EUROPEAN TRANSACTIONS ON ELECTRICAL POWER
Euro. Trans. Electr. Power 2005; 15: 109–121
Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/etep.39
Optimized conductors for XLPE cables with a large cross-section
Udo Fromm*,y
Deutsches Patent- und Markenamt, D-80297 München, Germany
SUMMARY
The focus of this article is the determination of the AC resistance of cable conductors. A simple routine
measurement method to measure the AC resistance of complete cable drums is described. An optimization method
based on model-simulations and measured values is provided. Copyright # 2005 John Wiley & Sons, Ltd.
key words:
AC resistance; optimized conductors; XLPE cables; skin effect; measurement method; high voltage
cables
1. INTRODUCTION
The current rating of HVAC (high-voltage alternating current) cables depends strongly on the AC
resistance of the conductor. The AC resistance Rac exceeds the DC resistance Rdc due to the skineffect: the AC current in the conductor generates an alternating magnetic field. This field induces a
voltage opposing the current in the centre and enhancing the current near the surface of the conductor.
The losses caused by the skin effect can be reduced by the use of segmented conductors. This was
suggested for the first time by Humphreys Milliken [1] in combination with insulated conductor
strands (Figure 1(a)).
In a segmented conductor the strands are arranged on layers. The pitch of the strands of each layer
causes every strand to change its path between the centre and the surface of the conductor. This leads
to loops formed by strands in adjacent layers (Figure 1(b)). The loop current is determined by the
induced voltage Uind and by the loop impedance. Conductors with unidirectional layers have a lower
induced voltage than conductors with contra-directional layers [2]. Therefore the AC losses can be
reduced by using unidirectional layers for uninsulated strands. When insulated strands are used the
additional AC losses are minimized.
When neglecting the losses due to the proximity effect, the international standard [3] defines a skin
effect factor ys as follows:
ys ¼
Rac
1
Rdc
*Correspondence to: Udo Fromm, Deutsches Patent- und Markenamt, D-80297 München, Germany.
y
E-mail: u.fromm@ieee.org
Copyright # 2005 John Wiley & Sons, Ltd.
ð1Þ
110
U. FROMM
Figure 1. Milliken conductor (a) as patented [1], (b) loop formed by the conductor strands of two adjacent layers.
For solid conductors with a circular cross-section the skin effect factor ys can exactly be calculated:
"
#
xs ð1Þ3=4 I0 ðxs ð1Þ3=4 Þ
1
ys ¼ Re
2
I1 ðxs ð1Þ3=4 Þ
ð2Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffi
20 f
xs ¼
Rdc
ð3Þ
where
There are many papers on AC resistance of conductors for oil impregnated cables. Improved
conductor designs are based on enamelled strands [1,4], layer insulation [5–7] or dynamically
balanced conductors [8].
On the other hand, the conductor strands of XLPE (cross-linked polyethylene) cables are not oil
impregnated which is reported to significantly enhance the AC resistance [2]. The compression forces
on the conductor are higher for XLPE cables due to the high shrinking stress of the extruded insulation.
This leads to a decreased contact resistance between the conductor strands, which causes higher AC
losses as compared to oil impregnated cables [9]. Many published data were derived from measurements on non-insulated conductors, so that they poorly represent the real situation in XLPE cables.
The aims of the paper are:
1. Model investigations to study the influence of selected design parameter
2. Presentation of a simple conductor optimization method
3. Description of a simple method to measure the AC conductor resistance.
2. THE MODEL
In the literature there are a number of papers describing a network model of the segmented conductor
[4,10,11]. A similar approach is used here. The aim is to determine the influence of the design parameter:
Copyright # 2005 John Wiley & Sons, Ltd.
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OPTIMIZED CONDUCTORS FOR XLPE CABLES
111
Figure 2. Network model.
*
*
*
*
number of wires per segment
number of segments
number of insulated wires per segment
the inner radius ri of a hollow conductor
on the AC resistance of the conductor. These results are discussed with respect to cost considerations.
The model describes a hollow conductor with an outer radius ra and an inner radius ri . The strands
are modelled as current paths with a resistance per length R, a self-inductance per length L, and with a
mutual inductance to the other strands, see Figure 2. The conductor consists of a number of segments.
Each segment consists of a number of layers. Every layer consists of a number of strands (Figure 2).
