CC BY
7
Electrical Insulation and Cable Engineering
UDC 621.319 https://doi.org/10.20998/2074-272X.202L3.08
G.V. Bezprozvannych, I.A. Kostiukov
A CALCULATION MODEL FOR DETERMINATION OF IMPEDANCE OF POWER HIGH VOLTAGE SINGLE-CORE CABLES WITH POLYMER INSULATION
Introduction. The wave parameters of power cables with polymer insulation differ significantly from the parameters of overhead lines and power transformers. As a result, there are more and more objects in electrical networks for which the occurrence of complex multi frequency transients, accompanied by dangerous overvoltages, should be expected. Purpose. To develop a computational model of the complex impedance of high-voltage single-core power cables of coaxial design required to determine the frequency dependencies of the active resistance and inductance of the conductive core and metal shield, taking into account the surface effect and proximity effect. Methodology. The method is based on solving a system of linear algebraic Kirchhoff equations (SLAE) for magnetically coupled contours. SLAE can be used to calculate conductors taking into account the skin effect and proximity effect. Practical value. The developed model is the basis for determining the characteristic impedance of high-voltage single-core power cables in a wide range of frequencies required to establish adequate criteria for evaluating the parameters of high-frequency effects critical for cross linked polyethylene insulation. References 16, table 1, figures 7.
Key words: power cables, complex resistance, magnetically coupled circuits, system of linear algebraic equations, conductor, screen, coefficient of irregularity of current distribution.
Запропонована чисельна розрахункова модель визначення активного опору та iHdyKmueHocmi cmpyMonpoeidmi жили й металевого екрану силових одножильних кабелiв коаксiальноi конструкци з урахуванням поверхневого ефекту та ефекту близькостi в широкому дiапазонi частоти. Виконано в залежностi вiд частоти порiвняння коефiцieнтiв нерiвномiрностi розподыу струму по перетину стрyмопровiдноi жили кабелю. Показано, що збыьшення товщини мiдного екрану призводить до збыьшення коефщента нерiвномiрностi розподыу струму по перерiзy екрану та активного опору кабелю для частоти 100 кГц при незмтному перерiзi стрyмопровiдноi жили. Розроблена модель е основою для визначення характеристичного iмпедансy силових високовольтних одножильних кабелiв в широкому дiапазонi частоти, необхiдного для встановлення адекватних критерпв оцтки параметрiв високочастотних впливiв, критичних для зшитоi полiетиленовоi iзоляцii. Бiбл. 16, табл. 1, рис. 7.
Ключовi слова: силовi кабел^ комплексний onip, магштозв'язаш контури, система лшшних алгебра!чних piBMHb, струмопроввд, екран, коефщент HepiBHoCTi розподшу струму.
Предложена численная расчетная модель определения активного сопротивления и индуктивности токопроводящей жилы и экрана силовых одножильных кабелей коаксиальной конструкции с учетом поверхностного эффекта и эффекта близости в широком диапазоне частоты. Выполнено в зависимости от частоты сравнение коэффициентов неравномерности распределения тока по сечению токопровода. Показано, что увеличение толщины медного экрана приводит к увеличению коэффициента неравномерности распределения тока по сечению экрана и активного сопротивления кабеля для частоты 100 кГц при неизменном сечении токопроводящей жилы. Разработанная модель является основой для определения характеристического импеданса силовых высоковольтных одножильных кабелей в широком диапазоне частоты, необходимого для установления адекватных критериев оценки параметров высокочастотных воздействий, критичных для сшитой полиэтиленовой изоляции. Библ. 16, табл. 1, рис. 7. Ключевые слова: силовые кабели, комплексное сопротивление, магнитосвязанные контуры, система линейных алгебраических уравнений, токопровод, экран, коэффициент неравномерности распределения тока.
Introduction. High-voltage power cables with thermosetting polyethylene insulation are critical components of power systems. Many investments are made in their production to ensure reliable operation [1-3]. Power high-voltage cables with insulation based on cross-linked polyethylene with voltage of 6-500 kV are characterized by increased values of electrical capacitance and reduced values of impedance (characteristic impedance) compared to cables with paper-impregnated insulation [4-8]. As a result, there are more and more objects in electrical networks for which complex multi-frequency transients should be expected, which are accompanied by dangerous overvoltages and currents [5-8]. The values of overvoltage and the duration of the transient are determined by the length of the cable line and the impedance of the high-voltage power cable. The duration of transients is tens and hundreds of microseconds, which corresponds to the frequency of such processes from units to tens and hundreds of kilohertz [9]. High-frequency components of current and voltage accelerate, in particular, the development of water
treeings in polymer insulation and can cause the development of electrical treeings, i.e. loss of electrical insulation strength of high-voltage power cables [10, 11]. The development of adequate criteria for assessing the parameters of high-frequency effects critical for cross-linked polyethylene insulation of high-voltage cables is based on monitoring of the electrical insulation state during operation, including the results of temporal reflectometry, which requires data on the complex resistance of conductive core and screen to determine the characteristic impedance of the power cable [12].
