CIRED2005 Session No 1

THERMAL ANALYSIS OF PARALLEL UNDERGROUND ENERGY CABLES J. Desmet*, D. Putman*, G. Vanalme*, R. Belmans†, D. Vandommelen†

*Hogeschool West-Vlaanderen, Dep PIH, Belgium jan.desmet@howest.be

†KULeuven, Dep ESAT-ELECTA, Belgium ronnie.belmans@esat.kuleuven.ac.be

1 INTRODUCTION A simple model for the calculation of cable temperature, in case of parallel underground cables, will be proposed and validated with test set ups for different cable topologies. The model takes into account the load conditions and also the influencing parameters such as cable type, cable geometry and environmental conditions. A practical test set up was built in order to investigate experimentally the thermal behaviour of underground cables. Sensitivity analysis is performed in order to evaluate the influence of the different boundary conditions, including the mutual influence of parallel cables. In a further analysis, temperature occurring in the cable for a certain distribution of heat generation is investigated by Flux2D finite element modelling. 2 EXPERIMENTS 2.1 Test Set-Up Measurements are performed on a medium voltage cable with specifications as given in Table 1. In order to measure the temperature in the cable (type EAXeCW20.8/36kV), thermocouple probes are penetrated into the cable. Different set ups are analysed, starting from three phases in ‘trefoil’ configuration in full ground, over to the same configuration in tubes both air filled and tubes filled with bentonite. At start up every five minutes the temperature is logged in order to determine the point of steady state.

15m 15m

F R

1

3

2

4

65

ABC

ABC

ABC

A

B

C

ABC

ABC

Fig. 1 Top view of cable positioning (first 15m in ground, next

15m in tube)

0,250,25

1,20

1,20

(a) first 15 meter of test set up

0,25

1,20

1,20

0,80

(b) last 15 meter of test set up

Fig. 2 Cross section view of cable positioning.

Table 1. Cable characteristics EAXeCW20.8/36kV Cross section (mm²) 240Conductor diameter (mm) 18.3Total thickness of insulation (mm) 9.1DC Resistance @ 20°C [Ω/km] 0.125AC Resistance @ 90°C / 50Hz [Ω/km] 0.161Capacity [µF/km] 0.29Inductance [mH/km] 0.37Current loading in ground trefoil/in line [A] 420/430Current loading in air trefoil/in line [A] 505/575

All measurements are performed with symmetrical and balanced sine wave currents but for different load conditions. Cable loading is done by creating a short circuit condition of a medium voltage transformer by the cable set (point 4, 5 or 6 in figure 1, depending of one, two or three parallel cable tests).

Fig. 3 Some detail photos of the construction of the test set up

CIRED2005 Session No 1

2.2 Measurements In a first measurement, the cable was loaded with a symmetrical balanced current of 420A each phase and three parallel cables connected. Temperature was logged for 24 hours or for 250 hours, depending on test set up. It is proven that end effects can be neglected [2]. A second and third measurement were performed with the same current, but this time for two parallel and one single cable in trefoil. Logging results of cable temperature are given in Fig. 4 en Fig. 5.

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

45.0

50.0

0:00

0:40

1:20

2:01

2:41

3:21

4:02

4:42

5:22

6:03

6:43

7:23

8:04

8:44

9:2410

:0510

:4511

:2512

:0612

:4613

:2614

:0714

:4715

:2716

:0816

:4817

:2918

:0918

:4919

:3020

:1020

:5021

:3122

:1122

:51

Time

Cab

le te

mpe

ratu

re [°

C]

1 cable2 cables3 cables

Fig. 4 Influence of the number of cables on the cable temperature

(measurements at outer cable, cable load: 420A)

0

10

20

30

40

50

60

70

10:2

1:27

19:3

4:32

4:47

:37

14:0

0:53

23:1

4:08

8:27

:18

17:4

0:33

2:53

:52

12:0

6:47

21:1

9:40

6:32

:12

15:5

0:11

1:09

:38

10:2

2:30

19:3

3:07

4:44

:05

13:5

4:43

23:0

5:20

8:15

:57

17:2

6:36

2:37

:15

11:4

8:11

20:5

8:50

6:09

:29

15:2

0:25

0:31

:04

9:42

:00

18:5

0:34

4:01

:13

13:1

1:51

22:2

2:31

7:33

:09

16:4

3:48

1:54

:27

11:0

5:06

20:1

5:45

5:26

:23

14:3

7:02

23:4

7:40

8:58

:20

Time

Cab

le te

mpe

ratu

re [°

C]

soilpolyurethaneconduit, filled with air

Fig. 5 Influence of the enviromental conditions on the cable

temperature (cable load: 336 A) 3 ANALYSIS 3.1 Electrical equations As known, power dissipation generates heat in electrical systems, so also power dissipation in energy cables will heat up cable. There is a direct link between dissipated power and cable resistance R. Also, cable resistance is temperature dependent according to

( )[ ]20120 −+= TRR α (1) where R20 [Ω/m] the reference resistance at 20°C, and T [°C] and α the temperature respectively temperature coefficient of the conductor material (αCu=0.0041 and αAl=0.0040). The cable series resistance is (2), where both skin and proximity effects are included and Rdc expressed per length units.