Within the layers there are the individual strands with the radius r0 :
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ra2 ri2
r0 ¼
n ns
ð4Þ
Every strand has the resistance per metre:
R¼
r02
ð5Þ
0
8
ð6Þ
and the self-inductance per metre:
L¼
The i-th strand is mutually coupled to the j-th strand by the mutual inductance M. Owing to symmetry
the model describes one segment only, e.g. the length of the inductance matrix is equal to the number
of strands per segment n. For the determination of the inductance matrix see Section 2.1. The voltages
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U. FROMM
across a length of the conductor strands and the currents in the strands can be described as vectors U
and I with:
0 1
0 1
i1
u1
B .. C
B .. C
B . C
B.C
B C
B C
C
B ij C
u
and
I
¼
ð7Þ
U¼B
B jC
B C
B .. C
B .. C
@ . A
@.A
un
in
where uj is the voltage per metre across the j-th strand and ij is the current through the j-th strand. Then
the voltage current relations can be written as the matrix equation:
U ¼ ðR þ j ! M Þ I
ð8Þ
Then the current distribution can be calculated assuming a given voltage drop (e.g. uj ¼ 1 V=m):
I ¼ ðR þ j ! MÞ1 U
ð9Þ
Then the AC resistance can be calculated:
Pn 2
R
j¼1 ij
Rac ¼ P
ns n 2
j¼1 ij ð10Þ
The ratio of AC resistance and DC resistance is a characteristic parameter of a conductor:
ys ¼
Rac
1
Rdc
ð11Þ
R
n ns
ð12Þ
where
Rdc ¼
2.1. Determination of the inductance matrix
For the determination of the mutual inductance matrix two cases are distinguished:
2.1.1. Inter-strand resistance within a layer is zero. In this case the layer within a segment behaves
like a solid metallic body. The strands are treated as line-like current paths in parallel to the conductor
axis. The elements of this inductance matrix M are (see Figure 3):
Mi;k ¼
ns X
0
a¼1
Copyright # 2005 John Wiley & Sons, Ltd.
8
þ
0 la r0
ln
2
r0
ð13Þ
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OPTIMIZED CONDUCTORS FOR XLPE CABLES
Figure 3. The inductance matrix of a conductor segment. Influence of the number of segments.
2.1.2. Inter-strand resistance of all strands within a layer is infinite. This simulates the case when all
strands within a layer are insulated (e.g. use of enamelled wires). Owing to the regular strand
arrangement within a layer, every strand of the layer carries the same current. This is taken into
account in the model by creating a modified inductance matrix M . First a matrix ML is derived from
the inductance matrix M. The elements of ML are the self-inductances of the strands of the considered
layer and the mutual inductances between the strands of the considered layer (see Figure 4). A
modified matrix ML is derived from the matrix ML . The relation between ML and ML is explained by
the following.
Owing to the helical arrangement of all strands of the same layer the mutual inductance to the
neighbouring strand is equal for all strands within this layer:
ML1
¼
ML1;2
¼
ML2;3
¼ ¼
MLs;1
s
X
1
MLs;1 þ
¼
MLk1;k
s
k¼2
!
ð14Þ
Figure 4. Construction of the modified inductance matrix.
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U. FROMM
The same reasoning applies for the mutual inductance between the strands with one strand in-between:
!
s
X
1
MLs1;1 þ MLs;2 þ
ML2 ¼ ML1;3 ¼ ML2;4 ¼ . . . MLs;2 ¼
ð15Þ
MLk2;k
s
k¼3
It can be generalized:
MLm
¼
ML1;1þm
¼
ML2;2þm
¼ ¼
MLs;m
m
s
X
1 X
¼
MLsmþp;p þ
MLkm;k
n p¼1
k¼mþ1
!
ð16Þ
The part ML of the inductance matrix M is then replaced by ML so that the modified inductance matrix
M is created.
2.2. Model verification
In order to verify the network model a solid cylindrical conductor is simulated as shown in Figure 5
(inter-strand resistance is zero). The current density S is plotted across the conductor radius. The
simulation is compared to the analytical solution (based on Bessel function):
SðrÞ ¼ I
p I0 ðqrÞ
2 ra I1 ðqrÞ
ð17Þ
where
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qp
ffiffiffiffiffiffiffi
q¼
1 2f 0 =
Figure 5. Current density as a function of the radius for a solid cylindrical conductor.