The normative and technical documentation gives the values of the resistance of the current-carrying core at DC at temperature of 20 °C and the inductance of the power single-core high-voltage cable of coaxial design depending on the spatial location in a three-phase cable line (triangle or plane) [2]. To determine the active resistance of the core during the flow of AC with frequency of 50 Hz, mathematical expressions are given
© G.V. Bezprozvannych, I.A. Kostiukov
that approximate the dependence of resistance due to the skin effect and the proximity effect of adjacent single-core cables [2, 13].
In [14] the analytical model of calculation of longitudinal complex resistance taking into account the proximity effect for a current-carrying core and inductance of a core and the metal screen without taking into account influence of frequency of a single-core power cable is presented. In [15], the intrinsic inductance of the metal screen is determined under the assumption of uniform distribution of the induced current in the screen of a single-core power cable.
Thus, obtaining the frequency dependencies of the active resistance and inductance of current-carrying cores and metal shields of high-voltage single-core power cables, taking into account the skin effect and proximity effect is an important and urgent problem, which until recently is insufficiently developed.
The goal of the research is to develop a computational model of complex resistance of highvoltage single-core power cables of coaxial design, necessary to determine the frequency dependencies of active resistance and inductance of conductive core and metal screen taking into account the skin effect and the proximity effect.
Numerical determination of the active resistance and inductance of the current line, taking into account the skin effect and the proximity effect. Divide the conductor into a number of parallel branches (Fig. 1) -current filaments. Each i-th branch has an active resistance R and an inductance Li, and due to the magnetic field is connected to the j-th branch. The mutual inductance between the i-th and the j-th branches is denoted as Mj.
x 10 ru m
4.99
4.995
5
5.005 5.01 rv, m
N1
(( + jœli )) + YjjvMj ■ij) = E , (1)
j *i
where the first term (j + jali )) determines the
voltage drop on the active resistance and the intrinsic inductance of the i-th branch, and all other components
Ni
Z (jcoMj ■ ij ) are the voltage drops on the mutual
j*i
inductance of the i-th branch with all other branches.
Equation (1) contains N1 terms with unknown currents of parallel branches: i1, i2, ... i, ... ij, ... iN. For each of N1 branches we write an equation similar to (1). We obtain a system of linear algebraic Kirchhoff equations (SLAE):
i
(j1 + jol )i1 +
j * N1 t
(r2 + J(l2 \i2 + ZR
oM
ij i j
= Eù
j*2
oM,
N1
(jN1 + jolN1 )iN1 + Z (J°Mij ^j ) = EN1. j * N1
Write SLAE (2) in matrix form:
ij
= E2
(2)
+ JOL1) J0M12
JoM21 (r2 + Jol2 )
J°M\N JoM2,N
fi \
fir \
.(3)
jaMN! jaMN,2 • •• (( + joLN )) vNJ yi,Nj
On the diagonal of the matrix of SLAE coefficients (3) there are the intrinsic resistances of the branches (active and inductive), outside the diagonal - the mutual inductive resistances of the branches.
In some cases, the values of EMF acting in the branches of the current line are the same - Es. Then SLAE (3) takes the form: f(( + joL) jwM 12 • • • joMXN IfA 21 ((2 + J0L2 )• • j®M2,N
J®Mm
JoMN,2 ■ ■ ■ (RN + JolN )J i
\ a \
N
E y
.(4)
SLAE (4) allows to determine the parameters of the complex resistance of current lines, taking into account the skin effect and the proximity effect.
The mutual inductance M12 between two coaxial ring contours with currents is calculated based on the analytical formula [16]:
Fig. 1. Layout of nodes along the cross section of the current line. The first node is in the center of the current line, the others are located radially. The line connecting the nodes determines the numbering order - counterclockwise
Suppose that in the i-th branch there is a source of variable electromotive force EMF Et of frequency f. Then for the closed contour of this branch we write the Kirchhoff equation:
M,
12
2 Woyln r ~
f k2 ^
1 - — 2
K (k )- E(k )
(5)
where rx, r2 are the radii of the current lines (for singlephase power cable of coaxial design r is the radius of the current-carrying core; r2 is the inner radius of the metal
cable screen, respectively); k =
4^
( + r2 ) + G2
where G
2
2
is the distance between the planes of the current lines along the axis of symmetry; K(k) and E(k) are functions of complete elliptic integrals of the first and second kind:
ni2 ni2
dP f ./, u2-2
K (k ) = J
0
■y/i - k2 Sin2 P
E(k) = J -/i - kz sinz P- dp .
