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Correction of the IEC Formula for Eddy-Current Loss Factor 397
Table 2 Thermal, electrical and dielectric properties of materials used for modeling
Material
kt e 20 r tan [W/(Km)] (m) (1/K) (–) (–) Aluminium at 20 °C 239 2.84 10 -^8 0.00403 –^ – Copper at 20 °C 385 1.7241 10 -^8 0.00393 –^ – XLPE 0.286 – – 2.5 0. PVC 0.167 – – – – Native soil 0.83 – – – –
3. IEC- AND FEM-BASED METHODS FOR CALCULATING THE CABLE AMPACITY
3.1. The current IEC 60287-based method
For three single-core cables installed directly in the soil without drying out, in trefoil
formation, whose screens and armourings are made from non-magnetic metals, in parallel,
bonded and earthed at both ends or at one end, the formula for the calculation of the cable
ampacity IC (in A) based on IEC 60287- 1 - 1 and IEC 60287- 2 - 1 reduces to [1, 21 ,2 8 ]:
- 5
, 1 1 1 2 3 4
, , 1 2 3 4 [ ( 1 ' ")( )]
R T T T T
W T T T T
I
C l
Ccp rs dl C
where Wd , l represents the dielectric losses per phase and unit length of a cable in W/m, T 1
is the thermal resistance of the layers between one conductor and the corresponding
metallic screen in Km/W, T 2 is the thermal resistance of the PVC bedding in Km/W, T 3
is the thermal resistance of the PVC oversheath in Km/W, and T 4 is the thermal resistance of
the surrounding soil in Km/W. The remaining parameters appearing in Equation (1) are
as follows: RC,l maximum AC resistance of one conductor at C , cp = 90 oC in /m (from
Table 1), 1 is the ratio of the losses due to circulating currents in one enlarged metallic
screen to the losses in one conductor, and 1 is the ratio of the losses due to eddy currents
in one enlarged metallic screen to the losses in one conductor. According to the IEC
60287 - 1 - 1 standard [1], 1 is equal to zero for metallic screens and armourings bonded
and earthed at both ends, and 1 is equal to zero for metallic screens and armourings
bonded and earthed at one end (or cross-bonded).
For three single-core cables in trefoil formation with metallic screens and armourings
bonded and earthed at both ends, the circulating-current loss factor 1 is given by [1,28]:
,
2
,
,
,
, 1 ( 2 )
R fM
fM
R
R
W
W
Cl se l
sel
Cl
sel
^ (^2 )
where Wse , l represents the losses due to circulating currents in an enlarged metallic screen
per phase and unit length of a cable in W/m, WC , l represents the losses in a conductor per
phase and unit length of a cable in W/m, Rse , l is the equivalent resistance of metallic
screen and armouring in parallel in /m, f = 50 Hz is the system frequency,
398 M. ŠUĆUROVIĆ, D. KLIMENTA, D. TASIĆ
− d se
s M
2 10 ln 7 (3)
is the mutual inductance between a conductor and an enlarged metallic screen in H/m; and
2 6 7
2 d 4 d 5 d d dse
is the mean diameter of an enlarged metallic screen in mm.
The AC resistance of any metallic screen at its maximum operating temperature s ,max (in
°C) per phase and unit length of a cable ( Rs , l in /m) is calculated using an effective cross-
section of the metallic screen (^ )^10 /^4
2 6 4
2 , 5
−
S^ seff =^ d − d^ (in m^2 ). The maximum operating
temperature s ,max in °C is estimated by means of the following formula:
1
4
,
2 , , , max , ln 2 d
d
k
R I
t XLPE
ClCT s Ccp
assuming Wd , l = 0 W/m for 33 kV cables.
The AC resistance of an armouring at its maximum operating temperature A ,max (in °C)
per phase and unit length of a cable ( RA , l in /m) is calculated using an effective cross-section
of the armouring (^ )^10 /^4
2 6 6
2 , 7
− S^ Aeff =^ d − d^ (in m^2 ). The maximum operating temperature
A ,max in °C is estimated by means of the following formula:
5
6
1 ,
4
,
2 , , ,max , ln
ln
2 d
d
d k
d
k
R I
tXLPE t PVC
ClCT A Ccp
assuming Wd , l = 0 W/m and ' 1 +" 1 = 0 for 33 kV cables.
For three single-core cables in trefoil formation with metallic screens and armourings
bonded and earthed at one end (or cross-bonded), the eddy-current loss factor 1 is given
by [1]:
= = ++ + 12
4 4 1 0 1 2 ,
,
,
, 1 1210
" (^) gs ( 1 ) se Cl
sel
Cl
se l C R
R
W
W
where Wse , l represents the losses due to eddy currents in an enlarged metallic screen per
phase and unit length of a cable in W/m,
3 1 7
- 74
7
− d d
C
se
gs^ ^ ,^ (^8 )
se
f
7
2
1 10
sl seff Al Aeff
sl seff Al Aeff se R S R S
R S R S
, , , ,
, , , ,
=^ is the electrical resistivity of^ an enlarged^ metallic screen at^ its
operating temperature in m, se =( d 7 − d 4 )/ 2 is the thickness of an enlarged metallic
400 M. ŠUĆUROVIĆ, D. KLIMENTA, D. TASIĆ
For the purpose of thermal analysis using the FEM in COMSOL 4.3, any cable of the type
Cu/XLPE/CTS/PVC/AWA/PVC 1/C 19/33 kV (BS 6622) needs to be represented by an
equivalent construction consisted of the copper conductor, XLPE insulation, equivalent
copper screen and PVC oversheath with outer diameters d 1 , d 4 , d 7 and d 8 , respectively. The
equivalent cable construction is based on the following IEC 60287- 1 - 1 standard instruction
[1]: “Where screening layers are present, for thermal calculations metallic tapes are
considered to be part of the conductor or sheath while semi-conducting layers (including
metallized carbon paper tapes) are considered as part of the insulation. The appropriate
component dimensions must be modified accordingly.” This means that the conductor and
insulation screens are added to XLPE insulation, and materials of the copper screen, PVC
bedding and aluminium armouring are modeled by the equivalent metallic screen having
thermal conductivity of copper. Since the PVC bedding is included in the equivalent metallic
screen, it means that the thermal resistance T 2 should be added to the thermal resistance T 3 ,
that is, an equivalent thermal conductivity of the PVC oversheath should be determined. In
that case, the equivalent thermal conductivity of the PVC oversheath k^ ' t , PVC in W/(Km) is
7
8
2 3
, ln 2 ( )
d
d
T T
k (^) tPVC
The volume power of heat sources in a conductor WC , v in W/m^3 is given by [29, 30 ]:
2 6 1
2 ,
,
, , 10
−
d
R I
S
W
W
ClC
Ceff
Cl Cv
where the cable ampacity IC can be replaced with the tabulated ampacity value IC , T , and
the diameter d 1 is in mm.