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OPTIMIZED CONDUCTORS FOR XLPE CABLES
115
The network model has its maximum at the radius of the centre of the outer wires, which is half a layer
thickness less than the conductor radius, see Figure 5. Otherwise the agreement seems acceptable.
3. SIMULATION RESULTS
3.1. Impact of the segment structure
For the model simulations a segment structure, as shown in Figure 6, was assumed. This structure was
described as a concentric structure in Reference [11]. The following cases were simulated:
*
*
*
*
*
no insulated strand (behaviour like a metallic tube)
outer layer consists of insulated strands
outer two layers consist of insulated strands
outer three layers consist of insulated strands
all strands are insulated
The impact of the number of segments on ys is shown in Figure 7. It is shown, that ys can be
remarkably reduced by using insulated strands. As reported in Reference [11] only marginal gains can
Figure 6. Wire arrangement within the simulated segment.
Figure 7. The ys -dependence on the fraction of insulated wires for various segment
numbers (ri ¼ 6 mm, ra ¼ 30 mmÞ:
Copyright # 2005 John Wiley & Sons, Ltd.
Euro. Trans. Electr. Power 2005; 15:109–121
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U. FROMM
Figure 8. The AC resistance of a conductor as a function of the inner conductor radius for different numbers of
insulated strands (ra ¼ 30 mm, 6 segments, 91 strands in 6 layers, 20 C).
be achieved by enhancing the number of segments and this only when using a high percentage of
insulated strands.
3.2. Impact of the inner radius
In a cylindrical conductor with a large diameter most of the current flows near the conductor surface
due to the skin depth of about 1 cm for copper at a frequency of 50 Hz. In the centre the current density
is low but the current has the opposite direction than when near the surface. Therefore the AC
conductor resistance does decrease somewhat for increased ri (up to about 15 mm at ra ¼ 30 mm) in the
model simulation without insulated strands, see Figure 8. For further enlarged ri the AC resistance
does increase. This means that the AC resistance has a minimum for a certain ri . This minimum clearly
appears in the model simulations for the conductor without strand insulation and for a conductor with
an outer insulated layer. An estimated function for the minimum AC resistance as mentioned above is
indicated in Figure 8.
A perfect segmental conductor leads to an almost even current distribution across the whole
conductor area so that the skin effect factor ys is close to zero. Then an increased ri would also increase
the AC resistance and no local minimum of the AC resistance exists. This can be seen for the simulation
results with three insulated layers and in the case of a complete strand insulation in Figure 8.
Figure 9 shows the same information as Figure 8 with the difference that the skin effect factor ys is
provided at the y-axis instead of the AC resistance. The value of ys contains information about the use
of the conductor material. A high skin effect factor ys indicates a poor conductor design. For ys ¼ 0 the
conductor material is used best.
Copyright # 2005 John Wiley & Sons, Ltd.
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OPTIMIZED CONDUCTORS FOR XLPE CABLES
117
Figure 9. The skin effect factor ys as a function of the inner conductor radius for different numbers of insulated
strands (ra ¼ 30 mm, 6 segments, 91 strands in 6 layers, 20 C).
4. DESIGN OF AN OPTIMIZED CONDUCTOR
Cost considerations show that the use of enamelled strands [12] is too expensive compared to the
positive effect of enhanced current carrying capacity. Therefore a practical segmental conductor will
probably have a low-cost insulation system within the segments like layer insulation [5–7] or low-cost
strand insulation [13–15].
In order to extrapolate the model simulations to practicable segment constructions it is assumed that
Figures 8 and 9 can be generalized. Any construction with a Rac ðri ¼ 0Þ and a ys ðri ¼ 0Þ follows the
same function Rac ðri Þ or ys ðri Þ. Then the optimization procedure is as follows:
(i) Rac is measured for a given ri and ra .
(ii) Based on a simulation as shown in Figure 8 an optimized ri is chosen.
(iii) The expected ys of the optimized conductor can be taken from Figure 9. A verifying measurement
can be used to check the accurateness of the first assumption.
5. THE MEASUREMENT OF THE 50 HZ CONDUCTOR RESISTANCE
There are basically two different methods to determine the AC conductor resistance. First the
calorimetric method. This method is based on the measurement of temperature differences due to the
conduction losses [10,16]. Calorimetric methods are time consuming. Second, there are the electric
resistance measurement methods [2,16–20]. These methods are based on the measurement of the
conductor current and the determination of the voltage contribution in phase with this current. All
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U. FROMM
electric methods depend very much on the absence of external magnetic fields or non-linear magnetic
materials.