0
The intrinsic inductance of a circular conductor of massive cross-section is calculated as the mutual inductance between two filamentary conductors of the same radius, located at a distance G from each other.
For a circular section of radius r, the geometric mean distance of the area of the circle from itself G is equal to:
G=-
(6)
where e = 2.71828... is the basis of natural logarithms.
As a result of solution (4), the currents in the branches I and the total current I^ of all parallel branches of the current line are determined.
After finding the total current of all parallel branches, the total complex resistance, its active R and reactive XL components, the equivalent inductance L of the current line are determined:
Z = -
I2nrv
R = Re(Z ) XL = Im(Z )
L =-
XL
(7)
where rv is the bending radius of the current line.
Results of numerical modelling. The axis of symmetry of the annular current line (Fig. 1) is on the left at a distance Z from the center of the current line. Passing the nodes in each layer begins on the outside of the annular current line, then passes on the inside and returns to the outside. The current tends to pass along the path of the smallest length (on the inner side of the current line), due to which its density is higher there (Fig. 1).
Figures 2-4 show the effect of frequency on the distribution of currents I in branches N (Fig. 2), active resistance Rc (Fig. 3), equivalent inductance Lc (Fig. 4) of a copper conductive core of a power cable with a cross section of 240 mm2 (Fig. 2,a; curves 1 in Fig. 3, 4) and 800 mm2 (Fig. 2,b; curves 2 in Fig. 3, 4), respectively.
With increasing frequency, the displacement of current on the surface of the current line increases. As a result, the effective resistance of the current line increases (Fig. 3).
Figure 5 shows a layout of the location of the nodes at the intersection of the conductive core and the screen of the power single-core cable of coaxial structure. The results of numerical calculation of frequency-dependent active resistance and inductance L of the power cable with voltage of 110 kV with a cross section of a copper conductive core of 240 mm2 and a copper screen with thickness of 1 mm are shown in Fig. 6, 7, respectively. The active resistance of the cable R consists of the resistance of the conductive core Rc and the metal screen Rs: R = Rc + Rs. The equivalent inductance of the cable L includes the equivalent inductance of the conductive core Lc, the equivalent screen inductance Ls and the mutual inductance between core and screen.
100
200
300
400
, 500
0 abs(I), A
o abs(I), A
10 10" 10" 10" 10" 10" 10"
101 10' 10' 10' 10' 10 10'
10 0 100 200 300 400 500 600 700 N 800 b
Fig. 2. Influence of frequency on current distribution on branches N of the power current line (curve 1 - frequency 50 Hz, curve 2 - 1 kHz, curve 3 - 10 kHz, curve 4 - 100 kHz, curve 5 - 1 MHz)
Rc, Ohm/m
10"1
10
10'
105 f, Hz 106
10 10 10 10 Fig. 3. Frequency dependencies of active resistance of copper power current line
5 4.8 4.6 4.4 4.2 4 3.8 3.6 3.4 3.2
x 10-5 Lc, H/m
HI! ÏÛ ::i 1 ! H | 4HH ilii H ilil
-H" iij 2 ..vu i 4IH Ijiji ________i..i. iii:
jj|
iii; iij....... —ï—i—5- \ -fit» jiiii --------i-—i— j-j-ji
10
10
10 10 10 f, Hz 10
Fig. 4. Frequency dependencies of equivalent inductance of copper power current line
r
s
cd
a b
Fig. 5. Layout of the nodes at the intersection of the conductive core and the screen with thickness of 1 mm (a) and 3 mm (b)
Rc+Rs, Ohm/m
....... 4IUCI.....1 -1-4-1411» 444l««l---444IUU--- !iI!|I(!Î4Îii{I! •t»l>»MIIII» ■ C--4-I4IC4C
■■■■fi-Hlffll.....1 -l-1-f-MI» —f Î-fHIffl- ■fi-f i-ïiiif ¿wliliii
....{..4-I4I4UI.....1 •■•41411» 4441UU— .(..4-I4IWÇ
.....Ï 1 ■ ,!"î,!,î!î!î
rhttèjtj* ii mi^:- -iÉiiiiia
....{.. 4.14 ^¿ü.....| jJ^fîlilJ -1-4-1411» 444ICUJ--- «■iiijjüK •M^Ioill)" ->--!-(-1411)1-- ■ C--4-I4IUC -C--4-I4IHC -1" 4-141««
■—i"4-»4l«H.....1 ■i ■i-ijiSf' ■fr-t-UHfl- ■I" ■ I" fr 1411»" ■ -I--4-I4IMC
/HlfllsT —r f'ïîitïïi" I'tflllili" ■rfrntS
■rpiiiira?