( )psdc YYRR ++= 1 (2) Both proximity and skin effect can increase the cable resistance. In a first consideration, we neglect harmonic currents, so current frequency remains at 50Hz. For cross sections up to 400mm² both skin and proximity effect have a minor effect, since the ratio

dcRf will

remain lower than 1000. Consequently, both proximity and skin effect are negligible in this case.

The dissipated power is calculated per conductor from the loaded rms-current, e.g. for phase A:

[ ]mWRIP AA /2= (3) The heat generated per unit length in the cable is calculated by

( ) phCBA RIIIQ 222 ++= (4)where Rph the phase conductor resistance [Ω/m] and neglecting both skin and proximity effect. 3.2 Thermal Equations Thermal calculations are made to estimate the maximum temperature in the cable. Subsequently, temperature distribution is calculated using a Flux2D finite element model. The conductors can be considered as heat sources. The radial thermal resistance for homogeneous material is given by:

πλ 2

ln

⋅⋅=

LdD

Rrad (5)

with D [m] the outer diameter of the considered cylindrical shell, d [m] the inner diameter, λ [W/mK] the thermal conductivity of the material and L [m] the length of the considered cable. The large difference in magnitude between longitudinal and radial thermal resistance justifies the assumption that the entire dissipated heat is carried out in radial direction [5]. With the parameters defined in the test set up, conductive temperature drop is calculated by (6).

πλ 2

ln.

⋅=∆ d

DQTconduct (6)

A good approach is found by considering the heat generating area as a circle in the centre of the cable section [2]. Consequently copper temperature is calculated from the ambient temperature and the temperature rise due to both

CIRED2005 Session No 1

convection, conduction and radiation. However, in our situation, in a first approach, we only consider conduction. Since Q is temperature dependent, Q and Tcu, are calculated for a new iteration step, thus proceeding until iteration converges. 3.3 Calculations A Matlab/Simulink® calculation was performed, based on an equivalent scheme as given in Fig 6. Heat dissipation is presented by a current source generating a given Q, while thermal resistivity (1/λ) is given by a resistance. The specific heat capacity cp will be presented by a capacity value in the electrical equivalent.

cable soil

1/λ s

1/λw

cp_s2 cp_w2

1/λca

cp_ca2cp_ca1

Q

cp_s1cp_w1

Fig. 6 Electrical equivalent of thermal behaviour In [2] it is proven that the heat generated by different cables can be taken as an equivalent heat circle in the centre of the three cables. Cable insulation can be taken as homogeneous and concentric distributed in different shields around this heat circle. Heat will be evacuated by those shields. Other calculations, using a thermal model for cables in trefoil, as given in [1,3,6,7,8] are also performed. Those calculations also give good results with respect to both Matlab/Simulink® and measurements.

0

10

20

30

40

50

60

0 2 4 6 8 10 12 14 16 18 20

Time [h]

Con

duct

or te

mpe

ratu

re [°

C]

measurement

simulation [k=0.8 W/(m.K); c=1.32E6 J/(m³K)]

simulation [k=0.6 W/(m.K); c=1.32E6 J/(m³K)]

simulation [k=0.7 W/(m.K); c=1.32E6 J/(m³K)]

Fig. 7 Influence of soil parameters on simulation results (1 cable,

460A), including measurement results 3.4 Simulations The mathematical model, as defined and discussed in paragraph 3.3 was validated with measurements as discussed in 2.2. Besides that, Flux2D finite element simulations ware performed in order to analyse both sensitivity parameters of configuration and geometry.

Color Shade ResultsQuantity : Temperature degrees C.Scale / Color12 / 16,7281516,72815 / 21,456321,4563 / 26,1844626,18446 / 30,9126130,91261 / 35,6407635,64076 / 40,3689240,36892 / 45,0970745,09707 / 49,8252249,82522 / 54,5533854,55338 / 59,2815359,28153 / 64,0096864,00968 / 68,7378468,73784 / 73,4659973,46599 / 78,1941578,19415 / 82,9222982,92229 / 87,65044

Fig. 8 Flux2D finite element simulations on 420A cable

50

60

70

80

0 0,5 1

mm

degrees C.

Fig. 9 Flux2D finite element simulations – temperature distribution Figure 8 gives temperature distribution over ground section in case of three parallel cables, while Figure 9 illustrates temperature over horizontal cross section in the center of the cables. The results gathered in the Flux2D finite element simulations are compared with both the mathematical model and measurements. Results are given and discussed in paragraph 4. Same simulation models are also used in order to analyse both the influence of different material constants and load conditions. Also the influence of cable position in function of cable temperature is analysed in case of the cable in tube. Results are given in both Fig. 10 and 11

Fig. 10 Flux2D finite element simulations on cable in tube. Thermal resistance between cable core and tube outer surface

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7

Distance from cable outer surface to inner boundary of tube [cm]

Ther

mal

resi

stan

ce [K

m/W

]

0

20

40

60

80

100

120

140

160

180

200

Tem

pera

ture

for s

peci

fic s

et u

p [°

C]

Fig. 11 Calculated value of both cable temperature and equivalent

thermal resistance in function of airgap between cable and inner surface of tube.