The volume power of heat sources located between a conductor and an equivalent
metallic screen Wd , v in W/m^3 is given by [ 29 , 30 ]:
2 6 1
2 4
, , ( ) 10
− −
d d
W
W
dl dv
where the diameters d 1 and d 4 are in mm.
The equivalent volume power of heat sources located between a semi-conducting
XLPE insulation screen and a PVC oversheath W se , v in W/m 3 is given by [ 29 , 30 ]:
2 6 4
2 7
2 1 , 2 6 4
2 7
1 , 2 6 4
2 7
, , ( ) 10
d d
R I
d d
W
d d
W
W
sel Cl ClC sev
where the cable ampacity IC can be replaced with the tabulated ampacity value IC , T ,
1 = 1 for the case when enlarged metallic screens are bonded and earthed at both ends,
1 = 1 for the case when enlarged metallic screens are bonded and earthed at one end,
and the diameters d 4 and d 7 are in mm. The loss factor 1 = 1 corresponds with Equation
( 2 ), while the loss factor 1 = 1 corresponds with Equation ( 7 ) or Equation ( 14 ).
The top side of the computational domain in Fig. 2 (a), i.e., the earth surface is modeled by
= rs (2 1 )
- constant temperature boundary condition, or
n ( − kt )= hc (− rs )
Correction of the IEC Formula for Eddy-Current Loss Factor 401
- convection boundary condition [ 29 , 30 ]. In addition, the left-hand, bottom, and right-
hand sides of the computational domain in Fig. 2 (a) are modeled by
n ( − k t )= 0
- adiabatic boundary condition [ 29 , 30 ]. In Equations ( 21 - 23 ), is the unknown temperature
of the earth surface in K; rs is the known temperature of the earth surface in K, or the
known temperature of the air along the earth surface in K in accordance with IEC 60287-
1 - 1 [1] and IEC TR 62095 [ 22 ]; n
is the outwards-oriented normal vector of the constant
temperature and convection boundaries; and hc = 250 W/(m^2 K) is the heat transfer coefficient
due to forced convection in accordance with IEC TR 62095 [ 22 ]. In addition, due to the
absence of the radiation boundary condition, Equation ( 16 ) is linear.
4. RESULTS AND DISCUSSION
The values of the thermal resistances and dielectric losses appearing in Equation (1)
are listed in Table 3. In addition to these values Table 3 contains the equivalent thermal
conductivity of the PVC oversheath calculated using Equation ( 17 ) for the purpose of
FEM-based steady-state thermal modeling in COMSOL 4.3.
Table 3 Thermal resistances, equivalent thermal conductivity and dielectric losses for
power cables of the type Cu/XLPE/CTS/PVC/AWA/PVC 1/C 19/33 kV (BS 6622)
SC,n / Ss,n T 1 T 2 T 3 T 4 k't,PVC Wd,l (mm^2 )/(mm^2 ) (Km/W) (Km/W) (Km/W) (Km/W) [W/(Km)] (W/m) 70/16 0.630 0.072 0.107 2.225 0.1 00 0. 95/16 0.573 0.068 0.107 2.198 0.102 0. 120/16 0.539 0.065 0.103 2.178 0.102 0. 150/25 0.493 0.068 0.102 2.147 0.1 00 0. 185/25 0.462 0.065 0.102 2.124 0.102 0. 240/25 0.411 0.062 0.097 2.101 0.102 0. 300/25 0.388 0.062 0.096 2.069 0.101 0. 400/35 0.354 0.058 0.094 2.038 0.103 0. 500/35 0.336 0.057 0.090 1.995 0.102 0. 630/35 0.298 0.053 0.088 1.959 0.104 0. 800/50 0.267 0.052 0.087 1.917 0.104 0. 1000/50 0.256 0.052 0.084 1.882 0.103 0.
Table 4 shows the circulating-current loss factor, volume powers of heat sources and
maximum conductor temperatures obtained for the tabulated ampacity values and other
service conditions taken from [20], and metallic screens and armourings bonded and
earthed at both ends.
Based on the maximum conductor temperatures from the last two columns of Table 4
(which are higher than the continuously permissible temperature of 90 °C), it is obvious
that there is something illogical about the cable ampacities provided by the manufacturer,
or about the metallic screen bonding design. In particular, this means that the volume
powers of heat sources in the conductors and equivalent metallic screens are not adequate
and the reason for this should be identified. In this regard, the volume powers of heat
sources in the XLPE insulations do not contribute to this illogicality.