■■■■(■■i-Hlffll.....1 -I--C-I4HI} -I--I-«-«-»!»}-- -C--4-I4IHC
Fig. 6. Frequency dependence of the active resistance of a power cable of single-core design of voltage of 110 kV
1IHS i jjiii .....hi-Htiii '"HitlS
"»W-MIÄ V" •VWi» —4-H-Û+MÎ —H—¿■■¡■««i
••••«■••> •»■»it« ••w Î4M1 .....»••¡•t-i">Hi ••••¡••¿•mttS
**"r*T*:7:ÏÏ! .....:":'?:?::: : —T: T :TTT
•Mm iiiii 2 • • • ••••• •Hill
The second coefficient KR is calculated through the ratio of the resistance of the current line at AC and DC.
Table 1 shows a comparative analysis of the coefficients of non-uniformity of current distribution at the intersection of conductive copper cores. The results of the calculations are identical.
Table 1
Comparative analysis of the coefficients of non-uniformity of current distribution across the current line section
Frequency, Hz Cross section of the copper current line 240 mm2 Cross section of the copper current line 800 mm2
Kf KR Kf Kr
50 1,0156 1,0156 1,1546 1,1545
100 1,0605 1,0605 1,4598 1,4598
1000 2,3581 2,3581 3,9658 3,9655
10000 6,5865 6,5865 9,2692 9,2686
100000 14,3079 14,3079 19,4337 19,4337
1000000 19,4332 19,4332 22,2853 22,2836
Fig. 7. Frequency dependence of the equivalent inductance of a power cable of single-core design of voltage of 110 kV
Increase the thickness of the copper screen 3 times (from 1 to 3 mm) leads to an increase in the coefficient of non-uniformity of current distribution across the screen by 1.25 and 4.48 times with constant value of the coefficient of non-uniformity of current distribution across the copper core section for frequency of 1 kHz and 100 kHz, respectively. For frequency of 1 kHz, the active resistance of a cable with a screen of smaller thickness is 1.4 times the active resistance of a cable with a screen of thicker thickness.
Checking the reliability of the results of numerical calculations is performed on the basis of comparing the coefficients of non-uniformity of the current distribution due to the skin effect and the proximity effect, along the cross section of the current line.
The first coefficient Kf is calculated through the ratio of heat output at AC and DC. The power of heat dissipation at AC is determined by the total power of heat dissipation of parallel branches taking into account the redistribution of current along the cross section of the conductive core. The power of heat dissipation at DC is determined by the uniform distribution of current across the cross section of the conductive core.
Conclusions.
For the first time, a computational numerical model for determining the complex resistance of high-voltage single-core power cables with polymer insulation is proposed, which allows to determine the active resistance and inductance of both conductive core and copper screen, taking into account the skin effect and proximity effect.
The developed model is the basis for determining the characteristic impedance of high-voltage single-core power cables in a wide range of frequencies required to establish adequate criteria for evaluating the parameters of high-frequency effects critical for cross-linked polyethylene insulation.
Based on the proposed computational numerical model, an appropriate technique can be developed to determine the frequency dependencies of the active resistance and inductance of a single-core power cable of coaxial design, which can be the basis for improving the accuracy of mathematical modelling of transients in cable lines arising at single-phase short circuits to the earth and other switchings in electric networks at bilateral grounding of the screen.
Conflict of interest. The authors declare that they have no conflicts of interest.
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Received 14.04.2021 Accepted 22.05.2021 Published 25.06.2021
G.V. Bezprozvannych1, Doctor of Technical Science, Professor,
1.A. Kostiukov1, PhD, Doctoral Student,
1 National Technical University «Kharkiv Polytechnic Institute»,
2, Kyrpychova Str., Kharkiv, 61002, Ukraine,
e-mail: bezprozvannych@kpi.kharkov.ua (Corresponding author),
iakostiukow@gmail.com
How to cite this article:
Bezprozvannych G.V., Kostiukov I.A. A calculation model for determination of impedance of power high voltage single-core cables with polymer insulation. Electrical Engineering & Electromechanics, 202i, no. 3, pp. 47-5i. doi: https://doi.org/i0.20998/2074-272X.202i.3.08.