The presented results are achieved with an electric method. Voltage and current were recorded with
a digitizer with a high sampling frequency ( > 1 MSa/s). The complex values of voltage and current are
the 50 Hz component of the sample data. Using the Fourier transform these values are:
pffiffiffi ns
2X
V¼
vr e2iðr1Þp=ns
ns r¼1
ð18Þ
The AC resistance can easily be calculated:
Rac ¼
jU j
cos ðargðUÞ argðIÞÞ
jI j
ð19Þ
This method can also be used for routine measurements of the AC resistance, see Figure 10. The
current is injected into the conductor on one end of the cable. At the other end of the cable the
conductor and the screen are shorted. The return current flows in the screen. Therefore there is no
increase of the AC resistance due to the current in the conductor of the neighbouring turn (proximity
effect). The magnetic field by the current in the conductor is compensated by the magnetic field of the
current in the screen. The voltage drop across the cable can be measured and the AC resistance can be
determined.
The phase resolution is affected by the sample rate. Assuming a sampling error of 1 sample the
phase resolution is
’ ¼
2 f
SR
ð20Þ
Figure 10. Principle for the Rac measurement on a delivery length [20].
Copyright # 2005 John Wiley & Sons, Ltd.
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OPTIMIZED CONDUCTORS FOR XLPE CABLES
119
Figure 11. The probability distributions of the measured AC resistances as functions
of the sample rate (2500 mm2 cross-section).
Therefore a minimal resolution of 0.1 requires a sample rate of 180 kSa/s. The finer the resolution, the
lower the spread between individual measurements as demonstrated in Figure 11.
For practical AC resistance measurements on power cables a sample rate of 1 MSa/s or higher seems
to be sufficient.
6. CONCLUSIONS
A method to optimize the conductor of XLPE cables with a large cross-section is described in detail.
Further, a simple measurement method is presented to determine the AC resistance of cable
conductors. This method can be used for routine measurements on complete cable drums.
7. LIST OF SYMBOLS AND ABBREVIATIONS
7.1. Symbols
a
f
i
ii
j
k
la
n
ns
index variable
frequency
index variable
current within the i-th strand
index variable
index variable
distance between the i-th strand of the segment a and the k-th strand
number of strands per segment
number of segments per conductor
Copyright # 2005 John Wiley & Sons, Ltd.
Euro. Trans. Electr. Power 2005; 15:109–121
120
ns
p
ra
ri
r0
s
ui
vr
xs
ys
I0
I1
L
Mi;k
Rac
Rdc
R
S
SR
V
’
0
I
M
M
ML
ML
U
U. FROMM
number of recorded samples
number of recorded periods
outer conductor radius
inner conductor radius
strand radius
number of strands per layer
voltage per length along the i-th strand
digitized time signal, and
parameter to calculate the skin effect factor ys defined in Equation (3)
skin effect factor
Bessel function of the first kind of the 0th order
Bessel function of the first kind of the 1st order
self-inductance of a strand per length
mutual inductance between the i-th strand and the k-th strand per length
AC resistance
DC resistance
strands resistance per length
complex current density
sample rate
complex rms value of the voltage or the current of the base frequency
phase resolution
magnetic permittivity of vacuum
resistivity of the conductor material
current vector
inductance matrix
modified inductance matrix
inductance matrix for the strands within a layer
modified inductance matrix for the strands within a layer taking into account the strand
insulation
voltage vector
7.2. Abbreviations
HVAC
XLPE
high-voltage alternating current
cross-linked polyethylene
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OPTIMIZED CONDUCTORS FOR XLPE CABLES
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AUTHOR’S BIOGRAPHY
Udo Fromm was born in Kossdorf, Germany in 1966. He received the Dipl.-Ing. degree in
Electrial Engineering from the Dresden University of Technology in 1991 and the PhD from
the Delft University of Technology in 1995. From 1996 to 2001 he worked for ABB
Corporate Research in the fields of high-voltage engineering, high-voltage insulation and
power transformers. From 2001 to 2003 he worked for ABB Energiekabel (now Südkabel
GmbH) in the field of conductor development. Results of this work are published here. In
2004 he joined the German Patent and Trade Mark Office as a patent examiner in the field of
electrical machines.
Copyright # 2005 John Wiley & Sons, Ltd.
Euro. Trans. Electr. Power 2005; 15:109–121