model is the combination of twosingle-stress life models, the Arrhenius model for thermallife and the Inverse power model for electrical life [7, 8].Expected life time of the cable whose temperature is T andelectric field in insulation is E can be calculated:

(11))(

00

0 cT bncT B

E E

e L L

where E 0 is value of electric field below which electricalaging is negligible, cT=1/T 0-1/T is conventional thermalstress, T 0 is reference temperature, n0 is voltage endurancecoefficient, L0 is life at T=T 0 and E=E 0, B= W/k , W isactivation energy of the main thermal degradation reaction,

k is Boltzmann constant, and b is parameter that rulessynergism between electrical and thermal stress. Theparameters of Arrhenius-IPM model for XLPE insulation aregiven in Table 1 [7]. During one step of daily stepwise-constant load cycle temperature of insulation varies.Therefore, different values of thermal life of cable insulationare obtained for each moment during the day. Loss-of-lifefraction relevant to the ith step of stepwise-constant loaddiagram (Fig. 1) is defined as:

(12)it

ii T L

dt LF

0 )(


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154 PRZEGL D ELEKTROTECHNICZNY, ISSN 0033-2097, R. 89 NR 5/2013

Table 1. Parameters of Arrhenius-IPM modelParameter Value

b [K] 4420B [K] 12430

n 0 [non-dimensional] 15E0 [kV/mm] 5

T0 [K] 293t 2

According to Miners cumulative damage theory, thesum of all loss-of-life fractions should yield 1 at failure. So,number of cycles to failure (number of days to failure incase of daily cycles) is:

(13)1

1

N

ii LF K

Design life of cable corresponds to certain design failureprobability. The cumulative probability distribution functionthat is commonly used for associating time to failureprobability in case of polymeric insulation for power cablesis the Weibulls one. Failure probability at mission time t p is:

(14)])/([ %631),,(

t p Lt

p eT E t P

where t is share parameter of cumulative probabilitydistribution function and L63% is failure-time with 63.2%probability. The relevant failure rate at time t p can beestimated through the following hazard function:

(15)

1

%63%63),,(

t

L

t

LT E t h pt p

Based on the equation (15) failure rate can becalculated for insulation of cable loaded by defined dailystepwise-constant cycle at the end of design life time.

Test example

The procedure described in the previous sections isapplied to HVAC XLPE insulated single-core cables [10]with aluminium conductors and copper wire screen.Maximum voltage of cables is 123 kV. The data aboutcables are shown in the Table 2. In Table 2, S c is conductorcross section, d c is diameter of conductor, Di is diameterabove the insulation, i is insulation thickness, Dc is outerdiameter of cable, and I R is rated current for consideredcable formation and used bonding method of metal screens.It is assumed that three single-core cables are laid in a flatformation and metal screens of cables are cross-bounded.

Table 2. Cable dataS c [mm ] 630 800 1000 1200 1400dc [mm] 29.8 33.7 37.9 42.8 46.4

Di [mm] 58.6 62.5 67.3 73.8 77.4i [mm] 13 13 13 13 13

Dc [mm2] 72.3 76.8 82 89.5 93.3

IR [A] 740 845 950 1025 1100

MCycle I 1.174 1.18 1.186 1.192 1.196Cycle II 1.146 1.15 1.154 1.159 1.162

Laying depth of cables is 1 m, ground temperature 20Cand distance between cables is d 1= Dk +70 mm. Design life ofcables is 30 years, while rated temperature of conductor is90C. Thermal resistivity and thermal capacity of cross-linked polyethylene are 3.5 Km/W and 2.410 6 J/(m 3K),respectively; thermal resistivity of soil 1 Km/W, and thermalcapacities of aluminium and copper are 2.510 6 J/(m 3K) and3.4510 6 J/(m 3K), respectively. It is assumed that theconstant electric field is equal to design values. Currentsthat correspond to each step of daily load cycle are given inTable 3.