CIRED2005 Session No 1

4 EVALUATION In order to evaluate results gathered by both simulations and calculations, results are compared to each other. Table 2 gives the steady state temperatures calculated and simulated, using Flux2D. Same material and boundary conditions are used and also given in Table 2.

Table 2. Calculated and Simulated cable temperatures I=420A each phase; ksoil = 0.6 W/(m.K); depth=1.2 m Flux2D Calculated 1 trefoil 122.5 °C 128.4 °C 2 trefoils 198.5 °C 201.3 °C 3 trefoils 331.0 °C 346.4 °C

The given results are very similar, however both temperatures are much higher than accepted by cable manufacturer and by standards. It’s a matter of fact that the results gathered by our calculations and simulations are estimated values for given conditions. Taking to account the correction factors as given in the standard NBN 259 (C33-112) for cable depth, thermal conductivity of ground and proximity of other cables, maximum current will be much lower. For those values of current, maximum cable temperature is determined again using both calculations and simulations. In Table 3 results are given. It is seen that in those conditions cable temperature remains under limitations as given by both standards and cable manufacturers.

Table3. Calculated and Simulated cable temperatures with corrections I=420A each phase; ksoil = 0.6 W/(m.K); depth=1.2 m

Temperature Correction Factor

Current each phase Flux2D Calculated

1 trefoil 1x0.98x0.84 346 A 77.7 °C 82.7 °C 2 trefoils 0.85x0.98x0.84 294 A 78.1°C 81.5 °C 3 trefoils 0.76x0.98x0.84 263 A 80.1°C 86.3 °C

SENSITIVITY ANALYSIS With the results gathered by calculations and measurements, it is found that different conditions can affect quite well results. So, sensitivity analysis has to be done in order to determinate the influence of different parameters, such as material specifications, soil moisture, ground materials e.o. Based on those analysis we can draw some conclusions concerning the influence and/or importance of different parameters. Especially for parameters who are not well known, such as moisture level of the soil, thermal conductivity of the soil. Due to the fact that those analysis can not be done by on the field tests, both simulations and calculations have to be done in order to analyse the influencing parameters.

0 50 100 150 200 250 300 350 400 450 5000

20

40

60

80

100

120

140

Time [h]

Tem

pera

ture

[°C]

Soil cylinder of 1.2 m

0.50.7511.251.52measuredSoil thermal conductivity [W/mK]

Fig 12. Influence of thermal conductivity of the soil

0 50 100 150 200 250 300 350 400 450 50010

20

30

40

50

60

70

Time [h]

Temperature [°C]

Soil cylinder of 1.2 m

00.10.20.30.40.5measuredMoisture fraction

Fig 13. Influence of moisture fraction of the soil CONCLUSIONS Changes of cable parameters have a relatively small influence on the conductor temperature, while environmental conditions such as thermal conductivity of soil (e.g. dry sand) affect conductor temperature highly. Next to a negative effect on the life span of the cable insulation, high cable temperatures result in a substantial increase of the cable losses, up to 4 % per 10 K raise in temperature. REFERENCES [1] Y.A. Cengel, Heat transfer, a practical approach, New York:

McGraw-Hill, 1998, p.418. [2] J. Desmet, D. Putman, F. D’Hulster, R. Belmans, “Thermal

analysis of the influence of non linear, unbalanced and asymmetric loads on current conducting capacity of LV cables,” presented at 2003 IEEE Bologna Power Tech Conference, Bologna Italy June 23-26th

[3] P. Caramia, G. Carpinelli, A. Russo, P. Verde, “Estimation of thermal useful life of MV/LV cables in presence of harmonics and moisture migration,” presented at 2003 IEEE Bologna Power Tech Conference, Bologna Italy June 23-26th

[4] VDI Wärmeatlas: Berechnungsblätter für den Wärmeübergang; ISBN 3-18-400415-5

[5] F. Donazzi, E. Occhhini, A. Seppi, “Soil thermal and hydrological characteristics in designing underground cables,” Proceedings IEEE Vol 126. No.6, June 1979, pp506-516

[6] G. Luoni, A. Morello and H.W. Holdup, "Calculation of the thermal resistance of buried cables throug conformal transformation," Proc.IEE, Vol.119, No.5, May 1972 , pp575-586

[7] F.C. Van Wormer, "An Improved Approximate Technique for Calculating Cable Temperature Transients ," Trans. Amer. Inst. Elect. Eng., Vol.74, part 3, April 1955 ; pp.277-280

[8] J.P. Holman, "Heat transfer," McGraw-Hill., Fifth edition, 1981