Table 3. Daily load cycle

Step. No. Time [h]I/Imax

Cycle I Cycle II1 00-04 0.26 0.262 04-08 0.70 0.913 08-12 1 14 12-16 0.83 0.835 16-20 0.96 0.986 20-24 0.52 0.52

Daily variation of conductor temperature for the Cycle Iof single-core XLPE cable with 1000 mm 2 cross-section isshown in Fig. 2 for different values of overload factor(maximum current relative to rated current). Numericalvalues of stepwise-constant cycle loadings shown in Fig. 1are given in Table 3. For overload factor ( I max/ I R) value of 1,difference between maximum and minimum temperatures isonly 16.8C (varies between 47.1C and 63.9C), while foroverload factor value of 1.4 this difference is 39.4C (variesbetween 82C and 121.4C). Also, from the Fig. 2 it can benoted that for overload factor value of 1.2, the maximumtemperature of conductor is close to 90C. Having in mindthat XLPE cable may be overloaded up to 105 C inemergency regime, from Fig. 2 can be concluded that theoverload factor for cycle I loading must be lower than 1.3.

Fig. 2. Temperature variation during daily cyclic loading I

The last row of Table 2 shows the results of cyclic ratingfactors (Cycle I and Cycle II) calculated by (5). As can beseen, these values vary in very narrow range between1.174 and 1.196 for the Cycle I, and between 1.146 and1.162 for the Cycle II. For 1000 mm 2 cross-section cyclicrating factors have values of 1.186 (Cycle I) and 1.154(Cycle II). For obtained daily variations of conductor

temperature, the expected life time of cable during cyclicloading (shown on Fig. 1) for different values of overloadfactor is calculated using relations (11)-(13).

Fig. 3 shows expected life time of single-core XLPEcables for the case of cycle loading I, with different cross-section areas in the case of daily cyclic loading as functionof overload factor. Having in mind that design life of cable is30 years, it is obvious that for daily cyclic diagram I cablescan be overloaded more than 28% without shortening thelife time. As previously noticed, according to the Fig. 2,overload factor of the cable for given daily cycle diagramand maximal temperature of 90C is slightly above 1.2.Fig. 3 shows that cable life time from the aspect ofelectrothermal aging, for given overload factor of 1.2, is

nearly 80 years. For cyclic rating factor of 1.186 calculatedaccording to relevant standard, life time from the aspect ofelectrothermal aging is approximately 90 years.


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PRZEGL D ELEKTROTECHNICZNY, ISSN 0033-2097, R. 89 NR 5/2013 155

Fig. 3. Life time of cables for cyclic loading I

Fig. 4 shows results of failure rate calculation at the endof design life time. As can be seen from this figure, foroverload factor of 1.186 or 1.2, values of failure rates at theend of design life time are very low. Having in mind all thefacts stated above, it can be concluded that consideredpower cable (with 1000 mm 2 cross-section) has thepotential for additional current load increase of 8.3%compared to the value obtained by the standard. Thisadditional current load increase is similar to ones whichcorrespond to other power cables with different cross-sections.

Fig. 4. Cables failure rate at the end of design life for cycle I

Fig. 5. Temperature variation during daily cyclic loading II

Fig. 6. Life time of cables for cyclic loading II

Fig. 7. Cables failure rate at the end of design life for cycle II

Fig. 5 illustrates the temperature variation during dailycycling loading II, while Fig. 6 and fig. 7 illustrate life time ofcables and cables failure rate at the end of design life forcycle II, respectively. The same analysis can be carried outfor the case of cycle loading II and also, the sameconclusion can be derived. The only difference between twocycles is that for the cycle loading II, additional current loadincrease is approximately 6% and for the cycle loading I itaccounts approximately 8%, compared to values obtainedby the standard.

ConclusionIn this paper, a transient temperature response

calculation of single-core XLPE cables buried in the ground,at two daily load cycles, is conducted. For the assumeddaily load cycles, cyclic rating factors and daily temperaturevariations at different values of overload factor aredetermined. On the basis of obtained results, it is shownthat rated temperature is attained at values of overloadfactors slightly higher than cyclic rating factor values. Usingthe Arrhenius-IPM electrothermal model, expected cable lifetime for different values of overload factors, respecting thedaily cable temperature variations, is determined. On thebasis of obtained results, it is concluded that overloadcapability of single-core XLPE cables, considering thedesign life time, is higher than one obtained by relevant IECstandard. It is shown that cables can be additionally loadedup to 8%, depending on the cycle.

1.0 1.1 1.2 1.3 1.41

10

100

1000

L i f

e T

i m

e [ y

e a r s

]

Imax

/IR

630 m m2

800 m m2

1000 mm2

1200 mm2

1400 mm2

1.0 1.1 1.2 1.3 1.41E-5

1E-4

1E-3

0.01

0.1

1

F a

i l u r e r a t e

[ 1 / y

e a r k m

]

Imax

/IR

630 mm2

800 mm2

1000 m m2

1200 m m2

1400 m m2

1.0 1.1 1.2 1.3 1.41

10

100

1000

L i f

e T

i m

e [ y

e a r s

]

Imax

/IR

630 mm2

800 mm2

1000 m m2

1200 m m2

1400 m m2

1.0 1.1 1.2 1.3 1.41E-5

1E-4

1E-3

0.01

0.1

1

F a

i l u r e r a t e

[ 1 / y

e a r

k m

]

Imax /I R

630 mm2

800 mm2

1000 m m2

1200 m m2

1400 m m2


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156 PRZEGL D ELEKTROTECHNICZNY, ISSN 0033-2097, R. 89 NR 5/2013

REFERENCES[1] Neher J.H., McGrath M.H., The Calculation of the Temperature

Rise and Load Capability of Cable Systems, AIEETransactions, Power Apparatus and Systems , 76 (1957), No. 3,752-772

[2] Anders G.J.,El-Kady M.A., Transient Rating of Buried PowerCables, Part 1: Historical Perspective and Mathematical Model, IEEE Trans. Power Delivery , 7 (1992), No.4, 1724-1734

[3] Anders G.J., Rating of Electric Power Cables in UnfavorableThermal Environment, New York, Wiley-IEEE Press , 2005

[4] Calculation of the Cyclic and Emergency Current Ratings ofCables, Part 1: Cyclic Rating Factor for Cables up to andIncluding 18/30 (36) kV, IEC Std. 60853-1 , 1985

[5] Calculation of the Cyclic and Emergency Current Ratings ofCables, Part 2: Cyclic Rating Factor of Cables Greater than18/30 (36) kV and Emergency Ratings for Cables of AllVoltages, IEC Std. 60853-2 , 1989

[6] Mazzanti G., Montanari G. C., A Comparisom between XLPEand EPR as Insulated Materials for HV Cables, , IEEE Trans.On Power Delivery , 12 (1997), No. 1, 15-28

[7] Mazzanti G., Analysis of the Combined Effects of Load Cycling,Thermal Transients and Electrothermal Stress on Life

Expectancy of High Voltage AC Cables, IEEE Trans. PowerDelivery , 22 (2007), No.4, 2000-2009

[8] Mazzanti G., The Combination of Electro-thermal Stress, LoadCycling and Thermal Transients and its Effects on the Life ofHigh Voltage AC Cables, IEEE Trans. On Dielectrics andElectrical Insulation , 16 (2009), No. 4, 1168-1179

[9] Abramowitz M., Stegun I., Handbook of MathematicalFunctions, New York, Dover Publications, INC. , 1972

[10] XLPE Land Cable Systems, ABB, Users Guide , 5 (2010)

Authors : MSc. Miodrag Stojanovic, University of Nis, Faculty ofElectronic Engineering, Aleksandra Medvedeva 14, 18000 Nis,Serbia, E-mail: [email protected]; prof. dr DraganTasic, University of Nis, Faculty of Electronic Engineering,

Aleksandra Medvedeva 14, 18000 Nis, Serbia, E-mail:[email protected]; dipl. ing. Aleksa Ristic, University ofNis, Faculty of Electronic Engineering, Aleksandra Medvedeva 14,18000 Nis, Serbia, E-mail: [email protected];

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Cyclic Current Ratings